Internet Research Task Force (IRTF) A. Faz-Hernandez
Request for Comments: 9380 Cloudflare, Inc.
Category: Informational S. Scott
ISSN: 2070-1721 Oso Security, Inc.
N. Sullivan
Cloudflare, Inc.
R. S. Wahby
Stanford University
C. A. Wood
Cloudflare, Inc.
August 2023
Hashing to Elliptic Curves
Abstract
This document specifies a number of algorithms for encoding or
hashing an arbitrary string to a point on an elliptic curve. This
document is a product of the Crypto Forum Research Group (CFRG) in
the IRTF.
Status of This Memo
This document is not an Internet Standards Track specification; it is
published for informational purposes.
This document is a product of the Internet Research Task Force
(IRTF). The IRTF publishes the results of Internet-related research
and development activities. These results might not be suitable for
deployment. This RFC represents the consensus of the Crypto Forum
Research Group of the Internet Research Task Force (IRTF). Documents
approved for publication by the IRSG are not candidates for any level
of Internet Standard; see Section 2 of RFC 7841.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
https://www.rfc-editor.org/info/rfc9380.
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Table of Contents
1. Introduction
1.1. Requirements Notation
2. Background
2.1. Elliptic Curves
2.2. Terminology
2.2.1. Mappings
2.2.2. Encodings
2.2.3. Random Oracle Encodings
2.2.4. Serialization
2.2.5. Domain Separation
3. Encoding Byte Strings to Elliptic Curves
3.1. Domain Separation Requirements
4. Utility Functions
4.1. The sgn0 Function
5. Hashing to a Finite Field
5.1. Efficiency Considerations in Extension Fields
5.2. hash_to_field Implementation
5.3. expand_message
5.3.1. expand_message_xmd
5.3.2. expand_message_xof
5.3.3. Using DSTs Longer than 255 Bytes
5.3.4. Defining Other expand_message Variants
6. Deterministic Mappings
6.1. Choosing a Mapping Function
6.2. Interface
6.3. Notation
6.4. Sign of the Resulting Point
6.5. Exceptional Cases
6.6. Mappings for Weierstrass Curves
6.6.1. Shallue-van de Woestijne Method
6.6.2. Simplified Shallue-van de Woestijne-Ulas Method
6.6.3. Simplified SWU for AB == 0
6.7. Mappings for Montgomery Curves
6.7.1. Elligator 2 Method
6.8. Mappings for Twisted Edwards Curves
6.8.1. Rational Maps from Montgomery to Twisted Edwards Curves
6.8.2. Elligator 2 Method
7. Clearing the Cofactor
8. Suites for Hashing
8.1. Implementing a Hash-to-Curve Suite
8.2. Suites for NIST P-256
8.3. Suites for NIST P-384
8.4. Suites for NIST P-521
8.5. Suites for curve25519 and edwards25519
8.6. Suites for curve448 and edwards448
8.7. Suites for secp256k1
8.8. Suites for BLS12-381
8.8.1. BLS12-381 G1
8.8.2. BLS12-381 G2
8.9. Defining a New Hash-to-Curve Suite
8.10. Suite ID Naming Conventions
9. IANA Considerations
10. Security Considerations
10.1. Properties of Encodings
10.2. Hashing Passwords
10.3. Constant-Time Requirements
10.4. encode_to_curve: Output Distribution and
Indifferentiability
10.5. hash_to_field Security
10.6. expand_message_xmd Security
10.7. Domain Separation for expand_message Variants
10.8. Target Security Levels
11. References
11.1. Normative References
11.2. Informative References
Appendix A. Related Work
Appendix B. Hashing to ristretto255
Appendix C. Hashing to decaf448
Appendix D. Rational Maps
D.1. Generic Mapping from Montgomery to Twisted Edwards
D.2. Mapping from Weierstrass to Montgomery
Appendix E. Isogeny Maps for Suites
E.1. 3-Isogeny Map for secp256k1
E.2. 11-Isogeny Map for BLS12-381 G1
E.3. 3-Isogeny Map for BLS12-381 G2
Appendix F. Straight-Line Implementations of Deterministic
Mappings
F.1. Shallue-van de Woestijne Method
F.2. Simplified SWU Method
F.2.1. sqrt_ratio Subroutine
F.3. Elligator 2 Method
Appendix G. Curve-Specific Optimized Sample Code
G.1. Interface and Projective Coordinate Systems
G.2. Elligator 2
G.2.1. curve25519 (q = 5 (mod 8), K = 1)
G.2.2. edwards25519
G.2.3. curve448 (q = 3 (mod 4), K = 1)
G.2.4. edwards448
G.2.5. Montgomery Curves with q = 3 (mod 4)
G.2.6. Montgomery Curves with q = 5 (mod 8)
G.3. Cofactor Clearing for BLS12-381 G2
Appendix H. Scripts for Parameter Generation
H.1. Finding Z for the Shallue-van de Woestijne Map
H.2. Finding Z for Simplified SWU
H.3. Finding Z for Elligator 2
Appendix I. sqrt and is_square Functions
I.1. sqrt for q = 3 (mod 4)
I.2. sqrt for q = 5 (mod 8)
I.3. sqrt for q = 9 (mod 16)
I.4. Constant-Time Tonelli-Shanks Algorithm
I.5. is_square for F = GF(p^2)
Appendix J. Suite Test Vectors
J.1. NIST P-256
J.1.1. P256_XMD:SHA-256_SSWU_RO_
J.1.2. P256_XMD:SHA-256_SSWU_NU_
J.2. NIST P-384
J.2.1. P384_XMD:SHA-384_SSWU_RO_
J.2.2. P384_XMD:SHA-384_SSWU_NU_
J.3. NIST P-521
J.3.1. P521_XMD:SHA-512_SSWU_RO_
J.3.2. P521_XMD:SHA-512_SSWU_NU_
J.4. curve25519
J.4.1. curve25519_XMD:SHA-512_ELL2_RO_
J.4.2. curve25519_XMD:SHA-512_ELL2_NU_
J.5. edwards25519
J.5.1. edwards25519_XMD:SHA-512_ELL2_RO_
J.5.2. edwards25519_XMD:SHA-512_ELL2_NU_
J.6. curve448
J.6.1. curve448_XOF:SHAKE256_ELL2_RO_
J.6.2. curve448_XOF:SHAKE256_ELL2_NU_
J.7. edwards448
J.7.1. edwards448_XOF:SHAKE256_ELL2_RO_
J.7.2. edwards448_XOF:SHAKE256_ELL2_NU_
J.8. secp256k1
J.8.1. secp256k1_XMD:SHA-256_SSWU_RO_
J.8.2. secp256k1_XMD:SHA-256_SSWU_NU_
J.9. BLS12-381 G1
J.9.1. BLS12381G1_XMD:SHA-256_SSWU_RO_
J.9.2. BLS12381G1_XMD:SHA-256_SSWU_NU_
J.10. BLS12-381 G2
J.10.1. BLS12381G2_XMD:SHA-256_SSWU_RO_
J.10.2. BLS12381G2_XMD:SHA-256_SSWU_NU_
Appendix K. Expand Test Vectors
K.1. expand_message_xmd(SHA-256)
K.2. expand_message_xmd(SHA-256) (Long DST)
K.3. expand_message_xmd(SHA-512)
K.4. expand_message_xof(SHAKE128)
K.5. expand_message_xof(SHAKE128) (Long DST)
K.6. expand_message_xof(SHAKE256)
Acknowledgements
Contributors
Authors' Addresses
1. Introduction
Many cryptographic protocols require a procedure that encodes an
arbitrary input, e.g., a password, to a point on an elliptic curve.
This procedure is known as hashing to an elliptic curve, where the
hashing procedure provides collision resistance and does not reveal
the discrete logarithm of the output point. Prominent examples of
cryptosystems that hash to elliptic curves include password-
authenticated key exchanges [BM92] [J96] [BMP00] [p1363.2], Identity-
Based Encryption [BF01], Boneh-Lynn-Shacham signatures [BLS01]
[BLS-SIG], Verifiable Random Functions [MRV99] [VRF], and Oblivious
Pseudorandom Functions [NR97] [OPRFs].
Unfortunately for implementors, the precise hash function that is
suitable for a given protocol implemented using a given elliptic
curve is often unclear from the protocol's description. Meanwhile,
an incorrect choice of hash function can have disastrous consequences
for security.
This document aims to bridge this gap by providing a comprehensive
set of recommended algorithms for a range of curve types. Each
algorithm conforms to a common interface: it takes as input an
arbitrary-length byte string and produces as output a point on an
elliptic curve. We provide implementation details for each
algorithm, describe the security rationale behind each
recommendation, and give guidance for elliptic curves that are not
explicitly covered. We also present optimized implementations for
internal functions used by these algorithms.
Readers wishing to quickly specify or implement a conforming hash
function should consult Section 8, which lists recommended hash-to-
curve suites and describes both how to implement an existing suite
and how to specify a new one.
This document does not specify probabilistic rejection sampling
methods, sometimes referred to as "try-and-increment" or "hunt-and-
peck," because the goal is to specify algorithms that can plausibly
be computed in constant time. Use of these probabilistic rejection
methods is NOT RECOMMENDED, because they have been a perennial cause
of side-channel vulnerabilities. See Dragonblood [VR20] as one
example of this problem in practice, and see Appendix A for an
informal description of rejection sampling methods and the timing
side-channels they introduce.
This document represents the consensus of the Crypto Forum Research
Group (CFRG).
1.1. Requirements Notation
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
2. Background
2.1. Elliptic Curves
The following is a brief definition of elliptic curves, with an
emphasis on important parameters and their relation to hashing to
curves. For further reference on elliptic curves, consult
[CFADLNV05] or [W08].
Let F be the finite field GF(q) of prime characteristic p > 3. (This
document does not consider elliptic curves over fields of
characteristic 2 or 3.) In most cases, F is a prime field, so q = p.
Otherwise, F is an extension field, so q = p^m for an integer m > 1.
This document writes elements of extension fields in a primitive
element or polynomial basis, i.e., as a vector of m elements of GF(p)
written in ascending order by degree. The entries of this vector are
indexed in ascending order starting from 1, i.e., x = (x_1, x_2, ...,
x_m). For example, if q = p^2 and the primitive element basis is (1,
I), then x = (a, b) corresponds to the element a + b * I, where x_1 =
a and x_2 = b. (Note that all choices of basis are isomorphic, but
certain choices may result in a more efficient implementation; this
document does not make any particular assumptions about choice of
basis.)
An elliptic curve E is specified by an equation in two variables and
a finite field F. An elliptic curve equation takes one of several
standard forms, including (but not limited to) Weierstrass,
Montgomery, and Edwards.
The curve E induces an algebraic group of order n, meaning that the
group has n distinct elements. (This document uses additive notation
for the elliptic curve group operation.) Elements of an elliptic
curve group are points with affine coordinates (x, y) satisfying the
curve equation, where x and y are elements of F. In addition, all
elliptic curve groups have a distinguished element, the identity
point, which acts as the identity element for the group operation.
On certain curves (including Weierstrass and Montgomery curves), the
identity point cannot be represented as an (x, y) coordinate pair.
For security reasons, cryptographic applications of elliptic curves
generally require using a (sub)group of prime order. Let G be such a
subgroup of the curve of prime order r, where n = h * r. In this
equation, h is an integer called the cofactor. An algorithm that
takes as input an arbitrary point on the curve E and produces as
output a point in the subgroup G of E is said to "clear the
cofactor." Such algorithms are discussed in Section 7.
Certain hash-to-curve algorithms restrict the form of the curve
equation, the characteristic of the field, or the parameters of the
curve. For each algorithm presented, this document lists the
relevant restrictions.
The table below summarizes quantities relevant to hashing to curves:
+========+=====================+=======================+
| Symbol | Meaning | Relevance |
+========+=====================+=======================+
| F,q,p | A finite field F of | For prime fields, |
| | characteristic p | q = p; otherwise, |
| | and #F = q = p^m. | q = p^m and m>1. |
+--------+---------------------+-----------------------+
| E | Elliptic curve. | E is specified by an |
| | | equation and a field |
| | | F. |
+--------+---------------------+-----------------------+
| n | Number of points on | n = h * r, for h and |
| | the elliptic curve | r defined below. |
| | E. | |
+--------+---------------------+-----------------------+
| G | A prime-order | G is a destination |
| | subgroup of the | group to which byte |
| | points on E. | strings are encoded. |
+--------+---------------------+-----------------------+
| r | Order of G. | r is a prime factor |
| | | of n (usually, the |
| | | largest such factor). |
+--------+---------------------+-----------------------+
| h | Cofactor, h >= 1. | h is an integer |
| | | satisfying n = h * r. |
+--------+---------------------+-----------------------+
Table 1: Summary of Symbols and Their Definitions
2.2. Terminology
In this section, we define important terms used throughout the
document.
2.2.1. Mappings
A mapping is a deterministic function from an element of the field F
to a point on an elliptic curve E defined over F.
In general, the set of all points that a mapping can produce over all
possible inputs may be only a subset of the points on an elliptic
curve (i.e., the mapping may not be surjective). In addition, a
mapping may output the same point for two or more distinct inputs
(i.e., the mapping may not be injective). For example, consider a
mapping from F to an elliptic curve having n points: if the number of
elements of F is not equal to n, then this mapping cannot be
bijective (i.e., both injective and surjective), since the mapping is
defined to be deterministic.
Mappings may also be invertible, meaning that there is an efficient
algorithm that, for any point P output by the mapping, outputs an x
in F such that applying the mapping to x outputs P. Some of the
mappings given in Section 6 are invertible, but this document does
not discuss inversion algorithms.
2.2.2. Encodings
Encodings are closely related to mappings. Like a mapping, an
encoding is a function that outputs a point on an elliptic curve. In
contrast to a mapping, however, the input to an encoding is an
arbitrary-length byte string.
This document constructs deterministic encodings by composing a hash
function Hf with a deterministic mapping. In particular, Hf takes as
input an arbitrary string and outputs an element of F. The
deterministic mapping takes that element as input and outputs a point
on an elliptic curve E defined over F. Since Hf takes arbitrary-
length byte strings as inputs, it cannot be injective: the set of
inputs is larger than the set of outputs, so there must be distinct
inputs that give the same output (i.e., there must be collisions).
Thus, any encoding built from Hf is also not injective.
Like mappings, encodings may be invertible, meaning that there is an
efficient algorithm that, for any point P output by the encoding,
outputs a string s such that applying the encoding to s outputs P.
However, the instantiation of Hf used by all encodings specified in
this document (Section 5) is not invertible; thus, those encodings
are also not invertible.
In some applications of hashing to elliptic curves, it is important
that encodings do not leak information through side channels. [VR20]
is one example of this type of leakage leading to a security
vulnerability. See Section 10.3 for further discussion.
2.2.3. Random Oracle Encodings
A random-oracle encoding satisfies a strong property: it can be
proved indifferentiable from a random oracle [MRH04] under a suitable
assumption.
Both constructions described in Section 3 are indifferentiable from
random oracles [MRH04] when instantiated following the guidelines in
this document. The constructions differ in their output
distributions: one gives a uniformly random point on the curve, the
other gives a point sampled from a nonuniform distribution.
A random-oracle encoding with a uniform output distribution is
suitable for use in many cryptographic protocols proven secure in the
random-oracle model. See Section 10.1 for further discussion.
2.2.4. Serialization
A procedure related to encoding is the conversion of an elliptic
curve point to a bit string. This is called serialization, and it is
typically used for compactly storing or transmitting points. The
inverse operation, deserialization, converts a bit string to an
elliptic curve point. For example, [SEC1] and [p1363a] give standard
methods for serialization and deserialization.
Deserialization is different from encoding in that only certain
strings (namely, those output by the serialization procedure) can be
deserialized. In contrast, this document is concerned with encodings
from arbitrary strings to elliptic curve points. This document does
not cover serialization or deserialization.
2.2.5. Domain Separation
Cryptographic protocols proven secure in the random-oracle model are
often analyzed under the assumption that the random oracle only
answers queries associated with that protocol (including queries made
by adversaries) [BR93]. In practice, this assumption does not hold
if two protocols use the same function to instantiate the random
oracle. Concretely, consider protocols P1 and P2 that query a
random-oracle RO: if P1 and P2 both query RO on the same value x, the
security analysis of one or both protocols may be invalidated.
A common way of addressing this issue is called domain separation,
which allows a single random oracle to simulate multiple, independent
oracles. This is effected by ensuring that each simulated oracle
sees queries that are distinct from those seen by all other simulated
oracles. For example, to simulate two oracles RO1 and RO2 given a
single oracle RO, one might define
RO1(x) := RO("RO1" || x)
RO2(x) := RO("RO2" || x)
where || is the concatenation operator. In this example, "RO1" and
"RO2" are called domain separation tags (DSTs); they ensure that
queries to RO1 and RO2 cannot result in identical queries to RO,
meaning that it is safe to treat RO1 and RO2 as independent oracles.
In general, domain separation requires defining a distinct injective
encoding for each oracle being simulated. In the above example,
"RO1" and "RO2" have the same length and thus satisfy this
requirement when used as prefixes. The algorithms specified in this
document take a different approach to ensuring injectivity; see
Sections 5.3 and 10.7 for more details.
3. Encoding Byte Strings to Elliptic Curves
This section presents a general framework and interface for encoding
byte strings to points on an elliptic curve. The constructions in
this section rely on three basic functions:
* The function hash_to_field hashes arbitrary-length byte strings to
a list of one or more elements of a finite field F; its
implementation is defined in Section 5.
hash_to_field(msg, count)
Input:
- msg, a byte string containing the message to hash.
- count, the number of elements of F to output.
Output:
- (u_0, ..., u_(count - 1)), a list of field elements.
Steps: defined in Section 5.
* The function map_to_curve calculates a point on the elliptic curve
E from an element of the finite field F over which E is defined.
Section 6 describes mappings for a range of curve families.
map_to_curve(u)
Input: u, an element of field F.
Output: Q, a point on the elliptic curve E.
Steps: defined in Section 6.
* The function clear_cofactor sends any point on the curve E to the
subgroup G of E. Section 7 describes methods to perform this
operation.
clear_cofactor(Q)
Input: Q, a point on the elliptic curve E.
Output: P, a point in G.
Steps: defined in Section 7.
The two encodings (Section 2.2.2) defined in this section have the
same interface and are both random-oracle encodings (Section 2.2.3).
Both are implemented as a composition of the three basic functions
above. The difference between the two is that their outputs are
sampled from different distributions:
* encode_to_curve is a nonuniform encoding from byte strings to
points in G. That is, the distribution of its output is not
uniformly random in G: the set of possible outputs of
encode_to_curve is only a fraction of the points in G, and some
points in this set are more likely to be output than others.
Section 10.4 gives a more precise definition of encode_to_curve's
output distribution.
encode_to_curve(msg)
Input: msg, an arbitrary-length byte string.
Output: P, a point in G.
Steps:
1. u = hash_to_field(msg, 1)
2. Q = map_to_curve(u[0])
3. P = clear_cofactor(Q)
4. return P
* hash_to_curve is a uniform encoding from byte strings to points in
G. That is, the distribution of its output is statistically close
to uniform in G.
This function is suitable for most applications requiring a random
oracle returning points in G, when instantiated with any of the
map_to_curve functions described in Section 6. See Section 10.1
for further discussion.
hash_to_curve(msg)
Input: msg, an arbitrary-length byte string.
Output: P, a point in G.
Steps:
1. u = hash_to_field(msg, 2)
2. Q0 = map_to_curve(u[0])
3. Q1 = map_to_curve(u[1])
4. R = Q0 + Q1 # Point addition
5. P = clear_cofactor(R)
6. return P
Each hash-to-curve suite in Section 8 instantiates one of these
encoding functions for a specific elliptic curve.
3.1. Domain Separation Requirements
All uses of the encoding functions defined in this document MUST
include domain separation (Section 2.2.5) to avoid interfering with
other uses of similar functionality.
Applications that instantiate multiple, independent instances of
either hash_to_curve or encode_to_curve MUST enforce domain
separation between those instances. This requirement applies in both
the case of multiple instances targeting the same curve and the case
of multiple instances targeting different curves. (This is because
the internal hash_to_field primitive (Section 5) requires domain
separation to guarantee independent outputs.)
Domain separation is enforced with a domain separation tag (DST),
which is a byte string constructed according to the following
requirements:
1. Tags MUST be supplied as the DST parameter to hash_to_field, as
described in Section 5.
2. Tags MUST have nonzero length. A minimum length of 16 bytes is
RECOMMENDED to reduce the chance of collisions with other
applications.
3. Tags SHOULD begin with a fixed identification string that is
unique to the application.
4. Tags SHOULD include a version number.
5. For applications that define multiple ciphersuites, each
ciphersuite's tag MUST be different. For this purpose, it is
RECOMMENDED to include a ciphersuite identifier in each tag.
6. For applications that use multiple encodings, to either the same
curve or different curves, each encoding MUST use a different
tag. For this purpose, it is RECOMMENDED to include the
encoding's Suite ID (Section 8) in the domain separation tag.
For independent encodings based on the same suite, each tag
SHOULD also include a distinct identifier, e.g., "ENC1" and
"ENC2".
As an example, consider a fictional application named Quux that
defines several different ciphersuites, each for a different curve.
A reasonable choice of tag is "QUUX-V<xx>-CS<yy>-<suiteID>", where
<xx> and <yy> are two-digit numbers indicating the version and
ciphersuite, respectively, and <suiteID> is the Suite ID of the
encoding used in ciphersuite <yy>.
As another example, consider a fictional application named Baz that
requires two independent random oracles to the same curve.
Reasonable choices of tags for these oracles are "BAZ-V<xx>-CS<yy>-
<suiteID>-ENC1" and "BAZ-V<xx>-CS<yy>-<suiteID>-ENC2", respectively,
where <xx>, <yy>, and <suiteID> are as described above.
The example tags given above are assumed to be ASCII-encoded byte
strings without null termination, which is the RECOMMENDED format.
Other encodings can be used, but in all cases the encoding as a
sequence of bytes MUST be specified unambiguously.
4. Utility Functions
Algorithms in this document use the utility functions described
below, plus standard arithmetic operations (addition, multiplication,
modular reduction, etc.) and elliptic curve point operations (point
addition and scalar multiplication).
For security, implementations of these functions SHOULD be constant
time: in brief, this means that execution time and memory access
patterns SHOULD NOT depend on the values of secret inputs,
intermediate values, or outputs. For such constant-time
implementations, all arithmetic, comparisons, and assignments MUST
also be implemented in constant time. Section 10.3 briefly discusses
constant-time security issues.
Guidance on implementing low-level operations (in constant time or
otherwise) is beyond the scope of this document; readers should
consult standard reference material [MOV96] [CFADLNV05].
* CMOV(a, b, c): If c is False, CMOV returns a; otherwise, it
returns b. For constant-time implementations, this operation must
run in a time that is independent of the value of c.
* AND, OR, NOT, and XOR are standard bitwise logical operators. For
constant-time implementations, short-circuit operators MUST be
avoided.
* is_square(x): This function returns True whenever the value x is a
square in the field F. By Euler's criterion, this function can be
calculated in constant time as
is_square(x) := { True, if x^((q - 1) / 2) is 0 or 1 in F;
{ False, otherwise.
In certain extension fields, is_square can be computed in constant
time more quickly than by the above exponentiation. [AR13] and
[S85] describe optimized methods for extension fields.
Appendix I.5 gives an optimized straight-line method for GF(p^2).
* sqrt(x): The sqrt operation is a multi-valued function, i.e.,
there exist two roots of x in the field F whenever x is square
(except when x = 0). To maintain compatibility across
implementations while allowing implementors leeway for
optimizations, this document does not require sqrt() to return a
particular value. Instead, as explained in Section 6.4, any
function that calls sqrt also specifies how to determine the
correct root.
The preferred way of computing square roots is to fix a
deterministic algorithm particular to F. We give several
algorithms in Appendix I.
* sgn0(x): This function returns either 0 or 1 indicating the "sign"
of x, where sgn0(x) == 1 just when x is "negative". (In other
words, this function always considers 0 to be positive.)
Section 4.1 defines this function and discusses its
implementation.
* inv0(x): This function returns the multiplicative inverse of x in
F, extended to all of F by fixing inv0(0) == 0. A straightforward
way to implement inv0 in constant time is to compute
inv0(x) := x^(q - 2).
Notice that on input 0, the output is 0 as required. Certain
fields may allow faster inversion methods; detailed discussion of
such methods is beyond the scope of this document.
* I2OSP and OS2IP: These functions are used to convert a byte string
to and from a non-negative integer as described in [RFC8017].
(Note that these functions operate on byte strings in big-endian
byte order.)
* a || b: denotes the concatenation of byte strings a and b. For
example, "ABC" || "DEF" == "ABCDEF".
* substr(str, sbegin, slen): For a byte string str, this function
returns the slen-byte substring starting at position sbegin;
positions are zero indexed. For example, substr("ABCDEFG", 2, 3)
== "CDE".
* len(str): For a byte string str, this function returns the length
of str in bytes. For example, len("ABC") == 3.
* strxor(str1, str2): For byte strings str1 and str2, strxor(str1,
str2) returns the bitwise XOR of the two strings. For example,
strxor("abc", "XYZ") == "9;9" (the strings in this example are
ASCII literals, but strxor is defined for arbitrary byte strings).
In this document, strxor is only applied to inputs of equal
length.
4.1. The sgn0 Function
This section defines a generic sgn0 implementation that applies to
any field F = GF(p^m). It also gives simplified implementations for
the cases F = GF(p) and F = GF(p^2).
The definition of the sgn0 function for extension fields relies on
the polynomial basis or vector representation of field elements, and
iterates over the entire vector representation of the input element.
As a result, sgn0 depends on the primitive polynomial used to define
the polynomial basis; see Section 8 for more information about this
basis, and see Section 2.1 for a discussion of representing elements
of extension fields as vectors.
sgn0(x)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
- p, the characteristic of F (see immediately above).
- m, the extension degree of F, m >= 1 (see immediately above).
Input: x, an element of F.
Output: 0 or 1.
Steps:
1. sign = 0
2. zero = 1
3. for i in (1, 2, ..., m):
4. sign_i = x_i mod 2
5. zero_i = x_i == 0
6. sign = sign OR (zero AND sign_i) # Avoid short-circuit logic ops
7. zero = zero AND zero_i
8. return sign
When m == 1, sgn0 can be significantly simplified:
sgn0_m_eq_1(x)
Input: x, an element of GF(p).
Output: 0 or 1.
Steps:
1. return x mod 2
The case m == 2 is only slightly more complicated:
sgn0_m_eq_2(x)
Input: x, an element of GF(p^2).
Output: 0 or 1.
Steps:
1. sign_0 = x_0 mod 2
2. zero_0 = x_0 == 0
3. sign_1 = x_1 mod 2
4. s = sign_0 OR (zero_0 AND sign_1) # Avoid short-circuit logic ops
5. return s
5. Hashing to a Finite Field
The hash_to_field function hashes a byte string msg of arbitrary
length into one or more elements of a field F. This function works
in two steps: it first hashes the input byte string to produce a
uniformly random byte string, and then interprets this byte string as
one or more elements of F.
For the first step, hash_to_field calls an auxiliary function
expand_message. This document defines two variants of
expand_message: one appropriate for hash functions like SHA-2
[FIPS180-4] or SHA-3 [FIPS202], and another appropriate for
extendable-output functions such as SHAKE128 [FIPS202]. Security
considerations for each expand_message variant are discussed below
(Sections 5.3.1 and 5.3.2).
Implementors MUST NOT use rejection sampling to generate a uniformly
random element of F, to ensure that the hash_to_field function is
amenable to constant-time implementation. The reason is that
rejection sampling procedures are difficult to implement in constant
time, and later well-meaning "optimizations" may silently render an
implementation non-constant-time. This means that any hash_to_field
function based on rejection sampling would be incompatible with
constant-time implementation.
The hash_to_field function is also suitable for securely hashing to
scalars. For example, when hashing to the scalar field for an
elliptic curve (sub)group with prime order r, it suffices to
instantiate hash_to_field with target field GF(r).
The hash_to_field function is designed to be indifferentiable from a
random oracle [MRH04] when expand_message (Section 5.3) is modeled as
a random oracle (see Section 10.5 for details about its
indifferentiability). Ensuring indifferentiability requires care; to
see why, consider a prime p that is close to 3/4 * 2^256. Reducing a
random 256-bit integer modulo this p yields a value that is in the
range [0, p / 3] with probability roughly 1/2, meaning that this
value is statistically far from uniform in [0, p - 1].
To control bias, hash_to_field instead uses random integers whose
length is at least ceil(log2(p)) + k bits, where k is the target
security level for the suite in bits. Reducing such integers mod p
gives bias at most 2^-k for any p; this bias is appropriate when
targeting k-bit security. For each such integer, hash_to_field uses
expand_message to obtain L uniform bytes, where
L = ceil((ceil(log2(p)) + k) / 8)
These uniform bytes are then interpreted as an integer via OS2IP.
For example, for a 255-bit prime p, and k = 128-bit security, L =
ceil((255 + 128) / 8) = 48 bytes.
Note that k is an upper bound on the security level for the
corresponding curve. See Section 10.8 for more details and
Section 8.9 for guidelines on choosing k for a given curve.
5.1. Efficiency Considerations in Extension Fields
The hash_to_field function described in this section is inefficient
for certain extension fields. Specifically, when hashing to an
element of the extension field GF(p^m), hash_to_field requires
expanding msg into m * L bytes (for L as defined above). For
extension fields where log2(p) is significantly smaller than the
security level k, this approach is inefficient: it requires
expand_message to output roughly m * log2(p) + m * k bits, whereas m
* log2(p) + k bytes suffices to generate an element of GF(p^m) with
bias at most 2^-k. In such cases, applications MAY use an
alternative hash_to_field function, provided it meets the following
security requirements:
* The function MUST output one or more field elements that are
uniformly random except with bias at most 2^-k.
* The function MUST NOT use rejection sampling.
* The function SHOULD be amenable to straight-line implementations.
For example, Pornin [P20] describes a method for hashing to
GF(9767^19) that meets these requirements while using fewer output
bits from expand_message than hash_to_field would for that field.
5.2. hash_to_field Implementation
The following procedure implements hash_to_field.
The expand_message parameter to this function MUST conform to the
requirements given in Section 5.3. Section 3.1 discusses the
REQUIRED method for constructing DST, the domain separation tag.
Note that hash_to_field may fail (ABORT) if expand_message fails.
hash_to_field(msg, count)
Parameters:
- DST, a domain separation tag (see Section 3.1).
- F, a finite field of characteristic p and order q = p^m.
- p, the characteristic of F (see immediately above).
- m, the extension degree of F, m >= 1 (see immediately above).
- L = ceil((ceil(log2(p)) + k) / 8), where k is the security
parameter of the suite (e.g., k = 128).
- expand_message, a function that expands a byte string and
domain separation tag into a uniformly random byte string
(see Section 5.3).
Input:
- msg, a byte string containing the message to hash.
- count, the number of elements of F to output.
Output:
- (u_0, ..., u_(count - 1)), a list of field elements.
Steps:
1. len_in_bytes = count * m * L
2. uniform_bytes = expand_message(msg, DST, len_in_bytes)
3. for i in (0, ..., count - 1):
4. for j in (0, ..., m - 1):
5. elm_offset = L * (j + i * m)
6. tv = substr(uniform_bytes, elm_offset, L)
7. e_j = OS2IP(tv) mod p
8. u_i = (e_0, ..., e_(m - 1))
9. return (u_0, ..., u_(count - 1))
5.3. expand_message
expand_message is a function that generates a uniformly random byte
string. It takes three arguments:
1. msg, a byte string containing the message to hash,
2. DST, a byte string that acts as a domain separation tag, and
3. len_in_bytes, the number of bytes to be generated.
This document defines the following two variants of expand_message:
* expand_message_xmd (Section 5.3.1) is appropriate for use with a
wide range of hash functions, including SHA-2 [FIPS180-4], SHA-3
[FIPS202], BLAKE2 [RFC7693], and others.
* expand_message_xof (Section 5.3.2) is appropriate for use with
extendable-output functions (XOFs), including functions in the
SHAKE [FIPS202] or BLAKE2X [BLAKE2X] families.
These variants should suffice for the vast majority of use cases, but
other variants are possible; Section 5.3.4 discusses requirements.
5.3.1. expand_message_xmd
The expand_message_xmd function produces a uniformly random byte
string using a cryptographic hash function H that outputs b bits.
For security, H MUST meet the following requirements:
* The number of bits output by H MUST be b >= 2 * k, where k is the
target security level in bits, and b MUST be divisible by 8. The
first requirement ensures k-bit collision resistance; the second
ensures uniformity of expand_message_xmd's output.
* H MAY be a Merkle-Damgaard hash function like SHA-2. In this
case, security holds when the underlying compression function is
modeled as a random oracle [CDMP05]. (See Section 10.6 for
discussion.)
* H MAY be a sponge-based hash function like SHA-3 or BLAKE2. In
this case, security holds when the inner function is modeled as a
random transformation or as a random permutation [BDPV08].
* Otherwise, H MUST be a hash function that has been proved
indifferentiable from a random oracle [MRH04] under a reasonable
cryptographic assumption.
SHA-2 [FIPS180-4] and SHA-3 [FIPS202] are typical and RECOMMENDED
choices. As an example, for the 128-bit security level, b >= 256
bits and either SHA-256 or SHA3-256 would be an appropriate choice.
The hash function H is assumed to work by repeatedly ingesting fixed-
length blocks of data. The length in bits of these blocks is called
the input block size (s). As examples, s = 1024 for SHA-512
[FIPS180-4] and s = 576 for SHA3-512 [FIPS202]. For correctness, H
requires b <= s.
The following procedure implements expand_message_xmd.
expand_message_xmd(msg, DST, len_in_bytes)
Parameters:
- H, a hash function (see requirements above).
- b_in_bytes, b / 8 for b the output size of H in bits.
For example, for b = 256, b_in_bytes = 32.
- s_in_bytes, the input block size of H, measured in bytes (see
discussion above). For example, for SHA-256, s_in_bytes = 64.
Input:
- msg, a byte string.
- DST, a byte string of at most 255 bytes.
See below for information on using longer DSTs.
- len_in_bytes, the length of the requested output in bytes,
not greater than the lesser of (255 * b_in_bytes) or 2^16-1.
Output:
- uniform_bytes, a byte string.
Steps:
1. ell = ceil(len_in_bytes / b_in_bytes)
2. ABORT if ell > 255 or len_in_bytes > 65535 or len(DST) > 255
3. DST_prime = DST || I2OSP(len(DST), 1)
4. Z_pad = I2OSP(0, s_in_bytes)
5. l_i_b_str = I2OSP(len_in_bytes, 2)
6. msg_prime = Z_pad || msg || l_i_b_str || I2OSP(0, 1) || DST_prime
7. b_0 = H(msg_prime)
8. b_1 = H(b_0 || I2OSP(1, 1) || DST_prime)
9. for i in (2, ..., ell):
10. b_i = H(strxor(b_0, b_(i - 1)) || I2OSP(i, 1) || DST_prime)
11. uniform_bytes = b_1 || ... || b_ell
12. return substr(uniform_bytes, 0, len_in_bytes)
Note that the string Z_pad (step 6) is prefixed to msg before
computing b_0 (step 7). This is necessary for security when H is a
Merkle-Damgaard hash, e.g., SHA-2 (see Section 10.6). Hashing this
additional data means that the cost of computing b_0 is higher than
the cost of simply computing H(msg). In most settings, this overhead
is negligible, because the cost of evaluating H is much less than the
other costs involved in hashing to a curve.
It is possible, however, to entirely avoid this overhead by taking
advantage of the fact that Z_pad depends only on H, and not on the
arguments to expand_message_xmd. To do so, first precompute and save
the internal state of H after ingesting Z_pad. Then, when computing
b_0, initialize H using the saved state. Further details are
implementation dependent and are beyond the scope of this document.
5.3.2. expand_message_xof
The expand_message_xof function produces a uniformly random byte
string using an extendable-output function (XOF) H. For security, H
MUST meet the following criteria:
* The collision resistance of H MUST be at least k bits.
* H MUST be an XOF that has been proved indifferentiable from a
random oracle under a reasonable cryptographic assumption.
The SHAKE XOF family [FIPS202] is a typical and RECOMMENDED choice.
As an example, for 128-bit security, SHAKE128 would be an appropriate
choice.
The following procedure implements expand_message_xof.
expand_message_xof(msg, DST, len_in_bytes)
Parameters:
- H(m, d), an extendable-output function that processes
input message m and returns d bytes.
Input:
- msg, a byte string.
- DST, a byte string of at most 255 bytes.
See below for information on using longer DSTs.
- len_in_bytes, the length of the requested output in bytes.
Output:
- uniform_bytes, a byte string.
Steps:
1. ABORT if len_in_bytes > 65535 or len(DST) > 255
2. DST_prime = DST || I2OSP(len(DST), 1)
3. msg_prime = msg || I2OSP(len_in_bytes, 2) || DST_prime
4. uniform_bytes = H(msg_prime, len_in_bytes)
5. return uniform_bytes
5.3.3. Using DSTs Longer than 255 Bytes
The expand_message variants defined in this section accept domain
separation tags of at most 255 bytes. If applications require a
domain separation tag longer than 255 bytes, e.g., because of
requirements imposed by an invoking protocol, implementors MUST
compute a short domain separation tag by hashing, as follows:
* For expand_message_xmd using hash function H, DST is computed as
DST = H("H2C-OVERSIZE-DST-" || a_very_long_DST)
* For expand_message_xof using extendable-output function H, DST is
computed as
DST = H("H2C-OVERSIZE-DST-" || a_very_long_DST, ceil(2 * k / 8))
Here, a_very_long_DST is the DST whose length is greater than 255
bytes, "H2C-OVERSIZE-DST-" is a 17-byte ASCII string literal, and k
is the target security level in bits.
5.3.4. Defining Other expand_message Variants
When defining a new expand_message variant, the most important
consideration is that hash_to_field models expand_message as a random
oracle. Thus, implementors SHOULD prove indifferentiability from a
random oracle under an appropriate assumption about the underlying
cryptographic primitives; see Section 10.5 for more information.
In addition, expand_message variants:
* MUST give collision resistance commensurate with the security
level of the target elliptic curve.
* MUST be built on primitives designed for use in applications
requiring cryptographic randomness. As examples, a secure stream
cipher is an appropriate primitive, whereas a Mersenne twister
pseudorandom number generator [MT98] is not.
* MUST NOT use rejection sampling.
* MUST give independent values for distinct (msg, DST, length)
inputs. Meeting this requirement is subtle. As a simplified
example, hashing msg || DST does not work, because in this case
distinct (msg, DST) pairs whose concatenations are equal will
return the same output (e.g., ("AB", "CDEF") and ("ABC", "DEF")).
The variants defined in this document use a suffix-free encoding
of DST to avoid this issue.
* MUST use the domain separation tag DST to ensure that invocations
of cryptographic primitives inside of expand_message are domain-
separated from invocations outside of expand_message. For
example, if the expand_message variant uses a hash function H, an
encoding of DST MUST be added either as a prefix or a suffix of
the input to each invocation of H. Adding DST as a suffix is the
RECOMMENDED approach.
* SHOULD read msg exactly once, for efficiency when msg is long.
In addition, each expand_message variant MUST specify a unique
EXP_TAG that identifies that variant in a Suite ID. See Section 8.10
for more information.
6. Deterministic Mappings
The mappings in this section are suitable for implementing either
nonuniform or uniform encodings using the constructions in Section 3.
Certain mappings restrict the form of the curve or its parameters.
For each mapping presented, this document lists the relevant
restrictions.
Note that mappings in this section are not interchangeable: different
mappings will almost certainly output different points when evaluated
on the same input.
6.1. Choosing a Mapping Function
This section gives brief guidelines on choosing a mapping function
for a given elliptic curve. Note that the suites given in Section 8
are recommended mappings for the respective curves.
If the target elliptic curve is a Montgomery curve (Section 6.7), the
Elligator 2 method (Section 6.7.1) is recommended. Similarly, if the
target elliptic curve is a twisted Edwards curve (Section 6.8), the
twisted Edwards Elligator 2 method (Section 6.8.2) is recommended.
The remaining cases are Weierstrass curves. For curves supported by
the Simplified Shallue-van de Woestijne-Ulas (SWU) method
(Section 6.6.2), that mapping is the recommended one. Otherwise, the
Simplified SWU method for AB == 0 (Section 6.6.3) is recommended if
the goal is best performance, while the Shallue-van de Woestijne
method (Section 6.6.1) is recommended if the goal is simplicity of
implementation. (The reason for this distinction is that the
Simplified SWU method for AB == 0 requires implementing an isogeny
map in addition to the mapping function, while the Shallue-van de
Woestijne method does not.)
The Shallue-van de Woestijne method (Section 6.6.1) works with any
curve and may be used in cases where a generic mapping is required.
Note, however, that this mapping is almost always more
computationally expensive than the curve-specific recommendations
above.
6.2. Interface
The generic interface shared by all mappings in this section is as
follows:
(x, y) = map_to_curve(u)
The input u and outputs x and y are elements of the field F. The
affine coordinates (x, y) specify a point on an elliptic curve
defined over F. Note, however, that the point (x, y) is not a
uniformly random point.
6.3. Notation
As a rough guide, the following conventions are used in pseudocode:
* All arithmetic operations are performed over a field F, unless
explicitly stated otherwise.
* u: the input to the mapping function. This is an element of F
produced by the hash_to_field function.
* (x, y), (s, t), (v, w): the affine coordinates of the point output
by the mapping. Indexed variables (e.g., x1, y2, ...) are used
for candidate values.
* tv1, tv2, ...: reusable temporary variables.
* c1, c2, ...: constant values, which can be computed in advance.
6.4. Sign of the Resulting Point
In general, elliptic curves have equations of the form y^2 = g(x).
The mappings in this section first identify an x such that g(x) is
square, then take a square root to find y. Since there are two
square roots when g(x) != 0, this may result in an ambiguity
regarding the sign of y.
When necessary, the mappings in this section resolve this ambiguity
by specifying the sign of the y-coordinate in terms of the input to
the mapping function. Two main reasons support this approach: first,
this covers elliptic curves over any field in a uniform way, and
second, it gives implementors leeway in optimizing square-root
implementations.
6.5. Exceptional Cases
Mappings may have exceptional cases, i.e., inputs u on which the
mapping is undefined. These cases must be handled carefully,
especially for constant-time implementations.
For each mapping in this section, we discuss the exceptional cases
and show how to handle them in constant time. Note that all
implementations SHOULD use inv0 (Section 4) to compute multiplicative
inverses, to avoid exceptional cases that result from attempting to
compute the inverse of 0.
6.6. Mappings for Weierstrass Curves
The mappings in this section apply to a target curve E defined by the
equation
y^2 = g(x) = x^3 + A * x + B
where 4 * A^3 + 27 * B^2 != 0.
6.6.1. Shallue-van de Woestijne Method
Shallue and van de Woestijne [SW06] describe a mapping that applies
to essentially any elliptic curve. (Note, however, that this mapping
is more expensive to evaluate than the other mappings in this
document.)
The parameterization given below is for Weierstrass curves; its
derivation is detailed in [W19]. This parameterization also works
for Montgomery curves (Section 6.7) and twisted Edwards curves
(Section 6.8) via the rational maps given in Appendix D: first,
evaluate the Shallue-van de Woestijne mapping to an equivalent
Weierstrass curve, then map that point to the target Montgomery or
twisted Edwards curve using the corresponding rational map.
Preconditions: A Weierstrass curve y^2 = x^3 + A * x + B.
Constants:
* A and B, the parameter of the Weierstrass curve.
* Z, a non-zero element of F meeting the below criteria.
Appendix H.1 gives a Sage script [SAGE] that outputs the
RECOMMENDED Z.
1. g(Z) != 0 in F.
2. -(3 * Z^2 + 4 * A) / (4 * g(Z)) != 0 in F.
3. -(3 * Z^2 + 4 * A) / (4 * g(Z)) is square in F.
4. At least one of g(Z) and g(-Z / 2) is square in F.
Sign of y: Inputs u and -u give the same x-coordinate for many values
of u. Thus, we set sgn0(y) == sgn0(u).
Exceptions: The exceptional cases for u occur when (1 + u^2 * g(Z)) *
(1 - u^2 * g(Z)) == 0. The restrictions on Z given above ensure that
implementations that use inv0 to invert this product are exception
free.
Operations:
1. tv1 = u^2 * g(Z)
2. tv2 = 1 + tv1
3. tv1 = 1 - tv1
4. tv3 = inv0(tv1 * tv2)
5. tv4 = sqrt(-g(Z) * (3 * Z^2 + 4 * A)) # can be precomputed
6. If sgn0(tv4) == 1, set tv4 = -tv4 # sgn0(tv4) MUST equal 0
7. tv5 = u * tv1 * tv3 * tv4
8. tv6 = -4 * g(Z) / (3 * Z^2 + 4 * A) # can be precomputed
9. x1 = -Z / 2 - tv5
10. x2 = -Z / 2 + tv5
11. x3 = Z + tv6 * (tv2^2 * tv3)^2
12. If is_square(g(x1)), set x = x1 and y = sqrt(g(x1))
13. Else If is_square(g(x2)), set x = x2 and y = sqrt(g(x2))
14. Else set x = x3 and y = sqrt(g(x3))
15. If sgn0(u) != sgn0(y), set y = -y
16. return (x, y)
Appendix F.1 gives an example straight-line implementation of this
mapping.
6.6.2. Simplified Shallue-van de Woestijne-Ulas Method
The function map_to_curve_simple_swu(u) implements a simplification
of the Shallue-van de Woestijne-Ulas mapping [U07] described by Brier
et al. [BCIMRT10], which they call the "simplified SWU" map. Wahby
and Boneh [WB19] generalize and optimize this mapping.
Preconditions: A Weierstrass curve y^2 = x^3 + A * x + B where A != 0
and B != 0.
Constants:
* A and B, the parameters of the Weierstrass curve.
* Z, an element of F meeting the below criteria. Appendix H.2 gives
a Sage script [SAGE] that outputs the RECOMMENDED Z. The criteria
are as follows:
1. Z is non-square in F,
2. Z != -1 in F,
3. the polynomial g(x) - Z is irreducible over F, and
4. g(B / (Z * A)) is square in F.
Sign of y: Inputs u and -u give the same x-coordinate. Thus, we set
sgn0(y) == sgn0(u).
Exceptions: The exceptional cases are values of u such that Z^2 * u^4
+ Z * u^2 == 0. This includes u == 0 and may include other values
that depend on Z. Implementations must detect this case and set x1 =
B / (Z * A), which guarantees that g(x1) is square by the condition
on Z given above.
Operations:
1. tv1 = inv0(Z^2 * u^4 + Z * u^2)
2. x1 = (-B / A) * (1 + tv1)
3. If tv1 == 0, set x1 = B / (Z * A)
4. gx1 = x1^3 + A * x1 + B
5. x2 = Z * u^2 * x1
6. gx2 = x2^3 + A * x2 + B
7. If is_square(gx1), set x = x1 and y = sqrt(gx1)
8. Else set x = x2 and y = sqrt(gx2)
9. If sgn0(u) != sgn0(y), set y = -y
10. return (x, y)
Appendix F.2 gives a general and optimized straight-line
implementation of this mapping. For more information on optimizing
this mapping, see Section 4 of [WB19] or the example code found at
[hash2curve-repo].
6.6.3. Simplified SWU for AB == 0
Wahby and Boneh [WB19] show how to adapt the Simplified SWU mapping
to Weierstrass curves having A == 0 or B == 0, which the mapping of
Section 6.6.2 does not support. (The case A == B == 0 is excluded
because y^2 = x^3 is not an elliptic curve.)
This method applies to curves like secp256k1 [SEC2] and to pairing-
friendly curves in the Barreto-Lynn-Scott family [BLS03], Barreto-
Naehrig family [BN05], and other families.
This method requires finding another elliptic curve E' given by the
equation
y'^2 = g'(x') = x'^3 + A' * x' + B'
that is isogenous to E and has A' != 0 and B' != 0. (See [WB19],
Appendix A, for one way of finding E' using [SAGE].) This isogeny
defines a map iso_map(x', y') given by a pair of rational functions.
iso_map takes as input a point on E' and produces as output a point
on E.
Once E' and iso_map are identified, this mapping works as follows: on
input u, first apply the Simplified SWU mapping to get a point on E',
then apply the isogeny map to that point to get a point on E.
Note that iso_map is a group homomorphism, meaning that point
addition commutes with iso_map. Thus, when using this mapping in the
hash_to_curve construction discussed in Section 3, one can effect a
small optimization by first mapping u0 and u1 to E', adding the
resulting points on E', and then applying iso_map to the sum. This
gives the same result while requiring only one evaluation of iso_map.
Preconditions: An elliptic curve E' with A' != 0 and B' != 0 that is
isogenous to the target curve E with isogeny map iso_map from E' to
E.
Helper functions:
* map_to_curve_simple_swu is the mapping of Section 6.6.2 to E'
* iso_map is the isogeny map from E' to E
Sign of y: For this map, the sign is determined by
map_to_curve_simple_swu. No further sign adjustments are necessary.
Exceptions: map_to_curve_simple_swu handles its exceptional cases.
Exceptional cases of iso_map are inputs that cause the denominator of
either rational function to evaluate to zero; such cases MUST return
the identity point on E.
Operations:
1. (x', y') = map_to_curve_simple_swu(u) # (x', y') is on E'
2. (x, y) = iso_map(x', y') # (x, y) is on E
3. return (x, y)
See [hash2curve-repo] or Section 4.3 of [WB19] for details on
implementing the isogeny map.
6.7. Mappings for Montgomery Curves
The mapping defined in this section applies to a target curve M
defined by the equation
K * t^2 = s^3 + J * s^2 + s
6.7.1. Elligator 2 Method
Bernstein, Hamburg, Krasnova, and Lange give a mapping that applies
to any curve with a point of order 2 [BHKL13], which they call
Elligator 2.
Preconditions: A Montgomery curve K * t^2 = s^3 + J * s^2 + s where
J != 0, K != 0, and (J^2 - 4) / K^2 is non-zero and non-square in F.
Constants:
* J and K, the parameters of the elliptic curve.
* Z, a non-square element of F. Appendix H.3 gives a Sage script
[SAGE] that outputs the RECOMMENDED Z.
Sign of t: This mapping fixes the sign of t as specified in [BHKL13].
No additional adjustment is required.
Exceptions: The exceptional case is Z * u^2 == -1, i.e., 1 + Z * u^2
== 0. Implementations must detect this case and set x1 = -(J / K).
Note that this can only happen when q = 3 (mod 4).
Operations:
1. x1 = -(J / K) * inv0(1 + Z * u^2)
2. If x1 == 0, set x1 = -(J / K)
3. gx1 = x1^3 + (J / K) * x1^2 + x1 / K^2
4. x2 = -x1 - (J / K)
5. gx2 = x2^3 + (J / K) * x2^2 + x2 / K^2
6. If is_square(gx1), set x = x1, y = sqrt(gx1) with sgn0(y) == 1.
7. Else set x = x2, y = sqrt(gx2) with sgn0(y) == 0.
8. s = x * K
9. t = y * K
10. return (s, t)
Appendix F.3 gives an example straight-line implementation of this
mapping. Appendix G.2 gives optimized straight-line procedures that
apply to specific classes of curves and base fields.
6.8. Mappings for Twisted Edwards Curves
Twisted Edwards curves (a class of curves that includes Edwards
curves) are given by the equation
a * v^2 + w^2 = 1 + d * v^2 * w^2
with a != 0, d != 0, and a != d [BBJLP08].
These curves are closely related to Montgomery curves (Section 6.7):
every twisted Edwards curve is birationally equivalent to a
Montgomery curve ([BBJLP08], Theorem 3.2). This equivalence yields
an efficient way of hashing to a twisted Edwards curve: first, hash
to an equivalent Montgomery curve, then transform the result into a
point on the twisted Edwards curve via a rational map. This method
of hashing to a twisted Edwards curve thus requires identifying a
corresponding Montgomery curve and rational map. We describe how to
identify such a curve and map immediately below.
6.8.1. Rational Maps from Montgomery to Twisted Edwards Curves
There are two ways to select a Montgomery curve and rational map for
use when hashing to a given twisted Edwards curve. The selected
Montgomery curve and rational map MUST be specified as part of the
hash-to-curve suite for a given twisted Edwards curve; see Section 8.
1. When hashing to a standardized twisted Edwards curve for which a
corresponding Montgomery form and rational map are also
standardized, the standard Montgomery form and rational map
SHOULD be used to ensure compatibility with existing software.
In certain cases, e.g., edwards25519 [RFC7748], the sign of the
rational map from the twisted Edwards curve to its corresponding
Montgomery curve is not given explicitly. In this case, the sign
MUST be fixed such that applying the rational map to the twisted
Edwards curve's base point yields the Montgomery curve's base
point with correct sign. (For edwards25519, see [RFC7748] and
[Err4730].)
When defining new twisted Edwards curves, a Montgomery equivalent
and rational map SHOULD also be specified, and the sign of the
rational map SHOULD be stated explicitly.
2. When hashing to a twisted Edwards curve that does not have a
standardized Montgomery form or rational map, the map given in
Appendix D SHOULD be used.
6.8.2. Elligator 2 Method
Preconditions: A twisted Edwards curve E and an equivalent Montgomery
curve M meeting the requirements in Section 6.8.1.
Helper functions:
* map_to_curve_elligator2 is the mapping of Section 6.7.1 to the
curve M.
* rational_map is a function that takes a point (s, t) on M and
returns a point (v, w) on E. This rational map should be chosen
as defined in Section 6.8.1.
Sign of t (and v): For this map, the sign is determined by
map_to_curve_elligator2. No further sign adjustments are required.
Exceptions: The exceptions for the Elligator 2 mapping are as given
in Section 6.7.1. The exceptions for the rational map are as given
in Section 6.8.1. No other exceptions are possible.
The following procedure implements the Elligator 2 mapping for a
twisted Edwards curve. (Note that the output point is denoted (v, w)
because it is a point on the target twisted Edwards curve.)
map_to_curve_elligator2_edwards(u)
Input: u, an element of F.
Output: (v, w), a point on E.
1. (s, t) = map_to_curve_elligator2(u) # (s, t) is on M
2. (v, w) = rational_map(s, t) # (v, w) is on E
3. return (v, w)
7. Clearing the Cofactor
The mappings of Section 6 always output a point on the elliptic
curve, i.e., a point in a group of order h * r (Section 2.1).
Obtaining a point in G may require a final operation commonly called
"clearing the cofactor," which takes as input any point on the curve
and produces as output a point in the prime-order (sub)group G
(Section 2.1).
The cofactor can always be cleared via scalar multiplication by h.
For elliptic curves where h = 1, i.e., the curves with a prime number
of points, no operation is required. This applies, for example, to
the NIST curves P-256, P-384, and P-521 [FIPS186-4].
In some cases, it is possible to clear the cofactor via a faster
method than scalar multiplication by h. These methods are equivalent
to (but usually faster than) multiplication by some scalar h_eff
whose value is determined by the method and the curve. Examples of
fast cofactor clearing methods include the following:
* For certain pairing-friendly curves having subgroup G2 over an
extension field, Scott et al. [SBCDK09] describe a method for fast
cofactor clearing that exploits an efficiently computable
endomorphism. Fuentes-Castañeda et al. [FKR11] propose an
alternative method that is sometimes more efficient. Budroni and
Pintore [BP17] give concrete instantiations of these methods for
Barreto-Lynn-Scott pairing-friendly curves [BLS03]. This method
is described for the specific case of BLS12-381 in Appendix G.3.
* Wahby and Boneh ([WB19], Section 5) describe a trick due to Scott
for fast cofactor clearing on any elliptic curve for which the
prime factorization of h and the structure of the elliptic curve
group meet certain conditions.
The clear_cofactor function is parameterized by a scalar h_eff.
Specifically,
clear_cofactor(P) := h_eff * P
where * represents scalar multiplication. When a curve does not
support a fast cofactor clearing method, h_eff = h and the cofactor
MUST be cleared via scalar multiplication.
When a curve admits a fast cofactor clearing method, clear_cofactor
MAY be evaluated either via that method or via scalar multiplication
by the equivalent h_eff; these two methods give the same result.
Note that in this case scalar multiplication by the cofactor h does
not generally give the same result as the fast method and MUST NOT be
used.
8. Suites for Hashing
This section lists recommended suites for hashing to standard
elliptic curves.
A hash-to-curve suite fully specifies the procedure for hashing byte
strings to points on a specific elliptic curve group. Section 8.1
describes how to implement a suite. Applications that require
hashing to an elliptic curve should use either an existing suite or a
new suite specified as described in Section 8.9.
All applications using a hash-to-curve suite MUST choose a domain
separation tag (DST) in accordance with the guidelines in
Section 3.1. In addition, applications whose security requires a
random oracle that returns uniformly random points on the target
curve MUST use a suite whose encoding type is hash_to_curve; see
Section 3 and immediately below for more information.
A hash-to-curve suite comprises the following parameters:
* Suite ID, a short name used to refer to a given suite.
Section 8.10 discusses the naming conventions for Suite IDs.
* encoding type, either uniform (hash_to_curve) or nonuniform
(encode_to_curve). See Section 3 for definitions of these
encoding types.
* E, the target elliptic curve over a field F.
* p, the characteristic of the field F.
* m, the extension degree of the field F. If m > 1, the suite MUST
also specify the polynomial basis used to represent extension
field elements.
* k, the target security level of the suite in bits. (See
Section 10.8 for discussion.)
* L, the length parameter for hash_to_field (Section 5).
* expand_message, one of the variants specified in Section 5.3 plus
any parameters required for the specified variant (for example, H,
the underlying hash function).
* f, a mapping function from Section 6.
* h_eff, the scalar parameter for clear_cofactor (Section 7).
In addition to the above parameters, the mapping f may require
additional parameters Z, M, rational_map, E', or iso_map. When
applicable, these MUST be specified.
The table below lists suites RECOMMENDED for some elliptic curves.
The corresponding parameters are given in the following subsections.
Applications instantiating cryptographic protocols whose security
analysis relies on a random oracle that outputs points with a uniform
distribution MUST NOT use a nonuniform encoding. Moreover,
applications that use a nonuniform encoding SHOULD carefully analyze
the security implications of nonuniformity. When the required
encoding is not clear, applications SHOULD use a uniform encoding for
security.
+==============+===================================+=========+
| E | Suites | Section |
+==============+===================================+=========+
| NIST P-256 | P256_XMD:SHA-256_SSWU_RO_ | 8.2 |
| | P256_XMD:SHA-256_SSWU_NU_ | |
+--------------+-----------------------------------+---------+
| NIST P-384 | P384_XMD:SHA-384_SSWU_RO_ | 8.3 |
| | P384_XMD:SHA-384_SSWU_NU_ | |
+--------------+-----------------------------------+---------+
| NIST P-521 | P521_XMD:SHA-512_SSWU_RO_ | 8.4 |
| | P521_XMD:SHA-512_SSWU_NU_ | |
+--------------+-----------------------------------+---------+
| curve25519 | curve25519_XMD:SHA-512_ELL2_RO_ | 8.5 |
| | curve25519_XMD:SHA-512_ELL2_NU_ | |
+--------------+-----------------------------------+---------+
| edwards25519 | edwards25519_XMD:SHA-512_ELL2_RO_ | 8.5 |
| | edwards25519_XMD:SHA-512_ELL2_NU_ | |
+--------------+-----------------------------------+---------+
| curve448 | curve448_XOF:SHAKE256_ELL2_RO_ | 8.6 |
| | curve448_XOF:SHAKE256_ELL2_NU_ | |
+--------------+-----------------------------------+---------+
| edwards448 | edwards448_XOF:SHAKE256_ELL2_RO_ | 8.6 |
| | edwards448_XOF:SHAKE256_ELL2_NU_ | |
+--------------+-----------------------------------+---------+
| secp256k1 | secp256k1_XMD:SHA-256_SSWU_RO_ | 8.7 |
| | secp256k1_XMD:SHA-256_SSWU_NU_ | |
+--------------+-----------------------------------+---------+
| BLS12-381 G1 | BLS12381G1_XMD:SHA-256_SSWU_RO_ | 8.8 |
| | BLS12381G1_XMD:SHA-256_SSWU_NU_ | |
+--------------+-----------------------------------+---------+
| BLS12-381 G2 | BLS12381G2_XMD:SHA-256_SSWU_RO_ | 8.8 |
| | BLS12381G2_XMD:SHA-256_SSWU_NU_ | |
+--------------+-----------------------------------+---------+
Table 2: Suites for hashing to elliptic curves.
8.1. Implementing a Hash-to-Curve Suite
A hash-to-curve suite requires the following functions. Note that
some of these require utility functions from Section 4.
1. Base field arithmetic operations for the target elliptic curve,
e.g., addition, multiplication, and square root.
2. Elliptic curve point operations for the target curve, e.g., point
addition and scalar multiplication.
3. The hash_to_field function; see Section 5. This includes the
expand_message variant (Section 5.3) and any constituent hash
function or XOF.
4. The suite-specified mapping function; see the corresponding
subsection of Section 6.
5. A cofactor clearing function; see Section 7. This may be
implemented as scalar multiplication by h_eff or as a faster
equivalent method.
6. The desired encoding function; see Section 3. This is either
hash_to_curve or encode_to_curve.
8.2. Suites for NIST P-256
This section defines ciphersuites for the NIST P-256 elliptic curve
[FIPS186-4].
P256_XMD:SHA-256_SSWU_RO_ is defined as follows:
* encoding type: hash_to_curve (Section 3)
* E: y^2 = x^3 + A * x + B, where
- A = -3
- B = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e2
7d2604b
* p: 2^256 - 2^224 + 2^192 + 2^96 - 1
* m: 1
* k: 128
* expand_message: expand_message_xmd (Section 5.3.1)
* H: SHA-256
* L: 48
* f: Simplified SWU method (Section 6.6.2)
* Z: -10
* h_eff: 1
P256_XMD:SHA-256_SSWU_NU_ is identical to P256_XMD:SHA-256_SSWU_RO_,
except that the encoding type is encode_to_curve (Section 3).
An optimized example implementation of the Simplified SWU mapping to
P-256 is given in Appendix F.2.
8.3. Suites for NIST P-384
This section defines ciphersuites for the NIST P-384 elliptic curve
[FIPS186-4].
P384_XMD:SHA-384_SSWU_RO_ is defined as follows:
* encoding type: hash_to_curve (Section 3)
* E: y^2 = x^3 + A * x + B, where
- A = -3
- B = 0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5
013875ac656398d8a2ed19d2a85c8edd3ec2aef
* p: 2^384 - 2^128 - 2^96 + 2^32 - 1
* m: 1
* k: 192
* expand_message: expand_message_xmd (Section 5.3.1)
* H: SHA-384
* L: 72
* f: Simplified SWU method (Section 6.6.2)
* Z: -12
* h_eff: 1
P384_XMD:SHA-384_SSWU_NU_ is identical to P384_XMD:SHA-384_SSWU_RO_,
except that the encoding type is encode_to_curve (Section 3).
An optimized example implementation of the Simplified SWU mapping to
P-384 is given in Appendix F.2.
8.4. Suites for NIST P-521
This section defines ciphersuites for the NIST P-521 elliptic curve
[FIPS186-4].
P521_XMD:SHA-512_SSWU_RO_ is defined as follows:
* encoding type: hash_to_curve (Section 3)
* E: y^2 = x^3 + A * x + B, where
- A = -3
- B = 0x51953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b4899
18ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451f
d46b503f00
* p: 2^521 - 1
* m: 1
* k: 256
* expand_message: expand_message_xmd (Section 5.3.1)
* H: SHA-512
* L: 98
* f: Simplified SWU method (Section 6.6.2)
* Z: -4
* h_eff: 1
P521_XMD:SHA-512_SSWU_NU_ is identical to P521_XMD:SHA-512_SSWU_RO_,
except that the encoding type is encode_to_curve (Section 3).
An optimized example implementation of the Simplified SWU mapping to
P-521 is given in Appendix F.2.
8.5. Suites for curve25519 and edwards25519
This section defines ciphersuites for curve25519 and edwards25519
[RFC7748]. Note that these ciphersuites MUST NOT be used when
hashing to ristretto255 [ristretto255-decaf448]. See Appendix B for
information on how to hash to that group.
curve25519_XMD:SHA-512_ELL2_RO_ is defined as follows:
* encoding type: hash_to_curve (Section 3)
* E: K * t^2 = s^3 + J * s^2 + s, where
- J = 486662
- K = 1
* p: 2^255 - 19
* m: 1
* k: 128
* expand_message: expand_message_xmd (Section 5.3.1)
* H: SHA-512
* L: 48
* f: Elligator 2 method (Section 6.7.1)
* Z: 2
* h_eff: 8
edwards25519_XMD:SHA-512_ELL2_RO_ is identical to curve25519_XMD:SHA-
512_ELL2_RO_, except for the following parameters:
* E: a * v^2 + w^2 = 1 + d * v^2 * w^2, where
- a = -1
- d = 0x52036cee2b6ffe738cc740797779e89800700a4d4141d8ab75eb4dca1
35978a3
* f: Twisted Edwards Elligator 2 method (Section 6.8.2)
* M: curve25519, defined in [RFC7748], Section 4.1
* rational_map: the birational maps defined in [RFC7748],
Section 4.1
curve25519_XMD:SHA-512_ELL2_NU_ is identical to curve25519_XMD:SHA-
512_ELL2_RO_, except that the encoding type is encode_to_curve
(Section 3).
edwards25519_XMD:SHA-512_ELL2_NU_ is identical to
edwards25519_XMD:SHA-512_ELL2_RO_, except that the encoding type is
encode_to_curve (Section 3).
Optimized example implementations of the above mappings are given in
Appendix G.2.1 and Appendix G.2.2.
8.6. Suites for curve448 and edwards448
This section defines ciphersuites for curve448 and edwards448
[RFC7748]. Note that these ciphersuites MUST NOT be used when
hashing to decaf448 [ristretto255-decaf448]. See Appendix C for
information on how to hash to that group.
curve448_XOF:SHAKE256_ELL2_RO_ is defined as follows:
* encoding type: hash_to_curve (Section 3)
* E: K * t^2 = s^3 + J * s^2 + s, where
- J = 156326
- K = 1
* p: 2^448 - 2^224 - 1
* m: 1
* k: 224
* expand_message: expand_message_xof (Section 5.3.2)
* H: SHAKE256
* L: 84
* f: Elligator 2 method (Section 6.7.1)
* Z: -1
* h_eff: 4
edwards448_XOF:SHAKE256_ELL2_RO_ is identical to
curve448_XOF:SHAKE256_ELL2_RO_, except for the following parameters:
* E: a * v^2 + w^2 = 1 + d * v^2 * w^2, where
- a = 1
- d = -39081
* f: Twisted Edwards Elligator 2 method (Section 6.8.2)
* M: curve448, defined in [RFC7748], Section 4.2
* rational_map: the 4-isogeny map defined in [RFC7748], Section 4.2
curve448_XOF:SHAKE256_ELL2_NU_ is identical to
curve448_XOF:SHAKE256_ELL2_RO_, except that the encoding type is
encode_to_curve (Section 3).
edwards448_XOF:SHAKE256_ELL2_NU_ is identical to
edwards448_XOF:SHAKE256_ELL2_RO_, except that the encoding type is
encode_to_curve (Section 3).
Optimized example implementations of the above mappings are given in
Appendix G.2.3 and Appendix G.2.4.
8.7. Suites for secp256k1
This section defines ciphersuites for the secp256k1 elliptic curve
[SEC2].
secp256k1_XMD:SHA-256_SSWU_RO_ is defined as follows:
* encoding type: hash_to_curve (Section 3)
* E: y^2 = x^3 + 7
* p: 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1
* m: 1
* k: 128
* expand_message: expand_message_xmd (Section 5.3.1)
* H: SHA-256
* L: 48
* f: Simplified SWU for AB == 0 (Section 6.6.3)
* Z: -11
* E': y'^2 = x'^3 + A' * x' + B', where
- A': 0x3f8731abdd661adca08a5558f0f5d272e953d363cb6f0e5d405447c01
a444533
- B': 1771
* iso_map: the 3-isogeny map from E' to E given in Appendix E.1
* h_eff: 1
secp256k1_XMD:SHA-256_SSWU_NU_ is identical to secp256k1_XMD:SHA-
256_SSWU_RO_, except that the encoding type is encode_to_curve
(Section 3).
An optimized example implementation of the Simplified SWU mapping to
the curve E' isogenous to secp256k1 is given in Appendix F.2.
8.8. Suites for BLS12-381
This section defines ciphersuites for groups G1 and G2 of the
BLS12-381 elliptic curve [BLS12-381].
8.8.1. BLS12-381 G1
BLS12381G1_XMD:SHA-256_SSWU_RO_ is defined as follows:
* encoding type: hash_to_curve (Section 3)
* E: y^2 = x^3 + 4
* p: 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f
6241eabfffeb153ffffb9feffffffffaaab
* m: 1
* k: 128
* expand_message: expand_message_xmd (Section 5.3.1)
* H: SHA-256
* L: 64
* f: Simplified SWU for AB == 0 (Section 6.6.3)
* Z: 11
* E': y'^2 = x'^3 + A' * x' + B', where
- A' = 0x144698a3b8e9433d693a02c96d4982b0ea985383ee66a8d8e8981aef
d881ac98936f8da0e0f97f5cf428082d584c1d
- B' = 0x12e2908d11688030018b12e8753eee3b2016c1f0f24f4070a0b9c14f
cef35ef55a23215a316ceaa5d1cc48e98e172be0
* iso_map: the 11-isogeny map from E' to E given in Appendix E.2
* h_eff: 0xd201000000010001
BLS12381G1_XMD:SHA-256_SSWU_NU_ is identical to BLS12381G1_XMD:SHA-
256_SSWU_RO_, except that the encoding type is encode_to_curve
(Section 3).
Note that the h_eff values for these suites are chosen for
compatibility with the fast cofactor clearing method described by
Scott ([WB19], Section 5).
An optimized example implementation of the Simplified SWU mapping to
the curve E' isogenous to BLS12-381 G1 is given in Appendix F.2.
8.8.2. BLS12-381 G2
BLS12381G2_XMD:SHA-256_SSWU_RO_ is defined as follows:
* encoding type: hash_to_curve (Section 3)
* E: y^2 = x^3 + 4 * (1 + I)
* base field F is GF(p^m), where
- p: 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6
b0f6241eabfffeb153ffffb9feffffffffaaab
- m: 2
- (1, I) is the basis for F, where I^2 + 1 == 0 in F
* k: 128
* expand_message: expand_message_xmd (Section 5.3.1)
* H: SHA-256
* L: 64
* f: Simplified SWU for AB == 0 (Section 6.6.3)
* Z: -(2 + I)
* E': y'^2 = x'^3 + A' * x' + B', where
- A' = 240 * I
- B' = 1012 * (1 + I)
* iso_map: the isogeny map from E' to E given in Appendix E.3
* h_eff: 0xbc69f08f2ee75b3584c6a0ea91b352888e2a8e9145ad7689986ff0315
08ffe1329c2f178731db956d82bf015d1212b02ec0ec69d7477c1ae954cbc06689
f6a359894c0adebbf6b4e8020005aaa95551
BLS12381G2_XMD:SHA-256_SSWU_NU_ is identical to BLS12381G2_XMD:SHA-
256_SSWU_RO_, except that the encoding type is encode_to_curve
(Section 3).
Note that the h_eff values for these suites are chosen for
compatibility with the fast cofactor clearing method described by
Budroni and Pintore ([BP17], Section 4.1) and are summarized in
Appendix G.3.
An optimized example implementation of the Simplified SWU mapping to
the curve E' isogenous to BLS12-381 G2 is given in Appendix F.2.
8.9. Defining a New Hash-to-Curve Suite
For elliptic curves not listed elsewhere in Section 8, a new hash-to-
curve suite can be defined by the following:
1. E, F, p, and m are determined by the elliptic curve and its base
field.
2. k is an upper bound on the target security level of the suite
(Section 10.8). A reasonable choice of k is ceil(log2(r) / 2),
where r is the order of the subgroup G of the curve E
(Section 2.1).
3. Choose encoding type, either hash_to_curve or encode_to_curve
(Section 3).
4. Compute L as described in Section 5.
5. Choose an expand_message variant from Section 5.3 plus any
underlying cryptographic primitives (e.g., a hash function H).
6. Choose a mapping following the guidelines in Section 6.1, and
select any required parameters for that mapping.
7. Choose h_eff to be either the cofactor of E or, if a fast
cofactor clearing method is to be used, a value appropriate to
that method as discussed in Section 7.
8. Construct a Suite ID following the guidelines in Section 8.10.
8.10. Suite ID Naming Conventions
Suite IDs MUST be constructed as follows:
CURVE_ID || "_" || HASH_ID || "_" || MAP_ID || "_" || ENC_VAR || "_"
The fields CURVE_ID, HASH_ID, MAP_ID, and ENC_VAR are ASCII-encoded
strings of at most 64 characters each. Fields MUST contain only
ASCII characters between 0x21 and 0x7E (inclusive), except that
underscore (i.e., 0x5F) is not allowed.
As indicated above, each field (including the last) is followed by an
underscore ("_", ASCII 0x5F). This helps to ensure that Suite IDs
are prefix free. Suite IDs MUST include the final underscore and
MUST NOT include any characters after the final underscore.
Suite ID fields MUST be chosen as follows:
* CURVE_ID: a human-readable representation of the target elliptic
curve.
* HASH_ID: a human-readable representation of the expand_message
function and any underlying hash primitives used in hash_to_field
(Section 5). This field MUST be constructed as follows:
EXP_TAG || ":" || HASH_NAME
EXP_TAG indicates the expand_message variant:
- "XMD" for expand_message_xmd (Section 5.3.1).
- "XOF" for expand_message_xof (Section 5.3.2).
HASH_NAME is a human-readable name for the underlying hash
primitive. As examples:
1. For expand_message_xof (Section 5.3.2) with SHAKE128, HASH_ID
is "XOF:SHAKE128".
2. For expand_message_xmd (Section 5.3.1) with SHA3-256, HASH_ID
is "XMD:SHA3-256".
Suites that use an alternative hash_to_field function that meets
the requirements in Section 5.1 MUST indicate this by appending a
tag identifying that function to the HASH_ID field, separated by a
colon (":", ASCII 0x3A).
* MAP_ID: a human-readable representation of the map_to_curve
function as defined in Section 6. These are defined as follows:
- "SVDW" for Shallue and van de Woestijne (Section 6.6.1).
- "SSWU" for Simplified SWU (Sections 6.6.2 and 6.6.3).
- "ELL2" for Elligator 2 (Sections 6.7.1 and 6.8.2).
* ENC_VAR: a string indicating the encoding type and other
information. The first two characters of this string indicate
whether the suite represents a hash_to_curve or an encode_to_curve
operation (Section 3), as follows:
- If ENC_VAR begins with "RO", the suite uses hash_to_curve.
- If ENC_VAR begins with "NU", the suite uses encode_to_curve.
- ENC_VAR MUST NOT begin with any other string.
ENC_VAR MAY also be used to encode other information used to
identify variants, for example, a version number. The RECOMMENDED
way to do so is to add one or more subfields separated by colons.
For example, "RO:V02" is an appropriate ENC_VAR value for the
second version of a uniform encoding suite, while
"RO:V02:FOO01:BAR17" might be used to indicate a variant of that
suite.
9. IANA Considerations
This document has no IANA actions.
10. Security Considerations
This section contains additional security considerations about the
hash-to-curve mechanisms described in this document.
10.1. Properties of Encodings
Each encoding type (Section 3) accepts an arbitrary byte string and
maps it to a point on the curve sampled from a distribution that
depends on the encoding type. It is important to note that using a
nonuniform encoding or directly evaluating one of the mappings of
Section 6 produces an output that is easily distinguished from a
uniformly random point. Applications that use a nonuniform encoding
SHOULD carefully analyze the security implications of nonuniformity.
When the required encoding is not clear, applications SHOULD use a
uniform encoding.
Both encodings given in Section 3 can output the identity element of
the group G. The probability that either encoding function outputs
the identity element is roughly 1/r for a random input, which is
negligible for cryptographically useful elliptic curves. Further, it
is computationally infeasible to find an input to either encoding
function whose corresponding output is the identity element. (Both
of these properties hold when the encoding functions are instantiated
with a hash_to_field function that follows all guidelines in
Section 5.) Protocols that use these encoding functions SHOULD NOT
add a special case to detect and "fix" the identity element.
When the hash_to_curve function (Section 3) is instantiated with a
hash_to_field function that is indifferentiable from a random oracle
(Section 5), the resulting function is indifferentiable from a random
oracle ([MRH04] [BCIMRT10] [FFSTV13] [LBB19] [H20]). In many cases,
such a function can be safely used in cryptographic protocols whose
security analysis assumes a random oracle that outputs uniformly
random points on an elliptic curve. As Ristenpart et al. discuss in
[RSS11], however, not all security proofs that rely on random oracles
continue to hold when those oracles are replaced by indifferentiable
functionalities. This limitation should be considered when analyzing
the security of protocols relying on the hash_to_curve function.
10.2. Hashing Passwords
When hashing passwords using any function described in this document,
an adversary who learns the output of the hash function (or
potentially any intermediate value, e.g., the output of
hash_to_field) may be able to carry out a dictionary attack. To
mitigate such attacks, it is recommended to first execute a more
costly key derivation function (e.g., PBKDF2 [RFC8018], scrypt
[RFC7914], or Argon2 [RFC9106]) on the password, then hash the output
of that function to the target elliptic curve. For collision
resistance, the hash underlying the key derivation function should be
chosen according to the guidelines listed in Section 5.3.1.
10.3. Constant-Time Requirements
Constant-time implementations of all functions in this document are
STRONGLY RECOMMENDED for all uses, to avoid leaking information via
side channels. It is especially important to use a constant-time
implementation when inputs to an encoding are secret values; in such
cases, constant-time implementations are REQUIRED for security
against timing attacks (e.g., [VR20]). When constant-time
implementations are required, all basic operations and utility
functions must be implemented in constant time, as discussed in
Section 4. In some applications (e.g., embedded systems), leakage
through other side channels (e.g., power or electromagnetic side
channels) may be pertinent. Defending against such leakage is
outside the scope of this document, because the nature of the leakage
and the appropriate defense depend on the application.
10.4. encode_to_curve: Output Distribution and Indifferentiability
The encode_to_curve function (Section 3) returns points sampled from
a distribution that is statistically far from uniform. This
distribution is bounded roughly as follows: first, it includes at
least one eighth of the points in G, and second, the probability of
points in the distribution varies by at most a factor of four. These
bounds hold when encode_to_curve is instantiated with any of the
map_to_curve functions in Section 6.
The bounds above are derived from several works in the literature.
Specifically:
* Shallue and van de Woestijne [SW06] and Fouque and Tibouchi [FT12]
derive bounds on the Shallue-van de Woestijne mapping
(Section 6.6.1).
* Fouque and Tibouchi [FT10] and Tibouchi [T14] derive bounds for
the Simplified SWU mapping (Sections 6.6.2 and 6.6.3).
* Bernstein et al. [BHKL13] derive bounds for the Elligator 2
mapping (Sections 6.7.1 and 6.8.2).
Indifferentiability of encode_to_curve follows from an argument
similar to the one given by Brier et al. [BCIMRT10]; we briefly
sketch this argument as follows. Consider an ideal random oracle
Hc() that samples from the distribution induced by the map_to_curve
function called by encode_to_curve, and assume for simplicity that
the target elliptic curve has cofactor 1 (a similar argument applies
for non-unity cofactors). Indifferentiability holds just if it is
possible to efficiently simulate the "inner" random oracle in
encode_to_curve, namely, hash_to_field. The simulator works as
follows: on a fresh query msg, the simulator queries Hc(msg) and
receives a point P in the image of map_to_curve (if msg is the same
as a prior query, the simulator just returns the value it gave in
response to that query). The simulator then computes the possible
preimages of P under map_to_curve, i.e., elements u of F such that
map_to_curve(u) == P (Tibouchi [T14] shows that this can be done
efficiently for the Shallue-van de Woestijne and Simplified SWU maps,
and Bernstein et al. show the same for Elligator 2). The simulator
selects one such preimage at random and returns this value as the
simulated output of the "inner" random oracle. By hypothesis, Hc()
samples from the distribution induced by map_to_curve on a uniformly
random input element of F, so this value is uniformly random and
induces the correct point P when passed through map_to_curve.
10.5. hash_to_field Security
The hash_to_field function, defined in Section 5, is indifferentiable
from a random oracle [MRH04] when expand_message (Section 5.3) is
modeled as a random oracle. Since indifferentiability proofs are
composable, this also holds when expand_message is proved
indifferentiable from a random oracle relative to an underlying
primitive that is modeled as a random oracle. When following the
guidelines in Section 5.3, both variants of expand_message defined in
that section meet this requirement (see also Section 10.6).
We very briefly sketch the indifferentiability argument for
hash_to_field. Notice that each integer mod p that hash_to_field
returns (i.e., each element of the vector representation of F) is a
member of an equivalence class of roughly 2^k integers of length
log2(p) + k bits, all of which are equal modulo p. For each integer
mod p that hash_to_field returns, the simulator samples one member of
this equivalence class at random and outputs the byte string returned
by I2OSP. (Notice that this is essentially the inverse of the
hash_to_field procedure.)
10.6. expand_message_xmd Security
The expand_message_xmd function, defined in Section 5.3.1, is
indifferentiable from a random oracle [MRH04] when one of the
following holds:
1. H is indifferentiable from a random oracle,
2. H is a sponge-based hash function whose inner function is modeled
as a random transformation or random permutation [BDPV08], or
3. H is a Merkle-Damgaard hash function whose compression function
is modeled as a random oracle [CDMP05].
For cases (1) and (2), the indifferentiability of expand_message_xmd
follows directly from the indifferentiability of H.
For case (3), i.e., where H is a Merkle-Damgaard hash function,
indifferentiability follows from [CDMP05], Theorem 5. In particular,
expand_message_xmd computes b_0 by prefixing the message with one
block of zeros plus auxiliary information (length, counter, and DST).
Then, each of the output blocks b_i, i >= 1 in expand_message_xmd is
the result of invoking H on a unique, prefix-free encoding of b_0.
This is true, first because the length of the input to all such
invocations is equal and fixed by the choice of H and DST, and second
because each such input has a unique suffix (because of the inclusion
of the counter byte I2OSP(i, 1)).
The essential difference between the construction discussed in
[CDMP05] and expand_message_xmd is that the latter hashes a counter
appended to strxor(b_0, b_(i - 1)) ({#hashtofield-expand-xmd}, step
10) rather than to b_0. This approach increases the Hamming distance
between inputs to different invocations of H, which reduces the
likelihood that nonidealities in H affect the distribution of the b_i
values.
We note that expand_message_xmd can be used to instantiate a general-
purpose indifferentiable functionality with variable-length output
based on any hash function meeting one of the above criteria.
Applications that use expand_message_xmd outside of hash_to_field
should ensure domain separation by picking a distinct value for DST.
10.7. Domain Separation for expand_message Variants
As discussed in Section 2.2.5, the purpose of domain separation is to
ensure that security analyses of cryptographic protocols that query
multiple independent random oracles remain valid even if all of these
random oracles are instantiated based on one underlying function H.
The expand_message variants in this document (Section 5.3) ensure
domain separation by appending a suffix-free-encoded domain
separation tag DST_prime to all strings hashed by H, an underlying
hash or extendable-output function. (Other expand_message variants
that follow the guidelines in Section 5.3.4 are expected to behave
similarly, but these should be analyzed on a case-by-case basis.)
For security, applications that use the same function H outside of
expand_message should enforce domain separation between those uses of
H and expand_message, and they should separate all of these from uses
of H in other applications.
This section suggests four methods for enforcing domain separation
from expand_message variants, explains how each method achieves
domain separation, and lists the situations in which each is
appropriate. These methods share a high-level structure: the
application designer fixes a tag DST_ext distinct from DST_prime and
augments calls to H with DST_ext. Each method augments calls to H
differently, and each may impose additional requirements on DST_ext.
These methods can be used to instantiate multiple domain-separated
functions (e.g., H1 and H2) by selecting distinct DST_ext values for
each (e.g., DST_ext1, DST_ext2).
1. (Suffix-only domain separation.) This method is useful when
domain-separating invocations of H from expand_message_xmd or
expand_message_xof. It is not appropriate for domain-separating
expand_message from HMAC-H [RFC2104]; for that purpose, see
method 4.
To instantiate a suffix-only domain-separated function Hso,
compute
Hso(msg) = H(msg || DST_ext)
DST_ext should be suffix-free encoded (e.g., by appending one
byte encoding the length of DST_ext) to make it infeasible to
find distinct (msg, DST_ext) pairs that hash to the same value.
This method ensures domain separation because all distinct
invocations of H have distinct suffixes, since DST_ext is
distinct from DST_prime.
2. (Prefix-suffix domain separation.) This method can be used in
the same cases as the suffix-only method.
To instantiate a prefix-suffix domain-separated function Hps,
compute
Hps(msg) = H(DST_ext || msg || I2OSP(0, 1))
DST_ext should be prefix-free encoded (e.g., by adding a one-byte
prefix that encodes the length of DST_ext) to make it infeasible
to find distinct (msg, DST_ext) pairs that hash to the same
value.
This method ensures domain separation because appending the byte
I2OSP(0, 1) ensures that inputs to H inside Hps are distinct from
those inside expand_message. Specifically, the final byte of
DST_prime encodes the length of DST, which is required to be
nonzero (Section 3.1, requirement 2), and DST_prime is always
appended to invocations of H inside expand_message.
3. (Prefix-only domain separation.) This method is only useful for
domain-separating invocations of H from expand_message_xmd. It
does not give domain separation for expand_message_xof or HMAC-H.
To instantiate a prefix-only domain-separated function Hpo,
compute
Hpo(msg) = H(DST_ext || msg)
In order for this method to give domain separation, DST_ext
should be at least b bits long, where b is the number of bits
output by the hash function H. In addition, at least one of the
first b bits must be nonzero. Finally, DST_ext should be prefix-
free encoded (e.g., by adding a one-byte prefix that encodes the
length of DST_ext) to make it infeasible to find distinct (msg,
DST_ext) pairs that hash to the same value.
This method ensures domain separation as follows. First, since
DST_ext contains at least one nonzero bit among its first b bits,
it is guaranteed to be distinct from the value Z_pad
(Section 5.3.1, step 4), which ensures that all inputs to H are
distinct from the input used to generate b_0 in
expand_message_xmd. Second, since DST_ext is at least b bits
long, it is almost certainly distinct from the values b_0 and
strxor(b_0, b_(i - 1)), and therefore all inputs to H are
distinct from the inputs used to generate b_i, i >= 1, with high
probability.
4. (XMD-HMAC domain separation.) This method is useful for domain-
separating invocations of H inside HMAC-H (i.e., HMAC [RFC2104]
instantiated with hash function H) from expand_message_xmd. It
also applies to HKDF-H (i.e., HKDF [RFC5869] instantiated with
hash function H), as discussed below.
Specifically, this method applies when HMAC-H is used with a non-
secret key to instantiate a random oracle based on a hash
function H (note that expand_message_xmd can also be used for
this purpose; see Section 10.6). When using HMAC-H with a high-
entropy secret key, domain separation is not necessary; see
discussion below.
To choose a non-secret HMAC key DST_key that ensures domain
separation from expand_message_xmd, compute
DST_key_preimage = "DERIVE-HMAC-KEY-" || DST_ext || I2OSP(0, 1)
DST_key = H(DST_key_preimage)
Then, to instantiate the random oracle Hro using HMAC-H, compute
Hro(msg) = HMAC-H(DST_key, msg)
The trailing zero byte in DST_key_preimage ensures that this
value is distinct from inputs to H inside expand_message_xmd
(because all such inputs have suffix DST_prime, which cannot end
with a zero byte as discussed above). This ensures domain
separation because, with overwhelming probability, all inputs to
H inside of HMAC-H using key DST_key have prefixes that are
distinct from the values Z_pad, b_0, and strxor(b_0, b_(i - 1))
inside of expand_message_xmd.
For uses of HMAC-H that instantiate a private random oracle by
fixing a high-entropy secret key, domain separation from
expand_message_xmd is not necessary. This is because, similarly
to the case above, all inputs to H inside HMAC-H using this
secret key almost certainly have distinct prefixes from all
inputs to H inside expand_message_xmd.
Finally, this method can be used with HKDF-H [RFC5869] by fixing
the salt input to HKDF-Extract to DST_key, computed as above.
This ensures domain separation for HKDF-Extract by the same
argument as for HMAC-H using DST_key. Moreover, assuming that
the input keying material (IKM) supplied to HKDF-Extract has
sufficiently high entropy (say, commensurate with the security
parameter), the HKDF-Expand step is domain-separated by the same
argument as for HMAC-H with a high-entropy secret key (since a
pseudorandom key is exactly that).
10.8. Target Security Levels
Each ciphersuite specifies a target security level (in bits) for the
underlying curve. This parameter ensures the corresponding
hash_to_field instantiation is conservative and correct. We stress
that this parameter is only an upper bound on the security level of
the curve and is neither a guarantee nor endorsement of its
suitability for a given application. Mathematical and cryptographic
advancements may reduce the effective security level for any curve.
11. References
11.1. Normative References
[Err4730] RFC Errata, "Erratum ID 4730", RFC 7748, July 2016,
<https://www.rfc-editor.org/errata/eid4730>.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/info/rfc2119>.
[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", RFC 7748, DOI 10.17487/RFC7748, January
2016, <https://www.rfc-editor.org/info/rfc7748>.
[RFC8017] Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
"PKCS #1: RSA Cryptography Specifications Version 2.2",
RFC 8017, DOI 10.17487/RFC8017, November 2016,
<https://www.rfc-editor.org/info/rfc8017>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/info/rfc8174>.
11.2. Informative References
[AFQTZ14] Aranha, D. F., Fouque, P.-A., Qian, C., Tibouchi, M., and
J. C. Zapalowicz, "Binary Elligator Squared", In Selected
Areas in Cryptography - SAC 2014, pages 20-37,
DOI 10.1007/978-3-319-13051-4_2, November 2014,
<https://doi.org/10.1007/978-3-319-13051-4_2>.
[AR13] Adj, G. and F. Rodríguez-Henríquez, "Square Root
Computation over Even Extension Fields", In IEEE
Transactions on Computers. vol 63 issue 11, pages
2829-2841, DOI 10.1109/TC.2013.145, November 2014,
<https://doi.org/10.1109/TC.2013.145>.
[BBJLP08] Bernstein, D. J., Birkner, P., Joye, M., Lange, T., and C.
Peters, "Twisted Edwards Curves", In AFRICACRYPT 2008,
pages 389-405, DOI 10.1007/978-3-540-68164-9_26, June
2008, <https://doi.org/10.1007/978-3-540-68164-9_26>.
[BCIMRT10] Brier, E., Coron, J.-S., Icart, T., Madore, D., Randriam,
H., and M. Tibouchi, "Efficient Indifferentiable Hashing
into Ordinary Elliptic Curves", In Advances in Cryptology
- CRYPTO 2010, pages 237-254,
DOI 10.1007/978-3-642-14623-7_13, August 2010,
<https://doi.org/10.1007/978-3-642-14623-7_13>.
[BDPV08] Bertoni, G., Daemen, J., Peeters, M., and G. Van Assche,
"On the Indifferentiability of the Sponge Construction",
In Advances in Cryptology - EUROCRYPT 2008, pages 181-197,
DOI 10.1007/978-3-540-78967-3_11, April 2008,
<https://doi.org/10.1007/978-3-540-78967-3_11>.
[BF01] Boneh, D. and M. Franklin, "Identity-Based Encryption from
the Weil Pairing", In Advances in Cryptology - CRYPTO
2001, pages 213-229, DOI 10.1007/3-540-44647-8_13, August
2001, <https://doi.org/10.1007/3-540-44647-8_13>.
[BHKL13] Bernstein, D. J., Hamburg, M., Krasnova, A., and T. Lange,
"Elligator: elliptic-curve points indistinguishable from
uniform random strings", In Proceedings of the 2013 ACM
SIGSAC Conference on Computer and Communications Security,
pages 967-980, DOI 10.1145/2508859.2516734, November 2013,
<https://doi.org/10.1145/2508859.2516734>.
[BLAKE2X] Aumasson, J.-P., Neves, S., Wilcox-O'Hearn, Z., and C.
Winnerlein, "BLAKE2X", December 2016,
<https://blake2.net/blake2x.pdf>.
[BLMP19] Bernstein, D. J., Lange, T., Martindale, C., and L. Panny,
"Quantum Circuits for the CSIDH: Optimizing Quantum
Evaluation of Isogenies", In Advances in Cryptology -
EUROCRYPT 2019, pages 409-441,
DOI 10.1007/978-3-030-17656-3, May 2019,
<https://doi.org/10.1007/978-3-030-17656-3_15>.
[BLS-SIG] Boneh, D., Gorbunov, S., Wahby, R. S., Wee, H., Wood, C.
A., and Z. Zhang, "BLS Signatures", Work in Progress,
Internet-Draft, draft-irtf-cfrg-bls-signature-05, 16 June
2022, <https://datatracker.ietf.org/doc/html/draft-irtf-
cfrg-bls-signature-05>.
[BLS01] Boneh, D., Lynn, B., and H. Shacham, "Short Signatures
from the Weil Pairing", In Journal of Cryptology, vol 17,
pages 297-319, DOI 10.1007/s00145-004-0314-9, July 2004,
<https://doi.org/10.1007/s00145-004-0314-9>.
[BLS03] Barreto, P. S. L. M., Lynn, B., and M. Scott,
"Constructing Elliptic Curves with Prescribed Embedding
Degrees", In Security in Communication Networks, pages
257-267, DOI 10.1007/3-540-36413-7_19, September 2002,
<https://doi.org/10.1007/3-540-36413-7_19>.
[BLS12-381]
Bowe, S., "BLS12-381: New zk-SNARK Elliptic Curve
Construction", March 2017,
<https://electriccoin.co/blog/new-snark-curve/>.
[BM92] Bellovin, S. M. and M. Merritt, "Encrypted key exchange:
password-based protocols secure against dictionary
attacks", In IEEE Symposium on Security and Privacy -
Oakland 1992, pages 72-84, DOI 10.1109/RISP.1992.213269,
May 1992, <https://doi.org/10.1109/RISP.1992.213269>.
[BMP00] Boyko, V., MacKenzie, P., and S. Patel, "Provably Secure
Password-Authenticated Key Exchange Using Diffie-Hellman",
In Advances in Cryptology - EUROCRYPT 2000, pages 156-171,
DOI 10.1007/3-540-45539-6_12, May 2000,
<https://doi.org/10.1007/3-540-45539-6_12>.
[BN05] Barreto, P. S. L. M. and M. Naehrig, "Pairing-Friendly
Elliptic Curves of Prime Order", In Selected Areas in
Cryptography 2005, pages 319-331, DOI 10.1007/11693383_22,
2006, <https://doi.org/10.1007/11693383_22>.
[BP17] Budroni, A. and F. Pintore, "Efficient hash maps to
\mathbb{G}_2 on BLS curves", Cryptology ePrint Archive,
Paper 2017/419, May 2017,
<https://eprint.iacr.org/2017/419>.
[BR93] Bellare, M. and P. Rogaway, "Random oracles are practical:
a paradigm for designing efficient protocols", In
Proceedings of the 1993 ACM Conference on Computer and
Communications Security, pages 62-73,
DOI 10.1145/168588.168596, December 1993,
<https://doi.org/10.1145/168588.168596>.
[C93] Cohen, H., "A Course in Computational Algebraic Number
Theory", Springer-Verlag, ISBN 9783642081422,
DOI 10.1007/978-3-662-02945-9, 1993,
<https://doi.org/10.1007/978-3-662-02945-9>.
[CDMP05] Coron, J.-S., Dodis, Y., Malinaud, C., and P. Puniya,
"Merkle-Damgård Revisited: How to Construct a Hash
Function", In Advances in Cryptology -- CRYPTO 2005, pages
430-448, DOI 10.1007/11535218_26, August 2005,
<https://doi.org/10.1007/11535218_26>.
[CFADLNV05]
Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T.,
Nguyen, K., and F. Vercauteren, "Handbook of Elliptic and
Hyperelliptic Curve Cryptography", Chapman and Hall / CRC,
ISBN 9781584885184, 2005,
<https://www.crcpress.com/9781584885184>.
[CK11] Couveignes, J.-M. and J.-G. Kammerer, "The geometry of
flex tangents to a cubic curve and its parameterizations",
In Journal of Symbolic Computation, vol 47 issue 3, pages
266-281, DOI 10.1016/j.jsc.2011.11.003, March 2012,
<https://doi.org/10.1016/j.jsc.2011.11.003>.
[F11] Farashahi, R. R., "Hashing into Hessian Curves", In
AFRICACRYPT 2011, pages 278-289,
DOI 10.1007/978-3-642-21969-6_17, July 2011,
<https://doi.org/10.1007/978-3-642-21969-6_17>.
[FFSTV13] Farashahi, R. R., Fouque, P.-A., Shparlinski, I. E.,
Tibouchi, M., and J. F. Voloch, "Indifferentiable
deterministic hashing to elliptic and hyperelliptic
curves", In Mathematics of Computation. vol 82, pages
491-512, DOI 10.1090/S0025-5718-2012-02606-8, 2013,
<https://doi.org/10.1090/S0025-5718-2012-02606-8>.
[FIPS180-4]
National Institute of Standards and Technology (NIST),
"Secure Hash Standard (SHS)", FIPS 180-4,
DOI 10.6028/NIST.FIPS.180-4, August 2015,
<https://nvlpubs.nist.gov/nistpubs/FIPS/
NIST.FIPS.180-4.pdf>.
[FIPS186-4]
National Institute of Standards and Technology (NIST),
"Digital Signature Standard (DSS)", FIPS 186-4,
DOI 10.6028/NIST.FIPS.186-4, July 2013,
<https://nvlpubs.nist.gov/nistpubs/FIPS/
NIST.FIPS.186-4.pdf>.
[FIPS202] National Institute of Standards and Technology (NIST),
"SHA-3 Standard: Permutation-Based Hash and Extendable-
Output Functions", FIPS 202, DOI 10.6028/NIST.FIPS.202,
August 2015, <https://nvlpubs.nist.gov/nistpubs/FIPS/
NIST.FIPS.202.pdf>.
[FJT13] Fouque, P.-A., Joux, A., and M. Tibouchi, "Injective
Encodings to Elliptic Curves", In ACISP 2013, pages
203-218, DOI 10.1007/978-3-642-39059-3_14, 2013,
<https://doi.org/10.1007/978-3-642-39059-3_14>.
[FKR11] Fuentes-Castañeda, L., Knapp, E., and F. Rodriguez-
Henriquez, "Faster Hashing to G2", In Selected Areas in
Cryptography, pages 412-430,
DOI 10.1007/978-3-642-28496-0_25, August 2011,
<https://doi.org/10.1007/978-3-642-28496-0_25>.
[FSV09] Farashahi, R. R., Shparlinski, I. E., and J. F. Voloch,
"On hashing into elliptic curves", In Journal of
Mathematical Cryptology, vol 3 no 4, pages 353-360,
DOI 10.1515/JMC.2009.022, March 2009,
<https://doi.org/10.1515/JMC.2009.022>.
[FT10] Fouque, P.-A. and M. Tibouchi, "Estimating the Size of the
Image of Deterministic Hash Functions to Elliptic Curves",
In Progress in Cryptology - LATINCRYPT 2010, pages 81-91,
DOI 10.1007/978-3-642-14712-8_5, August 2010,
<https://doi.org/10.1007/978-3-642-14712-8_5>.
[FT12] Fouque, P.-A. and M. Tibouchi, "Indifferentiable Hashing
to Barreto--Naehrig Curves", In Progress in Cryptology -
LATINCRYPT 2012, pages 1-17,
DOI 10.1007/978-3-642-33481-8_1, 2012,
<https://doi.org/10.1007/978-3-642-33481-8_1>.
[H20] Hamburg, M., "Indifferentiable hashing from Elligator 2",
Cryptology ePrint Archive, Paper 2020/1513, 2020,
<https://eprint.iacr.org/2020/1513>.
[hash2curve-repo]
"Hashing to Elliptic Curves", commit 664b135, June 2022,
<https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve>.
[Icart09] Icart, T., "How to Hash into Elliptic Curves", In Advances
in Cryptology - CRYPTO 2009, pages 303-316,
DOI 10.1007/978-3-642-03356-8_18, August 2009,
<https://doi.org/10.1007/978-3-642-03356-8_18>.
[J96] Jablon, D. P., "Strong password-only authenticated key
exchange", In SIGCOMM Computer Communication Review, vol
26 issue 5, pages 5-26, DOI 10.1145/242896.242897, October
1996, <https://doi.org/10.1145/242896.242897>.
[jubjub-fq]
"zkcrypto/jubjub - fq.rs", 2019,
<https://github.com/zkcrypto/jubjub/pull/18>.
[KLR10] Kammerer, J.-G., Lercier, R., and G. Renault, "Encoding
Points on Hyperelliptic Curves over Finite Fields in
Deterministic Polynomial Time", In Pairing-Based
Cryptography - Pairing 2010, pages 278-297,
DOI 10.1007/978-3-642-17455-1_18, 2010,
<https://doi.org/10.1007/978-3-642-17455-1_18>.
[L13] Langley, A., "Implementing Elligator for Curve25519",
December 2013, <https://www.imperialviolet.org/2013/12/25/
elligator.html>.
[LBB19] Lipp, B., Blanchet, B., and K. Bhargavan, "A Mechanised
Cryptographic Proof of the WireGuard Virtual Private
Network Protocol", In INRIA Research Report 9269, April
2019, <https://hal.inria.fr/hal-02100345/>.
[MOV96] Menezes, A. J., van Oorschot, P. C., and S. A. Vanstone,
"Handbook of Applied Cryptography", CRC Press,
ISBN 9780849385230, October 1996,
<http://cacr.uwaterloo.ca/hac/>.
[MRH04] Maurer, U., Renner, R., and C. Holenstein,
"Indifferentiability, Impossibility Results on Reductions,
and Applications to the Random Oracle Methodology", In TCC
2004: Theory of Cryptography, pages 21-39,
DOI 10.1007/978-3-540-24638-1_2, February 2004,
<https://doi.org/10.1007/978-3-540-24638-1_2>.
[MRV99] Micali, S., Rabin, M., and S. Vadhan, "Verifiable random
functions", 40th Annual Symposium on Foundations of
Computer Science (Cat. No.99CB37039), pages 120-130,
DOI 10.1109/SFFCS.1999.814584, October 1999,
<https://doi.org/10.1109/SFFCS.1999.814584>.
[MT98] Matsumoto, M. and T. Nishimura, "Mersenne twister: A
623-dimensionally equidistributed uniform pseudo-random
number generator", In ACM Transactions on Modeling and
Computer Simulation (TOMACS), vol 8 issue 1, pages 3-30,
DOI 10.1145/272991.272995, January 1998,
<https://doi.org/10.1145/272991.272995>.
[NR97] Naor, M. and O. Reingold, "Number-theoretic constructions
of efficient pseudo-random functions", In Proceedings 38th
Annual Symposium on Foundations of Computer Science, pages
458-467, DOI 10.1109/SFCS.1997.646134, October 1997,
<https://doi.org/10.1109/SFCS.1997.646134>.
[OPRFs] Davidson, A., Faz-Hernandez, A., Sullivan, N., and C. A.
Wood, "Oblivious Pseudorandom Functions (OPRFs) using
Prime-Order Groups", Work in Progress, Internet-Draft,
draft-irtf-cfrg-voprf-21, 21 February 2023,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
voprf-21>.
[p1363.2] IEEE, "IEEE Standard Specification for Password-Based
Public-Key Cryptography Techniques", IEEE 1363.2-2008,
September 2008,
<https://standards.ieee.org/standard/1363_2-2008.html>.
[p1363a] IEEE, "IEEE Standard Specifications for Public-Key
Cryptography - Amendment 1: Additional Techniques", IEEE
1363a-2004, March 2004,
<https://standards.ieee.org/standard/1363a-2004.html>.
[P20] Pornin, T., "Efficient Elliptic Curve Operations On
Microcontrollers With Finite Field Extensions", Cryptology
ePrint Archive, Paper 2020/009, 2020,
<https://eprint.iacr.org/2020/009>.
[RCB16] Renes, J., Costello, C., and L. Batina, "Complete Addition
Formulas for Prime Order Elliptic Curves", In Advances in
Cryptology - EUROCRYPT 2016, pages 403-428,
DOI 10.1007/978-3-662-49890-3_16, April 2016,
<https://doi.org/10.1007/978-3-662-49890-3_16>.
[RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
Hashing for Message Authentication", RFC 2104,
DOI 10.17487/RFC2104, February 1997,
<https://www.rfc-editor.org/info/rfc2104>.
[RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
Key Derivation Function (HKDF)", RFC 5869,
DOI 10.17487/RFC5869, May 2010,
<https://www.rfc-editor.org/info/rfc5869>.
[RFC7693] Saarinen, M., Ed. and J. Aumasson, "The BLAKE2
Cryptographic Hash and Message Authentication Code (MAC)",
RFC 7693, DOI 10.17487/RFC7693, November 2015,
<https://www.rfc-editor.org/info/rfc7693>.
[RFC7914] Percival, C. and S. Josefsson, "The scrypt Password-Based
Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914,
August 2016, <https://www.rfc-editor.org/info/rfc7914>.
[RFC8018] Moriarty, K., Ed., Kaliski, B., and A. Rusch, "PKCS #5:
Password-Based Cryptography Specification Version 2.1",
RFC 8018, DOI 10.17487/RFC8018, January 2017,
<https://www.rfc-editor.org/info/rfc8018>.
[RFC9106] Biryukov, A., Dinu, D., Khovratovich, D., and S.
Josefsson, "Argon2 Memory-Hard Function for Password
Hashing and Proof-of-Work Applications", RFC 9106,
DOI 10.17487/RFC9106, September 2021,
<https://www.rfc-editor.org/info/rfc9106>.
[ristretto255-decaf448]
de Valence, H., Grigg, J., Hamburg, M., Lovecruft, I.,
Tankersley, G., and F. Valsorda, "The ristretto255 and
decaf448 Groups", Work in Progress, Internet-Draft, draft-
irtf-cfrg-ristretto255-decaf448-07, 3 April 2023,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
ristretto255-decaf448-07>.
[RSS11] Ristenpart, T., Shacham, H., and T. Shrimpton, "Careful
with Composition: Limitations of the Indifferentiability
Framework", In Advances in Cryptology - EUROCRYPT 2011,
pages 487-506, DOI 10.1007/978-3-642-20465-4_27, May 2011,
<https://doi.org/10.1007/978-3-642-20465-4_27>.
[S05] SkaÅba, M., "Points on elliptic curves over finite
fields", In Acta Arithmetica, vol 117 no 3, pages 293-301,
DOI 10.4064/aa117-3-7, 2005,
<https://doi.org/10.4064/aa117-3-7>.
[S85] Schoof, R., "Elliptic curves over finite fields and the
computation of square roots mod p", In Mathematics of
Computation, vol 44 issue 170, pages 483-494,
DOI 10.1090/S0025-5718-1985-0777280-6, April 1985,
<https://doi.org/10.1090/S0025-5718-1985-0777280-6>.
[SAGE] The Sage Developers, "SageMath, the Sage Mathematics
Software System", <https://www.sagemath.org>.
[SBCDK09] Scott, M., Benger, N., Charlemagne, M., Dominguez Perez,
L. J., and E. J. Kachisa, "Fast Hashing to G2 on Pairing-
Friendly Curves", In Pairing-Based Cryptography - Pairing
2009, pages 102-113, DOI 10.1007/978-3-642-03298-1_8,
August 2009,
<https://doi.org/10.1007/978-3-642-03298-1_8>.
[SEC1] Standards for Efficient Cryptography Group (SECG), "SEC 1:
Elliptic Curve Cryptography", May 2009,
<http://www.secg.org/sec1-v2.pdf>.
[SEC2] Standards for Efficient Cryptography Group (SECG), "SEC 2:
Recommended Elliptic Curve Domain Parameters", January
2010, <http://www.secg.org/sec2-v2.pdf>.
[SS04] Schinzel, A. and M. SkaÅba, "On equations y^2 = x^n + k in
a finite field", In Bulletin Polish Academy of Sciences.
Mathematics, vol 52 no 3, pages 223-226,
DOI 10.4064/ba52-3-1, 2004,
<https://doi.org/10.4064/ba52-3-1>.
[SW06] Shallue, A. and C. E. van de Woestijne, "Construction of
Rational Points on Elliptic Curves over Finite Fields", In
Algorithmic Number Theory - ANTS 2006, pages 510-524,
DOI 10.1007/11792086_36, July 2006,
<https://doi.org/10.1007/11792086_36>.
[T14] Tibouchi, M., "Elligator Squared: Uniform Points on
Elliptic Curves of Prime Order as Uniform Random Strings",
In Financial Cryptography and Data Security - FC 2014,
pages 139-156, DOI 10.1007/978-3-662-45472-5_10, November
2014, <https://doi.org/10.1007/978-3-662-45472-5_10>.
[TK17] Tibouchi, M. and T. Kim, "Improved elliptic curve hashing
and point representation", In Designs, Codes, and
Cryptography, vol 82, pages 161-177,
DOI 10.1007/s10623-016-0288-2, January 2017,
<https://doi.org/10.1007/s10623-016-0288-2>.
[U07] Ulas, M., "Rational Points on Certain Hyperelliptic Curves
over Finite Fields", In Bulletin Polish Academy of
Science. Mathematics, vol 55 no 2, pages 97-104,
DOI 10.4064/ba55-2-1, July 2007,
<https://doi.org/10.4064/ba55-2-1>.
[VR20] Vanhoef, M. and E. Ronen, "Dragonblood: Analyzing the
Dragonfly Handshake of WPA3 and EAP-pwd", In IEEE
Symposium on Security & Privacy (SP), May 2020,
<https://eprint.iacr.org/2019/383>.
[VRF] Goldberg, S., Reyzin, L., Papadopoulos, D., and J. VÄelák,
"Verifiable Random Functions (VRFs)", Work in Progress,
Internet-Draft, draft-irtf-cfrg-vrf-15, 9 August 2022,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
vrf-15>.
[W08] Washington, L. C., "Elliptic Curves: Number Theory and
Cryptography, Second Edition", Chapman and Hall / CRC,
ISBN 9781420071467, April 2008,
<https://www.crcpress.com/9781420071467>.
[W19] Wahby, R. S., "An explicit, generic parameterization for
the Shallue--van de Woestijne map", commit e2a625f, March
2020, <https://github.com/cfrg/draft-irtf-cfrg-hash-to-
curve/blob/draft-irtf-cfrg-hash-to-curve-14/doc/
svdw_params.pdf>.
[WB19] Wahby, R. S. and D. Boneh, "Fast and simple constant-time
hashing to the BLS12-381 elliptic curve", In IACR
Transactions on Cryptographic Hardware and Embedded
Systems, vol 2019 issue 4, Cryptology ePrint Archive,
Paper 2019/403, DOI 10.13154/tches.v2019.i4.154-179,
August 2019, <https://eprint.iacr.org/2019/403>.
Appendix A. Related Work
The problem of mapping arbitrary bit strings to elliptic curve points
has been the subject of both practical and theoretical research.
This section briefly describes the background and research results
that underlie the recommendations in this document. This section is
provided for informational purposes only.
A naive but generally insecure method of mapping a string msg to a
point on an elliptic curve E having n points is to first fix a point
P that generates the elliptic curve group, and a hash function Hn
from bit strings to integers less than n; then compute Hn(msg) * P,
where the * operator represents scalar multiplication. The reason
this approach is insecure is that the resulting point has a known
discrete log relationship to P. Thus, except in cases where this
method is specified by the protocol, it must not be used; doing so
risks catastrophic security failures.
Boneh et al. [BLS01] describe an encoding method they call
MapToGroup, which works roughly as follows: first, use the input
string to initialize a pseudorandom number generator, then use the
generator to produce a value x in F. If x is the x-coordinate of a
point on the elliptic curve, output that point. Otherwise, generate
a new value x in F and try again. Since a random value x in F has
probability about 1/2 of corresponding to a point on the curve, the
expected number of tries is just two. However, the running time of
this method, which is generally referred to as a probabilistic try-
and-increment algorithm, depends on the input string. As such, it is
not safe to use in protocols sensitive to timing side channels, as
was exemplified by the Dragonblood attack [VR20].
Schinzel and Skalba [SS04] introduce a method of constructing
elliptic curve points deterministically, for a restricted class of
curves and a very small number of points. Skalba [S05] generalizes
this construction to more curves and more points on those curves.
Shallue and van de Woestijne [SW06] further generalize and simplify
Skalba's construction, yielding concretely efficient maps to a
constant fraction of the points on almost any curve. Fouque and
Tibouchi [FT12] give a parameterization of this mapping for Barreto-
Naehrig pairing-friendly curves [BN05].
Ulas [U07] describes a simpler version of the Shallue-van de
Woestijne map, and Brier et al. [BCIMRT10] give a further
simplification, which the authors call the "Simplified SWU" map.
That simplified map applies only to fields of characteristic p = 3
(mod 4); Wahby and Boneh [WB19] generalize to fields of any
characteristic and give further optimizations.
Boneh and Franklin give a deterministic algorithm mapping to certain
supersingular curves over fields of characteristic p = 2 (mod 3)
[BF01]. Icart gives another deterministic algorithm that maps to any
curve over a field of characteristic p = 2 (mod 3) [Icart09].
Several extensions and generalizations follow this work, including
[FSV09], [FT10], [KLR10], [F11], and [CK11].
Following the work of Farashahi [F11], Fouque et al. [FJT13] describe
a mapping to curves over fields of characteristic p = 3 (mod 4)
having a number of points divisible by 4. Bernstein et al. [BHKL13]
optimize this mapping and describe a related mapping that they call
"Elligator 2," which applies to any curve over a field of odd
characteristic having a point of order 2. This includes Curve25519
and Curve448, both of which are CFRG-recommended curves [RFC7748].
Bernstein et al. [BLMP19] extend the Elligator 2 map to a class of
supersingular curves over fields of characteristic p = 3 (mod 4).
An important caveat regarding all of the above deterministic mapping
functions is that none of them map to the entire curve, but rather to
some fraction of the points. This means that they cannot be used
directly to construct a random oracle that outputs points on the
curve.
Brier et al. [BCIMRT10] give two solutions to this problem. The
first, which Brier et al. prove applies to Icart's method, computes
f(H0(msg)) + f(H1(msg)) for two distinct hash functions H0 and H1
from bit strings to F and a mapping f from F to the elliptic curve E.
The second, which applies to essentially all deterministic mappings
but is more costly, computes f(H0(msg)) + H2(msg) * P, where P is a
generator of the elliptic curve group, H2 is a hash from bit strings
to integers modulo r, and r is the order of the elliptic curve group.
Farashahi et al. [FFSTV13] improve the analysis of the first method,
showing that it applies to essentially all deterministic mappings.
Tibouchi and Kim [TK17] further refine the analysis and describe
additional optimizations.
Complementary to the problem of mapping from bit strings to elliptic
curve points, Bernstein et al. [BHKL13] study the problem of mapping
from elliptic curve points to uniformly random bit strings, giving
solutions for a class of curves that includes Montgomery and twisted
Edwards curves. Tibouchi [T14] and Aranha et al. [AFQTZ14]
generalize these results. This document does not deal with this
complementary problem.
Appendix B. Hashing to ristretto255
ristretto255 [ristretto255-decaf448] provides a prime-order group
based on curve25519 [RFC7748]. This section describes
hash_to_ristretto255, which implements a random-oracle encoding to
this group that has a uniform output distribution (Section 2.2.3) and
the same security properties and interface as the hash_to_curve
function (Section 3).
The ristretto255 API defines a one-way map ([ristretto255-decaf448],
Section 4.3.4); this section refers to that map as ristretto255_map.
The hash_to_ristretto255 function MUST be instantiated with an
expand_message function that conforms to the requirements given in
Section 5.3. In addition, it MUST use a domain separation tag
constructed as described in Section 3.1, and all domain separation
recommendations given in Section 10.7 apply when implementing
protocols that use hash_to_ristretto255.
hash_to_ristretto255(msg)
Parameters:
- DST, a domain separation tag (see discussion above).
- expand_message, a function that expands a byte string and
domain separation tag into a uniformly random byte string
(see discussion above).
- ristretto255_map, the one-way map from the ristretto255 API.
Input: msg, an arbitrary-length byte string.
Output: P, an element of the ristretto255 group.
Steps:
1. uniform_bytes = expand_message(msg, DST, 64)
2. P = ristretto255_map(uniform_bytes)
3. return P
Since hash_to_ristretto255 is not a hash-to-curve suite, it does not
have a Suite ID. If a similar identifier is needed, it MUST be
constructed following the guidelines in Section 8.10, with the
following parameters:
* CURVE_ID: "ristretto255"
* HASH_ID: as described in Section 8.10
* MAP_ID: "R255MAP"
* ENC_VAR: "RO"
For example, if expand_message is expand_message_xmd using SHA-512,
the REQUIRED identifier is:
ristretto255_XMD:SHA-512_R255MAP_RO_
Appendix C. Hashing to decaf448
Similar to ristretto255, decaf448 [ristretto255-decaf448] provides a
prime-order group based on curve448 [RFC7748]. This section
describes hash_to_decaf448, which implements a random-oracle encoding
to this group that has a uniform output distribution (Section 2.2.3)
and the same security properties and interface as the hash_to_curve
function (Section 3).
The decaf448 API defines a one-way map ([ristretto255-decaf448],
Section 5.3.4); this section refers to that map as decaf448_map.
The hash_to_decaf448 function MUST be instantiated with an
expand_message function that conforms to the requirements given in
Section 5.3. In addition, it MUST use a domain separation tag
constructed as described in Section 3.1, and all domain separation
recommendations given in Section 10.7 apply when implementing
protocols that use hash_to_decaf448.
hash_to_decaf448(msg)
Parameters:
- DST, a domain separation tag (see discussion above).
- expand_message, a function that expands a byte string and
domain separation tag into a uniformly random byte string
(see discussion above).
- decaf448_map, the one-way map from the decaf448 API.
Input: msg, an arbitrary-length byte string.
Output: P, an element of the decaf448 group.
Steps:
1. uniform_bytes = expand_message(msg, DST, 112)
2. P = decaf448_map(uniform_bytes)
3. return P
Since hash_to_decaf448 is not a hash-to-curve suite, it does not have
a Suite ID. If a similar identifier is needed, it MUST be
constructed following the guidelines in Section 8.10, with the
following parameters:
* CURVE_ID: "decaf448"
* HASH_ID: as described in Section 8.10
* MAP_ID: "D448MAP"
* ENC_VAR: "RO"
For example, if expand_message is expand_message_xof using SHAKE256,
the REQUIRED identifier is:
decaf448_XOF:SHAKE256_D448MAP_RO_
Appendix D. Rational Maps
This section gives rational maps that can be used when hashing to
twisted Edwards or Montgomery curves.
Given a twisted Edwards curve, Appendix D.1 shows how to derive a
corresponding Montgomery curve and how to map from that curve to the
twisted Edwards curve. This mapping may be used when hashing to
twisted Edwards curves as described in Section 6.8.
Given a Montgomery curve, Appendix D.2 shows how to derive a
corresponding Weierstrass curve and how to map from that curve to the
Montgomery curve. This mapping can be used to hash to Montgomery or
twisted Edwards curves via the Shallue-van de Woestijne method
(Section 6.6.1) or Simplified SWU method (Section 6.6.2), as follows:
* For Montgomery curves, first map to the Weierstrass curve, then
convert to Montgomery coordinates via the mapping.
* For twisted Edwards curves, compose the mapping from Weierstrass
to Montgomery with the mapping from Montgomery to twisted Edwards
(Appendix D.1) to obtain a Weierstrass curve and a mapping to the
target twisted Edwards curve. Map to this Weierstrass curve, then
convert to Edwards coordinates via the mapping.
D.1. Generic Mapping from Montgomery to Twisted Edwards
This section gives a generic birational map between twisted Edwards
and Montgomery curves.
The map in this section is a simplified version of the map given in
[BBJLP08], Theorem 3.2. Specifically, this section's map handles
exceptional cases in a simplified way that is geared towards hashing
to a twisted Edwards curve's prime-order subgroup.
The twisted Edwards curve
a * v^2 + w^2 = 1 + d * v^2 * w^2
is birationally equivalent to the Montgomery curve
K * t^2 = s^3 + J * s^2 + s
which has the form required by the Elligator 2 mapping of
Section 6.7.1. The coefficients of the Montgomery curve are
* J = 2 * (a + d) / (a - d)
* K = 4 / (a - d)
The rational map from the point (s, t) on the above Montgomery curve
to the point (v, w) on the twisted Edwards curve is given by
* v = s / t
* w = (s - 1) / (s + 1)
This mapping is undefined when t == 0 or s == -1, i.e., when the
denominator of either of the above rational functions is zero.
Implementations MUST detect exceptional cases and return the value
(v, w) = (0, 1), which is the identity point on all twisted Edwards
curves.
The following straight-line implementation of the above rational map
handles the exceptional cases.
monty_to_edw_generic(s, t)
Input: (s, t), a point on the curve K * t^2 = s^3 + J * s^2 + s.
Output: (v, w), a point on an equivalent twisted Edwards curve.
1. tv1 = s + 1
2. tv2 = tv1 * t # (s + 1) * t
3. tv2 = inv0(tv2) # 1 / ((s + 1) * t)
4. v = tv2 * tv1 # 1 / t
5. v = v * s # s / t
6. w = tv2 * t # 1 / (s + 1)
7. tv1 = s - 1
8. w = w * tv1 # (s - 1) / (s + 1)
9. e = tv2 == 0
10. w = CMOV(w, 1, e) # handle exceptional case
11. return (v, w)
For completeness, we also give the inverse relations. (Note that
this map is not required when hashing to twisted Edwards curves.)
The coefficients of the twisted Edwards curve corresponding to the
above Montgomery curve are
* a = (J + 2) / K
* d = (J - 2) / K
The rational map from the point (v, w) on the twisted Edwards curve
to the point (s, t) on the Montgomery curve is given by
* s = (1 + w) / (1 - w)
* t = (1 + w) / (v * (1 - w))
The mapping is undefined when v == 0 or w == 1. When the goal is to
map into the prime-order subgroup of the Montgomery curve, it
suffices to return the identity point on the Montgomery curve in the
exceptional cases.
D.2. Mapping from Weierstrass to Montgomery
The rational map from the point (s, t) on the Montgomery curve
K * t^2 = s^3 + J * s^2 + s
to the point (x, y) on the equivalent Weierstrass curve
y^2 = x^3 + A * x + B
is given by
* A = (3 - J^2) / (3 * K^2)
* B = (2 * J^3 - 9 * J) / (27 * K^3)
* x = (3 * s + J) / (3 * K)
* y = t / K
The inverse map, from the point (x, y) to the point (s, t), is given
by
* s = (3 * K * x - J) / 3
* t = y * K
This mapping can be used to apply the Shallue-van de Woestijne method
(Section 6.6.1) or Simplified SWU method (Section 6.6.2) to
Montgomery curves.
Appendix E. Isogeny Maps for Suites
This section specifies the isogeny maps for the secp256k1 and
BLS12-381 suites listed in Section 8.
These maps are given in terms of affine coordinates. Wahby and Boneh
([WB19], Section 4.3) show how to evaluate these maps in a projective
coordinate system (Appendix G.1), which avoids modular inversions.
Refer to [hash2curve-repo] for a Sage [SAGE] script that constructs
these isogenies.
E.1. 3-Isogeny Map for secp256k1
This section specifies the isogeny map for the secp256k1 suite listed
in Section 8.7.
The 3-isogeny map from (x', y') on E' to (x, y) on E is given by the
following rational functions:
* x = x_num / x_den, where
- x_num = k_(1,3) * x'^3 + k_(1,2) * x'^2 + k_(1,1) * x' +
k_(1,0)
- x_den = x'^2 + k_(2,1) * x' + k_(2,0)
* y = y' * y_num / y_den, where
- y_num = k_(3,3) * x'^3 + k_(3,2) * x'^2 + k_(3,1) * x' +
k_(3,0)
- y_den = x'^3 + k_(4,2) * x'^2 + k_(4,1) * x' + k_(4,0)
The constants used to compute x_num are as follows:
* k_(1,0) =
0x8e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38daaaaa8c7
* k_(1,1) =
0x7d3d4c80bc321d5b9f315cea7fd44c5d595d2fc0bf63b92dfff1044f17c6581
* k_(1,2) =
0x534c328d23f234e6e2a413deca25caece4506144037c40314ecbd0b53d9dd262
* k_(1,3) =
0x8e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38daaaaa88c
The constants used to compute x_den are as follows:
* k_(2,0) =
0xd35771193d94918a9ca34ccbb7b640dd86cd409542f8487d9fe6b745781eb49b
* k_(2,1) =
0xedadc6f64383dc1df7c4b2d51b54225406d36b641f5e41bbc52a56612a8c6d14
The constants used to compute y_num are as follows:
* k_(3,0) =
0x4bda12f684bda12f684bda12f684bda12f684bda12f684bda12f684b8e38e23c
* k_(3,1) =
0xc75e0c32d5cb7c0fa9d0a54b12a0a6d5647ab046d686da6fdffc90fc201d71a3
* k_(3,2) =
0x29a6194691f91a73715209ef6512e576722830a201be2018a765e85a9ecee931
* k_(3,3) =
0x2f684bda12f684bda12f684bda12f684bda12f684bda12f684bda12f38e38d84
The constants used to compute y_den are as follows:
* k_(4,0) =
0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffff93b
* k_(4,1) =
0x7a06534bb8bdb49fd5e9e6632722c2989467c1bfc8e8d978dfb425d2685c2573
* k_(4,2) =
0x6484aa716545ca2cf3a70c3fa8fe337e0a3d21162f0d6299a7bf8192bfd2a76f
E.2. 11-Isogeny Map for BLS12-381 G1
The 11-isogeny map from (x', y') on E' to (x, y) on E is given by the
following rational functions:
* x = x_num / x_den, where
- x_num = k_(1,11) * x'^11 + k_(1,10) * x'^10 + k_(1,9) * x'^9 +
... + k_(1,0)
- x_den = x'^10 + k_(2,9) * x'^9 + k_(2,8) * x'^8 + ... + k_(2,0)
* y = y' * y_num / y_den, where
- y_num = k_(3,15) * x'^15 + k_(3,14) * x'^14 + k_(3,13) * x'^13
+ ... + k_(3,0)
- y_den = x'^15 + k_(4,14) * x'^14 + k_(4,13) * x'^13 + ... +
k_(4,0)
The constants used to compute x_num are as follows:
* k_(1,0) = 0x11a05f2b1e833340b809101dd99815856b303e88a2d7005ff2627b
56cdb4e2c85610c2d5f2e62d6eaeac1662734649b7
* k_(1,1) = 0x17294ed3e943ab2f0588bab22147a81c7c17e75b2f6a8417f565e3
3c70d1e86b4838f2a6f318c356e834eef1b3cb83bb
* k_(1,2) = 0xd54005db97678ec1d1048c5d10a9a1bce032473295983e56878e50
1ec68e25c958c3e3d2a09729fe0179f9dac9edcb0
* k_(1,3) = 0x1778e7166fcc6db74e0609d307e55412d7f5e4656a8dbf25f1b332
89f1b330835336e25ce3107193c5b388641d9b6861
* k_(1,4) = 0xe99726a3199f4436642b4b3e4118e5499db995a1257fb3f086eeb6
5982fac18985a286f301e77c451154ce9ac8895d9
* k_(1,5) = 0x1630c3250d7313ff01d1201bf7a74ab5db3cb17dd952799b9ed3ab
9097e68f90a0870d2dcae73d19cd13c1c66f652983
* k_(1,6) = 0xd6ed6553fe44d296a3726c38ae652bfb11586264f0f8ce19008e21
8f9c86b2a8da25128c1052ecaddd7f225a139ed84
* k_(1,7) = 0x17b81e7701abdbe2e8743884d1117e53356de5ab275b4db1a682c6
2ef0f2753339b7c8f8c8f475af9ccb5618e3f0c88e
* k_(1,8) = 0x80d3cf1f9a78fc47b90b33563be990dc43b756ce79f5574a2c596c
928c5d1de4fa295f296b74e956d71986a8497e317
* k_(1,9) = 0x169b1f8e1bcfa7c42e0c37515d138f22dd2ecb803a0c5c99676314
baf4bb1b7fa3190b2edc0327797f241067be390c9e
* k_(1,10) = 0x10321da079ce07e272d8ec09d2565b0dfa7dccdde6787f96d50af
36003b14866f69b771f8c285decca67df3f1605fb7b
* k_(1,11) = 0x6e08c248e260e70bd1e962381edee3d31d79d7e22c837bc23c0bf
1bc24c6b68c24b1b80b64d391fa9c8ba2e8ba2d229
The constants used to compute x_den are as follows:
* k_(2,0) = 0x8ca8d548cff19ae18b2e62f4bd3fa6f01d5ef4ba35b48ba9c95886
17fc8ac62b558d681be343df8993cf9fa40d21b1c
* k_(2,1) = 0x12561a5deb559c4348b4711298e536367041e8ca0cf0800c0126c2
588c48bf5713daa8846cb026e9e5c8276ec82b3bff
* k_(2,2) = 0xb2962fe57a3225e8137e629bff2991f6f89416f5a718cd1fca64e0
0b11aceacd6a3d0967c94fedcfcc239ba5cb83e19
* k_(2,3) = 0x3425581a58ae2fec83aafef7c40eb545b08243f16b1655154cca8a
bc28d6fd04976d5243eecf5c4130de8938dc62cd8
* k_(2,4) = 0x13a8e162022914a80a6f1d5f43e7a07dffdfc759a12062bb8d6b44
e833b306da9bd29ba81f35781d539d395b3532a21e
* k_(2,5) = 0xe7355f8e4e667b955390f7f0506c6e9395735e9ce9cad4d0a43bce
f24b8982f7400d24bc4228f11c02df9a29f6304a5
* k_(2,6) = 0x772caacf16936190f3e0c63e0596721570f5799af53a1894e2e073
062aede9cea73b3538f0de06cec2574496ee84a3a
* k_(2,7) = 0x14a7ac2a9d64a8b230b3f5b074cf01996e7f63c21bca68a81996e1
cdf9822c580fa5b9489d11e2d311f7d99bbdcc5a5e
* k_(2,8) = 0xa10ecf6ada54f825e920b3dafc7a3cce07f8d1d7161366b74100da
67f39883503826692abba43704776ec3a79a1d641
* k_(2,9) = 0x95fc13ab9e92ad4476d6e3eb3a56680f682b4ee96f7d03776df533
978f31c1593174e4b4b7865002d6384d168ecdd0a
The constants used to compute y_num are as follows:
* k_(3,0) = 0x90d97c81ba24ee0259d1f094980dcfa11ad138e48a869522b52af6
c956543d3cd0c7aee9b3ba3c2be9845719707bb33
* k_(3,1) = 0x134996a104ee5811d51036d776fb46831223e96c254f383d0f9063
43eb67ad34d6c56711962fa8bfe097e75a2e41c696
* k_(3,2) = 0xcc786baa966e66f4a384c86a3b49942552e2d658a31ce2c344be4b
91400da7d26d521628b00523b8dfe240c72de1f6
* k_(3,3) = 0x1f86376e8981c217898751ad8746757d42aa7b90eeb791c09e4a3e
c03251cf9de405aba9ec61deca6355c77b0e5f4cb
* k_(3,4) = 0x8cc03fdefe0ff135caf4fe2a21529c4195536fbe3ce50b879833fd
221351adc2ee7f8dc099040a841b6daecf2e8fedb
* k_(3,5) = 0x16603fca40634b6a2211e11db8f0a6a074a7d0d4afadb7bd76505c
3d3ad5544e203f6326c95a807299b23ab13633a5f0
* k_(3,6) = 0x4ab0b9bcfac1bbcb2c977d027796b3ce75bb8ca2be184cb5231413
c4d634f3747a87ac2460f415ec961f8855fe9d6f2
* k_(3,7) = 0x987c8d5333ab86fde9926bd2ca6c674170a05bfe3bdd81ffd038da
6c26c842642f64550fedfe935a15e4ca31870fb29
* k_(3,8) = 0x9fc4018bd96684be88c9e221e4da1bb8f3abd16679dc26c1e8b6e6
a1f20cabe69d65201c78607a360370e577bdba587
* k_(3,9) = 0xe1bba7a1186bdb5223abde7ada14a23c42a0ca7915af6fe06985e7
ed1e4d43b9b3f7055dd4eba6f2bafaaebca731c30
* k_(3,10) = 0x19713e47937cd1be0dfd0b8f1d43fb93cd2fcbcb6caf493fd1183
e416389e61031bf3a5cce3fbafce813711ad011c132
* k_(3,11) = 0x18b46a908f36f6deb918c143fed2edcc523559b8aaf0c2462e6bf
e7f911f643249d9cdf41b44d606ce07c8a4d0074d8e
* k_(3,12) = 0xb182cac101b9399d155096004f53f447aa7b12a3426b08ec02710
e807b4633f06c851c1919211f20d4c04f00b971ef8
* k_(3,13) = 0x245a394ad1eca9b72fc00ae7be315dc757b3b080d4c158013e663
2d3c40659cc6cf90ad1c232a6442d9d3f5db980133
* k_(3,14) = 0x5c129645e44cf1102a159f748c4a3fc5e673d81d7e86568d9ab0f
5d396a7ce46ba1049b6579afb7866b1e715475224b
* k_(3,15) = 0x15e6be4e990f03ce4ea50b3b42df2eb5cb181d8f84965a3957add
4fa95af01b2b665027efec01c7704b456be69c8b604
The constants used to compute y_den are as follows:
* k_(4,0) = 0x16112c4c3a9c98b252181140fad0eae9601a6de578980be6eec323
2b5be72e7a07f3688ef60c206d01479253b03663c1
* k_(4,1) = 0x1962d75c2381201e1a0cbd6c43c348b885c84ff731c4d59ca4a103
56f453e01f78a4260763529e3532f6102c2e49a03d
* k_(4,2) = 0x58df3306640da276faaae7d6e8eb15778c4855551ae7f310c35a5d
d279cd2eca6757cd636f96f891e2538b53dbf67f2
* k_(4,3) = 0x16b7d288798e5395f20d23bf89edb4d1d115c5dbddbcd30e123da4
89e726af41727364f2c28297ada8d26d98445f5416
* k_(4,4) = 0xbe0e079545f43e4b00cc912f8228ddcc6d19c9f0f69bbb0542eda0
fc9dec916a20b15dc0fd2ededda39142311a5001d
* k_(4,5) = 0x8d9e5297186db2d9fb266eaac783182b70152c65550d881c5ecd87
b6f0f5a6449f38db9dfa9cce202c6477faaf9b7ac
* k_(4,6) = 0x166007c08a99db2fc3ba8734ace9824b5eecfdfa8d0cf8ef5dd365
bc400a0051d5fa9c01a58b1fb93d1a1399126a775c
* k_(4,7) = 0x16a3ef08be3ea7ea03bcddfabba6ff6ee5a4375efa1f4fd7feb34f
d206357132b920f5b00801dee460ee415a15812ed9
* k_(4,8) = 0x1866c8ed336c61231a1be54fd1d74cc4f9fb0ce4c6af5920abc575
0c4bf39b4852cfe2f7bb9248836b233d9d55535d4a
* k_(4,9) = 0x167a55cda70a6e1cea820597d94a84903216f763e13d87bb530859
2e7ea7d4fbc7385ea3d529b35e346ef48bb8913f55
* k_(4,10) = 0x4d2f259eea405bd48f010a01ad2911d9c6dd039bb61a6290e591b
36e636a5c871a5c29f4f83060400f8b49cba8f6aa8
* k_(4,11) = 0xaccbb67481d033ff5852c1e48c50c477f94ff8aefce42d28c0f9a
88cea7913516f968986f7ebbea9684b529e2561092
* k_(4,12) = 0xad6b9514c767fe3c3613144b45f1496543346d98adf02267d5cee
f9a00d9b8693000763e3b90ac11e99b138573345cc
* k_(4,13) = 0x2660400eb2e4f3b628bdd0d53cd76f2bf565b94e72927c1cb748d
f27942480e420517bd8714cc80d1fadc1326ed06f7
* k_(4,14) = 0xe0fa1d816ddc03e6b24255e0d7819c171c40f65e273b853324efc
d6356caa205ca2f570f13497804415473a1d634b8f
E.3. 3-Isogeny Map for BLS12-381 G2
The 3-isogeny map from (x', y') on E' to (x, y) on E is given by the
following rational functions:
* x = x_num / x_den, where
- x_num = k_(1,3) * x'^3 + k_(1,2) * x'^2 + k_(1,1) * x' +
k_(1,0)
- x_den = x'^2 + k_(2,1) * x' + k_(2,0)
* y = y' * y_num / y_den, where
- y_num = k_(3,3) * x'^3 + k_(3,2) * x'^2 + k_(3,1) * x' +
k_(3,0)
- y_den = x'^3 + k_(4,2) * x'^2 + k_(4,1) * x' + k_(4,0)
The constants used to compute x_num are as follows:
* k_(1,0) = 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b5842
3c50ae15d5c2638e343d9c71c6238aaaaaaaa97d6 + 0x5c759507e8e333ebb5b7
a9a47d7ed8532c52d39fd3a042a88b58423c50ae15d5c2638e343d9c71c6238aaa
aaaaa97d6 * I
* k_(1,1) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c
6b4f20a4181472aaa9cb8d555526a9ffffffffc71a * I
* k_(1,2) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c
6b4f20a4181472aaa9cb8d555526a9ffffffffc71e + 0x8ab05f8bdd54cde1909
37e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ff
ffffffe38d * I
* k_(1,3) = 0x171d6541fa38ccfaed6dea691f5fb614cb14b4e7f4e810aa22d610
8f142b85757098e38d0f671c7188e2aaaaaaaa5ed1
The constants used to compute x_den are as follows:
* k_(2,0) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2
a0f6b0f6241eabfffeb153ffffb9feffffffffaa63 * I
* k_(2,1) = 0xc + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf
6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa9f * I
The constants used to compute y_num are as follows:
* k_(3,0) = 0x1530477c7ab4113b59a4c18b076d11930f7da5d4a07f649bf54439
d87d27e500fc8c25ebf8c92f6812cfc71c71c6d706 + 0x1530477c7ab4113b59a
4c18b076d11930f7da5d4a07f649bf54439d87d27e500fc8c25ebf8c92f6812cfc
71c71c6d706 * I
* k_(3,1) = 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b5842
3c50ae15d5c2638e343d9c71c6238aaaaaaaa97be * I
* k_(3,2) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c
6b4f20a4181472aaa9cb8d555526a9ffffffffc71c + 0x8ab05f8bdd54cde1909
37e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ff
ffffffe38f * I
* k_(3,3) = 0x124c9ad43b6cf79bfbf7043de3811ad0761b0f37a1e26286b0e977
c69aa274524e79097a56dc4bd9e1b371c71c718b10
The constants used to compute y_den are as follows:
* k_(4,0) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2
a0f6b0f6241eabfffeb153ffffb9feffffffffa8fb + 0x1a0111ea397fe69a4b1
ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9fef
fffffffa8fb * I
* k_(4,1) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2
a0f6b0f6241eabfffeb153ffffb9feffffffffa9d3 * I
* k_(4,2) = 0x12 + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512b
f6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa99 * I
Appendix F. Straight-Line Implementations of Deterministic Mappings
This section gives straight-line implementations of the mappings of
Section 6. These implementations are generic, i.e., they are defined
for any curve and field. Appendix G gives further implementations
that are optimized for specific classes of curves and fields.
F.1. Shallue-van de Woestijne Method
This section gives a straight-line implementation of the Shallue-van
de Woestijne method for any Weierstrass curve of the form given in
Section 6.6. See Section 6.6.1 for information on the constants used
in this mapping.
Note that the constant c3 below MUST be chosen such that sgn0(c3) =
0. In other words, if the square-root computation returns a value cx
such that sgn0(cx) = 1, set c3 = -cx; otherwise, set c3 = cx.
map_to_curve_svdw(u)
Input: u, an element of F.
Output: (x, y), a point on E.
Constants:
1. c1 = g(Z)
2. c2 = -Z / 2
3. c3 = sqrt(-g(Z) * (3 * Z^2 + 4 * A)) # sgn0(c3) MUST equal 0
4. c4 = -4 * g(Z) / (3 * Z^2 + 4 * A)
Steps:
1. tv1 = u^2
2. tv1 = tv1 * c1
3. tv2 = 1 + tv1
4. tv1 = 1 - tv1
5. tv3 = tv1 * tv2
6. tv3 = inv0(tv3)
7. tv4 = u * tv1
8. tv4 = tv4 * tv3
9. tv4 = tv4 * c3
10. x1 = c2 - tv4
11. gx1 = x1^2
12. gx1 = gx1 + A
13. gx1 = gx1 * x1
14. gx1 = gx1 + B
15. e1 = is_square(gx1)
16. x2 = c2 + tv4
17. gx2 = x2^2
18. gx2 = gx2 + A
19. gx2 = gx2 * x2
20. gx2 = gx2 + B
21. e2 = is_square(gx2) AND NOT e1 # Avoid short-circuit logic ops
22. x3 = tv2^2
23. x3 = x3 * tv3
24. x3 = x3^2
25. x3 = x3 * c4
26. x3 = x3 + Z
27. x = CMOV(x3, x1, e1) # x = x1 if gx1 is square, else x = x3
28. x = CMOV(x, x2, e2) # x = x2 if gx2 is square and gx1 is not
29. gx = x^2
30. gx = gx + A
31. gx = gx * x
32. gx = gx + B
33. y = sqrt(gx)
34. e3 = sgn0(u) == sgn0(y)
35. y = CMOV(-y, y, e3) # Select correct sign of y
36. return (x, y)
F.2. Simplified SWU Method
This section gives a straight-line implementation of the Simplified
SWU method for any Weierstrass curve of the form given in
Section 6.6. See Section 6.6.2 for information on the constants used
in this mapping.
This optimized, straight-line procedure applies to any base field.
The sqrt_ratio subroutine is defined in Appendix F.2.1.
map_to_curve_simple_swu(u)
Input: u, an element of F.
Output: (x, y), a point on E.
Steps:
1. tv1 = u^2
2. tv1 = Z * tv1
3. tv2 = tv1^2
4. tv2 = tv2 + tv1
5. tv3 = tv2 + 1
6. tv3 = B * tv3
7. tv4 = CMOV(Z, -tv2, tv2 != 0)
8. tv4 = A * tv4
9. tv2 = tv3^2
10. tv6 = tv4^2
11. tv5 = A * tv6
12. tv2 = tv2 + tv5
13. tv2 = tv2 * tv3
14. tv6 = tv6 * tv4
15. tv5 = B * tv6
16. tv2 = tv2 + tv5
17. x = tv1 * tv3
18. (is_gx1_square, y1) = sqrt_ratio(tv2, tv6)
19. y = tv1 * u
20. y = y * y1
21. x = CMOV(x, tv3, is_gx1_square)
22. y = CMOV(y, y1, is_gx1_square)
23. e1 = sgn0(u) == sgn0(y)
24. y = CMOV(-y, y, e1)
25. x = x / tv4
26. return (x, y)
F.2.1. sqrt_ratio Subroutine
This section defines three variants of the sqrt_ratio subroutine used
by the above procedure. The first variant can be used with any
field; the others are optimized versions for specific fields.
The routines given in this section depend on the constant Z from the
Simplified SWU map. For correctness, sqrt_ratio and
map_to_curve_simple_swu MUST use the same value for Z.
F.2.1.1. sqrt_ratio for Any Field
sqrt_ratio(u, v)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
- Z, the constant from the Simplified SWU map.
Input: u and v, elements of F, where v != 0.
Output: (b, y), where
b = True and y = sqrt(u / v) if (u / v) is square in F, and
b = False and y = sqrt(Z * (u / v)) otherwise.
Constants:
1. c1, the largest integer such that 2^c1 divides q - 1.
2. c2 = (q - 1) / (2^c1) # Integer arithmetic
3. c3 = (c2 - 1) / 2 # Integer arithmetic
4. c4 = 2^c1 - 1 # Integer arithmetic
5. c5 = 2^(c1 - 1) # Integer arithmetic
6. c6 = Z^c2
7. c7 = Z^((c2 + 1) / 2)
Procedure:
1. tv1 = c6
2. tv2 = v^c4
3. tv3 = tv2^2
4. tv3 = tv3 * v
5. tv5 = u * tv3
6. tv5 = tv5^c3
7. tv5 = tv5 * tv2
8. tv2 = tv5 * v
9. tv3 = tv5 * u
10. tv4 = tv3 * tv2
11. tv5 = tv4^c5
12. isQR = tv5 == 1
13. tv2 = tv3 * c7
14. tv5 = tv4 * tv1
15. tv3 = CMOV(tv2, tv3, isQR)
16. tv4 = CMOV(tv5, tv4, isQR)
17. for i in (c1, c1 - 1, ..., 2):
18. tv5 = i - 2
19. tv5 = 2^tv5
20. tv5 = tv4^tv5
21. e1 = tv5 == 1
22. tv2 = tv3 * tv1
23. tv1 = tv1 * tv1
24. tv5 = tv4 * tv1
25. tv3 = CMOV(tv2, tv3, e1)
26. tv4 = CMOV(tv5, tv4, e1)
27. return (isQR, tv3)
F.2.1.2. Optimized sqrt_ratio for q = 3 mod 4
sqrt_ratio_3mod4(u, v)
Parameters:
- F, a finite field of characteristic p and order q = p^m,
where q = 3 mod 4.
- Z, the constant from the Simplified SWU map.
Input: u and v, elements of F, where v != 0.
Output: (b, y), where
b = True and y = sqrt(u / v) if (u / v) is square in F, and
b = False and y = sqrt(Z * (u / v)) otherwise.
Constants:
1. c1 = (q - 3) / 4 # Integer arithmetic
2. c2 = sqrt(-Z)
Procedure:
1. tv1 = v^2
2. tv2 = u * v
3. tv1 = tv1 * tv2
4. y1 = tv1^c1
5. y1 = y1 * tv2
6. y2 = y1 * c2
7. tv3 = y1^2
8. tv3 = tv3 * v
9. isQR = tv3 == u
10. y = CMOV(y2, y1, isQR)
11. return (isQR, y)
F.2.1.3. Optimized sqrt_ratio for q = 5 mod 8
sqrt_ratio_5mod8(u, v)
Parameters:
- F, a finite field of characteristic p and order q = p^m,
where q = 5 mod 8.
- Z, the constant from the Simplified SWU map.
Input: u and v, elements of F, where v != 0.
Output: (b, y), where
b = True and y = sqrt(u / v) if (u / v) is square in F, and
b = False and y = sqrt(Z * (u / v)) otherwise.
Constants:
1. c1 = (q - 5) / 8
2. c2 = sqrt(-1)
3. c3 = sqrt(Z / c2)
Steps:
1. tv1 = v^2
2. tv2 = tv1 * v
3. tv1 = tv1^2
4. tv2 = tv2 * u
5. tv1 = tv1 * tv2
6. y1 = tv1^c1
7. y1 = y1 * tv2
8. tv1 = y1 * c2
9. tv2 = tv1^2
10. tv2 = tv2 * v
11. e1 = tv2 == u
12. y1 = CMOV(y1, tv1, e1)
13. tv2 = y1^2
14. tv2 = tv2 * v
15. isQR = tv2 == u
16. y2 = y1 * c3
17. tv1 = y2 * c2
18. tv2 = tv1^2
19. tv2 = tv2 * v
20. tv3 = Z * u
21. e2 = tv2 == tv3
22. y2 = CMOV(y2, tv1, e2)
23. y = CMOV(y2, y1, isQR)
24. return (isQR, y)
F.3. Elligator 2 Method
This section gives a straight-line implementation of the Elligator 2
method for any Montgomery curve of the form given in Section 6.7.
See Section 6.7.1 for information on the constants used in this
mapping.
Appendix G.2 gives optimized straight-line procedures that apply to
specific classes of curves and base fields, including curve25519 and
curve448 [RFC7748].
map_to_curve_elligator2(u)
Input: u, an element of F.
Output: (s, t), a point on M.
Constants:
1. c1 = J / K
2. c2 = 1 / K^2
Steps:
1. tv1 = u^2
2. tv1 = Z * tv1 # Z * u^2
3. e1 = tv1 == -1 # exceptional case: Z * u^2 == -1
4. tv1 = CMOV(tv1, 0, e1) # if tv1 == -1, set tv1 = 0
5. x1 = tv1 + 1
6. x1 = inv0(x1)
7. x1 = -c1 * x1 # x1 = -(J / K) / (1 + Z * u^2)
8. gx1 = x1 + c1
9. gx1 = gx1 * x1
10. gx1 = gx1 + c2
11. gx1 = gx1 * x1 # gx1 = x1^3 + (J / K) * x1^2 + x1 / K^2
12. x2 = -x1 - c1
13. gx2 = tv1 * gx1
14. e2 = is_square(gx1) # If is_square(gx1)
15. x = CMOV(x2, x1, e2) # then x = x1, else x = x2
16. y2 = CMOV(gx2, gx1, e2) # then y2 = gx1, else y2 = gx2
17. y = sqrt(y2)
18. e3 = sgn0(y) == 1
19. y = CMOV(y, -y, e2 XOR e3) # fix sign of y
20. s = x * K
21. t = y * K
22. return (s, t)
Appendix G. Curve-Specific Optimized Sample Code
This section gives sample implementations optimized for some of the
elliptic curves listed in Section 8. Sample Sage code [SAGE] for
each algorithm can also be found in [hash2curve-repo].
G.1. Interface and Projective Coordinate Systems
The sample code in this section uses a different interface than the
mappings of Section 6. Specifically, each mapping function in this
section has the following signature:
(xn, xd, yn, yd) = map_to_curve(u)
The resulting affine point (x, y) is given by (xn / xd, yn / yd).
The reason for this modified interface is that it enables further
optimizations when working with points in a projective coordinate
system. This is desirable, for example, when the resulting point
will be immediately multiplied by a scalar, since most scalar
multiplication algorithms operate on projective points.
Projective coordinates are also useful when implementing random-
oracle encodings (Section 3). One reason is that, in general, point
addition is faster using projective coordinates. Another reason is
that, for Weierstrass curves, projective coordinates allow using
complete addition formulas [RCB16]. This is especially convenient
when implementing a constant-time encoding, because it eliminates the
need for a special case when Q0 == Q1, which incomplete addition
formulas usually do not handle.
The following are two commonly used projective coordinate systems and
the corresponding conversions:
* A point (X, Y, Z) in homogeneous projective coordinates
corresponds to the affine point (x, y) = (X / Z, Y / Z); the
inverse conversion is given by (X, Y, Z) = (x, y, 1). To convert
(xn, xd, yn, yd) to homogeneous projective coordinates, compute
(X, Y, Z) = (xn * yd, yn * xd, xd * yd).
* A point (X', Y', Z') in Jacobian projective coordinates
corresponds to the affine point (x, y) = (X' / Z'^2, Y' / Z'^3);
the inverse conversion is given by (X', Y', Z') = (x, y, 1). To
convert (xn, xd, yn, yd) to Jacobian projective coordinates,
compute (X', Y', Z') = (xn * xd * yd^2, yn * yd^2 * xd^3, xd *
yd).
G.2. Elligator 2
G.2.1. curve25519 (q = 5 (mod 8), K = 1)
The following is a straight-line implementation of Elligator 2 for
curve25519 [RFC7748] as specified in Section 8.5.
This implementation can also be used for any Montgomery curve with K
= 1 over GF(q) where q = 5 (mod 8).
map_to_curve_elligator2_curve25519(u)
Input: u, an element of F.
Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
point on curve25519.
Constants:
1. c1 = (q + 3) / 8 # Integer arithmetic
2. c2 = 2^c1
3. c3 = sqrt(-1)
4. c4 = (q - 5) / 8 # Integer arithmetic
Steps:
1. tv1 = u^2
2. tv1 = 2 * tv1
3. xd = tv1 + 1 # Nonzero: -1 is square (mod p), tv1 is not
4. x1n = -J # x1 = x1n / xd = -J / (1 + 2 * u^2)
5. tv2 = xd^2
6. gxd = tv2 * xd # gxd = xd^3
7. gx1 = J * tv1 # x1n + J * xd
8. gx1 = gx1 * x1n # x1n^2 + J * x1n * xd
9. gx1 = gx1 + tv2 # x1n^2 + J * x1n * xd + xd^2
10. gx1 = gx1 * x1n # x1n^3 + J * x1n^2 * xd + x1n * xd^2
11. tv3 = gxd^2
12. tv2 = tv3^2 # gxd^4
13. tv3 = tv3 * gxd # gxd^3
14. tv3 = tv3 * gx1 # gx1 * gxd^3
15. tv2 = tv2 * tv3 # gx1 * gxd^7
16. y11 = tv2^c4 # (gx1 * gxd^7)^((p - 5) / 8)
17. y11 = y11 * tv3 # gx1 * gxd^3 * (gx1 * gxd^7)^((p - 5) / 8)
18. y12 = y11 * c3
19. tv2 = y11^2
20. tv2 = tv2 * gxd
21. e1 = tv2 == gx1
22. y1 = CMOV(y12, y11, e1) # If g(x1) is square, this is its sqrt
23. x2n = x1n * tv1 # x2 = x2n / xd = 2 * u^2 * x1n / xd
24. y21 = y11 * u
25. y21 = y21 * c2
26. y22 = y21 * c3
27. gx2 = gx1 * tv1 # g(x2) = gx2 / gxd = 2 * u^2 * g(x1)
28. tv2 = y21^2
29. tv2 = tv2 * gxd
30. e2 = tv2 == gx2
31. y2 = CMOV(y22, y21, e2) # If g(x2) is square, this is its sqrt
32. tv2 = y1^2
33. tv2 = tv2 * gxd
34. e3 = tv2 == gx1
35. xn = CMOV(x2n, x1n, e3) # If e3, x = x1, else x = x2
36. y = CMOV(y2, y1, e3) # If e3, y = y1, else y = y2
37. e4 = sgn0(y) == 1 # Fix sign of y
38. y = CMOV(y, -y, e3 XOR e4)
39. return (xn, xd, y, 1)
G.2.2. edwards25519
The following is a straight-line implementation of Elligator 2 for
edwards25519 [RFC7748] as specified in Section 8.5. The subroutine
map_to_curve_elligator2_curve25519 is defined in Appendix G.2.1.
Note that the sign of the constant c1 below is chosen as specified in
Section 6.8.1, i.e., applying the rational map to the edwards25519
base point yields the curve25519 base point (see erratum [Err4730]).
map_to_curve_elligator2_edwards25519(u)
Input: u, an element of F.
Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
point on edwards25519.
Constants:
1. c1 = sqrt(-486664) # sgn0(c1) MUST equal 0
Steps:
1. (xMn, xMd, yMn, yMd) = map_to_curve_elligator2_curve25519(u)
2. xn = xMn * yMd
3. xn = xn * c1
4. xd = xMd * yMn # xn / xd = c1 * xM / yM
5. yn = xMn - xMd
6. yd = xMn + xMd # (n / d - 1) / (n / d + 1) = (n - d) / (n + d)
7. tv1 = xd * yd
8. e = tv1 == 0
9. xn = CMOV(xn, 0, e)
10. xd = CMOV(xd, 1, e)
11. yn = CMOV(yn, 1, e)
12. yd = CMOV(yd, 1, e)
13. return (xn, xd, yn, yd)
G.2.3. curve448 (q = 3 (mod 4), K = 1)
The following is a straight-line implementation of Elligator 2 for
curve448 [RFC7748] as specified in Section 8.6.
This implementation can also be used for any Montgomery curve with K
= 1 over GF(q) where q = 3 (mod 4).
map_to_curve_elligator2_curve448(u)
Input: u, an element of F.
Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
point on curve448.
Constants:
1. c1 = (q - 3) / 4 # Integer arithmetic
Steps:
1. tv1 = u^2
2. e1 = tv1 == 1
3. tv1 = CMOV(tv1, 0, e1) # If Z * u^2 == -1, set tv1 = 0
4. xd = 1 - tv1
5. x1n = -J
6. tv2 = xd^2
7. gxd = tv2 * xd # gxd = xd^3
8. gx1 = -J * tv1 # x1n + J * xd
9. gx1 = gx1 * x1n # x1n^2 + J * x1n * xd
10. gx1 = gx1 + tv2 # x1n^2 + J * x1n * xd + xd^2
11. gx1 = gx1 * x1n # x1n^3 + J * x1n^2 * xd + x1n * xd^2
12. tv3 = gxd^2
13. tv2 = gx1 * gxd # gx1 * gxd
14. tv3 = tv3 * tv2 # gx1 * gxd^3
15. y1 = tv3^c1 # (gx1 * gxd^3)^((p - 3) / 4)
16. y1 = y1 * tv2 # gx1 * gxd * (gx1 * gxd^3)^((p - 3) / 4)
17. x2n = -tv1 * x1n # x2 = x2n / xd = -1 * u^2 * x1n / xd
18. y2 = y1 * u
19. y2 = CMOV(y2, 0, e1)
20. tv2 = y1^2
21. tv2 = tv2 * gxd
22. e2 = tv2 == gx1
23. xn = CMOV(x2n, x1n, e2) # If e2, x = x1, else x = x2
24. y = CMOV(y2, y1, e2) # If e2, y = y1, else y = y2
25. e3 = sgn0(y) == 1 # Fix sign of y
26. y = CMOV(y, -y, e2 XOR e3)
27. return (xn, xd, y, 1)
G.2.4. edwards448
The following is a straight-line implementation of Elligator 2 for
edwards448 [RFC7748] as specified in Section 8.6. The subroutine
map_to_curve_elligator2_curve448 is defined in Appendix G.2.3.
map_to_curve_elligator2_edwards448(u)
Input: u, an element of F.
Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
point on edwards448.
Steps:
1. (xn, xd, yn, yd) = map_to_curve_elligator2_curve448(u)
2. xn2 = xn^2
3. xd2 = xd^2
4. xd4 = xd2^2
5. yn2 = yn^2
6. yd2 = yd^2
7. xEn = xn2 - xd2
8. tv2 = xEn - xd2
9. xEn = xEn * xd2
10. xEn = xEn * yd
11. xEn = xEn * yn
12. xEn = xEn * 4
13. tv2 = tv2 * xn2
14. tv2 = tv2 * yd2
15. tv3 = 4 * yn2
16. tv1 = tv3 + yd2
17. tv1 = tv1 * xd4
18. xEd = tv1 + tv2
19. tv2 = tv2 * xn
20. tv4 = xn * xd4
21. yEn = tv3 - yd2
22. yEn = yEn * tv4
23. yEn = yEn - tv2
24. tv1 = xn2 + xd2
25. tv1 = tv1 * xd2
26. tv1 = tv1 * xd
27. tv1 = tv1 * yn2
28. tv1 = -2 * tv1
29. yEd = tv2 + tv1
30. tv4 = tv4 * yd2
31. yEd = yEd + tv4
32. tv1 = xEd * yEd
33. e = tv1 == 0
34. xEn = CMOV(xEn, 0, e)
35. xEd = CMOV(xEd, 1, e)
36. yEn = CMOV(yEn, 1, e)
37. yEd = CMOV(yEd, 1, e)
38. return (xEn, xEd, yEn, yEd)
G.2.5. Montgomery Curves with q = 3 (mod 4)
The following is a straight-line implementation of Elligator 2 that
applies to any Montgomery curve defined over GF(q) where q = 3 (mod
4).
For curves where K = 1, the implementation given in Appendix G.2.3
gives identical results with slightly reduced cost.
map_to_curve_elligator2_3mod4(u)
Input: u, an element of F.
Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
point on the target curve.
Constants:
1. c1 = (q - 3) / 4 # Integer arithmetic
2. c2 = K^2
Steps:
1. tv1 = u^2
2. e1 = tv1 == 1
3. tv1 = CMOV(tv1, 0, e1) # If Z * u^2 == -1, set tv1 = 0
4. xd = 1 - tv1
5. xd = xd * K
6. x1n = -J # x1 = x1n / xd = -J / (K * (1 + 2 * u^2))
7. tv2 = xd^2
8. gxd = tv2 * xd
9. gxd = gxd * c2 # gxd = xd^3 * K^2
10. gx1 = x1n * K
11. tv3 = xd * J
12. tv3 = gx1 + tv3 # x1n * K + xd * J
13. gx1 = gx1 * tv3 # K^2 * x1n^2 + J * K * x1n * xd
14. gx1 = gx1 + tv2 # K^2 * x1n^2 + J * K * x1n * xd + xd^2
15. gx1 = gx1 * x1n # K^2 * x1n^3 + J * K * x1n^2 * xd + x1n * xd^2
16. tv3 = gxd^2
17. tv2 = gx1 * gxd # gx1 * gxd
18. tv3 = tv3 * tv2 # gx1 * gxd^3
19. y1 = tv3^c1 # (gx1 * gxd^3)^((q - 3) / 4)
20. y1 = y1 * tv2 # gx1 * gxd * (gx1 * gxd^3)^((q - 3) / 4)
21. x2n = -tv1 * x1n # x2 = x2n / xd = -1 * u^2 * x1n / xd
22. y2 = y1 * u
23. y2 = CMOV(y2, 0, e1)
24. tv2 = y1^2
25. tv2 = tv2 * gxd
26. e2 = tv2 == gx1
27. xn = CMOV(x2n, x1n, e2) # If e2, x = x1, else x = x2
28. xn = xn * K
29. y = CMOV(y2, y1, e2) # If e2, y = y1, else y = y2
30. e3 = sgn0(y) == 1 # Fix sign of y
31. y = CMOV(y, -y, e2 XOR e3)
32. y = y * K
33. return (xn, xd, y, 1)
G.2.6. Montgomery Curves with q = 5 (mod 8)
The following is a straight-line implementation of Elligator 2 that
applies to any Montgomery curve defined over GF(q) where q = 5 (mod
8).
For curves where K = 1, the implementation given in Appendix G.2.1
gives identical results with slightly reduced cost.
map_to_curve_elligator2_5mod8(u)
Input: u, an element of F.
Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
point on the target curve.
Constants:
1. c1 = (q + 3) / 8 # Integer arithmetic
2. c2 = 2^c1
3. c3 = sqrt(-1)
4. c4 = (q - 5) / 8 # Integer arithmetic
5. c5 = K^2
Steps:
1. tv1 = u^2
2. tv1 = 2 * tv1
3. xd = tv1 + 1 # Nonzero: -1 is square (mod p), tv1 is not
4. xd = xd * K
5. x1n = -J # x1 = x1n / xd = -J / (K * (1 + 2 * u^2))
6. tv2 = xd^2
7. gxd = tv2 * xd
8. gxd = gxd * c5 # gxd = xd^3 * K^2
9. gx1 = x1n * K
10. tv3 = xd * J
11. tv3 = gx1 + tv3 # x1n * K + xd * J
12. gx1 = gx1 * tv3 # K^2 * x1n^2 + J * K * x1n * xd
13. gx1 = gx1 + tv2 # K^2 * x1n^2 + J * K * x1n * xd + xd^2
14. gx1 = gx1 * x1n # K^2 * x1n^3 + J * K * x1n^2 * xd + x1n * xd^2
15. tv3 = gxd^2
16. tv2 = tv3^2 # gxd^4
17. tv3 = tv3 * gxd # gxd^3
18. tv3 = tv3 * gx1 # gx1 * gxd^3
19. tv2 = tv2 * tv3 # gx1 * gxd^7
20. y11 = tv2^c4 # (gx1 * gxd^7)^((q - 5) / 8)
21. y11 = y11 * tv3 # gx1 * gxd^3 * (gx1 * gxd^7)^((q - 5) / 8)
22. y12 = y11 * c3
23. tv2 = y11^2
24. tv2 = tv2 * gxd
25. e1 = tv2 == gx1
26. y1 = CMOV(y12, y11, e1) # If g(x1) is square, this is its sqrt
27. x2n = x1n * tv1 # x2 = x2n / xd = 2 * u^2 * x1n / xd
28. y21 = y11 * u
29. y21 = y21 * c2
30. y22 = y21 * c3
31. gx2 = gx1 * tv1 # g(x2) = gx2 / gxd = 2 * u^2 * g(x1)
32. tv2 = y21^2
33. tv2 = tv2 * gxd
34. e2 = tv2 == gx2
35. y2 = CMOV(y22, y21, e2) # If g(x2) is square, this is its sqrt
36. tv2 = y1^2
37. tv2 = tv2 * gxd
38. e3 = tv2 == gx1
39. xn = CMOV(x2n, x1n, e3) # If e3, x = x1, else x = x2
40. xn = xn * K
41. y = CMOV(y2, y1, e3) # If e3, y = y1, else y = y2
42. e4 = sgn0(y) == 1 # Fix sign of y
43. y = CMOV(y, -y, e3 XOR e4)
44. y = y * K
45. return (xn, xd, y, 1)
G.3. Cofactor Clearing for BLS12-381 G2
The curve BLS12-381, whose parameters are defined in Section 8.8.2,
admits an efficiently computable endomorphism, psi, that can be used
to speed up cofactor clearing for G2 [SBCDK09] [FKR11] [BP17] (see
also Section 7). This section implements the endomorphism psi and a
fast cofactor clearing method described by Budroni and Pintore
[BP17].
The functions in this section operate on points whose coordinates are
represented as ratios, i.e., (xn, xd, yn, yd) corresponds to the
point (xn / xd, yn / yd); see Appendix G.1 for further discussion of
projective coordinates. When points are represented in affine
coordinates, one can simply ignore the denominators (xd == 1 and
yd == 1).
The following function computes the Frobenius endomorphism for an
element of F = GF(p^2) with basis (1, I), where I^2 + 1 == 0 in F.
(This is the base field of the elliptic curve E defined in
Section 8.8.2.)
frobenius(x)
Input: x, an element of GF(p^2).
Output: a, an element of GF(p^2).
Notation: x = x0 + I * x1, where x0 and x1 are elements of GF(p).
Steps:
1. a = x0 - I * x1
2. return a
The following function computes the endomorphism psi for points on
the elliptic curve E defined in Section 8.8.2.
psi(xn, xd, yn, yd)
Input: P, a point (xn / xd, yn / yd) on the curve E (see above).
Output: Q, a point on the same curve.
Constants:
1. c1 = 1 / (1 + I)^((p - 1) / 3) # in GF(p^2)
2. c2 = 1 / (1 + I)^((p - 1) / 2) # in GF(p^2)
Steps:
1. qxn = c1 * frobenius(xn)
2. qxd = frobenius(xd)
3. qyn = c2 * frobenius(yn)
4. qyd = frobenius(yd)
5. return (qxn, qxd, qyn, qyd)
The following function efficiently computes psi(psi(P)).
psi2(xn, xd, yn, yd)
Input: P, a point (xn / xd, yn / yd) on the curve E (see above).
Output: Q, a point on the same curve.
Constants:
1. c1 = 1 / 2^((p - 1) / 3) # in GF(p^2)
Steps:
1. qxn = c1 * xn
2. qyn = -yn
3. return (qxn, xd, qyn, yd)
The following function maps any point on the elliptic curve E
(Section 8.8.2) into the prime-order subgroup G2. This function
returns a point equal to h_eff * P, where h_eff is the parameter
given in Section 8.8.2.
clear_cofactor_bls12381_g2(P)
Input: P, a point (xn / xd, yn / yd) on the curve E (see above).
Output: Q, a point in the subgroup G2 of BLS12-381.
Constants:
1. c1 = -15132376222941642752 # the BLS parameter for BLS12-381
# i.e., -0xd201000000010000
Notation: in this procedure, + and - represent elliptic curve point
addition and subtraction, respectively, and * represents scalar
multiplication.
Steps:
1. t1 = c1 * P
2. t2 = psi(P)
3. t3 = 2 * P
4. t3 = psi2(t3)
5. t3 = t3 - t2
6. t2 = t1 + t2
7. t2 = c1 * t2
8. t3 = t3 + t2
9. t3 = t3 - t1
10. Q = t3 - P
11. return Q
Appendix H. Scripts for Parameter Generation
This section gives Sage scripts [SAGE] used to generate parameters
for the mappings of Section 6.
H.1. Finding Z for the Shallue-van de Woestijne Map
The below function outputs an appropriate Z for the Shallue-van de
Woestijne map (Section 6.6.1).
# Arguments:
# - F, a field object, e.g., F = GF(2^521 - 1)
# - A and B, the coefficients of the curve y^2 = x^3 + A * x + B
def find_z_svdw(F, A, B, init_ctr=1):
g = lambda x: F(x)^3 + F(A) * F(x) + F(B)
h = lambda Z: -(F(3) * Z^2 + F(4) * A) / (F(4) * g(Z))
# NOTE: if init_ctr=1 fails to find Z, try setting it to F.gen()
ctr = init_ctr
while True:
for Z_cand in (F(ctr), F(-ctr)):
# Criterion 1:
# g(Z) != 0 in F.
if g(Z_cand) == F(0):
continue
# Criterion 2:
# -(3 * Z^2 + 4 * A) / (4 * g(Z)) != 0 in F.
if h(Z_cand) == F(0):
continue
# Criterion 3:
# -(3 * Z^2 + 4 * A) / (4 * g(Z)) is square in F.
if not is_square(h(Z_cand)):
continue
# Criterion 4:
# At least one of g(Z) and g(-Z / 2) is square in F.
if is_square(g(Z_cand)) or is_square(g(-Z_cand / F(2))):
return Z_cand
ctr += 1
H.2. Finding Z for Simplified SWU
The below function outputs an appropriate Z for the Simplified SWU
map (Section 6.6.2).
# Arguments:
# - F, a field object, e.g., F = GF(2^521 - 1)
# - A and B, the coefficients of the curve y^2 = x^3 + A * x + B
def find_z_sswu(F, A, B):
R.<xx> = F[] # Polynomial ring over F
g = xx^3 + F(A) * xx + F(B) # y^2 = g(x) = x^3 + A * x + B
ctr = F.gen()
while True:
for Z_cand in (F(ctr), F(-ctr)):
# Criterion 1: Z is non-square in F.
if is_square(Z_cand):
continue
# Criterion 2: Z != -1 in F.
if Z_cand == F(-1):
continue
# Criterion 3: g(x) - Z is irreducible over F.
if not (g - Z_cand).is_irreducible():
continue
# Criterion 4: g(B / (Z * A)) is square in F.
if is_square(g(B / (Z_cand * A))):
return Z_cand
ctr += 1
H.3. Finding Z for Elligator 2
The below function outputs an appropriate Z for the Elligator 2 map
(Section 6.7.1).
# Argument:
# - F, a field object, e.g., F = GF(2^255 - 19)
def find_z_ell2(F):
ctr = F.gen()
while True:
for Z_cand in (F(ctr), F(-ctr)):
# Z must be a non-square in F.
if is_square(Z_cand):
continue
return Z_cand
ctr += 1
Appendix I. sqrt and is_square Functions
This section defines special-purpose sqrt functions for the three
most common cases, q = 3 (mod 4), q = 5 (mod 8), and q = 9 (mod 16),
plus a generic constant-time algorithm that works for any prime
modulus.
In addition, it gives an optimized is_square method for GF(p^2).
I.1. sqrt for q = 3 (mod 4)
sqrt_3mod4(x)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
Input: x, an element of F.
Output: z, an element of F such that (z^2) == x, if x is square in F.
Constants:
1. c1 = (q + 1) / 4 # Integer arithmetic
Procedure:
1. return x^c1
I.2. sqrt for q = 5 (mod 8)
sqrt_5mod8(x)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
Input: x, an element of F.
Output: z, an element of F such that (z^2) == x, if x is square in F.
Constants:
1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F
2. c2 = (q + 3) / 8 # Integer arithmetic
Procedure:
1. tv1 = x^c2
2. tv2 = tv1 * c1
3. e = (tv1^2) == x
4. z = CMOV(tv2, tv1, e)
5. return z
I.3. sqrt for q = 9 (mod 16)
sqrt_9mod16(x)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
Input: x, an element of F.
Output: z, an element of F such that (z^2) == x, if x is square in F.
Constants:
1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F
2. c2 = sqrt(c1) in F, i.e., (c2^2) == c1 in F
3. c3 = sqrt(-c1) in F, i.e., (c3^2) == -c1 in F
4. c4 = (q + 7) / 16 # Integer arithmetic
Procedure:
1. tv1 = x^c4
2. tv2 = c1 * tv1
3. tv3 = c2 * tv1
4. tv4 = c3 * tv1
5. e1 = (tv2^2) == x
6. e2 = (tv3^2) == x
7. tv1 = CMOV(tv1, tv2, e1) # Select tv2 if (tv2^2) == x
8. tv2 = CMOV(tv4, tv3, e2) # Select tv3 if (tv3^2) == x
9. e3 = (tv2^2) == x
10. z = CMOV(tv1, tv2, e3) # Select the sqrt from tv1 and tv2
11. return z
I.4. Constant-Time Tonelli-Shanks Algorithm
This algorithm is a constant-time version of the classic Tonelli-
Shanks algorithm ([C93], Algorithm 1.5.1) due to Sean Bowe, Jack
Grigg, and Eirik Ogilvie-Wigley [jubjub-fq], adapted and optimized by
Michael Scott.
This algorithm applies to GF(p) for any p. Note, however, that the
special-purpose algorithms given in the prior sections are faster,
when they apply.
sqrt_ts_ct(x)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
Input x, an element of F.
Output: z, an element of F such that z^2 == x, if x is square in F.
Constants:
1. c1, the largest integer such that 2^c1 divides q - 1.
2. c2 = (q - 1) / (2^c1) # Integer arithmetic
3. c3 = (c2 - 1) / 2 # Integer arithmetic
4. c4, a non-square value in F
5. c5 = c4^c2 in F
Procedure:
1. z = x^c3
2. t = z * z
3. t = t * x
4. z = z * x
5. b = t
6. c = c5
7. for i in (c1, c1 - 1, ..., 2):
8. for j in (1, 2, ..., i - 2):
9. b = b * b
10. e = b == 1
11. zt = z * c
12. z = CMOV(zt, z, e)
13. c = c * c
14. tt = t * c
15. t = CMOV(tt, t, e)
16. b = t
17. return z
I.5. is_square for F = GF(p^2)
The following is_square method applies to any field F = GF(p^2) with
basis (1, I) represented as described in Section 2.1, i.e., an
element x = (x_1, x_2) = x_1 + x_2 * I.
Other optimizations of this type are possible in other extension
fields; see, for example, [AR13] for more information.
is_square(x)
Parameters:
- F, an extension field of characteristic p and order q = p^2
with basis (1, I).
Input: x, an element of F.
Output: True if x is square in F, and False otherwise.
Constants:
1. c1 = (p - 1) / 2 # Integer arithmetic
Procedure:
1. tv1 = x_1^2
2. tv2 = I * x_2
3. tv2 = tv2^2
4. tv1 = tv1 - tv2
5. tv1 = tv1^c1
6. e1 = tv1 != -1 # Note: -1 in F
7. return e1
Appendix J. Suite Test Vectors
This section gives test vectors for each suite defined in Section 8.
The test vectors in this section were generated using code that is
available from [hash2curve-repo].
Each test vector in this section lists values computed by the
appropriate encoding function, with variable names defined as in
Section 3. For example, for a suite whose encoding type is random
oracle, the test vector gives the value for msg, u, Q0, Q1, and the
output point P.
J.1. NIST P-256
J.1.1. P256_XMD:SHA-256_SSWU_RO_
suite = P256_XMD:SHA-256_SSWU_RO_
dst = QUUX-V01-CS02-with-P256_XMD:SHA-256_SSWU_RO_
msg =
P.x = 2c15230b26dbc6fc9a37051158c95b79656e17a1a920b11394ca91
c44247d3e4
P.y = 8a7a74985cc5c776cdfe4b1f19884970453912e9d31528c060be9a
b5c43e8415
u[0] = ad5342c66a6dd0ff080df1da0ea1c04b96e0330dd89406465eeba1
1582515009
u[1] = 8c0f1d43204bd6f6ea70ae8013070a1518b43873bcd850aafa0a9e
220e2eea5a
Q0.x = ab640a12220d3ff283510ff3f4b1953d09fad35795140b1c5d64f3
13967934d5
Q0.y = dccb558863804a881d4fff3455716c836cef230e5209594ddd33d8
5c565b19b1
Q1.x = 51cce63c50d972a6e51c61334f0f4875c9ac1cd2d3238412f84e31
da7d980ef5
Q1.y = b45d1a36d00ad90e5ec7840a60a4de411917fbe7c82c3949a6e699
e5a1b66aac
msg = abc
P.x = 0bb8b87485551aa43ed54f009230450b492fead5f1cc91658775da
c4a3388a0f
P.y = 5c41b3d0731a27a7b14bc0bf0ccded2d8751f83493404c84a88e71
ffd424212e
u[0] = afe47f2ea2b10465cc26ac403194dfb68b7f5ee865cda61e9f3e07
a537220af1
u[1] = 379a27833b0bfe6f7bdca08e1e83c760bf9a338ab335542704edcd
69ce9e46e0
Q0.x = 5219ad0ddef3cc49b714145e91b2f7de6ce0a7a7dc7406c7726c7e
373c58cb48
Q0.y = 7950144e52d30acbec7b624c203b1996c99617d0b61c2442354301
b191d93ecf
Q1.x = 019b7cb4efcfeaf39f738fe638e31d375ad6837f58a852d032ff60
c69ee3875f
Q1.y = 589a62d2b22357fed5449bc38065b760095ebe6aeac84b01156ee4
252715446e
msg = abcdef0123456789
P.x = 65038ac8f2b1def042a5df0b33b1f4eca6bff7cb0f9c6c15268118
64e544ed80
P.y = cad44d40a656e7aff4002a8de287abc8ae0482b5ae825822bb870d
6df9b56ca3
u[0] = 0fad9d125a9477d55cf9357105b0eb3a5c4259809bf87180aa01d6
51f53d312c
u[1] = b68597377392cd3419d8fcc7d7660948c8403b19ea78bbca4b133c
9d2196c0fb
Q0.x = a17bdf2965eb88074bc01157e644ed409dac97cfcf0c61c998ed0f
a45e79e4a2
Q0.y = 4f1bc80c70d411a3cc1d67aeae6e726f0f311639fee560c7f5a664
554e3c9c2e
Q1.x = 7da48bb67225c1a17d452c983798113f47e438e4202219dd0715f8
419b274d66
Q1.y = b765696b2913e36db3016c47edb99e24b1da30e761a8a3215dc0ec
4d8f96e6f9
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 4be61ee205094282ba8a2042bcb48d88dfbb609301c49aa8b07853
3dc65a0b5d
P.y = 98f8df449a072c4721d241a3b1236d3caccba603f916ca680f4539
d2bfb3c29e
u[0] = 3bbc30446f39a7befad080f4d5f32ed116b9534626993d2cc5033f
6f8d805919
u[1] = 76bb02db019ca9d3c1e02f0c17f8baf617bbdae5c393a81d9ce11e
3be1bf1d33
Q0.x = c76aaa823aeadeb3f356909cb08f97eee46ecb157c1f56699b5efe
bddf0e6398
Q0.y = 776a6f45f528a0e8d289a4be12c4fab80762386ec644abf2bffb9b
627e4352b1
Q1.x = 418ac3d85a5ccc4ea8dec14f750a3a9ec8b85176c95a7022f39182
6794eb5a75
Q1.y = fd6604f69e9d9d2b74b072d14ea13050db72c932815523305cb9e8
07cc900aff
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 457ae2981f70ca85d8e24c308b14db22f3e3862c5ea0f652ca38b5
e49cd64bc5
P.y = ecb9f0eadc9aeed232dabc53235368c1394c78de05dd96893eefa6
2b0f4757dc
u[0] = 4ebc95a6e839b1ae3c63b847798e85cb3c12d3817ec6ebc10af6ee
51adb29fec
u[1] = 4e21af88e22ea80156aff790750121035b3eefaa96b425a8716e0d
20b4e269ee
Q0.x = d88b989ee9d1295df413d4456c5c850b8b2fb0f5402cc5c4c7e815
412e926db8
Q0.y = bb4a1edeff506cf16def96afff41b16fc74f6dbd55c2210e5b8f01
1ba32f4f40
Q1.x = a281e34e628f3a4d2a53fa87ff973537d68ad4fbc28d3be5e8d9f6
a2571c5a4b
Q1.y = f6ed88a7aab56a488100e6f1174fa9810b47db13e86be999644922
961206e184
J.1.2. P256_XMD:SHA-256_SSWU_NU_
suite = P256_XMD:SHA-256_SSWU_NU_
dst = QUUX-V01-CS02-with-P256_XMD:SHA-256_SSWU_NU_
msg =
P.x = f871caad25ea3b59c16cf87c1894902f7e7b2c822c3d3f73596c5a
ce8ddd14d1
P.y = 87b9ae23335bee057b99bac1e68588b18b5691af476234b8971bc4
f011ddc99b
u[0] = b22d487045f80e9edcb0ecc8d4bf77833e2bf1f3a54004d7df1d57
f4802d311f
Q.x = f871caad25ea3b59c16cf87c1894902f7e7b2c822c3d3f73596c5a
ce8ddd14d1
Q.y = 87b9ae23335bee057b99bac1e68588b18b5691af476234b8971bc4
f011ddc99b
msg = abc
P.x = fc3f5d734e8dce41ddac49f47dd2b8a57257522a865c124ed02b92
b5237befa4
P.y = fe4d197ecf5a62645b9690599e1d80e82c500b22ac705a0b421fac
7b47157866
u[0] = c7f96eadac763e176629b09ed0c11992225b3a5ae99479760601cb
d69c221e58
Q.x = fc3f5d734e8dce41ddac49f47dd2b8a57257522a865c124ed02b92
b5237befa4
Q.y = fe4d197ecf5a62645b9690599e1d80e82c500b22ac705a0b421fac
7b47157866
msg = abcdef0123456789
P.x = f164c6674a02207e414c257ce759d35eddc7f55be6d7f415e2cc17
7e5d8faa84
P.y = 3aa274881d30db70485368c0467e97da0e73c18c1d00f34775d012
b6fcee7f97
u[0] = 314e8585fa92068b3ea2c3bab452d4257b38be1c097d58a2189045
6c2929614d
Q.x = f164c6674a02207e414c257ce759d35eddc7f55be6d7f415e2cc17
7e5d8faa84
Q.y = 3aa274881d30db70485368c0467e97da0e73c18c1d00f34775d012
b6fcee7f97
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 324532006312be4f162614076460315f7a54a6f85544da773dc659
aca0311853
P.y = 8d8197374bcd52de2acfefc8a54fe2c8d8bebd2a39f16be9b710e4
b1af6ef883
u[0] = 752d8eaa38cd785a799a31d63d99c2ae4261823b4a367b133b2c66
27f48858ab
Q.x = 324532006312be4f162614076460315f7a54a6f85544da773dc659
aca0311853
Q.y = 8d8197374bcd52de2acfefc8a54fe2c8d8bebd2a39f16be9b710e4
b1af6ef883
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 5c4bad52f81f39c8e8de1260e9a06d72b8b00a0829a8ea004a610b
0691bea5d9
P.y = c801e7c0782af1f74f24fc385a8555da0582032a3ce038de637ccd
cb16f7ef7b
u[0] = 0e1527840b9df2dfbef966678ff167140f2b27c4dccd884c25014d
ce0e41dfa3
Q.x = 5c4bad52f81f39c8e8de1260e9a06d72b8b00a0829a8ea004a610b
0691bea5d9
Q.y = c801e7c0782af1f74f24fc385a8555da0582032a3ce038de637ccd
cb16f7ef7b
J.2. NIST P-384
J.2.1. P384_XMD:SHA-384_SSWU_RO_
suite = P384_XMD:SHA-384_SSWU_RO_
dst = QUUX-V01-CS02-with-P384_XMD:SHA-384_SSWU_RO_
msg =
P.x = eb9fe1b4f4e14e7140803c1d99d0a93cd823d2b024040f9c067a8e
ca1f5a2eeac9ad604973527a356f3fa3aeff0e4d83
P.y = 0c21708cff382b7f4643c07b105c2eaec2cead93a917d825601e63
c8f21f6abd9abc22c93c2bed6f235954b25048bb1a
u[0] = 25c8d7dc1acd4ee617766693f7f8829396065d1b447eedb155871f
effd9c6653279ac7e5c46edb7010a0e4ff64c9f3b4
u[1] = 59428be4ed69131df59a0c6a8e188d2d4ece3f1b2a3a02602962b4
7efa4d7905945b1e2cc80b36aa35c99451073521ac
Q0.x = e4717e29eef38d862bee4902a7d21b44efb58c464e3e1f0d03894d
94de310f8ffc6de86786dd3e15a1541b18d4eb2846
Q0.y = 6b95a6e639822312298a47526bb77d9cd7bcf76244c991c8cd7007
5e2ee6e8b9a135c4a37e3c0768c7ca871c0ceb53d4
Q1.x = 509527cfc0750eedc53147e6d5f78596c8a3b7360e0608e2fab056
3a1670d58d8ae107c9f04bcf90e89489ace5650efd
Q1.y = 33337b13cb35e173fdea4cb9e8cce915d836ff57803dbbeb7998aa
49d17df2ff09b67031773039d09fbd9305a1566bc4
msg = abc
P.x = e02fc1a5f44a7519419dd314e29863f30df55a514da2d655775a81
d413003c4d4e7fd59af0826dfaad4200ac6f60abe1
P.y = 01f638d04d98677d65bef99aef1a12a70a4cbb9270ec55248c0453
0d8bc1f8f90f8a6a859a7c1f1ddccedf8f96d675f6
u[0] = 53350214cb6bef0b51abb791b1c4209a2b4c16a0c67e1ab1401017
fad774cd3b3f9a8bcdf7f6229dd8dd5a075cb149a0
u[1] = c0473083898f63e03f26f14877a2407bd60c75ad491e7d26cbc6cc
5ce815654075ec6b6898c7a41d74ceaf720a10c02e
Q0.x = fc853b69437aee9a19d5acf96a4ee4c5e04cf7b53406dfaa2afbdd
7ad2351b7f554e4bbc6f5db4177d4d44f933a8f6ee
Q0.y = 7e042547e01834c9043b10f3a8221c4a879cb156f04f72bfccab0c
047a304e30f2aa8b2e260d34c4592c0c33dd0c6482
Q1.x = 57912293709b3556b43a2dfb137a315d256d573b82ded120ef8c78
2d607c05d930d958e50cb6dc1cc480b9afc38c45f1
Q1.y = de9387dab0eef0bda219c6f168a92645a84665c4f2137c14270fb4
24b7532ff84843c3da383ceea24c47fa343c227bb8
msg = abcdef0123456789
P.x = bdecc1c1d870624965f19505be50459d363c71a699a496ab672f9a
5d6b78676400926fbceee6fcd1780fe86e62b2aa89
P.y = 57cf1f99b5ee00f3c201139b3bfe4dd30a653193778d89a0accc5e
0f47e46e4e4b85a0595da29c9494c1814acafe183c
u[0] = aab7fb87238cf6b2ab56cdcca7e028959bb2ea599d34f68484139d
de85ec6548a6e48771d17956421bdb7790598ea52e
u[1] = 26e8d833552d7844d167833ca5a87c35bcfaa5a0d86023479fb28e
5cd6075c18b168bf1f5d2a0ea146d057971336d8d1
Q0.x = 0ceece45b73f89844671df962ad2932122e878ad2259e650626924
e4e7f132589341dec1480ebcbbbe3509d11fb570b7
Q0.y = fafd71a3115298f6be4ae5c6dfc96c400cfb55760f185b7b03f3fa
45f3f91eb65d27628b3c705cafd0466fafa54883ce
Q1.x = dea1be8d3f9be4cbf4fab9d71d549dde76875b5d9b876832313a08
3ec81e528cbc2a0a1d0596b3bcb0ba77866b129776
Q1.y = eb15fe71662214fb03b65541f40d3eb0f4cf5c3b559f647da138c9
f9b7484c48a08760e02c16f1992762cb7298fa52cf
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 03c3a9f401b78c6c36a52f07eeee0ec1289f178adf78448f43a385
0e0456f5dd7f7633dd31676d990eda32882ab486c0
P.y = cc183d0d7bdfd0a3af05f50e16a3f2de4abbc523215bf57c848d5e
a662482b8c1f43dc453a93b94a8026db58f3f5d878
u[0] = 04c00051b0de6e726d228c85bf243bf5f4789efb512b22b498cde3
821db9da667199b74bd5a09a79583c6d353a3bb41c
u[1] = 97580f218255f899f9204db64cd15e6a312cb4d8182375d1e5157c
8f80f41d6a1a4b77fb1ded9dce56c32058b8d5202b
Q0.x = 051a22105e0817a35d66196338c8d85bd52690d79bba373ead8a86
dd9899411513bb9f75273f6483395a7847fb21edb4
Q0.y = f168295c1bbcff5f8b01248e9dbc885335d6d6a04aea960f7384f7
46ba6502ce477e624151cc1d1392b00df0f5400c06
Q1.x = 6ad7bc8ed8b841efd8ad0765c8a23d0b968ec9aa360a558ff33500
f164faa02bee6c704f5f91507c4c5aad2b0dc5b943
Q1.y = 47313cc0a873ade774048338fc34ca5313f96bbf6ae22ac6ef475d
85f03d24792dc6afba8d0b4a70170c1b4f0f716629
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 7b18d210b1f090ac701f65f606f6ca18fb8d081e3bc6cbd937c560
4325f1cdea4c15c10a54ef303aabf2ea58bd9947a4
P.y = ea857285a33abb516732915c353c75c576bf82ccc96adb63c094dd
e580021eddeafd91f8c0bfee6f636528f3d0c47fd2
u[0] = 480cb3ac2c389db7f9dac9c396d2647ae946db844598971c26d1af
d53912a1491199c0a5902811e4b809c26fcd37a014
u[1] = d28435eb34680e148bf3908536e42231cba9e1f73ae2c6902a222a
89db5c49c97db2f8fa4d4cd6e424b17ac60bdb9bb6
Q0.x = 42e6666f505e854187186bad3011598d9278b9d6e3e4d2503c3d23
6381a56748dec5d139c223129b324df53fa147c4df
Q0.y = 8ee51dbda46413bf621838cc935d18d617881c6f33f3838a79c767
a1e5618e34b22f79142df708d2432f75c7366c8512
Q1.x = 4ff01ceeba60484fa1bc0d825fe1e5e383d8f79f1e5bb78e5fb26b
7a7ef758153e31e78b9d60ce75c5e32e43869d4e12
Q1.y = 0f84b978fac8ceda7304b47e229d6037d32062e597dc7a9b95bcd9
af441f3c56c619a901d21635f9ec6ab4710b9fcd0e
J.2.2. P384_XMD:SHA-384_SSWU_NU_
suite = P384_XMD:SHA-384_SSWU_NU_
dst = QUUX-V01-CS02-with-P384_XMD:SHA-384_SSWU_NU_
msg =
P.x = de5a893c83061b2d7ce6a0d8b049f0326f2ada4b966dc7e7292725
6b033ef61058029a3bfb13c1c7ececd6641881ae20
P.y = 63f46da6139785674da315c1947e06e9a0867f5608cf24724eb379
3a1f5b3809ee28eb21a0c64be3be169afc6cdb38ca
u[0] = bc7dc1b2cdc5d588a66de3276b0f24310d4aca4977efda7d6272e1
be25187b001493d267dc53b56183c9e28282368e60
Q.x = de5a893c83061b2d7ce6a0d8b049f0326f2ada4b966dc7e7292725
6b033ef61058029a3bfb13c1c7ececd6641881ae20
Q.y = 63f46da6139785674da315c1947e06e9a0867f5608cf24724eb379
3a1f5b3809ee28eb21a0c64be3be169afc6cdb38ca
msg = abc
P.x = 1f08108b87e703c86c872ab3eb198a19f2b708237ac4be53d7929f
b4bd5194583f40d052f32df66afe5249c9915d139b
P.y = 1369dc8d5bf038032336b989994874a2270adadb67a7fcc32f0f88
24bc5118613f0ac8de04a1041d90ff8a5ad555f96c
u[0] = 9de6cf41e6e41c03e4a7784ac5c885b4d1e49d6de390b3cdd5a1ac
5dd8c40afb3dfd7bb2686923bab644134483fc1926
Q.x = 1f08108b87e703c86c872ab3eb198a19f2b708237ac4be53d7929f
b4bd5194583f40d052f32df66afe5249c9915d139b
Q.y = 1369dc8d5bf038032336b989994874a2270adadb67a7fcc32f0f88
24bc5118613f0ac8de04a1041d90ff8a5ad555f96c
msg = abcdef0123456789
P.x = 4dac31ec8a82ee3c02ba2d7c9fa431f1e59ffe65bf977b948c59e1
d813c2d7963c7be81aa6db39e78ff315a10115c0d0
P.y = 845333cdb5702ad5c525e603f302904d6fc84879f0ef2ee2014a6b
13edd39131bfd66f7bd7cdc2d9ccf778f0c8892c3f
u[0] = 84e2d430a5e2543573e58e368af41821ca3ccc97baba7e9aab51a8
4543d5a0298638a22ceee6090d9d642921112af5b7
Q.x = 4dac31ec8a82ee3c02ba2d7c9fa431f1e59ffe65bf977b948c59e1
d813c2d7963c7be81aa6db39e78ff315a10115c0d0
Q.y = 845333cdb5702ad5c525e603f302904d6fc84879f0ef2ee2014a6b
13edd39131bfd66f7bd7cdc2d9ccf778f0c8892c3f
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 13c1f8c52a492183f7c28e379b0475486718a7e3ac1dfef39283b9
ce5fb02b73f70c6c1f3dfe0c286b03e2af1af12d1d
P.y = 57e101887e73e40eab8963324ed16c177d55eb89f804ec9df06801
579820420b5546b579008df2145fd770f584a1a54c
u[0] = 504e4d5a529333b9205acaa283107bd1bffde753898f7744161f7d
d19ba57fbb6a64214a2e00ddd2613d76cd508ddb30
Q.x = 13c1f8c52a492183f7c28e379b0475486718a7e3ac1dfef39283b9
ce5fb02b73f70c6c1f3dfe0c286b03e2af1af12d1d
Q.y = 57e101887e73e40eab8963324ed16c177d55eb89f804ec9df06801
579820420b5546b579008df2145fd770f584a1a54c
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = af129727a4207a8cb9e9dce656d88f79fce25edbcea350499d65e9
bf1204537bdde73c7cefb752a6ed5ebcd44e183302
P.y = ce68a3d5e161b2e6a968e4ddaa9e51504ad1516ec170c7eef3ca6b
5327943eca95d90b23b009ba45f58b72906f2a99e2
u[0] = 7b01ce9b8c5a60d9fbc202d6dde92822e46915d8c17e03fcb92ece
1ed6074d01e149fc9236def40d673de903c1d4c166
Q.x = af129727a4207a8cb9e9dce656d88f79fce25edbcea350499d65e9
bf1204537bdde73c7cefb752a6ed5ebcd44e183302
Q.y = ce68a3d5e161b2e6a968e4ddaa9e51504ad1516ec170c7eef3ca6b
5327943eca95d90b23b009ba45f58b72906f2a99e2
J.3. NIST P-521
J.3.1. P521_XMD:SHA-512_SSWU_RO_
suite = P521_XMD:SHA-512_SSWU_RO_
dst = QUUX-V01-CS02-with-P521_XMD:SHA-512_SSWU_RO_
msg =
P.x = 00fd767cebb2452030358d0e9cf907f525f50920c8f607889a6a35
680727f64f4d66b161fafeb2654bea0d35086bec0a10b30b14adef
3556ed9f7f1bc23cecc9c088
P.y = 0169ba78d8d851e930680322596e39c78f4fe31b97e57629ef6460
ddd68f8763fd7bd767a4e94a80d3d21a3c2ee98347e024fc73ee1c
27166dc3fe5eeef782be411d
u[0] = 01e5f09974e5724f25286763f00ce76238c7a6e03dc396600350ee
2c4135fb17dc555be99a4a4bae0fd303d4f66d984ed7b6a3ba3860
93752a855d26d559d69e7e9e
u[1] = 00ae593b42ca2ef93ac488e9e09a5fe5a2f6fb330d18913734ff60
2f2a761fcaaf5f596e790bcc572c9140ec03f6cccc38f767f1c197
5a0b4d70b392d95a0c7278aa
Q0.x = 00b70ae99b6339fffac19cb9bfde2098b84f75e50ac1e80d6acb95
4e4534af5f0e9c4a5b8a9c10317b8e6421574bae2b133b4f2b8c6c
e4b3063da1d91d34fa2b3a3c
Q0.y = 007f368d98a4ddbf381fb354de40e44b19e43bb11a1278759f4ea7
b485e1b6db33e750507c071250e3e443c1aaed61f2c28541bb54b1
b456843eda1eb15ec2a9b36e
Q1.x = 01143d0e9cddcdacd6a9aafe1bcf8d218c0afc45d4451239e821f5
d2a56df92be942660b532b2aa59a9c635ae6b30e803c45a6ac8714
32452e685d661cd41cf67214
Q1.y = 00ff75515df265e996d702a5380defffab1a6d2bc232234c7bcffa
433cd8aa791fbc8dcf667f08818bffa739ae25773b32073213cae9
a0f2a917a0b1301a242dda0c
msg = abc
P.x = 002f89a1677b28054b50d15e1f81ed6669b5a2158211118ebdef8a
6efc77f8ccaa528f698214e4340155abc1fa08f8f613ef14a04371
7503d57e267d57155cf784a4
P.y = 010e0be5dc8e753da8ce51091908b72396d3deed14ae166f66d8eb
f0a4e7059ead169ea4bead0232e9b700dd380b316e9361cfdba55a
08c73545563a80966ecbb86d
u[0] = 003d00c37e95f19f358adeeaa47288ec39998039c3256e13c2a4c0
0a7cb61a34c8969472960150a27276f2390eb5e53e47ab193351c2
d2d9f164a85c6a5696d94fe8
u[1] = 01f3cbd3df3893a45a2f1fecdac4d525eb16f345b03e2820d69bc5
80f5cbe9cb89196fdf720ef933c4c0361fcfe29940fd0db0a5da6b
afb0bee8876b589c41365f15
Q0.x = 01b254e1c99c835836f0aceebba7d77750c48366ecb07fb658e4f5
b76e229ae6ca5d271bb0006ffcc42324e15a6d3daae587f9049de2
dbb0494378ffb60279406f56
Q0.y = 01845f4af72fc2b1a5a2fe966f6a97298614288b456cfc385a425b
686048b25c952fbb5674057e1eb055d04568c0679a8e2dda3158dc
16ac598dbb1d006f5ad915b0
Q1.x = 007f08e813c620e527c961b717ffc74aac7afccb9158cebc347d57
15d5c2214f952c97e194f11d114d80d3481ed766ac0a3dba3eb73f
6ff9ccb9304ad10bbd7b4a36
Q1.y = 0022468f92041f9970a7cc025d71d5b647f822784d29ca7b3bc3b0
829d6bb8581e745f8d0cc9dc6279d0450e779ac2275c4c3608064a
d6779108a7828ebd9954caeb
msg = abcdef0123456789
P.x = 006e200e276a4a81760099677814d7f8794a4a5f3658442de63c18
d2244dcc957c645e94cb0754f95fcf103b2aeaf94411847c24187b
89fb7462ad3679066337cbc4
P.y = 001dd8dfa9775b60b1614f6f169089d8140d4b3e4012949b52f98d
b2deff3e1d97bf73a1fa4d437d1dcdf39b6360cc518d8ebcc0f899
018206fded7617b654f6b168
u[0] = 00183ee1a9bbdc37181b09ec336bcaa34095f91ef14b66b1485c16
6720523dfb81d5c470d44afcb52a87b704dbc5c9bc9d0ef524dec2
9884a4795f55c1359945baf3
u[1] = 00504064fd137f06c81a7cf0f84aa7e92b6b3d56c2368f0a08f447
76aa8930480da1582d01d7f52df31dca35ee0a7876500ece3d8fe0
293cd285f790c9881c998d5e
Q0.x = 0021482e8622aac14da60e656043f79a6a110cbae5012268a62dd6
a152c41594549f373910ebed170ade892dd5a19f5d687fae7095a4
61d583f8c4295f7aaf8cd7da
Q0.y = 0177e2d8c6356b7de06e0b5712d8387d529b848748e54a8bc0ef5f
1475aa569f8f492fa85c3ad1c5edc51faf7911f11359bfa2a12d2e
f0bd73df9cb5abd1b101c8b1
Q1.x = 00abeafb16fdbb5eb95095678d5a65c1f293291dfd20a3751dbe05
d0a9bfe2d2eef19449fe59ec32cdd4a4adc3411177c0f2dffd0159
438706159a1bbd0567d9b3d0
Q1.y = 007cc657f847db9db651d91c801741060d63dab4056d0a1d3524e2
eb0e819954d8f677aa353bd056244a88f00017e00c3ce8beeedb43
82d83d74418bd48930c6c182
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 01b264a630bd6555be537b000b99a06761a9325c53322b65bdc41b
f196711f9708d58d34b3b90faf12640c27b91c70a507998e559406
48caa8e71098bf2bc8d24664
P.y = 01ea9f445bee198b3ee4c812dcf7b0f91e0881f0251aab272a1220
1fd89b1a95733fd2a699c162b639e9acdcc54fdc2f6536129b6beb
0432be01aa8da02df5e59aaa
u[0] = 0159871e222689aad7694dc4c3480a49807b1eedd9c8cb4ae1b219
d5ba51655ea5b38e2e4f56b36bf3e3da44a7b139849d28f598c816
fe1bc7ed15893b22f63363c3
u[1] = 004ef0cffd475152f3858c0a8ccbdf7902d8261da92744e98df9b7
fadb0a5502f29c5086e76e2cf498f47321434a40b1504911552ce4
4ad7356a04e08729ad9411f5
Q0.x = 0005eac7b0b81e38727efcab1e375f6779aea949c3e409b53a1d37
aa2acbac87a7e6ad24aafbf3c52f82f7f0e21b872e88c55e17b7fa
21ce08a94ea2121c42c2eb73
Q0.y = 00a173b6a53a7420dbd61d4a21a7c0a52de7a5c6ce05f31403bef7
47d16cc8604a039a73bdd6e114340e55dacd6bea8e217ffbadfb8c
292afa3e1b2afc839a6ce7bb
Q1.x = 01881e3c193a69e4d88d8180a6879b74782a0bc7e529233e9f84bf
7f17d2f319c36920ffba26f9e57a1e045cc7822c834c239593b6e1
42a694aa00c757b0db79e5e8
Q1.y = 01558b16d396d866e476e001f2dd0758927655450b84e12f154032
c7c2a6db837942cd9f44b814f79b4d729996ced61eec61d85c6751
39cbffe3fbf071d2c21cfecb
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 00c12bc3e28db07b6b4d2a2b1167ab9e26fc2fa85c7b0498a17b03
47edf52392856d7e28b8fa7a2dd004611159505835b687ecf1a764
857e27e9745848c436ef3925
P.y = 01cd287df9a50c22a9231beb452346720bb163344a41c5f5a24e83
35b6ccc595fd436aea89737b1281aecb411eb835f0b939073fdd1d
d4d5a2492e91ef4a3c55bcbd
u[0] = 0033d06d17bc3b9a3efc081a05d65805a14a3050a0dd4dfb488461
8eb5c73980a59c5a246b18f58ad022dd3630faa22889fbb8ba1593
466515e6ab4aeb7381c26334
u[1] = 0092290ab99c3fea1a5b8fb2ca49f859994a04faee3301cefab312
d34227f6a2d0c3322cf76861c6a3683bdaa2dd2a6daa5d6906c663
e065338b2344d20e313f1114
Q0.x = 00041f6eb92af8777260718e4c22328a7d74203350c6c8f5794d99
d5789766698f459b83d5068276716f01429934e40af3d1111a2278
0b1e07e72238d2207e5386be
Q0.y = 001c712f0182813942b87cab8e72337db017126f52ed797dd23458
4ac9ae7e80dfe7abea11db02cf1855312eae1447dbaecc9d7e8c88
0a5e76a39f6258074e1bc2e0
Q1.x = 0125c0b69bcf55eab49280b14f707883405028e05c927cd7625d4e
04115bd0e0e6323b12f5d43d0d6d2eff16dbcf244542f84ec05891
1260dc3bb6512ab5db285fbd
Q1.y = 008bddfb803b3f4c761458eb5f8a0aee3e1f7f68e9d7424405fa69
172919899317fb6ac1d6903a432d967d14e0f80af63e7035aaae0c
123e56862ce969456f99f102
J.3.2. P521_XMD:SHA-512_SSWU_NU_
suite = P521_XMD:SHA-512_SSWU_NU_
dst = QUUX-V01-CS02-with-P521_XMD:SHA-512_SSWU_NU_
msg =
P.x = 01ec604b4e1e3e4c7449b7a41e366e876655538acf51fd40d08b97
be066f7d020634e906b1b6942f9174b417027c953d75fb6ec64b8c
ee2a3672d4f1987d13974705
P.y = 00944fc439b4aad2463e5c9cfa0b0707af3c9a42e37c5a57bb4ecd
12fef9fb21508568aedcdd8d2490472df4bbafd79081c81e99f4da
3286eddf19be47e9c4cf0e91
u[0] = 01e4947fe62a4e47792cee2798912f672fff820b2556282d9843b4
b465940d7683a986f93ccb0e9a191fbc09a6e770a564490d2a4ae5
1b287ca39f69c3d910ba6a4f
Q.x = 01ec604b4e1e3e4c7449b7a41e366e876655538acf51fd40d08b97
be066f7d020634e906b1b6942f9174b417027c953d75fb6ec64b8c
ee2a3672d4f1987d13974705
Q.y = 00944fc439b4aad2463e5c9cfa0b0707af3c9a42e37c5a57bb4ecd
12fef9fb21508568aedcdd8d2490472df4bbafd79081c81e99f4da
3286eddf19be47e9c4cf0e91
msg = abc
P.x = 00c720ab56aa5a7a4c07a7732a0a4e1b909e32d063ae1b58db5f0e
b5e09f08a9884bff55a2bef4668f715788e692c18c1915cd034a6b
998311fcf46924ce66a2be9a
P.y = 003570e87f91a4f3c7a56be2cb2a078ffc153862a53d5e03e5dad5
bccc6c529b8bab0b7dbb157499e1949e4edab21cf5d10b782bc1e9
45e13d7421ad8121dbc72b1d
u[0] = 0019b85ef78596efc84783d42799e80d787591fe7432dee1d9fa2b
7651891321be732ddf653fa8fefa34d86fb728db569d36b5b6ed39
83945854b2fc2dc6a75aa25b
Q.x = 00c720ab56aa5a7a4c07a7732a0a4e1b909e32d063ae1b58db5f0e
b5e09f08a9884bff55a2bef4668f715788e692c18c1915cd034a6b
998311fcf46924ce66a2be9a
Q.y = 003570e87f91a4f3c7a56be2cb2a078ffc153862a53d5e03e5dad5
bccc6c529b8bab0b7dbb157499e1949e4edab21cf5d10b782bc1e9
45e13d7421ad8121dbc72b1d
msg = abcdef0123456789
P.x = 00bcaf32a968ff7971b3bbd9ce8edfbee1309e2019d7ff373c3838
7a782b005dce6ceffccfeda5c6511c8f7f312f343f3a891029c585
8f45ee0bf370aba25fc990cc
P.y = 00923517e767532d82cb8a0b59705eec2b7779ce05f9181c7d5d5e
25694ef8ebd4696343f0bc27006834d2517215ecf79482a84111f5
0c1bae25044fe1dd77744bbd
u[0] = 01dba0d7fa26a562ee8a9014ebc2cca4d66fd9de036176aca8fc11
ef254cd1bc208847ab7701dbca7af328b3f601b11a1737a899575a
5c14f4dca5aaca45e9935e07
Q.x = 00bcaf32a968ff7971b3bbd9ce8edfbee1309e2019d7ff373c3838
7a782b005dce6ceffccfeda5c6511c8f7f312f343f3a891029c585
8f45ee0bf370aba25fc990cc
Q.y = 00923517e767532d82cb8a0b59705eec2b7779ce05f9181c7d5d5e
25694ef8ebd4696343f0bc27006834d2517215ecf79482a84111f5
0c1bae25044fe1dd77744bbd
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 001ac69014869b6c4ad7aa8c443c255439d36b0e48a0f57b03d6fe
9c40a66b4e2eaed2a93390679a5cc44b3a91862b34b673f0e92c83
187da02bf3db967d867ce748
P.y = 00d5603d530e4d62b30fccfa1d90c2206654d74291c1db1c25b86a
051ee3fffc294e5d56f2e776853406bd09206c63d40f37ad882952
4cf89ad70b5d6e0b4a3b7341
u[0] = 00844da980675e1244cb209dcf3ea0aabec23bd54b2cda69fff86e
b3acc318bf3d01bae96e9cd6f4c5ceb5539df9a7ad7fcc5e9d5469
6081ba9782f3a0f6d14987e3
Q.x = 001ac69014869b6c4ad7aa8c443c255439d36b0e48a0f57b03d6fe
9c40a66b4e2eaed2a93390679a5cc44b3a91862b34b673f0e92c83
187da02bf3db967d867ce748
Q.y = 00d5603d530e4d62b30fccfa1d90c2206654d74291c1db1c25b86a
051ee3fffc294e5d56f2e776853406bd09206c63d40f37ad882952
4cf89ad70b5d6e0b4a3b7341
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 01801de044c517a80443d2bd4f503a9e6866750d2f94a22970f62d
721f96e4310e4a828206d9cdeaa8f2d476705cc3bbc490a6165c68
7668f15ec178a17e3d27349b
P.y = 0068889ea2e1442245fe42bfda9e58266828c0263119f35a61631a
3358330f3bb84443fcb54fcd53a1d097fccbe310489b74ee143fc2
938959a83a1f7dd4a6fd395b
u[0] = 01aab1fb7e5cd44ba4d9f32353a383cb1bb9eb763ed40b32bdd5f6
66988970205998c0e44af6e2b5f6f8e48e969b3f649cae3c6ab463
e1b274d968d91c02f00cce91
Q.x = 01801de044c517a80443d2bd4f503a9e6866750d2f94a22970f62d
721f96e4310e4a828206d9cdeaa8f2d476705cc3bbc490a6165c68
7668f15ec178a17e3d27349b
Q.y = 0068889ea2e1442245fe42bfda9e58266828c0263119f35a61631a
3358330f3bb84443fcb54fcd53a1d097fccbe310489b74ee143fc2
938959a83a1f7dd4a6fd395b
J.4. curve25519
J.4.1. curve25519_XMD:SHA-512_ELL2_RO_
suite = curve25519_XMD:SHA-512_ELL2_RO_
dst = QUUX-V01-CS02-with-curve25519_XMD:SHA-512_ELL2_RO_
msg =
P.x = 2de3780abb67e861289f5749d16d3e217ffa722192d16bbd9d1bfb
9d112b98c0
P.y = 3b5dc2a498941a1033d176567d457845637554a2fe7a3507d21abd
1c1bd6e878
u[0] = 005fe8a7b8fef0a16c105e6cadf5a6740b3365e18692a9c05bfbb4
d97f645a6a
u[1] = 1347edbec6a2b5d8c02e058819819bee177077c9d10a4ce165aab0
fd0252261a
Q0.x = 36b4df0c864c64707cbf6cf36e9ee2c09a6cb93b28313c169be295
61bb904f98
Q0.y = 6cd59d664fb58c66c892883cd0eb792e52055284dac3907dd756b4
5d15c3983d
Q1.x = 3fa114783a505c0b2b2fbeef0102853c0b494e7757f2a089d0daae
7ed9a0db2b
Q1.y = 76c0fe7fec932aaafb8eefb42d9cbb32eb931158f469ff3050af15
cfdbbeff94
msg = abc
P.x = 2b4419f1f2d48f5872de692b0aca72cc7b0a60915dd70bde432e82
6b6abc526d
P.y = 1b8235f255a268f0a6fa8763e97eb3d22d149343d495da1160eff9
703f2d07dd
u[0] = 49bed021c7a3748f09fa8cdfcac044089f7829d3531066ac9e74e0
994e05bc7d
u[1] = 5c36525b663e63389d886105cee7ed712325d5a97e60e140aba7e2
ce5ae851b6
Q0.x = 16b3d86e056b7970fa00165f6f48d90b619ad618791661b7b5e1ec
78be10eac1
Q0.y = 4ab256422d84c5120b278cbdfc4e1facc5baadffeccecf8ee9bf39
46106d50ca
Q1.x = 7ec29ddbf34539c40adfa98fcb39ec36368f47f30e8f888cc7e86f
4d46e0c264
Q1.y = 10d1abc1cae2d34c06e247f2141ba897657fb39f1080d54f09ce0a
f128067c74
msg = abcdef0123456789
P.x = 68ca1ea5a6acf4e9956daa101709b1eee6c1bb0df1de3b90d46023
82a104c036
P.y = 2a375b656207123d10766e68b938b1812a4a6625ff83cb8d5e86f5
8a4be08353
u[0] = 6412b7485ba26d3d1b6c290a8e1435b2959f03721874939b21782d
f17323d160
u[1] = 24c7b46c1c6d9a21d32f5707be1380ab82db1054fde82865d5c9e3
d968f287b2
Q0.x = 71de3dadfe268872326c35ac512164850860567aea0e7325e6b91a
98f86533ad
Q0.y = 26a08b6e9a18084c56f2147bf515414b9b63f1522e1b6c5649f7d4
b0324296ec
Q1.x = 5704069021f61e41779e2ba6b932268316d6d2a6f064f997a22fef
16d1eaeaca
Q1.y = 50483c7540f64fb4497619c050f2c7fe55454ec0f0e79870bb4430
2e34232210
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 096e9c8bae6c06b554c1ee69383bb0e82267e064236b3a30608d4e
d20b73ac5a
P.y = 1eb5a62612cafb32b16c3329794645b5b948d9f8ffe501d4e26b07
3fef6de355
u[0] = 5e123990f11bbb5586613ffabdb58d47f64bb5f2fa115f8ea8df01
88e0c9e1b5
u[1] = 5e8553eb00438a0bb1e7faa59dec6d8087f9c8011e5fb8ed9df31c
b6c0d4ac19
Q0.x = 7a94d45a198fb5daa381f45f2619ab279744efdd8bd8ed587fc5b6
5d6cea1df0
Q0.y = 67d44f85d376e64bb7d713585230cdbfafc8e2676f7568e0b6ee59
361116a6e1
Q1.x = 30506fb7a32136694abd61b6113770270debe593027a968a01f271
e146e60c18
Q1.y = 7eeee0e706b40c6b5174e551426a67f975ad5a977ee2f01e8e20a6
d612458c3b
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 1bc61845a138e912f047b5e70ba9606ba2a447a4dade024c8ef3dd
42b7bbc5fe
P.y = 623d05e47b70e25f7f1d51dda6d7c23c9a18ce015fe3548df596ea
9e38c69bf1
u[0] = 20f481e85da7a3bf60ac0fb11ed1d0558fc6f941b3ac5469aa8b56
ec883d6d7d
u[1] = 017d57fd257e9a78913999a23b52ca988157a81b09c5442501d07f
ed20869465
Q0.x = 02d606e2699b918ee36f2818f2bc5013e437e673c9f9b9cdc15fd0
c5ee913970
Q0.y = 29e9dc92297231ef211245db9e31767996c5625dfbf92e1c8107ef
887365de1e
Q1.x = 38920e9b988d1ab7449c0fa9a6058192c0c797bb3d42ac34572434
1a1aa98745
Q1.y = 24dcc1be7c4d591d307e89049fd2ed30aae8911245a9d8554bf603
2e5aa40d3d
J.4.2. curve25519_XMD:SHA-512_ELL2_NU_
suite = curve25519_XMD:SHA-512_ELL2_NU_
dst = QUUX-V01-CS02-with-curve25519_XMD:SHA-512_ELL2_NU_
msg =
P.x = 1bb913f0c9daefa0b3375378ffa534bda5526c97391952a7789eb9
76edfe4d08
P.y = 4548368f4f983243e747b62a600840ae7c1dab5c723991f85d3a97
68479f3ec4
u[0] = 608d892b641f0328523802a6603427c26e55e6f27e71a91a478148
d45b5093cd
Q.x = 51125222da5e763d97f3c10fcc92ea6860b9ccbbd2eb1285728f56
6721c1e65b
Q.y = 343d2204f812d3dfc5304a5808c6c0d81a903a5d228b342442aa3c
9ba5520a3d
msg = abc
P.x = 7c22950b7d900fa866334262fcaea47a441a578df43b894b4625c9
b450f9a026
P.y = 5547bc00e4c09685dcbc6cb6765288b386d8bdcb595fa5a6e3969e
08097f0541
u[0] = 46f5b22494bfeaa7f232cc8d054be68561af50230234d7d1d63d1d
9abeca8da5
Q.x = 7d56d1e08cb0ccb92baf069c18c49bb5a0dcd927eff8dcf75ca921
ef7f3e6eeb
Q.y = 404d9a7dc25c9c05c44ab9a94590e7c3fe2dcec74533a0b24b188a
5d5dacf429
msg = abcdef0123456789
P.x = 31ad08a8b0deeb2a4d8b0206ca25f567ab4e042746f792f4b7973f
3ae2096c52
P.y = 405070c28e78b4fa269427c82827261991b9718bd6c6e95d627d70
1a53c30db1
u[0] = 235fe40c443766ce7e18111c33862d66c3b33267efa50d50f9e8e5
d252a40aaa
Q.x = 3fbe66b9c9883d79e8407150e7c2a1c8680bee496c62fabe4619a7
2b3cabe90f
Q.y = 08ec476147c9a0a3ff312d303dbbd076abb7551e5fce82b48ab14b
433f8d0a7b
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 027877759d155b1997d0d84683a313eb78bdb493271d935b622900
459d52ceaa
P.y = 54d691731a53baa30707f4a87121d5169fb5d587d70fb0292b5830
dedbec4c18
u[0] = 001e92a544463bda9bd04ddbe3d6eed248f82de32f522669efc5dd
ce95f46f5b
Q.x = 227e0bb89de700385d19ec40e857db6e6a3e634b1c32962f370d26
f84ff19683
Q.y = 5f86ff3851d262727326a32c1bf7655a03665830fa7f1b8b1e5a09
d85bc66e4a
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 5fd892c0958d1a75f54c3182a18d286efab784e774d1e017ba2fb2
52998b5dc1
P.y = 750af3c66101737423a4519ac792fb93337bd74ee751f19da4cf1e
94f4d6d0b8
u[0] = 1a68a1af9f663592291af987203393f707305c7bac9c8d63d6a729
bdc553dc19
Q.x = 3bcd651ee54d5f7b6013898aab251ee8ecc0688166fce6e9548d38
472f6bd196
Q.y = 1bb36ad9197299f111b4ef21271c41f4b7ecf5543db8bb5931307e
bdb2eaa465
J.5. edwards25519
J.5.1. edwards25519_XMD:SHA-512_ELL2_RO_
suite = edwards25519_XMD:SHA-512_ELL2_RO_
dst = QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_RO_
msg =
P.x = 3c3da6925a3c3c268448dcabb47ccde5439559d9599646a8260e47
b1e4822fc6
P.y = 09a6c8561a0b22bef63124c588ce4c62ea83a3c899763af26d7953
02e115dc21
u[0] = 03fef4813c8cb5f98c6eef88fae174e6e7d5380de2b007799ac7ee
712d203f3a
u[1] = 780bdddd137290c8f589dc687795aafae35f6b674668d92bf92ae7
93e6a60c75
Q0.x = 6549118f65bb617b9e8b438decedc73c496eaed496806d3b2eb9ee
60b88e09a7
Q0.y = 7315bcc8cf47ed68048d22bad602c6680b3382a08c7c5d3f439a97
3fb4cf9feb
Q1.x = 31dcfc5c58aa1bee6e760bf78cbe71c2bead8cebb2e397ece0f37a
3da19c9ed2
Q1.y = 7876d81474828d8a5928b50c82420b2bd0898d819e9550c5c82c39
fc9bafa196
msg = abc
P.x = 608040b42285cc0d72cbb3985c6b04c935370c7361f4b7fbdb1ae7
f8c1a8ecad
P.y = 1a8395b88338f22e435bbd301183e7f20a5f9de643f11882fb237f
88268a5531
u[0] = 5081955c4141e4e7d02ec0e36becffaa1934df4d7a270f70679c78
f9bd57c227
u[1] = 005bdc17a9b378b6272573a31b04361f21c371b256252ae5463119
aa0b925b76
Q0.x = 5c1525bd5d4b4e034512949d187c39d48e8cd84242aa4758956e4a
dc7d445573
Q0.y = 2bf426cf7122d1a90abc7f2d108befc2ef415ce8c2d09695a74072
40faa01f29
Q1.x = 37b03bba828860c6b459ddad476c83e0f9285787a269df2156219b
7e5c86210c
Q1.y = 285ebf5412f84d0ad7bb4e136729a9ffd2195d5b8e73c0dc85110c
e06958f432
msg = abcdef0123456789
P.x = 6d7fabf47a2dc03fe7d47f7dddd21082c5fb8f86743cd020f3fb14
7d57161472
P.y = 53060a3d140e7fbcda641ed3cf42c88a75411e648a1add71217f70
ea8ec561a6
u[0] = 285ebaa3be701b79871bcb6e225ecc9b0b32dff2d60424b4c50642
636a78d5b3
u[1] = 2e253e6a0ef658fedb8e4bd6a62d1544fd6547922acb3598ec6b36
9760b81b31
Q0.x = 3ac463dd7fddb773b069c5b2b01c0f6b340638f54ee3bd92d452fc
ec3015b52d
Q0.y = 7b03ba1e8db9ec0b390d5c90168a6a0b7107156c994c674b61fe69
6cbeb46baf
Q1.x = 0757e7e904f5e86d2d2f4acf7e01c63827fde2d363985aa7432106
f1b3a444ec
Q1.y = 50026c96930a24961e9d86aa91ea1465398ff8e42015e2ec1fa397
d416f6a1c0
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 5fb0b92acedd16f3bcb0ef83f5c7b7a9466b5f1e0d8d217421878e
a3686f8524
P.y = 2eca15e355fcfa39d2982f67ddb0eea138e2994f5956ed37b7f72e
ea5e89d2f7
u[0] = 4fedd25431c41f2a606952e2945ef5e3ac905a42cf64b8b4d4a83c
533bf321af
u[1] = 02f20716a5801b843987097a8276b6d869295b2e11253751ca72c1
09d37485a9
Q0.x = 703e69787ea7524541933edf41f94010a201cc841c1cce60205ec3
8513458872
Q0.y = 32bb192c4f89106466f0874f5fd56a0d6b6f101cb714777983336c
159a9bec75
Q1.x = 0c9077c5c31720ed9413abe59bf49ce768506128d810cb882435aa
90f713ef6b
Q1.y = 7d5aec5210db638c53f050597964b74d6dda4be5b54fa73041bf90
9ccb3826cb
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 0efcfde5898a839b00997fbe40d2ebe950bc81181afbd5cd6b9618
aa336c1e8c
P.y = 6dc2fc04f266c5c27f236a80b14f92ccd051ef1ff027f26a07f8c0
f327d8f995
u[0] = 6e34e04a5106e9bd59f64aba49601bf09d23b27f7b594e56d5de06
df4a4ea33b
u[1] = 1c1c2cb59fc053f44b86c5d5eb8c1954b64976d0302d3729ff66e8
4068f5fd96
Q0.x = 21091b2e3f9258c7dfa075e7ae513325a94a3d8a28e1b1cb3b5b6f
5d65675592
Q0.y = 41a33d324c89f570e0682cdf7bdb78852295daf8084c669f2cc969
2896ab5026
Q1.x = 4c07ec48c373e39a23bd7954f9e9b66eeab9e5ee1279b867b3d531
5aa815454f
Q1.y = 67ccac7c3cb8d1381242d8d6585c57eabaddbb5dca5243a68a8aeb
5477d94b3a
J.5.2. edwards25519_XMD:SHA-512_ELL2_NU_
suite = edwards25519_XMD:SHA-512_ELL2_NU_
dst = QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_NU_
msg =
P.x = 1ff2b70ecf862799e11b7ae744e3489aa058ce805dd323a936375a
84695e76da
P.y = 222e314d04a4d5725e9f2aff9fb2a6b69ef375a1214eb19021ceab
2d687f0f9b
u[0] = 7f3e7fb9428103ad7f52db32f9df32505d7b427d894c5093f7a0f0
374a30641d
Q.x = 42836f691d05211ebc65ef8fcf01e0fb6328ec9c4737c26050471e
50803022eb
Q.y = 22cb4aaa555e23bd460262d2130d6a3c9207aa8bbb85060928beb2
63d6d42a95
msg = abc
P.x = 5f13cc69c891d86927eb37bd4afc6672360007c63f68a33ab423a3
aa040fd2a8
P.y = 67732d50f9a26f73111dd1ed5dba225614e538599db58ba30aaea1
f5c827fa42
u[0] = 09cfa30ad79bd59456594a0f5d3a76f6b71c6787b04de98be5cd20
1a556e253b
Q.x = 333e41b61c6dd43af220c1ac34a3663e1cf537f996bab50ab66e33
c4bd8e4e19
Q.y = 51b6f178eb08c4a782c820e306b82c6e273ab22e258d972cd0c511
787b2a3443
msg = abcdef0123456789
P.x = 1dd2fefce934ecfd7aae6ec998de088d7dd03316aa1847198aecf6
99ba6613f1
P.y = 2f8a6c24dd1adde73909cada6a4a137577b0f179d336685c4a955a
0a8e1a86fb
u[0] = 475ccff99225ef90d78cc9338e9f6a6bb7b17607c0c4428937de75
d33edba941
Q.x = 55186c242c78e7d0ec5b6c9553f04c6aeef64e69ec2e824472394d
a32647cfc6
Q.y = 5b9ea3c265ee42256a8f724f616307ef38496ef7eba391c08f99f3
bea6fa88f0
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 35fbdc5143e8a97afd3096f2b843e07df72e15bfca2eaf6879bf97
c5d3362f73
P.y = 2af6ff6ef5ebba128b0774f4296cb4c2279a074658b083b8dcca91
f57a603450
u[0] = 049a1c8bd51bcb2aec339f387d1ff51428b88d0763a91bcdf69298
14ac95d03d
Q.x = 024b6e1621606dca8071aa97b43dce4040ca78284f2a527dcf5d0f
bfac2b07e7
Q.y = 5102353883d739bdc9f8a3af650342b171217167dcce34f8db5720
8ec1dfdbf2
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 6e5e1f37e99345887fc12111575fc1c3e36df4b289b8759d23af14
d774b66bff
P.y = 2c90c3d39eb18ff291d33441b35f3262cdd307162cc97c31bfcc7a
4245891a37
u[0] = 3cb0178a8137cefa5b79a3a57c858d7eeeaa787b2781be4a362a2f
0750d24fa0
Q.x = 3e6368cff6e88a58e250c54bd27d2c989ae9b3acb6067f2651ad28
2ab8c21cd9
Q.y = 38fb39f1566ca118ae6c7af42810c0bb9767ae5960abb5a8ca7925
30bfb9447d
J.6. curve448
J.6.1. curve448_XOF:SHAKE256_ELL2_RO_
suite = curve448_XOF:SHAKE256_ELL2_RO_
dst = QUUX-V01-CS02-with-curve448_XOF:SHAKE256_ELL2_RO_
msg =
P.x = 5ea5ff623d27c75e73717514134e73e419f831a875ca9e82915fdf
c7069d0a9f8b532cfb32b1d8dd04ddeedbe3fa1d0d681c01e825d6
a9ea
P.y = afadd8de789f8f8e3516efbbe313a7eba364c939ecba00dabf4ced
5c563b18e70a284c17d8f46b564c4e6ce11784a3825d9411166221
28c1
u[0] = c704c7b3d3b36614cf3eedd0324fe6fe7d1402c50efd16cff89ff6
3f50938506280d3843478c08e24f7842f4e3ef45f6e3c4897f9d97
6148
u[1] = c25427dc97fff7a5ad0a78654e2c6c27b1c1127b5b53c7950cd1fd
6edd2703646b25f341e73deedfebf022d1d3cecd02b93b4d585ead
3ed7
Q0.x = 3ba318806f89c19cc019f51e33eb6b8c038dab892e858ce7c7f2c2
ac58618d06146a5fef31e49af49588d4d3db1bcf02bd4e4a733e37
065d
Q0.y = b30b4cfc2fd14d9d4b70456c0f5c6f6070be551788893d570e7955
675a20f6c286d01d6e90d2fb500d2efb8f4e18db7f8268bb9b7fbc
5975
Q1.x = f03a48cf003f63be61ca055fec87c750434da07a15f8aa6210389f
f85943b5166484339c8bea1af9fc571313d35ed2fbb779408b760c
4cbd
Q1.y = 23943a33b2954dc54b76a8222faf5b7e18405a41f5ecc61bf1b8df
1f9cbfad057307ed0c7b721f19c0390b8ee3a2dec223671f9ff905
fda7
msg = abc
P.x = 9b2f7ce34878d7cebf34c582db14958308ea09366d1ec71f646411
d3de0ae564d082b06f40cd30dfc08d9fb7cb21df390cf207806ad9
d0e4
P.y = 138a0eef0a4993ea696152ed7db61f7ddb4e8100573591e7466d61
c0c568ecaec939e36a84d276f34c402526d8989a96e99760c4869e
d633
u[0] = 2dd95593dfee26fe0d218d3d9a0a23d9e1a262fd1d0b602483d084
15213e75e2db3c69b0a5bc89e71bcefc8c723d2b6a0cf263f02ad2
aa70
u[1] = 272e4c79a1290cc6d2bc4f4f9d31bf7fbe956ca303c04518f117d7
7c0e9d850796fc3e1e2bcb9c75e8eaaded5e150333cae993186804
7c9d
Q0.x = 26714783887ec444fbade9ae350dc13e8d5a64150679232560726a
73d36e28bd56766d7d0b0899d79c8d1c889ae333f601c57532ff3c
4f09
Q0.y = 080e486f8f5740dbbe82305160cab9fac247b0b22a54d961de6750
37c3036fa68464c8756478c322ae0aeb9ba386fe626cebb0bcca46
840c
Q1.x = 0d9741d10421691a8ebc7778b5f623260fdf8b28ae28d776efcb8e
0d5fbb65139a2f828617835f527cb2ca24a8f5fc8e84378343c43d
096d
Q1.y = 54f4c499bf3d5b154511913f9615bd914969b65cfb74508d7ae5a1
69e9595b7cbcab9a1485e07b2ce426e4fbed052f03842c4313b7db
e39a
msg = abcdef0123456789
P.x = f54ecd14b85a50eeeee0618452df3a75be7bfba11da5118774ae4e
a55ac204e153f77285d780c4acee6c96abe3577a0c0b00be6e790c
f194
P.y = 935247a64bf78c107069943c7e3ecc52acb27ce4a3230407c83573
41685ea2152e8c3da93f8cd77da1bddb5bb759c6e7ae7d516dced4
2850
u[0] = 6aab71a38391639f27e49eae8b1cb6b7172a1f478190ece293957e
7cdb2391e7cc1c4261970d9c1bbf9c3915438f74fbd7eb5cd4d4d1
7ace
u[1] = c80b8380ca47a3bcbf76caa75cef0e09f3d270d5ee8f676cde11ae
df41aaca6741bd81a86232bd336ccb42efad39f06542bc06a67b65
909e
Q0.x = 946d91bd50c90ef70743e0dd194bddd68bb630f4e67e5b93e15a9b
94e62cb85134467993501759525c1f4fdbf06f10ddaf817847d735
e062
Q0.y = 185cf511262ec1e9b3c3cbdc015ab93df4e71cbe87766917d81c9f
3419d480407c1462385122c84982d4dae60c3ae4acce0089e37ad6
5934
Q1.x = 01778f4797b717cd6f83c193b2dfb92a1606a36ede941b0f6ab0ac
71ad0eac756d17604bf054398887da907e41065d3595f178ae802f
2087
Q1.y = b4ca727d0bda895e0eee7eb3cbc28710fa2e90a73b568cae26bd7c
2e73b70a9fa0affe1096f0810198890ed65d8935886b6e60dc4c56
9dc6
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 5bd67c4f88adf6beb10f7e0d0054659776a55c97b809ec8b310172
9e104fd0f684e103792f267fd87cc4afc25a073956ef4f268fb028
24d5
P.y = da1f5cb16a352719e4cb064cf47ba72aeba7752d03e8ca2c56229f
419b4ef378785a5af1a53dd7ab4d467c1f92f7b139b3752faf29c9
6432
u[0] = cb5c27e51f9c18ee8ffdb6be230f4eb4f2c2481963b2293484f08d
a2241c1ff59f80978e6defe9d70e34abba2fcbe12dc3a1eb2c5d3d
2e4a
u[1] = c895e8afecec5466e126fa70fc4aa784b8009063afb10e3ee06a9b
22318256aa8693b0c85b955cf2d6540b8ed71e729af1b8d5ca3b11
6cd7
Q0.x = c2d275826d6ad55e41a22318f6b6240f1f862a2e231120ff41eadb
ec319756032e8cef2a7ac6c10214fa0608c17fcaf61ec2694a8a2b
358b
Q0.y = 93d2e092762b135509840e609d413200df800d99da91d8b8284066
6cac30e7a3520adbaa4b089bfdc86132e42729f651d022f4782502
f12c
Q1.x = 3c0880ece7244036e9a45944a85599f9809d772f770cc237ac41b2
1aa71615e4f3bb08f64fca618896e4f6cf5bd92e16b89d2cf6e195
6bfb
Q1.y = 45cce4beb96505cac5976b3d2673641e9bcd18d3462bbb453d293e
5282740a6389cfeae610adc7bd425c728541ceec83fcc999164af4
3fb5
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = ea441c10b3636ecedd5c0dfcae96384cc40de8390a0ab648765b45
08da12c586d55dc981275776507ebca0e4d1bcaa302bb69dcfa31b
3451
P.y = fee0192d49bcc0c28d954763c2cbe739b9265c4bebe3883803c649
71220cfda60b9ac99ad986cd908c0534b260b5cfca46f6c2b0f3f2
1bda
u[0] = 8cba93a007bb2c801b1769e026b1fa1640b14a34cf3029db3c7fd6
392745d6fec0f7870b5071d6da4402cedbbde28ae4e50ab30e1049
a238
u[1] = 4223746145069e4b8a981acc3404259d1a2c3ecfed5d864798a89d
45f81a2c59e2d40eb1d5f0fe11478cbb2bb30246dd388cb932ad7b
b330
Q0.x = 4321ab02a9849128691e9b80a5c5576793a218de14885fddccb91f
17ceb1646ea00a28b69ad211e1f14f17739612dbde3782319bdf00
9689
Q0.y = 1b8a7b539519eec0ea9f7a46a43822e16cba39a439733d6847ac44
a806b8adb3e1a75ea48a1228b8937ba85c6cb6ee01046e10cad895
3b1e
Q1.x = 126d744da6a14fddec0f78a9cee4571c1320ac7645b600187812e4
d7021f98fc4703732c54daec787206e1f34d9dbbf4b292c68160b8
bfbd
Q1.y = 136eebe6020f2389d448923899a1a38a4c8ad74254e0686e91c4f9
3c1f8f8e1bd619ffb7c1281467882a9c957d22d50f65c5b72b2aee
11af
J.6.2. curve448_XOF:SHAKE256_ELL2_NU_
suite = curve448_XOF:SHAKE256_ELL2_NU_
dst = QUUX-V01-CS02-with-curve448_XOF:SHAKE256_ELL2_NU_
msg =
P.x = b65e8dbb279fd656f926f68d463b13ca7a982b32f5da9c7cc58afc
f6199e4729863fb75ca9ae3c95c6887d95a5102637a1c5c40ff0aa
fadc
P.y = ea1ea211cf29eca11c057fe8248181591a19f6ac51d45843a65d4b
b8b71bc83a64c771ed7686218a278ef1c5d620f3d26b5316218864
5453
u[0] = 242c70f74eac8184116c71630d284cf8a742fc463e710545847ff6
4d8e9161cb9f599728a18a32dbd8b67c3bec5d64c9b1d2f2cde7b5
888d
Q.x = e6304424de5af3f556d3e645600530c53ad949891c3e60ba041dd5
f68a93901beff8440164477d348c13d28e27bfcd360c44c80b4c7d
4cea
Q.y = 4160a8f2043a347185406a6a7e50973b98b82edbdfa3209b0e1c90
118e10eeb45045b0990d4b2b0708a30eca17df40ad53c9100f20c1
0b44
msg = abc
P.x = 51aceca4fa95854bbaba58d8a5e17a86c07acadef32e1188cafda2
6232131800002cc2f27c7aec454e5e0c615bddffb7df6a5f7f0f14
793f
P.y = c590c9246eb28b08dee816d608ef233ea5d76e305dc458774a1e1b
d880387e6734219e2018e4aa50a49486dce0ba8740065da37e6cf5
212c
u[0] = ef6dcb75b696d325fb36d66b104700df1480c4c17ea9190d447eee
1e7e4c9b7f36bbfb8ba7ba7c4cb6b07fed16531c1ac7a26a3618b4
0b34
Q.x = de0dc93df9ce7953452f20e270699c1e7dacd5d571c226d77f53b7
e3053d16f8a81b1601efb362054e973c8e733b663af93f00cb81ba
f130
Q.y = 8c5bdec6fa6690905f6eff966b0f98f5a8161493bd04976684d4ec
1f4512fa8743d86860b2ff2c5d67e9c145fd906f2cb89ff812c6b9
883f
msg = abcdef0123456789
P.x = c6d65987f146b8d0cb5d2c44e1872ac3af1f458f6a8bd8c232ffe8
b9d09496229a5a27f350eb7d97305bcc4e0f38328718352e8e3129
ed71
P.y = 4d2f901bf333fdc4135b954f20d59207e9f6a4ecf88ce5af11c892
b44f79766ec4ecc9f60d669b95ca8940f39b1b7044140ac2040c1b
f659
u[0] = 3012ba5d9b3bb648e4613833a26ecaeadb3e8c8bba07fc90ac3da0
375769289c44d3dc87474b23df7f45f9a4030892cda689e343aeee
a6ad
Q.x = dc29532761f03c24d57f530da4c24acc4c676d185becaa89fcc083
266541fb7f10ecec91dac64a34cd988274633ae25c4d784aee52de
47a8
Q.y = a5f6da11259c69f2e07fce6a7b6afec4c25bd2df83426765f9c070
4111da24c6a0550d5c7aac7d648d55f7640d50be99c926195e852a
daac
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 9b8d008863beb4a02fb9e4efefd2eba867307fb1c7ce01746115d3
2e1db551bb254e8e3e4532d5c74a83949a69a60519ecc9178083cb
e943
P.y = 346a1fca454d1e67c628437c270ec0f0c4256bb774fe6c0e49de70
04ff6d9199e2cd99d8f7575a96aafc4dc8db1811ba0a44317581f4
1371
u[0] = fe952ac0149f92436bba12ea2e542aa226f4fc074d79ff462c41b3
27968a649a495a8a93b6c3044af2273456abb5e166ce4fb8c9b10c
8c2e
Q.x = 512803d89f59c57376e6570cd54c4e901643e089cd9456f549daa4
372b8b52679860b68aa8bedfaa88970f15ab6098d5f252083ac98a
58c9
Q.y = 3d9b6593c7941a20d76161c9a171f1e507495a08f03dfcae33a2ac
3602698e46a74d1039b583c984036f590eaa43d20ba5aada3ffb55
2f77
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 8746dc34799112d1f20acda9d7f722c9abb29b1fb6b7e9e5669838
43c20bd7c9bfad21b45c5166b808d2f5d44e188f1fdaf29cdee8a7
2e4c
P.y = 7c1293484c9287c298a1a0600c64347eee8530acf563cd8705e057
28274d8cd8101835f8003b6f3b78b5beb28f5be188a3d7bce1ec5a
36b1
u[0] = afd3d7ad9d819be7561706e050d4f30b634b203387ab682739365f
62cd7393ca2cf18cd07a3d3af8dd163f043ac7457c2eb145b4a561
70a9
Q.x = 08aed6480793218034fd3b3b0867943d7e0bd1b6f76b4929e0885b
d082b84d4449341da6038bb08229ad9eb7d518dff2c7ea50148e70
a4db
Q.y = e00d32244561ebd4b5f4ef70fcac75a06416be0a1c1b304e7bd361
a6a6586915bb902a323eaf73cf7738e70d34282f61485395ab2833
d2c1
J.7. edwards448
J.7.1. edwards448_XOF:SHAKE256_ELL2_RO_
suite = edwards448_XOF:SHAKE256_ELL2_RO_
dst = QUUX-V01-CS02-with-edwards448_XOF:SHAKE256_ELL2_RO_
msg =
P.x = 73036d4a88949c032f01507005c133884e2f0d81f9a950826245dd
a9e844fc78186c39daaa7147ead3e462cff60e9c6340b58134480b
4d17
P.y = 94c1d61b43728e5d784ef4fcb1f38e1075f3aef5e99866911de5a2
34f1aafdc26b554344742e6ba0420b71b298671bbeb2b773661863
4610
u[0] = 0847c5ebf957d3370b1f98fde499fb3e659996d9fc9b5707176ade
785ba72cd84b8a5597c12b1024be5f510fa5ba99642c4cec7f3f69
d3e7
u[1] = f8cbd8a7ae8c8deed071f3ac4b93e7cfcb8f1eac1645d699fd6d38
81cb295a5d3006d9449ed7cad412a77a1fe61e84a9e41d59ef384d
6f9a
Q0.x = c08177330869db17fb81a5e6e53b36d29086d806269760f2e4caba
a4015f5dbadb7ca2ba594d96a89d0ca4f0944489e1ef393d53db85
096f
Q0.y = 02e894598c050eeb7195f5791f1a5f65da3776b7534be37640bcbf
95d4b915bd22333c50387583507169708fbd7bea0d7aa385dcc614
be9c
Q1.x = 770877fd3b6c5503398157b68a9d3609f585f40e1ebebdd69bb0e4
d3d9aa811995ce75333fdadfa50db886a35959cc59cffd5c9710da
ca25
Q1.y = b27fef77aa6231fbbc27538fa90eaca8abd03eb1e62fdae4ec5e82
8117c3b8b3ff8c34d0a6e6d79fff16d339b94ae8ede33331d5b464
c792
msg = abc
P.x = 4e0158acacffa545adb818a6ed8e0b870e6abc24dfc1dc45cf9a05
2e98469275d9ff0c168d6a5ac7ec05b742412ee090581f12aa398f
9f8c
P.y = 894d3fa437b2d2e28cdc3bfaade035430f350ec5239b6b406b5501
da6f6d6210ff26719cad83b63e97ab26a12df6dec851d6bf38e294
af9a
u[0] = 04d975cd938ab49be3e81703d6a57cca84ed80d2ff6d4756d3f229
47fb5b70ab0231f0087cbfb4b7cae73b41b0c9396b356a4831d9a1
4322
u[1] = 2547ca887ac3db7b5fad3a098aa476e90078afe1358af6c63d677d
6edfd2100bc004e0f5db94dd2560fc5b308e223241d00488c9ca6b
0ef2
Q0.x = 7544612a97f4419c94ab0f621a1ee8ccf46c6657b8e0778ec9718b
f4b41bc774487ad87d9b1e617aa49d3a4dd35a3cf57cd390ebf042
9952
Q0.y = d3ab703e60267d796b485bb58a28f934bd0133a6d1bbdfeda5277f
a293310be262d7f653a5adffa608c37ed45c0e6008e54a16e1a342
e4df
Q1.x = 6262f18d064bc131ade1b8bbcf1cbdf984f4f88153fcc9f94c888a
f35d5e41aae84c12f169a55d8abf06e6de6c5b23079e587a58cf73
303e
Q1.y = 6d57589e901abe7d947c93ab02c307ad9093ed9a83eb0b6e829fb7
318d590381ca25f3cc628a36a924a9ddfcf3cbedf94edf3b338ea7
7403
msg = abcdef0123456789
P.x = 2c25b4503fadc94b27391933b557abdecc601c13ed51c5de683894
84f93dbd6c22e5f962d9babf7a39f39f994312f8ca23344847e1fb
f176
P.y = d5e6f5350f430e53a110f5ac7fcc82a96cb865aeca982029522d32
601e41c042a9dfbdfbefa2b0bdcdc3bc58cca8a7cd546803083d3a
8548
u[0] = 10659ce25588db4e4be6f7c791a79eb21a7f24aaaca76a6ca3b83b
80aaf95aa328fe7d569a1ac99f9cd216edf3915d72632f1a8b990e
250c
u[1] = 9243e5b6c480683fd533e81f4a778349a309ce00bd163a29eb9fa8
dbc8f549242bef33e030db21cffacd408d2c4264b93e476c6a8590
e7aa
Q0.x = 1457b60c12e00e47ceb3ce64b57e7c3c61636475443d704a8e2b2a
b0a5ac7e4b3909435416784e16e19929c653b1bdcd9478a8e5331c
a9ae
Q0.y = 935d9f75f7a0babbc39c0a1c3b412518ed8a24bc2c4886722fb4b7
d4a747af98e4e2528c75221e2dffd3424abb436e10539a74caaafa
3ea3
Q1.x = b44d9e34211b4028f24117e856585ed81448f3c8b934987a1c5939
c86048737a08d85934fec6b3c2ef9f09cbd365cf22744f2e4ce697
62a4
Q1.y = dc996c1736f4319868f897d9a27c45b02dd3bc6b7ca356a039606e
5406e131a0bbe8238208b327b00853e8af84b58b13443e70542556
3323
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = a1861a9464ae31249a0e60bf38791f3663049a3f5378998499a832
92e159a2fecff838eb9bc6939e5c6ae76eb074ad4aae39b55b72ca
0b9a
P.y = 580a2798c5b904f8adfec5bd29fb49b4633cd9f8c2935eb4a0f12e
5dfa0285680880296bb729c6405337525fb5ed3dff930c137314f6
0401
u[0] = c80390020e578f009ead417029eff6cd0926110922db63ab98395e
3bdfdd5d8a65b1a2b8d495dc8c5e59b7f3518731f7dfc0f93ace5d
ee4b
u[1] = 1c4dc6653a445bbef2add81d8e90a6c8591a788deb91d0d3f1519a
2e4a460313041b77c1b0817f2e80b388e5c3e49f37d787dc1f85e4
324a
Q0.x = 9d355251e245e4b13ed4ea3e5a3c55bf9b7211f1704771f2e1d8f1
a65610c468b1cf70c6c2ce30dcaad54ad9e5439471ec554b862ec8
875a
Q0.y = 6689ba36a242af69ac2aadb955d15e982d9b04f5d77f7609ebf742
9587feb7e5ce27490b9c72114509f89565122074e46a614d7fd7c8
00bd
Q1.x = c4b3d3ad4d2d62739a62989532992c1081e9474a201085b4616da5
706cab824693b9fb428a201bcd1639a4588cc43b9eb841dbca7421
9b1f
Q1.y = 265286f5dee8f3d894b5649da8565b58e96b4cfd44b462a2883ea6
4dbcda21a00706ea3fea53fc2d769084b0b74589e91d0384d71189
09fb
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 987c5ac19dd4b47835466a50b2d9feba7c8491b8885a04edf577e1
5a9f2c98b203ec2cd3e5390b3d20bba0fa6fc3eecefb5029a31723
4401
P.y = 5e273fcfff6b007bb6771e90509275a71ff1480c459ded26fc7b10
664db0a68aaa98bc7ecb07e49cf05b80ae5ac653fbdd14276bbd35
ccbc
u[0] = 163c79ab0210a4b5e4f44fb19437ea965bf5431ab233ef16606f0b
03c5f16a3feb7d46a5a675ce8f606e9c2bf74ee5336c54a1e54919
f13f
u[1] = f99666bde4995c4088333d6c2734687e815f80a99c6da02c47df4b
51f6c9d9ed466b4fecf7d9884990a8e0d0be6907fa437e0b1a27f4
9265
Q0.x = d1a5eba4a332514b69760948af09ceaeddbbb9fd4cb1f19b78349c
2ee4cf9ee86dbcf9064659a4a0566fe9c34d90aec86f0801edc131
ad9b
Q0.y = 5d0a75a3014c3269c33b1b5da80706a4f097893461df286353484d
8031cd607c98edc2a846c77a841f057c7251eb45077853c7b20595
7e52
Q1.x = 69583b00dc6b2aced6ffa44630cc8c8cd0dd0649f57588dd0fb1da
ad2ce132e281d01e3f25ccd3f405be759975c6484268bfe8f5e5f2
3c30
Q1.y = 8418484035f60bdccf48cb488634c2dfb40272123435f7e654fb6f
254c6c42e7e38f1fa79a637a168a28de6c275232b704f9ded0ff76
dd94
J.7.2. edwards448_XOF:SHAKE256_ELL2_NU_
suite = edwards448_XOF:SHAKE256_ELL2_NU_
dst = QUUX-V01-CS02-with-edwards448_XOF:SHAKE256_ELL2_NU_
msg =
P.x = eb5a1fc376fd73230af2de0f3374087cc7f279f0460114cf0a6c12
d6d044c16de34ec2350c34b26bf110377655ab77936869d085406a
f71e
P.y = df5dcea6d42e8f494b279a500d09e895d26ac703d75ca6d118e8ca
58bf6f608a2a383f292fce1563ff995dce75aede1fdc8e7c0c737a
e9ad
u[0] = 1368aefc0416867ea2cfc515416bcbeecc9ec81c4ecbd52ccdb91e
06996b3f359bc930eef6743c7a2dd7adb785bc7093ed044efed950
86d7
Q.x = 4b2abf8c0fca49d027c2a81bf73bb5990e05f3e76c7ba137cc0b89
415ccd55ce7f191cc0c11b0560c1cdc2a8085dd56996079e05a3cd
8dde
Q.y = 82532f5b0cb3bfb8542d3228d055bfe61129dbeae8bace80cf61f1
7725e8ec8226a24f0e687f78f01da88e3b2715194a03dca7c0a96b
bf04
msg = abc
P.x = 4623a64bceaba3202df76cd8b6e3daf70164f3fcbda6d6e340f7fa
b5cdf89140d955f722524f5fe4d968fef6ba2853ff4ea086c2f67d
8110
P.y = abaac321a169761a8802ab5b5d10061fec1a83c670ac6bc9595470
0317ee5f82870120e0e2c5a21b12a0c7ad17ebd343363604c4bcec
afd1
u[0] = cda3b0ecfe054c4077007d7300969ec24f4c741300b630ec9188eb
ab31a5ae0065612ee22d9f793733179ffc2e10c53ca5b539057aaf
dc2f
Q.x = b1ca5bef2f157673a210f56c9b0039db8399e4749585abac64f831
f74ed1ec5f591928976c687c06d57686bacb98440e77af878349cd
f2d2
Q.y = 5bbfd6a3730d517b03c3cd9e2eed94af12891334ec090e0495c2ed
c588e9e10b6f63b03a62076808cbcd6da95adfb5af76c136b2d42e
0dac
msg = abcdef0123456789
P.x = e9eb562e76db093baa43a31b7edd04ec4aadcef3389a7b9c58a19c
f87f8ae3d154e134b6b3ed45847a741e33df51903da681629a4b8b
cc2e
P.y = 0cf6606927ad7eb15dbc193993bc7e4dda744b311a8ec4274c8f73
8f74f605934582474c79260f60280fe35bd37d4347e59184cbfa12
cbc4
u[0] = d36bae98351512c382c7a3e1eba22497574f11fef9867901b1a270
0b39fa2cd0d38ed4380387a99162b7ba0240c743f0532ef60d577c
413d
Q.x = 958a51e2f02e0dfd3930709010d5d16f869adb9d8a8f7c01139911
d206c20cdb7bfb40ee33ba30536a99f49362fa7633d0f417fc3914
fe21
Q.y = f4307a36ab6612fa97501497f01afa109733ce85875935551c3ca9
0f0fa7e0097a8640bb7e5dbcc38ab32b23b748790f2261f2c44c3b
f3ba
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 122a3234d34b26c69749f23356452bf9501efa2d94859d5ef741fe
f024156d9d191a03a2ad24c38186f93e02d05572575968b083d8a3
9738
P.y = ddf55e74eb4414c2c1fa4aa6bc37c4ab470a3fed6bb5af1e435703
09b162fb61879bb15f9ea49c712efd42d0a71666430f9f0d4a2050
5050
u[0] = 5945744d27122f89da3daf76ab4db9616053df64e25d30ec9a0066
7ee6710240579c1db8f8ef3386f3f4f413cfb325ac14094d582026
a971
Q.x = e7e1f2d13548ac2c8fcd346e4c63606545bf93652011721e83ac3b
64226f77a8823d3881e164bc6ca45505b236e8e3721c028052fcc9
ade5
Q.y = 7e0f340501bf25f018b9d374c2acbdd43c07261d85a6ef3c855113
d4e023634db59a87b8fab9efe04ed1fee302c8a4994e83bdda32bd
9c0b
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 221704949b1ce1ab8dd174dc9b8c56fcffa27179569ce9219c0c2f
e183d3d23343a4c42a0e2e9d6b9d0feb1df3883ec489b6671d1fa6
4089
P.y = ebdecfdc87142d1a919034bf22ecfad934c9a85effff14b594ae2c
00943ca62a39d6ee3be9df0bb504ce8a9e1669bc6959c42ad6a1d3
b686
u[0] = 1192e378043f01cedc7ea0209321519213b0184ea0d8575816bcd9
182a367823e1eecc2faf1df8f79b24027a4b9bfa208cd320e79bef
06ea
Q.x = 0fd3bb833c1d7a5b319d1d4117406a23b9aece976186ecb18a11a6
35e6fbdb920d47e04762b1f2a8c59d2f8435d0fdefe501f544cda2
3dbf
Q.y = f13b0dad4d5eeb120f2443ac4392f8096a1396f5014ec2a3506a34
7fef8076a7282035cf619599b1919cf29df5ce87711c11688aab77
00a6
J.8. secp256k1
J.8.1. secp256k1_XMD:SHA-256_SSWU_RO_
suite = secp256k1_XMD:SHA-256_SSWU_RO_
dst = QUUX-V01-CS02-with-secp256k1_XMD:SHA-256_SSWU_RO_
msg =
P.x = c1cae290e291aee617ebaef1be6d73861479c48b841eaba9b7b585
2ddfeb1346
P.y = 64fa678e07ae116126f08b022a94af6de15985c996c3a91b64c406
a960e51067
u[0] = 6b0f9910dd2ba71c78f2ee9f04d73b5f4c5f7fc773a701abea1e57
3cab002fb3
u[1] = 1ae6c212e08fe1a5937f6202f929a2cc8ef4ee5b9782db68b0d579
9fd8f09e16
Q0.x = 74519ef88b32b425a095e4ebcc84d81b64e9e2c2675340a720bb1a
1857b99f1e
Q0.y = c174fa322ab7c192e11748beed45b508e9fdb1ce046dee9c2cd3a2
a86b410936
Q1.x = 44548adb1b399263ded3510554d28b4bead34b8cf9a37b4bd0bd2b
a4db87ae63
Q1.y = 96eb8e2faf05e368efe5957c6167001760233e6dd2487516b46ae7
25c4cce0c6
msg = abc
P.x = 3377e01eab42db296b512293120c6cee72b6ecf9f9205760bd9ff1
1fb3cb2c4b
P.y = 7f95890f33efebd1044d382a01b1bee0900fb6116f94688d487c6c
7b9c8371f6
u[0] = 128aab5d3679a1f7601e3bdf94ced1f43e491f544767e18a4873f3
97b08a2b61
u[1] = 5897b65da3b595a813d0fdcc75c895dc531be76a03518b044daaa0
f2e4689e00
Q0.x = 07dd9432d426845fb19857d1b3a91722436604ccbbbadad8523b8f
c38a5322d7
Q0.y = 604588ef5138cffe3277bbd590b8550bcbe0e523bbaf1bed4014a4
67122eb33f
Q1.x = e9ef9794d15d4e77dde751e06c182782046b8dac05f8491eb88764
fc65321f78
Q1.y = cb07ce53670d5314bf236ee2c871455c562dd76314aa41f012919f
e8e7f717b3
msg = abcdef0123456789
P.x = bac54083f293f1fe08e4a70137260aa90783a5cb84d3f35848b324
d0674b0e3a
P.y = 4436476085d4c3c4508b60fcf4389c40176adce756b398bdee27bc
a19758d828
u[0] = ea67a7c02f2cd5d8b87715c169d055a22520f74daeb080e6180958
380e2f98b9
u[1] = 7434d0d1a500d38380d1f9615c021857ac8d546925f5f2355319d8
23a478da18
Q0.x = 576d43ab0260275adf11af990d130a5752704f7947862876172080
8862544b5d
Q0.y = 643c4a7fb68ae6cff55edd66b809087434bbaff0c07f3f9ec4d49b
b3c16623c3
Q1.x = f89d6d261a5e00fe5cf45e827b507643e67c2a947a20fd9ad71039
f8b0e29ff8
Q1.y = b33855e0cc34a9176ead91c6c3acb1aacb1ce936d563bc1cee1dcf
fc806caf57
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = e2167bc785333a37aa562f021f1e881defb853839babf52a7f72b1
02e41890e9
P.y = f2401dd95cc35867ffed4f367cd564763719fbc6a53e969fb8496a
1e6685d873
u[0] = eda89a5024fac0a8207a87e8cc4e85aa3bce10745d501a30deb873
41b05bcdf5
u[1] = dfe78cd116818fc2c16f3837fedbe2639fab012c407eac9dfe9245
bf650ac51d
Q0.x = 9c91513ccfe9520c9c645588dff5f9b4e92eaf6ad4ab6f1cd720d1
92eb58247a
Q0.y = c7371dcd0134412f221e386f8d68f49e7fa36f9037676e163d4a06
3fbf8a1fb8
Q1.x = 10fee3284d7be6bd5912503b972fc52bf4761f47141a0015f1c6ae
36848d869b
Q1.y = 0b163d9b4bf21887364332be3eff3c870fa053cf508732900fc69a
6eb0e1b672
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = e3c8d35aaaf0b9b647e88a0a0a7ee5d5bed5ad38238152e4e6fd8c
1f8cb7c998
P.y = 8446eeb6181bf12f56a9d24e262221cc2f0c4725c7e3803024b588
8ee5823aa6
u[0] = 8d862e7e7e23d7843fe16d811d46d7e6480127a6b78838c277bca1
7df6900e9f
u[1] = 68071d2530f040f081ba818d3c7188a94c900586761e9115efa47a
e9bd847938
Q0.x = b32b0ab55977b936f1e93fdc68cec775e13245e161dbfe556bbb1f
72799b4181
Q0.y = 2f5317098360b722f132d7156a94822641b615c91f8663be691698
70a12af9e8
Q1.x = 148f98780f19388b9fa93e7dc567b5a673e5fca7079cd9cdafd719
82ec4c5e12
Q1.y = 3989645d83a433bc0c001f3dac29af861f33a6fd1e04f4b36873f5
bff497298a
J.8.2. secp256k1_XMD:SHA-256_SSWU_NU_
suite = secp256k1_XMD:SHA-256_SSWU_NU_
dst = QUUX-V01-CS02-with-secp256k1_XMD:SHA-256_SSWU_NU_
msg =
P.x = a4792346075feae77ac3b30026f99c1441b4ecf666ded19b7522cf
65c4c55c5b
P.y = 62c59e2a6aeed1b23be5883e833912b08ba06be7f57c0e9cdc663f
31639ff3a7
u[0] = 0137fcd23bc3da962e8808f97474d097a6c8aa2881fceef4514173
635872cf3b
Q.x = a4792346075feae77ac3b30026f99c1441b4ecf666ded19b7522cf
65c4c55c5b
Q.y = 62c59e2a6aeed1b23be5883e833912b08ba06be7f57c0e9cdc663f
31639ff3a7
msg = abc
P.x = 3f3b5842033fff837d504bb4ce2a372bfeadbdbd84a1d2b678b6e1
d7ee426b9d
P.y = 902910d1fef15d8ae2006fc84f2a5a7bda0e0407dc913062c3a493
c4f5d876a5
u[0] = e03f894b4d7caf1a50d6aa45cac27412c8867a25489e32c5ddeb50
3229f63a2e
Q.x = 3f3b5842033fff837d504bb4ce2a372bfeadbdbd84a1d2b678b6e1
d7ee426b9d
Q.y = 902910d1fef15d8ae2006fc84f2a5a7bda0e0407dc913062c3a493
c4f5d876a5
msg = abcdef0123456789
P.x = 07644fa6281c694709f53bdd21bed94dab995671e4a8cd1904ec4a
a50c59bfdf
P.y = c79f8d1dad79b6540426922f7fbc9579c3018dafeffcd4552b1626
b506c21e7b
u[0] = e7a6525ae7069ff43498f7f508b41c57f80563c1fe4283510b3224
46f32af41b
Q.x = 07644fa6281c694709f53bdd21bed94dab995671e4a8cd1904ec4a
a50c59bfdf
Q.y = c79f8d1dad79b6540426922f7fbc9579c3018dafeffcd4552b1626
b506c21e7b
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = b734f05e9b9709ab631d960fa26d669c4aeaea64ae62004b9d34f4
83aa9acc33
P.y = 03fc8a4a5a78632e2eb4d8460d69ff33c1d72574b79a35e402e801
f2d0b1d6ee
u[0] = d97cf3d176a2f26b9614a704d7d434739d194226a706c886c5c3c3
9806bc323c
Q.x = b734f05e9b9709ab631d960fa26d669c4aeaea64ae62004b9d34f4
83aa9acc33
Q.y = 03fc8a4a5a78632e2eb4d8460d69ff33c1d72574b79a35e402e801
f2d0b1d6ee
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 17d22b867658977b5002dbe8d0ee70a8cfddec3eec50fb93f36136
070fd9fa6c
P.y = e9178ff02f4dab73480f8dd590328aea99856a7b6cc8e5a6cdf289
ecc2a51718
u[0] = a9ffbeee1d6e41ac33c248fb3364612ff591b502386c1bf6ac4aaf
1ea51f8c3b
Q.x = 17d22b867658977b5002dbe8d0ee70a8cfddec3eec50fb93f36136
070fd9fa6c
Q.y = e9178ff02f4dab73480f8dd590328aea99856a7b6cc8e5a6cdf289
ecc2a51718
J.9. BLS12-381 G1
J.9.1. BLS12381G1_XMD:SHA-256_SSWU_RO_
suite = BLS12381G1_XMD:SHA-256_SSWU_RO_
dst = QUUX-V01-CS02-with-BLS12381G1_XMD:SHA-256_SSWU_RO_
msg =
P.x = 052926add2207b76ca4fa57a8734416c8dc95e24501772c8142787
00eed6d1e4e8cf62d9c09db0fac349612b759e79a1
P.y = 08ba738453bfed09cb546dbb0783dbb3a5f1f566ed67bb6be0e8c6
7e2e81a4cc68ee29813bb7994998f3eae0c9c6a265
u[0] = 0ba14bd907ad64a016293ee7c2d276b8eae71f25a4b941eece7b0d
89f17f75cb3ae5438a614fb61d6835ad59f29c564f
u[1] = 019b9bd7979f12657976de2884c7cce192b82c177c80e0ec604436
a7f538d231552f0d96d9f7babe5fa3b19b3ff25ac9
Q0.x = 11a3cce7e1d90975990066b2f2643b9540fa40d6137780df4e753a
8054d07580db3b7f1f03396333d4a359d1fe3766fe
Q0.y = 0eeaf6d794e479e270da10fdaf768db4c96b650a74518fc67b04b0
3927754bac66f3ac720404f339ecdcc028afa091b7
Q1.x = 160003aaf1632b13396dbad518effa00fff532f604de1a7fc2082f
f4cb0afa2d63b2c32da1bef2bf6c5ca62dc6b72f9c
Q1.y = 0d8bb2d14e20cf9f6036152ed386d79189415b6d015a20133acb4e
019139b94e9c146aaad5817f866c95d609a361735e
msg = abc
P.x = 03567bc5ef9c690c2ab2ecdf6a96ef1c139cc0b2f284dca0a9a794
3388a49a3aee664ba5379a7655d3c68900be2f6903
P.y = 0b9c15f3fe6e5cf4211f346271d7b01c8f3b28be689c8429c85b67
af215533311f0b8dfaaa154fa6b88176c229f2885d
u[0] = 0d921c33f2bad966478a03ca35d05719bdf92d347557ea166e5bba
579eea9b83e9afa5c088573c2281410369fbd32951
u[1] = 003574a00b109ada2f26a37a91f9d1e740dffd8d69ec0c35e1e9f4
652c7dba61123e9dd2e76c655d956e2b3462611139
Q0.x = 125435adce8e1cbd1c803e7123f45392dc6e326d292499c2c45c58
65985fd74fe8f042ecdeeec5ecac80680d04317d80
Q0.y = 0e8828948c989126595ee30e4f7c931cbd6f4570735624fd25aef2
fa41d3f79cfb4b4ee7b7e55a8ce013af2a5ba20bf2
Q1.x = 11def93719829ecda3b46aa8c31fc3ac9c34b428982b898369608e
4f042babee6c77ab9218aad5c87ba785481eff8ae4
Q1.y = 0007c9cef122ccf2efd233d6eb9bfc680aa276652b0661f4f820a6
53cec1db7ff69899f8e52b8e92b025a12c822a6ce6
msg = abcdef0123456789
P.x = 11e0b079dea29a68f0383ee94fed1b940995272407e3bb916bbf26
8c263ddd57a6a27200a784cbc248e84f357ce82d98
P.y = 03a87ae2caf14e8ee52e51fa2ed8eefe80f02457004ba4d486d6aa
1f517c0889501dc7413753f9599b099ebcbbd2d709
u[0] = 062d1865eb80ebfa73dcfc45db1ad4266b9f3a93219976a3790ab8
d52d3e5f1e62f3b01795e36834b17b70e7b76246d4
u[1] = 0cdc3e2f271f29c4ff75020857ce6c5d36008c9b48385ea2f2bf6f
96f428a3deb798aa033cd482d1cdc8b30178b08e3a
Q0.x = 08834484878c217682f6d09a4b51444802fdba3d7f2df9903a0dda
db92130ebbfa807fffa0eabf257d7b48272410afff
Q0.y = 0b318f7ecf77f45a0f038e62d7098221d2dbbca2a394164e2e3fe9
53dc714ac2cde412d8f2d7f0c03b259e6795a2508e
Q1.x = 158418ed6b27e2549f05531a8281b5822b31c3bf3144277fbb977f
8d6e2694fedceb7011b3c2b192f23e2a44b2bd106e
Q1.y = 1879074f344471fac5f839e2b4920789643c075792bec5af4282c7
3f7941cda5aa77b00085eb10e206171b9787c4169f
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 15f68eaa693b95ccb85215dc65fa81038d69629f70aeee0d0f677c
f22285e7bf58d7cb86eefe8f2e9bc3f8cb84fac488
P.y = 1807a1d50c29f430b8cafc4f8638dfeeadf51211e1602a5f184443
076715f91bb90a48ba1e370edce6ae1062f5e6dd38
u[0] = 010476f6a060453c0b1ad0b628f3e57c23039ee16eea5e71bb87c3
b5419b1255dc0e5883322e563b84a29543823c0e86
u[1] = 0b1a912064fb0554b180e07af7e787f1f883a0470759c03c1b6509
eb8ce980d1670305ae7b928226bb58fdc0a419f46e
Q0.x = 0cbd7f84ad2c99643fea7a7ac8f52d63d66cefa06d9a56148e58b9
84b3dd25e1f41ff47154543343949c64f88d48a710
Q0.y = 052c00e4ed52d000d94881a5638ae9274d3efc8bc77bc0e5c650de
04a000b2c334a9e80b85282a00f3148dfdface0865
Q1.x = 06493fb68f0d513af08be0372f849436a787e7b701ae31cb964d96
8021d6ba6bd7d26a38aaa5a68e8c21a6b17dc8b579
Q1.y = 02e98f2ccf5802b05ffaac7c20018bc0c0b2fd580216c4aa2275d2
909dc0c92d0d0bdc979226adeb57a29933536b6bb4
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 082aabae8b7dedb0e78aeb619ad3bfd9277a2f77ba7fad20ef6aab
dc6c31d19ba5a6d12283553294c1825c4b3ca2dcfe
P.y = 05b84ae5a942248eea39e1d91030458c40153f3b654ab7872d779a
d1e942856a20c438e8d99bc8abfbf74729ce1f7ac8
u[0] = 0a8ffa7447f6be1c5a2ea4b959c9454b431e29ccc0802bc052413a
9c5b4f9aac67a93431bd480d15be1e057c8a08e8c6
u[1] = 05d487032f602c90fa7625dbafe0f4a49ef4a6b0b33d7bb349ff4c
f5410d297fd6241876e3e77b651cfc8191e40a68b7
Q0.x = 0cf97e6dbd0947857f3e578231d07b309c622ade08f2c08b32ff37
2bd90db19467b2563cc997d4407968d4ac80e154f8
Q0.y = 127f0cddf2613058101a5701f4cb9d0861fd6c2a1b8e0afe194fcc
f586a3201a53874a2761a9ab6d7220c68661a35ab3
Q1.x = 092f1acfa62b05f95884c6791fba989bbe58044ee6355d100973bf
9553ade52b47929264e6ae770fb264582d8dce512a
Q1.y = 028e6d0169a72cfedb737be45db6c401d3adfb12c58c619c82b93a
5dfcccef12290de530b0480575ddc8397cda0bbebf
J.9.2. BLS12381G1_XMD:SHA-256_SSWU_NU_
suite = BLS12381G1_XMD:SHA-256_SSWU_NU_
dst = QUUX-V01-CS02-with-BLS12381G1_XMD:SHA-256_SSWU_NU_
msg =
P.x = 184bb665c37ff561a89ec2122dd343f20e0f4cbcaec84e3c3052ea
81d1834e192c426074b02ed3dca4e7676ce4ce48ba
P.y = 04407b8d35af4dacc809927071fc0405218f1401a6d15af775810e
4e460064bcc9468beeba82fdc751be70476c888bf3
u[0] = 156c8a6a2c184569d69a76be144b5cdc5141d2d2ca4fe341f011e2
5e3969c55ad9e9b9ce2eb833c81a908e5fa4ac5f03
Q.x = 11398d3b324810a1b093f8e35aa8571cced95858207e7f49c4fd74
656096d61d8a2f9a23cdb18a4dd11cd1d66f41f709
Q.y = 19316b6fb2ba7717355d5d66a361899057e1e84a6823039efc7bec
cefe09d023fb2713b1c415fcf278eb0c39a89b4f72
msg = abc
P.x = 009769f3ab59bfd551d53a5f846b9984c59b97d6842b20a2c565ba
a167945e3d026a3755b6345df8ec7e6acb6868ae6d
P.y = 1532c00cf61aa3d0ce3e5aa20c3b531a2abd2c770a790a26138183
03c6b830ffc0ecf6c357af3317b9575c567f11cd2c
u[0] = 147e1ed29f06e4c5079b9d14fc89d2820d32419b990c1c7bb7dbea
2a36a045124b31ffbde7c99329c05c559af1c6cc82
Q.x = 1998321bc27ff6d71df3051b5aec12ff47363d81a5e9d2dff55f44
4f6ca7e7d6af45c56fd029c58237c266ef5cda5254
Q.y = 034d274476c6307ae584f951c82e7ea85b84f72d28f4d647173235
6121af8d62a49bc263e8eb913a6cf6f125995514ee
msg = abcdef0123456789
P.x = 1974dbb8e6b5d20b84df7e625e2fbfecb2cdb5f77d5eae5fb2955e
5ce7313cae8364bc2fff520a6c25619739c6bdcb6a
P.y = 15f9897e11c6441eaa676de141c8d83c37aab8667173cbe1dfd6de
74d11861b961dccebcd9d289ac633455dfcc7013a3
u[0] = 04090815ad598a06897dd89bcda860f25837d54e897298ce31e694
7378134d3761dc59a572154963e8c954919ecfa82d
Q.x = 17d502fa43bd6a4cad2859049a0c3ecefd60240d129be65da271a4
c03a9c38fa78163b9d2a919d2beb57df7d609b4919
Q.y = 109019902ae93a8732abecf2ff7fecd2e4e305eb91f41c9c3267f1
6b6c19de138c7272947f25512745da6c466cdfd1ac
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 0a7a047c4a8397b3446450642c2ac64d7239b61872c9ae7a59707a
8f4f950f101e766afe58223b3bff3a19a7f754027c
P.y = 1383aebba1e4327ccff7cf9912bda0dbc77de048b71ef8c8a81111
d71dc33c5e3aa6edee9cf6f5fe525d50cc50b77cc9
u[0] = 08dccd088ca55b8bfbc96fb50bb25c592faa867a8bb78d4e94a8cc
2c92306190244532e91feba2b7fed977e3c3bb5a1f
Q.x = 112eb92dd2b3aa9cd38b08de4bef603f2f9fb0ca226030626a9a2e
47ad1e9847fe0a5ed13766c339e38f514bba143b21
Q.y = 17542ce2f8d0a54f2c5ba8c4b14e10b22d5bcd7bae2af3c965c8c8
72b571058c720eac448276c99967ded2bf124490e1
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 0e7a16a975904f131682edbb03d9560d3e48214c9986bd50417a77
108d13dc957500edf96462a3d01e62dc6cd468ef11
P.y = 0ae89e677711d05c30a48d6d75e76ca9fb70fe06c6dd6ff988683d
89ccde29ac7d46c53bb97a59b1901abf1db66052db
u[0] = 0dd824886d2123a96447f6c56e3a3fa992fbfefdba17b6673f9f63
0ff19e4d326529db37e1c1be43f905bf9202e0278d
Q.x = 1775d400a1bacc1c39c355da7e96d2d1c97baa9430c4a3476881f8
521c09a01f921f592607961efc99c4cd46bd78ca19
Q.y = 1109b5d59f65964315de65a7a143e86eabc053104ed289cf480949
317a5685fad7254ff8e7fe6d24d3104e5d55ad6370
J.10. BLS12-381 G2
J.10.1. BLS12381G2_XMD:SHA-256_SSWU_RO_
suite = BLS12381G2_XMD:SHA-256_SSWU_RO_
dst = QUUX-V01-CS02-with-BLS12381G2_XMD:SHA-256_SSWU_RO_
msg =
P.x = 0141ebfbdca40eb85b87142e130ab689c673cf60f1a3e98d693352
66f30d9b8d4ac44c1038e9dcdd5393faf5c41fb78a
+ I * 05cb8437535e20ecffaef7752baddf98034139c38452458baeefab
379ba13dff5bf5dd71b72418717047f5b0f37da03d
P.y = 0503921d7f6a12805e72940b963c0cf3471c7b2a524950ca195d11
062ee75ec076daf2d4bc358c4b190c0c98064fdd92
+ I * 12424ac32561493f3fe3c260708a12b7c620e7be00099a974e259d
dc7d1f6395c3c811cdd19f1e8dbf3e9ecfdcbab8d6
u[0] = 03dbc2cce174e91ba93cbb08f26b917f98194a2ea08d1cce75b2b9
cc9f21689d80bd79b594a613d0a68eb807dfdc1cf8
+ I * 05a2acec64114845711a54199ea339abd125ba38253b70a92c876d
f10598bd1986b739cad67961eb94f7076511b3b39a
u[1] = 02f99798e8a5acdeed60d7e18e9120521ba1f47ec090984662846b
c825de191b5b7641148c0dbc237726a334473eee94
+ I * 145a81e418d4010cc027a68f14391b30074e89e60ee7a22f87217b
2f6eb0c4b94c9115b436e6fa4607e95a98de30a435
Q0.x = 019ad3fc9c72425a998d7ab1ea0e646a1f6093444fc6965f1cad5a
3195a7b1e099c050d57f45e3fa191cc6d75ed7458c
+ I * 171c88b0b0efb5eb2b88913a9e74fe111a4f68867b59db252ce586
8af4d1254bfab77ebde5d61cd1a86fb2fe4a5a1c1d
Q0.y = 0ba10604e62bdd9eeeb4156652066167b72c8d743b050fb4c1016c
31b505129374f76e03fa127d6a156213576910fef3
+ I * 0eb22c7a543d3d376e9716a49b72e79a89c9bfe9feee8533ed931c
bb5373dde1fbcd7411d8052e02693654f71e15410a
Q1.x = 113d2b9cd4bd98aee53470b27abc658d91b47a78a51584f3d4b950
677cfb8a3e99c24222c406128c91296ef6b45608be
+ I * 13855912321c5cb793e9d1e88f6f8d342d49c0b0dbac613ee9e17e
3c0b3c97dfbb5a49cc3fb45102fdbaf65e0efe2632
Q1.y = 0fd3def0b7574a1d801be44fde617162aa2e89da47f464317d9bb5
abc3a7071763ce74180883ad7ad9a723a9afafcdca
+ I * 056f617902b3c0d0f78a9a8cbda43a26b65f602f8786540b9469b0
60db7b38417915b413ca65f875c130bebfaa59790c
msg = abc
P.x = 02c2d18e033b960562aae3cab37a27ce00d80ccd5ba4b7fe0e7a21
0245129dbec7780ccc7954725f4168aff2787776e6
+ I * 139cddbccdc5e91b9623efd38c49f81a6f83f175e80b06fc374de9
eb4b41dfe4ca3a230ed250fbe3a2acf73a41177fd8
P.y = 1787327b68159716a37440985269cf584bcb1e621d3a7202be6ea0
5c4cfe244aeb197642555a0645fb87bf7466b2ba48
+ I * 00aa65dae3c8d732d10ecd2c50f8a1baf3001578f71c694e03866e
9f3d49ac1e1ce70dd94a733534f106d4cec0eddd16
u[0] = 15f7c0aa8f6b296ab5ff9c2c7581ade64f4ee6f1bf18f55179ff44
a2cf355fa53dd2a2158c5ecb17d7c52f63e7195771
+ I * 01c8067bf4c0ba709aa8b9abc3d1cef589a4758e09ef53732d670f
d8739a7274e111ba2fcaa71b3d33df2a3a0c8529dd
u[1] = 187111d5e088b6b9acfdfad078c4dacf72dcd17ca17c82be35e79f
8c372a693f60a033b461d81b025864a0ad051a06e4
+ I * 08b852331c96ed983e497ebc6dee9b75e373d923b729194af8e72a
051ea586f3538a6ebb1e80881a082fa2b24df9f566
Q0.x = 12b2e525281b5f4d2276954e84ac4f42cf4e13b6ac4228624e1776
0faf94ce5706d53f0ca1952f1c5ef75239aeed55ad
+ I * 05d8a724db78e570e34100c0bc4a5fa84ad5839359b40398151f37
cff5a51de945c563463c9efbdda569850ee5a53e77
Q0.y = 02eacdc556d0bdb5d18d22f23dcb086dd106cad713777c7e640794
3edbe0b3d1efe391eedf11e977fac55f9b94f2489c
+ I * 04bbe48bfd5814648d0b9e30f0717b34015d45a861425fabc1ee06
fdfce36384ae2c808185e693ae97dcde118f34de41
Q1.x = 19f18cc5ec0c2f055e47c802acc3b0e40c337256a208001dde14b2
5afced146f37ea3d3ce16834c78175b3ed61f3c537
+ I * 15b0dadc256a258b4c68ea43605dffa6d312eef215c19e6474b3e1
01d33b661dfee43b51abbf96fee68fc6043ac56a58
Q1.y = 05e47c1781286e61c7ade887512bd9c2cb9f640d3be9cf87ea0bad
24bd0ebfe946497b48a581ab6c7d4ca74b5147287f
+ I * 19f98db2f4a1fcdf56a9ced7b320ea9deecf57c8e59236b0dc21f6
ee7229aa9705ce9ac7fe7a31c72edca0d92370c096
msg = abcdef0123456789
P.x = 121982811d2491fde9ba7ed31ef9ca474f0e1501297f68c298e9f4
c0028add35aea8bb83d53c08cfc007c1e005723cd0
+ I * 190d119345b94fbd15497bcba94ecf7db2cbfd1e1fe7da034d26cb
ba169fb3968288b3fafb265f9ebd380512a71c3f2c
P.y = 05571a0f8d3c08d094576981f4a3b8eda0a8e771fcdcc8ecceaf13
56a6acf17574518acb506e435b639353c2e14827c8
+ I * 0bb5e7572275c567462d91807de765611490205a941a5a6af3b169
1bfe596c31225d3aabdf15faff860cb4ef17c7c3be
u[0] = 0313d9325081b415bfd4e5364efaef392ecf69b087496973b22930
3e1816d2080971470f7da112c4eb43053130b785e1
+ I * 062f84cb21ed89406890c051a0e8b9cf6c575cf6e8e18ecf63ba86
826b0ae02548d83b483b79e48512b82a6c0686df8f
u[1] = 1739123845406baa7be5c5dc74492051b6d42504de008c635f3535
bb831d478a341420e67dcc7b46b2e8cba5379cca97
+ I * 01897665d9cb5db16a27657760bbea7951f67ad68f8d55f7113f24
ba6ddd82caef240a9bfa627972279974894701d975
Q0.x = 0f48f1ea1318ddb713697708f7327781fb39718971d72a9245b973
1faaca4dbaa7cca433d6c434a820c28b18e20ea208
+ I * 06051467c8f85da5ba2540974758f7a1e0239a5981de441fdd8768
0a995649c211054869c50edbac1f3a86c561ba3162
Q0.y = 168b3d6df80069dbbedb714d41b32961ad064c227355e1ce5fac8e
105de5e49d77f0c64867f3834848f152497eb76333
+ I * 134e0e8331cee8cb12f9c2d0742714ed9eee78a84d634c9a95f6a7
391b37125ed48bfc6e90bf3546e99930ff67cc97bc
Q1.x = 004fd03968cd1c99a0dd84551f44c206c84dcbdb78076c5bfee24e
89a92c8508b52b88b68a92258403cbe1ea2da3495f
+ I * 1674338ea298281b636b2eb0fe593008d03171195fd6dcd4531e8a
1ed1f02a72da238a17a635de307d7d24aa2d969a47
Q1.y = 0dc7fa13fff6b12558419e0a1e94bfc3cfaf67238009991c5f24ee
94b632c3d09e27eca329989aee348a67b50d5e236c
+ I * 169585e164c131103d85324f2d7747b23b91d66ae5d947c449c819
4a347969fc6bbd967729768da485ba71868df8aed2
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 19a84dd7248a1066f737cc34502ee5555bd3c19f2ecdb3c7d9e24d
c65d4e25e50d83f0f77105e955d78f4762d33c17da
+ I * 0934aba516a52d8ae479939a91998299c76d39cc0c035cd18813be
c433f587e2d7a4fef038260eef0cef4d02aae3eb91
P.y = 14f81cd421617428bc3b9fe25afbb751d934a00493524bc4e06563
5b0555084dd54679df1536101b2c979c0152d09192
+ I * 09bcccfa036b4847c9950780733633f13619994394c23ff0b32fa6
b795844f4a0673e20282d07bc69641cee04f5e5662
u[0] = 025820cefc7d06fd38de7d8e370e0da8a52498be9b53cba9927b2e
f5c6de1e12e12f188bbc7bc923864883c57e49e253
+ I * 034147b77ce337a52e5948f66db0bab47a8d038e712123bb381899
b6ab5ad20f02805601e6104c29df18c254b8618c7b
u[1] = 0930315cae1f9a6017c3f0c8f2314baa130e1cf13f6532bff0a8a1
790cd70af918088c3db94bda214e896e1543629795
+ I * 10c4df2cacf67ea3cb3108b00d4cbd0b3968031ebc8eac4b1ebcef
e84d6b715fde66bef0219951ece29d1facc8a520ef
Q0.x = 09eccbc53df677f0e5814e3f86e41e146422834854a224bf5a83a5
0e4cc0a77bfc56718e8166ad180f53526ea9194b57
+ I * 0c3633943f91daee715277bd644fba585168a72f96ded64fc5a384
cce4ec884a4c3c30f08e09cd2129335dc8f67840ec
Q0.y = 0eb6186a0457d5b12d132902d4468bfeb7315d83320b6c32f1c875
f344efcba979952b4aa418589cb01af712f98cc555
+ I * 119e3cf167e69eb16c1c7830e8df88856d48be12e3ff0a40791a5c
d2f7221311d4bf13b1847f371f467357b3f3c0b4c7
Q1.x = 0eb3aabc1ddfce17ff18455fcc7167d15ce6b60ddc9eb9b59f8d40
ab49420d35558686293d046fc1e42f864b7f60e381
+ I * 198bdfb19d7441ebcca61e8ff774b29d17da16547d2c10c273227a
635cacea3f16826322ae85717630f0867539b5ed8b
Q1.y = 0aaf1dee3adf3ed4c80e481c09b57ea4c705e1b8d25b897f0ceeec
3990748716575f92abff22a1c8f4582aff7b872d52
+ I * 0d058d9061ed27d4259848a06c96c5ca68921a5d269b078650c882
cb3c2bd424a8702b7a6ee4e0ead9982baf6843e924
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 01a6ba2f9a11fa5598b2d8ace0fbe0a0eacb65deceb476fbbcb64f
d24557c2f4b18ecfc5663e54ae16a84f5ab7f62534
+ I * 11fca2ff525572795a801eed17eb12785887c7b63fb77a42be46ce
4a34131d71f7a73e95fee3f812aea3de78b4d01569
P.y = 0b6798718c8aed24bc19cb27f866f1c9effcdbf92397ad6448b5c9
db90d2b9da6cbabf48adc1adf59a1a28344e79d57e
+ I * 03a47f8e6d1763ba0cad63d6114c0accbef65707825a511b251a66
0a9b3994249ae4e63fac38b23da0c398689ee2ab52
u[0] = 190b513da3e66fc9a3587b78c76d1d132b1152174d0b83e3c11140
66392579a45824c5fa17649ab89299ddd4bda54935
+ I * 12ab625b0fe0ebd1367fe9fac57bb1168891846039b4216b9d9400
7b674de2d79126870e88aeef54b2ec717a887dcf39
u[1] = 0e6a42010cf435fb5bacc156a585e1ea3294cc81d0ceb81924d950
40298380b164f702275892cedd81b62de3aba3f6b5
+ I * 117d9a0defc57a33ed208428cb84e54c85a6840e7648480ae42883
8989d25d97a0af8e3255be62b25c2a85630d2dddd8
Q0.x = 17cadf8d04a1a170f8347d42856526a24cc466cb2ddfd506cff011
91666b7f944e31244d662c904de5440516a2b09004
+ I * 0d13ba91f2a8b0051cf3279ea0ee63a9f19bc9cb8bfcc7d78b3cbd
8cc4fc43ba726774b28038213acf2b0095391c523e
Q0.y = 17ef19497d6d9246fa94d35575c0f8d06ee02f21a284dbeaa78768
cb1e25abd564e3381de87bda26acd04f41181610c5
+ I * 12c3c913ba4ed03c24f0721a81a6be7430f2971ffca8fd1729aafe
496bb725807531b44b34b59b3ae5495e5a2dcbd5c8
Q1.x = 16ec57b7fe04c71dfe34fb5ad84dbce5a2dbbd6ee085f1d8cd17f4
5e8868976fc3c51ad9eeda682c7869024d24579bfd
+ I * 13103f7aace1ae1420d208a537f7d3a9679c287208026e4e3439ab
8cd534c12856284d95e27f5e1f33eec2ce656533b0
Q1.y = 0958b2c4c2c10fcef5a6c59b9e92c4a67b0fae3e2e0f1b6b5edad9
c940b8f3524ba9ebbc3f2ceb3cfe377655b3163bd7
+ I * 0ccb594ed8bd14ca64ed9cb4e0aba221be540f25dd0d6ba15a4a4b
e5d67bcf35df7853b2d8dad3ba245f1ea3697f66aa
J.10.2. BLS12381G2_XMD:SHA-256_SSWU_NU_
suite = BLS12381G2_XMD:SHA-256_SSWU_NU_
dst = QUUX-V01-CS02-with-BLS12381G2_XMD:SHA-256_SSWU_NU_
msg =
P.x = 00e7f4568a82b4b7dc1f14c6aaa055edf51502319c723c4dc2688c
7fe5944c213f510328082396515734b6612c4e7bb7
+ I * 126b855e9e69b1f691f816e48ac6977664d24d99f8724868a18418
6469ddfd4617367e94527d4b74fc86413483afb35b
P.y = 0caead0fd7b6176c01436833c79d305c78be307da5f6af6c133c47
311def6ff1e0babf57a0fb5539fce7ee12407b0a42
+ I * 1498aadcf7ae2b345243e281ae076df6de84455d766ab6fcdaad71
fab60abb2e8b980a440043cd305db09d283c895e3d
u[0] = 07355d25caf6e7f2f0cb2812ca0e513bd026ed09dda65b177500fa
31714e09ea0ded3a078b526bed3307f804d4b93b04
+ I * 02829ce3c021339ccb5caf3e187f6370e1e2a311dec9b753631170
63ab2015603ff52c3d3b98f19c2f65575e99e8b78c
Q.x = 18ed3794ad43c781816c523776188deafba67ab773189b8f18c49b
c7aa841cd81525171f7a5203b2a340579192403bef
+ I * 0727d90785d179e7b5732c8a34b660335fed03b913710b60903cf4
954b651ed3466dc3728e21855ae822d4a0f1d06587
Q.y = 00764a5cf6c5f61c52c838523460eb2168b5a5b43705e19cb612e0
06f29b717897facfd15dd1c8874c915f6d53d0342d
+ I * 19290bb9797c12c1d275817aa2605ebe42275b66860f0e4d04487e
bc2e47c50b36edd86c685a60c20a2bd584a82b011a
msg = abc
P.x = 108ed59fd9fae381abfd1d6bce2fd2fa220990f0f837fa30e0f279
14ed6e1454db0d1ee957b219f61da6ff8be0d6441f
+ I * 0296238ea82c6d4adb3c838ee3cb2346049c90b96d602d7bb1b469
b905c9228be25c627bffee872def773d5b2a2eb57d
P.y = 033f90f6057aadacae7963b0a0b379dd46750c1c94a6357c99b65f
63b79e321ff50fe3053330911c56b6ceea08fee656
+ I * 153606c417e59fb331b7ae6bce4fbf7c5190c33ce9402b5ebe2b70
e44fca614f3f1382a3625ed5493843d0b0a652fc3f
u[0] = 138879a9559e24cecee8697b8b4ad32cced053138ab913b9987277
2dc753a2967ed50aabc907937aefb2439ba06cc50c
+ I * 0a1ae7999ea9bab1dcc9ef8887a6cb6e8f1e22566015428d220b7e
ec90ffa70ad1f624018a9ad11e78d588bd3617f9f2
Q.x = 0f40e1d5025ecef0d850aa0bb7bbeceab21a3d4e85e6bee857805b
09693051f5b25428c6be343edba5f14317fcc30143
+ I * 02e0d261f2b9fee88b82804ec83db330caa75fbb12719cfa71ccce
1c532dc4e1e79b0a6a281ed8d3817524286c8bc04c
Q.y = 0cf4a4adc5c66da0bca4caddc6a57ecd97c8252d7526a8ff478e0d
fed816c4d321b5c3039c6683ae9b1e6a3a38c9c0ae
+ I * 11cad1646bb3768c04be2ab2bbe1f80263b7ff6f8f9488f5bc3b68
50e5a3e97e20acc583613c69cf3d2bfe8489744ebb
msg = abcdef0123456789
P.x = 038af300ef34c7759a6caaa4e69363cafeed218a1f207e93b2c70d
91a1263d375d6730bd6b6509dcac3ba5b567e85bf3
+ I * 0da75be60fb6aa0e9e3143e40c42796edf15685cafe0279afd2a67
c3dff1c82341f17effd402e4f1af240ea90f4b659b
P.y = 19b148cbdf163cf0894f29660d2e7bfb2b68e37d54cc83fd4e6e62
c020eaa48709302ef8e746736c0e19342cc1ce3df4
+ I * 0492f4fed741b073e5a82580f7c663f9b79e036b70ab3e51162359
cec4e77c78086fe879b65ca7a47d34374c8315ac5e
u[0] = 18c16fe362b7dbdfa102e42bdfd3e2f4e6191d479437a59db4eb71
6986bf08ee1f42634db66bde97d6c16bbfd342b3b8
+ I * 0e37812ce1b146d998d5f92bdd5ada2a31bfd63dfe18311aa91637
b5f279dd045763166aa1615e46a50d8d8f475f184e
Q.x = 13a9d4a738a85c9f917c7be36b240915434b58679980010499b9ae
8d7a1bf7fbe617a15b3cd6060093f40d18e0f19456
+ I * 16fa88754e7670366a859d6f6899ad765bf5a177abedb2740aacc9
252c43f90cd0421373fbd5b2b76bb8f5c4886b5d37
Q.y = 0a7fa7d82c46797039398253e8765a4194100b330dfed6d7fbb46d
6fbf01e222088779ac336e3675c7a7a0ee05bbb6e3
+ I * 0c6ee170ab766d11fa9457cef53253f2628010b2cffc102b3b2835
1eb9df6c281d3cfc78e9934769d661b72a5265338d
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
P.x = 0c5ae723be00e6c3f0efe184fdc0702b64588fe77dda152ab13099
a3bacd3876767fa7bbad6d6fd90b3642e902b208f9
+ I * 12c8c05c1d5fc7bfa847f4d7d81e294e66b9a78bc9953990c35894
5e1f042eedafce608b67fdd3ab0cb2e6e263b9b1ad
P.y = 04e77ddb3ede41b5ec4396b7421dd916efc68a358a0d7425bddd25
3547f2fb4830522358491827265dfc5bcc1928a569
+ I * 11c624c56dbe154d759d021eec60fab3d8b852395a89de497e4850
4366feedd4662d023af447d66926a28076813dd646
u[0] = 08d4a0997b9d52fecf99427abb721f0fa779479963315fe21c6445
250de7183e3f63bfdf86570da8929489e421d4ee95
+ I * 16cb4ccad91ec95aab070f22043916cd6a59c4ca94097f7f510043
d48515526dc8eaaea27e586f09151ae613688d5a89
Q.x = 0a08b2f639855dfdeaaed972702b109e2241a54de198b2b4cd12ad
9f88fa419a6086a58d91fc805de812ea29bee427c2
+ I * 04a7442e4cb8b42ef0f41dac9ee74e65ecad3ce0851f0746dc4756
8b0e7a8134121ed09ba054509232c49148aef62cda
Q.y = 05d60b1f04212b2c87607458f71d770f43973511c260f0540eef3a
565f42c7ce59aa1cea684bb2a7bcab84acd2f36c8c
+ I * 1017aa5747ba15505ece266a86b0ca9c712f41a254b76ca04094ca
442ce45ecd224bd5544cd16685d0d1b9d156dd0531
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
P.x = 0ea4e7c33d43e17cc516a72f76437c4bf81d8f4eac69ac355d3bf9
b71b8138d55dc10fd458be115afa798b55dac34be1
+ I * 1565c2f625032d232f13121d3cfb476f45275c303a037faa255f9d
a62000c2c864ea881e2bcddd111edc4a3c0da3e88d
P.y = 043b6f5fe4e52c839148dc66f2b3751e69a0f6ebb3d056d6465d50
d4108543ecd956e10fa1640dfd9bc0030cc2558d28
+ I * 0f8991d2a1ad662e7b6f58ab787947f1fa607fce12dde171bc1790
3b012091b657e15333e11701edcf5b63ba2a561247
u[0] = 03f80ce4ff0ca2f576d797a3660e3f65b274285c054feccc3215c8
79e2c0589d376e83ede13f93c32f05da0f68fd6a10
+ I * 006488a837c5413746d868d1efb7232724da10eca410b07d8b505b
9363bdccf0a1fc0029bad07d65b15ccfe6dd25e20d
Q.x = 19592c812d5a50c5601062faba14c7d670711745311c879de1235a
0a11c75aab61327bf2d1725db07ec4d6996a682886
+ I * 0eef4fa41ddc17ed47baf447a2c498548f3c72a02381313d13bef9
16e240b61ce125539090d62d9fbb14a900bf1b8e90
Q.y = 1260d6e0987eae96af9ebe551e08de22b37791d53f4db9e0d59da7
36e66699735793e853e26362531fe4adf99c1883e3
+ I * 0dbace5df0a4ac4ac2f45d8fdf8aee45484576fdd6efc4f98ab9b9
f4112309e628255e183022d98ea5ed6e47ca00306c
Appendix K. Expand Test Vectors
This section gives test vectors for expand_message variants specified
in Section 5.3. The test vectors in this section were generated
using code that is available from [hash2curve-repo].
Each test vector in this section lists the expand_message name, hash
function, and DST, along with a series of tuples of the function
inputs (msg and len_in_bytes), output (uniform_bytes), and
intermediate values (dst_prime and msg_prime). DST and msg are
represented as ASCII strings. Intermediate and output values are
represented as byte strings in hexadecimal.
K.1. expand_message_xmd(SHA-256)
name = expand_message_xmd
DST = QUUX-V01-CS02-with-expander-SHA256-128
hash = SHA256
k = 128
msg =
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413235362d31323826
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000002000515555582d5630312d43533032
2d776974682d657870616e6465722d5348413235362d31323826
uniform_bytes = 68a985b87eb6b46952128911f2a4412bbc302a9d759667f8
7f7a21d803f07235
msg = abc
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413235362d31323826
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000616263002000515555582d5630312d43
5330322d776974682d657870616e6465722d5348413235362d3132
3826
uniform_bytes = d8ccab23b5985ccea865c6c97b6e5b8350e794e603b4b979
02f53a8a0d605615
msg = abcdef0123456789
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413235362d31323826
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000061626364656630313233343536373839
002000515555582d5630312d435330322d776974682d657870616e
6465722d5348413235362d31323826
uniform_bytes = eff31487c770a893cfb36f912fbfcbff40d5661771ca4b2c
b4eafe524333f5c1
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413235362d31323826
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000713132385f7171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171002000515555582d5630312d435330322d77
6974682d657870616e6465722d5348413235362d31323826
uniform_bytes = b23a1d2b4d97b2ef7785562a7e8bac7eed54ed6e97e29aa5
1bfe3f12ddad1ff9
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413235362d31323826
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000613531325f6161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161002000515555582d5630312d
435330322d776974682d657870616e6465722d5348413235362d31
323826
uniform_bytes = 4623227bcc01293b8c130bf771da8c298dede7383243dc09
93d2d94823958c4c
msg =
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413235362d31323826
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000008000515555582d5630312d43533032
2d776974682d657870616e6465722d5348413235362d31323826
uniform_bytes = af84c27ccfd45d41914fdff5df25293e221afc53d8ad2ac0
6d5e3e29485dadbee0d121587713a3e0dd4d5e69e93eb7cd4f5df4
cd103e188cf60cb02edc3edf18eda8576c412b18ffb658e3dd6ec8
49469b979d444cf7b26911a08e63cf31f9dcc541708d3491184472
c2c29bb749d4286b004ceb5ee6b9a7fa5b646c993f0ced
msg = abc
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413235362d31323826
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000616263008000515555582d5630312d43
5330322d776974682d657870616e6465722d5348413235362d3132
3826
uniform_bytes = abba86a6129e366fc877aab32fc4ffc70120d8996c88aee2
fe4b32d6c7b6437a647e6c3163d40b76a73cf6a5674ef1d890f95b
664ee0afa5359a5c4e07985635bbecbac65d747d3d2da7ec2b8221
b17b0ca9dc8a1ac1c07ea6a1e60583e2cb00058e77b7b72a298425
cd1b941ad4ec65e8afc50303a22c0f99b0509b4c895f40
msg = abcdef0123456789
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413235362d31323826
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000061626364656630313233343536373839
008000515555582d5630312d435330322d776974682d657870616e
6465722d5348413235362d31323826
uniform_bytes = ef904a29bffc4cf9ee82832451c946ac3c8f8058ae97d8d6
29831a74c6572bd9ebd0df635cd1f208e2038e760c4994984ce73f
0d55ea9f22af83ba4734569d4bc95e18350f740c07eef653cbb9f8
7910d833751825f0ebefa1abe5420bb52be14cf489b37fe1a72f7d
e2d10be453b2c9d9eb20c7e3f6edc5a60629178d9478df
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413235362d31323826
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000713132385f7171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171008000515555582d5630312d435330322d77
6974682d657870616e6465722d5348413235362d31323826
uniform_bytes = 80be107d0884f0d881bb460322f0443d38bd222db8bd0b0a
5312a6fedb49c1bbd88fd75d8b9a09486c60123dfa1d73c1cc3169
761b17476d3c6b7cbbd727acd0e2c942f4dd96ae3da5de368d26b3
2286e32de7e5a8cb2949f866a0b80c58116b29fa7fabb3ea7d520e
e603e0c25bcaf0b9a5e92ec6a1fe4e0391d1cdbce8c68a
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413235362d31323826
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000613531325f6161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161008000515555582d5630312d
435330322d776974682d657870616e6465722d5348413235362d31
323826
uniform_bytes = 546aff5444b5b79aa6148bd81728704c32decb73a3ba76e9
e75885cad9def1d06d6792f8a7d12794e90efed817d96920d72889
6a4510864370c207f99bd4a608ea121700ef01ed879745ee3e4cee
f777eda6d9e5e38b90c86ea6fb0b36504ba4a45d22e86f6db5dd43
d98a294bebb9125d5b794e9d2a81181066eb954966a487
K.2. expand_message_xmd(SHA-256) (Long DST)
name = expand_message_xmd
DST = QUUX-V01-CS02-with-expander-SHA256-128-long-DST-111111
111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111
1111111111111111111111111111111111111111
hash = SHA256
k = 128
msg =
len_in_bytes = 0x20
DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
fb4d16c0a23620
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000002000412717974da474d0f8c420f320
ff81e8432adb7c927d9bd082b4fb4d16c0a23620
uniform_bytes = e8dc0c8b686b7ef2074086fbdd2f30e3f8bfbd3bdf177f73
f04b97ce618a3ed3
msg = abc
len_in_bytes = 0x20
DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
fb4d16c0a23620
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000616263002000412717974da474d0f8c4
20f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620
uniform_bytes = 52dbf4f36cf560fca57dedec2ad924ee9c266341d8f3d6af
e5171733b16bbb12
msg = abcdef0123456789
len_in_bytes = 0x20
DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
fb4d16c0a23620
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000061626364656630313233343536373839
002000412717974da474d0f8c420f320ff81e8432adb7c927d9bd0
82b4fb4d16c0a23620
uniform_bytes = 35387dcf22618f3728e6c686490f8b431f76550b0b2c61cb
c1ce7001536f4521
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x20
DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
fb4d16c0a23620
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000713132385f7171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171002000412717974da474d0f8c420f320ff81
e8432adb7c927d9bd082b4fb4d16c0a23620
uniform_bytes = 01b637612bb18e840028be900a833a74414140dde0c4754c
198532c3a0ba42bc
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x20
DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
fb4d16c0a23620
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000613531325f6161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161002000412717974da474d0f8
c420f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620
uniform_bytes = 20cce7033cabc5460743180be6fa8aac5a103f56d481cf36
9a8accc0c374431b
msg =
len_in_bytes = 0x80
DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
fb4d16c0a23620
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000008000412717974da474d0f8c420f320
ff81e8432adb7c927d9bd082b4fb4d16c0a23620
uniform_bytes = 14604d85432c68b757e485c8894db3117992fc57e0e136f7
1ad987f789a0abc287c47876978e2388a02af86b1e8d1342e5ce4f
7aaa07a87321e691f6fba7e0072eecc1218aebb89fb14a0662322d
5edbd873f0eb35260145cd4e64f748c5dfe60567e126604bcab1a3
ee2dc0778102ae8a5cfd1429ebc0fa6bf1a53c36f55dfc
msg = abc
len_in_bytes = 0x80
DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
fb4d16c0a23620
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000616263008000412717974da474d0f8c4
20f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620
uniform_bytes = 1a30a5e36fbdb87077552b9d18b9f0aee16e80181d5b951d
0471d55b66684914aef87dbb3626eaabf5ded8cd0686567e503853
e5c84c259ba0efc37f71c839da2129fe81afdaec7fbdc0ccd4c794
727a17c0d20ff0ea55e1389d6982d1241cb8d165762dbc39fb0cee
4474d2cbbd468a835ae5b2f20e4f959f56ab24cd6fe267
msg = abcdef0123456789
len_in_bytes = 0x80
DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
fb4d16c0a23620
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000061626364656630313233343536373839
008000412717974da474d0f8c420f320ff81e8432adb7c927d9bd0
82b4fb4d16c0a23620
uniform_bytes = d2ecef3635d2397f34a9f86438d772db19ffe9924e28a1ca
f6f1c8f15603d4028f40891044e5c7e39ebb9b31339979ff33a424
9206f67d4a1e7c765410bcd249ad78d407e303675918f20f26ce6d
7027ed3774512ef5b00d816e51bfcc96c3539601fa48ef1c07e494
bdc37054ba96ecb9dbd666417e3de289d4f424f502a982
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x80
DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
fb4d16c0a23620
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000713132385f7171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171008000412717974da474d0f8c420f320ff81
e8432adb7c927d9bd082b4fb4d16c0a23620
uniform_bytes = ed6e8c036df90111410431431a232d41a32c86e296c05d42
6e5f44e75b9a50d335b2412bc6c91e0a6dc131de09c43110d9180d
0a70f0d6289cb4e43b05f7ee5e9b3f42a1fad0f31bac6a625b3b5c
50e3a83316783b649e5ecc9d3b1d9471cb5024b7ccf40d41d1751a
04ca0356548bc6e703fca02ab521b505e8e45600508d32
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x80
DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
fb4d16c0a23620
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
0000000000000000000000613531325f6161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161008000412717974da474d0f8
c420f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620
uniform_bytes = 78b53f2413f3c688f07732c10e5ced29a17c6a16f717179f
fbe38d92d6c9ec296502eb9889af83a1928cd162e845b0d3c5424e
83280fed3d10cffb2f8431f14e7a23f4c68819d40617589e4c4116
9d0b56e0e3535be1fd71fbb08bb70c5b5ffed953d6c14bf7618b35
fc1f4c4b30538236b4b08c9fbf90462447a8ada60be495
K.3. expand_message_xmd(SHA-512)
name = expand_message_xmd
DST = QUUX-V01-CS02-with-expander-SHA512-256
hash = SHA512
k = 256
msg =
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000002000515555
582d5630312d435330322d776974682d657870616e6465722d5348
413531322d32353626
uniform_bytes = 6b9a7312411d92f921c6f68ca0b6380730a1a4d982c50721
1a90964c394179ba
msg = abc
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000616263002000
515555582d5630312d435330322d776974682d657870616e646572
2d5348413531322d32353626
uniform_bytes = 0da749f12fbe5483eb066a5f595055679b976e93abe9be6f
0f6318bce7aca8dc
msg = abcdef0123456789
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000616263646566
30313233343536373839002000515555582d5630312d435330322d
776974682d657870616e6465722d5348413531322d32353626
uniform_bytes = 087e45a86e2939ee8b91100af1583c4938e0f5fc6c9db4b1
07b83346bc967f58
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000713132385f71
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
71717171717171717171717171717171717171002000515555582d
5630312d435330322d776974682d657870616e6465722d53484135
31322d32353626
uniform_bytes = 7336234ee9983902440f6bc35b348352013becd88938d2af
ec44311caf8356b3
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000613531325f61
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161610020
00515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
uniform_bytes = 57b5f7e766d5be68a6bfe1768e3c2b7f1228b3e4b3134956
dd73a59b954c66f4
msg =
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000008000515555
582d5630312d435330322d776974682d657870616e6465722d5348
413531322d32353626
uniform_bytes = 41b037d1734a5f8df225dd8c7de38f851efdb45c372887be
655212d07251b921b052b62eaed99b46f72f2ef4cc96bfaf254ebb
bec091e1a3b9e4fb5e5b619d2e0c5414800a1d882b62bb5cd1778f
098b8eb6cb399d5d9d18f5d5842cf5d13d7eb00a7cff859b605da6
78b318bd0e65ebff70bec88c753b159a805d2c89c55961
msg = abc
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000616263008000
515555582d5630312d435330322d776974682d657870616e646572
2d5348413531322d32353626
uniform_bytes = 7f1dddd13c08b543f2e2037b14cefb255b44c83cc397c178
6d975653e36a6b11bdd7732d8b38adb4a0edc26a0cef4bb4521713
5456e58fbca1703cd6032cb1347ee720b87972d63fbf232587043e
d2901bce7f22610c0419751c065922b488431851041310ad659e4b
23520e1772ab29dcdeb2002222a363f0c2b1c972b3efe1
msg = abcdef0123456789
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000616263646566
30313233343536373839008000515555582d5630312d435330322d
776974682d657870616e6465722d5348413531322d32353626
uniform_bytes = 3f721f208e6199fe903545abc26c837ce59ac6fa45733f1b
aaf0222f8b7acb0424814fcb5eecf6c1d38f06e9d0a6ccfbf85ae6
12ab8735dfdf9ce84c372a77c8f9e1c1e952c3a61b7567dd069301
6af51d2745822663d0c2367e3f4f0bed827feecc2aaf98c949b5ed
0d35c3f1023d64ad1407924288d366ea159f46287e61ac
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000713132385f71
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
71717171717171717171717171717171717171008000515555582d
5630312d435330322d776974682d657870616e6465722d53484135
31322d32353626
uniform_bytes = b799b045a58c8d2b4334cf54b78260b45eec544f9f2fb5bd
12fb603eaee70db7317bf807c406e26373922b7b8920fa29142703
dd52bdf280084fb7ef69da78afdf80b3586395b433dc66cde048a2
58e476a561e9deba7060af40adf30c64249ca7ddea79806ee5beb9
a1422949471d267b21bc88e688e4014087a0b592b695ed
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
msg_prime = 0000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000613531325f61
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161610080
00515555582d5630312d435330322d776974682d657870616e6465
722d5348413531322d32353626
uniform_bytes = 05b0bfef265dcee87654372777b7c44177e2ae4c13a27f10
3340d9cd11c86cb2426ffcad5bd964080c2aee97f03be1ca18e30a
1f14e27bc11ebbd650f305269cc9fb1db08bf90bfc79b42a952b46
daf810359e7bc36452684784a64952c343c52e5124cd1f71d474d5
197fefc571a92929c9084ffe1112cf5eea5192ebff330b
K.4. expand_message_xof(SHAKE128)
name = expand_message_xof
DST = QUUX-V01-CS02-with-expander-SHAKE128
hash = SHAKE128
k = 128
msg =
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
msg_prime = 0020515555582d5630312d435330322d776974682d657870616e
6465722d5348414b4531323824
uniform_bytes = 86518c9cd86581486e9485aa74ab35ba150d1c75c88e26b7
043e44e2acd735a2
msg = abc
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
msg_prime = 6162630020515555582d5630312d435330322d776974682d6578
70616e6465722d5348414b4531323824
uniform_bytes = 8696af52a4d862417c0763556073f47bc9b9ba43c99b5053
05cb1ec04a9ab468
msg = abcdef0123456789
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
msg_prime = 616263646566303132333435363738390020515555582d563031
2d435330322d776974682d657870616e6465722d5348414b453132
3824
uniform_bytes = 912c58deac4821c3509dbefa094df54b34b8f5d01a191d1d
3108a2c89077acca
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
msg_prime = 713132385f717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717100
20515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
uniform_bytes = 1adbcc448aef2a0cebc71dac9f756b22e51839d348e031e6
3b33ebb50faeaf3f
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
msg_prime = 613531325f616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
61616161610020515555582d5630312d435330322d776974682d65
7870616e6465722d5348414b4531323824
uniform_bytes = df3447cc5f3e9a77da10f819218ddf31342c310778e0e4ef
72bbaecee786a4fe
msg =
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
msg_prime = 0080515555582d5630312d435330322d776974682d657870616e
6465722d5348414b4531323824
uniform_bytes = 7314ff1a155a2fb99a0171dc71b89ab6e3b2b7d59e38e644
19b8b6294d03ffee42491f11370261f436220ef787f8f76f5b26bd
cd850071920ce023f3ac46847744f4612b8714db8f5db83205b2e6
25d95afd7d7b4d3094d3bdde815f52850bb41ead9822e08f22cf41
d615a303b0d9dde73263c049a7b9898208003a739a2e57
msg = abc
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
msg_prime = 6162630080515555582d5630312d435330322d776974682d6578
70616e6465722d5348414b4531323824
uniform_bytes = c952f0c8e529ca8824acc6a4cab0e782fc3648c563ddb00d
a7399f2ae35654f4860ec671db2356ba7baa55a34a9d7f79197b60
ddae6e64768a37d699a78323496db3878c8d64d909d0f8a7de4927
dcab0d3dbbc26cb20a49eceb0530b431cdf47bc8c0fa3e0d88f53b
318b6739fbed7d7634974f1b5c386d6230c76260d5337a
msg = abcdef0123456789
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
msg_prime = 616263646566303132333435363738390080515555582d563031
2d435330322d776974682d657870616e6465722d5348414b453132
3824
uniform_bytes = 19b65ee7afec6ac06a144f2d6134f08eeec185f1a890fe34
e68f0e377b7d0312883c048d9b8a1d6ecc3b541cb4987c26f45e0c
82691ea299b5e6889bbfe589153016d8131717ba26f07c3c14ffbe
f1f3eff9752e5b6183f43871a78219a75e7000fbac6a7072e2b83c
790a3a5aecd9d14be79f9fd4fb180960a3772e08680495
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
msg_prime = 713132385f717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717100
80515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
uniform_bytes = ca1b56861482b16eae0f4a26212112362fcc2d76dcc80c93
c4182ed66c5113fe41733ed68be2942a3487394317f3379856f482
2a611735e50528a60e7ade8ec8c71670fec6661e2c59a09ed36386
513221688b35dc47e3c3111ee8c67ff49579089d661caa29db1ef1
0eb6eace575bf3dc9806e7c4016bd50f3c0e2a6481ee6d
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4531323824
msg_prime = 613531325f616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
61616161610080515555582d5630312d435330322d776974682d65
7870616e6465722d5348414b4531323824
uniform_bytes = 9d763a5ce58f65c91531b4100c7266d479a5d9777ba76169
3d052acd37d149e7ac91c796a10b919cd74a591a1e38719fb91b72
03e2af31eac3bff7ead2c195af7d88b8bc0a8adf3d1e90ab9bed6d
dc2b7f655dd86c730bdeaea884e73741097142c92f0e3fc1811b69
9ba593c7fbd81da288a29d423df831652e3a01a9374999
K.5. expand_message_xof(SHAKE128) (Long DST)
name = expand_message_xof
DST = QUUX-V01-CS02-with-expander-SHAKE128-long-DST-11111111
111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111
1111111111111111111111111111111111111111
hash = SHAKE128
k = 128
msg =
len_in_bytes = 0x20
DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
msg_prime = 0020acb9736c0867fdfbd6385519b90fc8c034b5af04a9589732
12950132d035792f20
uniform_bytes = 827c6216330a122352312bccc0c8d6e7a146c5257a776dbd
9ad9d75cd880fc53
msg = abc
len_in_bytes = 0x20
DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
msg_prime = 6162630020acb9736c0867fdfbd6385519b90fc8c034b5af04a9
58973212950132d035792f20
uniform_bytes = 690c8d82c7213b4282c6cb41c00e31ea1d3e2005f93ad19b
bf6da40f15790c5c
msg = abcdef0123456789
len_in_bytes = 0x20
DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
msg_prime = 616263646566303132333435363738390020acb9736c0867fdfb
d6385519b90fc8c034b5af04a958973212950132d035792f20
uniform_bytes = 979e3a15064afbbcf99f62cc09fa9c85028afcf3f825eb07
11894dcfc2f57057
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x20
DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
msg_prime = 713132385f717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717100
20acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
uniform_bytes = c5a9220962d9edc212c063f4f65b609755a1ed96e62f9db5
d1fd6adb5a8dc52b
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x20
DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
msg_prime = 613531325f616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
61616161610020acb9736c0867fdfbd6385519b90fc8c034b5af04
a958973212950132d035792f20
uniform_bytes = f7b96a5901af5d78ce1d071d9c383cac66a1dfadb508300e
c6aeaea0d62d5d62
msg =
len_in_bytes = 0x80
DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
msg_prime = 0080acb9736c0867fdfbd6385519b90fc8c034b5af04a9589732
12950132d035792f20
uniform_bytes = 3890dbab00a2830be398524b71c2713bbef5f4884ac2e6f0
70b092effdb19208c7df943dc5dcbaee3094a78c267ef276632ee2
c8ea0c05363c94b6348500fae4208345dd3475fe0c834c2beac7fa
7bc181692fb728c0a53d809fc8111495222ce0f38468b11becb15b
32060218e285c57a60162c2c8bb5b6bded13973cd41819
msg = abc
len_in_bytes = 0x80
DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
msg_prime = 6162630080acb9736c0867fdfbd6385519b90fc8c034b5af04a9
58973212950132d035792f20
uniform_bytes = 41b7ffa7a301b5c1441495ebb9774e2a53dbbf4e54b9a1af
6a20fd41eafd69ef7b9418599c5545b1ee422f363642b01d4a5344
9313f68da3e49dddb9cd25b97465170537d45dcbdf92391b5bdff3
44db4bd06311a05bca7dcd360b6caec849c299133e5c9194f4e15e
3e23cfaab4003fab776f6ac0bfae9144c6e2e1c62e7d57
msg = abcdef0123456789
len_in_bytes = 0x80
DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
msg_prime = 616263646566303132333435363738390080acb9736c0867fdfb
d6385519b90fc8c034b5af04a958973212950132d035792f20
uniform_bytes = 55317e4a21318472cd2290c3082957e1242241d9e0d04f47
026f03401643131401071f01aa03038b2783e795bdfa8a3541c194
ad5de7cb9c225133e24af6c86e748deb52e560569bd54ef4dac034
65111a3a44b0ea490fb36777ff8ea9f1a8a3e8e0de3cf0880b4b2f
8dd37d3a85a8b82375aee4fa0e909f9763319b55778e71
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x80
DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
msg_prime = 713132385f717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717100
80acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
uniform_bytes = 19fdd2639f082e31c77717ac9bb032a22ff0958382b2dbb3
9020cdc78f0da43305414806abf9a561cb2d0067eb2f7bc544482f
75623438ed4b4e39dd9e6e2909dd858bd8f1d57cd0fce2d3150d90
aa67b4498bdf2df98c0100dd1a173436ba5d0df6be1defb0b2ce55
ccd2f4fc05eb7cb2c019c35d5398b85adc676da4238bc7
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x80
DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
0132d035792f20
msg_prime = 613531325f616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
61616161610080acb9736c0867fdfbd6385519b90fc8c034b5af04
a958973212950132d035792f20
uniform_bytes = 945373f0b3431a103333ba6a0a34f1efab2702efde41754c
4cb1d5216d5b0a92a67458d968562bde7fa6310a83f53dda138368
0a276a283438d58ceebfa7ab7ba72499d4a3eddc860595f63c93b1
c5e823ea41fc490d938398a26db28f61857698553e93f0574eb8c5
017bfed6249491f9976aaa8d23d9485339cc85ca329308
K.6. expand_message_xof(SHAKE256)
name = expand_message_xof
DST = QUUX-V01-CS02-with-expander-SHAKE256
hash = SHAKE256
k = 256
msg =
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
msg_prime = 0020515555582d5630312d435330322d776974682d657870616e
6465722d5348414b4532353624
uniform_bytes = 2ffc05c48ed32b95d72e807f6eab9f7530dd1c2f013914c8
fed38c5ccc15ad76
msg = abc
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
msg_prime = 6162630020515555582d5630312d435330322d776974682d6578
70616e6465722d5348414b4532353624
uniform_bytes = b39e493867e2767216792abce1f2676c197c0692aed06156
0ead251821808e07
msg = abcdef0123456789
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
msg_prime = 616263646566303132333435363738390020515555582d563031
2d435330322d776974682d657870616e6465722d5348414b453235
3624
uniform_bytes = 245389cf44a13f0e70af8665fe5337ec2dcd138890bb7901
c4ad9cfceb054b65
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
msg_prime = 713132385f717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717100
20515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
uniform_bytes = 719b3911821e6428a5ed9b8e600f2866bcf23c8f0515e52d
6c6c019a03f16f0e
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x20
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
msg_prime = 613531325f616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
61616161610020515555582d5630312d435330322d776974682d65
7870616e6465722d5348414b4532353624
uniform_bytes = 9181ead5220b1963f1b5951f35547a5ea86a820562287d6c
a4723633d17ccbbc
msg =
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
msg_prime = 0080515555582d5630312d435330322d776974682d657870616e
6465722d5348414b4532353624
uniform_bytes = 7a1361d2d7d82d79e035b8880c5a3c86c5afa719478c007d
96e6c88737a3f631dd74a2c88df79a4cb5e5d9f7504957c70d669e
c6bfedc31e01e2bacc4ff3fdf9b6a00b17cc18d9d72ace7d6b81c2
e481b4f73f34f9a7505dccbe8f5485f3d20c5409b0310093d5d649
2dea4e18aa6979c23c8ea5de01582e9689612afbb353df
msg = abc
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
msg_prime = 6162630080515555582d5630312d435330322d776974682d6578
70616e6465722d5348414b4532353624
uniform_bytes = a54303e6b172909783353ab05ef08dd435a558c3197db0c1
32134649708e0b9b4e34fb99b92a9e9e28fc1f1d8860d85897a8e0
21e6382f3eea10577f968ff6df6c45fe624ce65ca25932f679a42a
404bc3681efe03fcd45ef73bb3a8f79ba784f80f55ea8a3c367408
f30381299617f50c8cf8fbb21d0f1e1d70b0131a7b6fbe
msg = abcdef0123456789
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
msg_prime = 616263646566303132333435363738390080515555582d563031
2d435330322d776974682d657870616e6465722d5348414b453235
3624
uniform_bytes = e42e4d9538a189316e3154b821c1bafb390f78b2f010ea40
4e6ac063deb8c0852fcd412e098e231e43427bd2be1330bb47b403
9ad57b30ae1fc94e34993b162ff4d695e42d59d9777ea18d3848d9
d336c25d2acb93adcad009bcfb9cde12286df267ada283063de0bb
1505565b2eb6c90e31c48798ecdc71a71756a9110ff373
msg = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
msg_prime = 713132385f717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717171
717171717171717171717171717171717171717171717171717100
80515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
uniform_bytes = 4ac054dda0a38a65d0ecf7afd3c2812300027c8789655e47
aecf1ecc1a2426b17444c7482c99e5907afd9c25b991990490bb9c
686f43e79b4471a23a703d4b02f23c669737a886a7ec28bddb92c3
a98de63ebf878aa363a501a60055c048bea11840c4717beae7eee2
8c3cfa42857b3d130188571943a7bd747de831bd6444e0
msg = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
len_in_bytes = 0x80
DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
722d5348414b4532353624
msg_prime = 613531325f616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
616161616161616161616161616161616161616161616161616161
61616161610080515555582d5630312d435330322d776974682d65
7870616e6465722d5348414b4532353624
uniform_bytes = 09afc76d51c2cccbc129c2315df66c2be7295a231203b8ab
2dd7f95c2772c68e500bc72e20c602abc9964663b7a03a389be128
c56971ce81001a0b875e7fd17822db9d69792ddf6a23a151bf4700
79c518279aef3e75611f8f828994a9988f4a8a256ddb8bae161e65
8d5a2a09bcfe839c6396dc06ee5c8ff3c22d3b1f9deb7e
Acknowledgements
The authors would like to thank Adam Langley for his detailed writeup
of Elligator 2 with Curve25519 [L13]; Dan Boneh, Benjamin Lipp,
Christopher Patton, and Leonid Reyzin for educational discussions;
and David Benjamin, Daniel Bourdrez, Frank Denis, Sean Devlin, Justin
Drake, Bjoern Haase, Mike Hamburg, Dan Harkins, Daira Hopwood, Thomas
Icart, Andy Polyakov, Thomas Pornin, Mamy Ratsimbazafy, Michael
Scott, Filippo Valsorda, and Mathy Vanhoef for helpful reviews and
feedback.
Contributors
Sharon Goldberg
Boston University
Email: goldbe@cs.bu.edu
Ela Lee
Royal Holloway, University of London
Email: Ela.Lee.2010@live.rhul.ac.uk
Michele Orru
Email: michele.orru@ens.fr
Authors' Addresses
Armando Faz-Hernandez
Cloudflare, Inc.
101 Townsend St
San Francisco, CA 94107
United States of America
Email: armfazh@cloudflare.com
Sam Scott
Oso Security, Inc.
335 Madison Ave
New York, NY 10017
United States of America
Email: sam.scott89@gmail.com
Nick Sullivan
Cloudflare, Inc.
101 Townsend St
San Francisco, CA 94107
United States of America
Email: nicholas.sullivan@gmail.com
Riad S. Wahby
Stanford University
Email: rsw@cs.stanford.edu
Christopher A. Wood
Cloudflare, Inc.
101 Townsend St
San Francisco, CA 94107
United States of America
Email: caw@heapingbits.net
