Independent Submission D. Atkins
Request for Comments: 9021 Veridify Security
Category: Informational May 2021
ISSN: 2070-1721
Use of the Walnut Digital Signature Algorithm with CBOR Object Signing
and Encryption (COSE)
Abstract
This document specifies the conventions for using the Walnut Digital
Signature Algorithm (WalnutDSA) for digital signatures with the CBOR
Object Signing and Encryption (COSE) syntax. WalnutDSA is a
lightweight, quantum-resistant signature scheme based on Group
Theoretic Cryptography with implementation and computational
efficiency of signature verification in constrained environments,
even on 8- and 16-bit platforms.
The goal of this publication is to document a way to use the
lightweight, quantum-resistant WalnutDSA signature algorithm in COSE
in a way that would allow multiple developers to build compatible
implementations. As of this publication, the security properties of
WalnutDSA have not been evaluated by the IETF and its use has not
been endorsed by the IETF.
WalnutDSA and the Walnut Digital Signature Algorithm are trademarks
of Veridify Security Inc.
Status of This Memo
This document is not an Internet Standards Track specification; it is
published for informational purposes.
This is a contribution to the RFC Series, independently of any other
RFC stream. The RFC Editor has chosen to publish this document at
its discretion and makes no statement about its value for
implementation or deployment. Documents approved for publication by
the RFC Editor 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/rfc9021.
Copyright Notice
Copyright (c) 2021 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(https://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document.
Table of Contents
1. Introduction
1.1. Motivation
1.2. Trademark Notice
2. Terminology
3. WalnutDSA Algorithm Overview
4. WalnutDSA Algorithm Identifiers
5. Security Considerations
5.1. Implementation Security Considerations
5.2. Method Security Considerations
6. IANA Considerations
6.1. COSE Algorithms Registry Entry
6.2. COSE Key Types Registry Entry
6.3. COSE Key Type Parameters Registry Entries
6.3.1. WalnutDSA Parameter: N
6.3.2. WalnutDSA Parameter: q
6.3.3. WalnutDSA Parameter: t-values
6.3.4. WalnutDSA Parameter: matrix 1
6.3.5. WalnutDSA Parameter: permutation 1
6.3.6. WalnutDSA Parameter: matrix 2
7. References
7.1. Normative References
7.2. Informative References
Acknowledgments
Author's Address
1. Introduction
This document specifies the conventions for using the Walnut Digital
Signature Algorithm (WalnutDSA) [WALNUTDSA] for digital signatures
with the CBOR Object Signing and Encryption (COSE) syntax [RFC8152].
WalnutDSA is a Group Theoretic signature scheme [GTC] where signature
validation is both computationally and space efficient, even on very
small processors. Unlike many hash-based signatures, there is no
state required and no limit on the number of signatures that can be
made. WalnutDSA private and public keys are relatively small;
however, the signatures are larger than RSA and Elliptic Curve
Cryptography (ECC), but still smaller than most all other quantum-
resistant schemes (including all hash-based schemes).
COSE provides a lightweight method to encode structured data.
WalnutDSA is a lightweight, quantum-resistant digital signature
algorithm. The goal of this specification is to document a method to
leverage WalnutDSA in COSE in a way that would allow multiple
developers to build compatible implementations.
As with all cryptosystems, the initial versions of WalnutDSA
underwent significant cryptanalysis, and, in some cases, identified
potential issues. For more discussion on this topic, a summary of
all published cryptanalysis can be found in Section 5.2. Validated
issues were addressed by reparameterization in updated versions of
WalnutDSA. Although the IETF has neither evaluated the security
properties of WalnutDSA nor endorsed WalnutDSA as of this
publication, this document provides a method to use WalnutDSA in
conjunction with IETF protocols. As always, users of any security
algorithm are advised to research the security properties of the
algorithm and make their own judgment about the risks involved.
1.1. Motivation
Recent advances in cryptanalysis [BH2013] and progress in the
development of quantum computers [NAS2019] pose a threat to widely
deployed digital signature algorithms. As a result, there is a need
to prepare for a day that cryptosystems such as RSA and DSA, which
depend on discrete logarithm and factoring, cannot be depended upon.
If large-scale quantum computers are ever built, these computers will
be able to break many of the public key cryptosystems currently in
use. A post-quantum cryptosystem [PQC] is a system that is secure
against quantum computers that have more than a trivial number of
quantum bits (qubits). It is open to conjecture when it will be
feasible to build such computers; however, RSA, DSA, the Elliptic
Curve Digital Signature Algorithm (ECDSA), and the Edwards-Curve
Digital Signature Algorithm (EdDSA) are all vulnerable if large-scale
quantum computers come to pass.
WalnutDSA does not depend on the difficulty of discrete logarithms or
factoring. As a result, this algorithm is considered to be resistant
to post-quantum attacks.
Today, RSA and ECDSA are often used to digitally sign software
updates. Unfortunately, implementations of RSA and ECDSA can be
relatively large, and verification can take a significant amount of
time on some very small processors. Therefore, we desire a digital
signature scheme that verifies faster with less code. Moreover, in
preparation for a day when RSA, DSA, and ECDSA cannot be depended
upon, a digital signature algorithm is needed that will remain secure
even if there are significant cryptanalytic advances or a large-scale
quantum computer is invented. WalnutDSA, specified in [WALNUTSPEC],
is a quantum-resistant algorithm that addresses these requirements.
1.2. Trademark Notice
WalnutDSA and the Walnut Digital Signature Algorithm are trademarks
of Veridify Security Inc.
2. Terminology
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.
3. WalnutDSA Algorithm Overview
This specification makes use of WalnutDSA signatures as described in
[WALNUTDSA] and more concretely specified in [WALNUTSPEC]. WalnutDSA
is a Group Theoretic cryptographic signature scheme that leverages
infinite group theory as the basis of its security and maps that to a
one-way evaluation of a series of matrices over small finite fields
with permuted multiplicants based on the group input. WalnutDSA
leverages the SHA2-256 and SHA2-512 one-way hash algorithms [SHA2] in
a hash-then-sign process.
WalnutDSA is based on a one-way function, E-multiplication, which is
an action on the infinite group. A single E-multiplication step
takes as input a matrix and permutation, a generator in the group,
and a set of T-values (entries in the finite field) and outputs a new
matrix and permutation. To process a long string of generators (like
a WalnutDSA signature), E-multiplication is iterated over each
generator. Due to its structure, E-multiplication is extremely easy
to implement.
In addition to being quantum resistant, the two main benefits of
using WalnutDSA are that the verification implementation is very
small and WalnutDSA signature verification is extremely fast, even on
very small processors (including 16- and even 8-bit
microcontrollers). This lends it well to use in constrained and/or
time-sensitive environments.
WalnutDSA has several parameters required to process a signature.
The main parameters are N and q. The parameter N defines the size of
the group by defining the number of strands in use and implies
working in an NxN matrix. The parameter q defines the number of
elements in the finite field. Signature verification also requires a
set of T-values, which is an ordered list of N entries in the finite
field F_q.
A WalnutDSA signature is just a string of generators in the infinite
group, packed into a byte string.
4. WalnutDSA Algorithm Identifiers
The CBOR Object Signing and Encryption (COSE) syntax [RFC8152]
supports two signature algorithm schemes. This specification makes
use of the signature with appendix scheme for WalnutDSA signatures.
The signature value is a large byte string. The byte string is
designed for easy parsing, and it includes a length (number of
generators) and type codes that indirectly provide all of the
information that is needed to parse the byte string during signature
validation.
When using a COSE key for this algorithm, the following checks are
made:
* The "kty" field MUST be present, and it MUST be "WalnutDSA".
* If the "alg" field is present, it MUST be "WalnutDSA".
* If the "key_ops" field is present, it MUST include "sign" when
creating a WalnutDSA signature.
* If the "key_ops" field is present, it MUST include "verify" when
verifying a WalnutDSA signature.
* If the "kid" field is present, it MAY be used to identify the
WalnutDSA Key.
5. Security Considerations
5.1. Implementation Security Considerations
Implementations MUST protect the private keys. Use of a hardware
security module (HSM) is one way to protect the private keys.
Compromising the private keys may result in the ability to forge
signatures. As a result, when a private key is stored on non-
volatile media or stored in a virtual machine environment, care must
be taken to preserve confidentiality and integrity.
The generation of private keys relies on random numbers. The use of
inadequate pseudorandom number generators (PRNGs) to generate these
values can result in little or no security. An attacker may find it
much easier to reproduce the PRNG environment that produced the keys,
searching the resulting small set of possibilities, rather than brute
force searching the whole key space. The generation of quality
random numbers is difficult, and [RFC4086] offers important guidance
in this area.
The generation of WalnutDSA signatures also depends on random
numbers. While the consequences of an inadequate PRNG to generate
these values are much less severe than the generation of private
keys, the guidance in [RFC4086] remains important.
5.2. Method Security Considerations
The Walnut Digital Signature Algorithm has undergone significant
cryptanalysis since it was first introduced, and several weaknesses
were found in early versions of the method, resulting in the
description of several attacks with exponential computational
complexity. A full writeup of all the analysis can be found in
[WalnutDSAAnalysis]. In summary, the original suggested parameters
(N=8, q=32) were too small, leading to many of these exponential-
growth attacks being practical. However, current parameters render
these attacks impractical. The following paragraphs summarize the
analysis and how the current parameters defeat all the previous
attacks.
First, the team of Hart et al. found a universal forgery attack based
on a group-factoring problem that runs in O(q^((N-1)/2)) with a
memory complexity of log_2(q) N^2 q^((N-1)/2). With parameters N=10
and q=M31 (the Mersenne prime 2^31 - 1), the runtime is 2^139 and
memory complexity is 2^151. W. Beullens found a modification of this
attack but its runtime is even longer.
Next, Beullens and Blackburn found several issues with the original
method and parameters. First, they used a Pollard-Rho attack and
discovered the original public key space was too small.
Specifically, they require that q^(N(N-1)-1) > 2^(2*Security Level).
One can clearly see that (N=10, q=M31) provides 128-bit security and
(N=10, q=M61) provides 256-bit security.
Beullens and Blackburn also found two issues with the original
message encoder of WalnutDSA. First, the original encoder was non-
injective, which reduced the available signature space. This was
repaired in an update. Second, they pointed out that the dimension
of the vector space generated by the encoder was too small.
Specifically, they require that q^dimension > 2^(2*Security Level).
With N=10, the current encoder produces a dimension of 66, which
clearly provides sufficient security with q=M31 or q=M61.
The final issue discovered by Beullens and Blackburn was a process to
theoretically "reverse" E-multiplication. First, their process
requires knowing the initial matrix and permutation (which are known
for WalnutDSA). But more importantly, their process runs at
O(q^((N-1)/2)), which for (N=10, q=M31) is greater than 2^128.
A team at Steven's Institute leveraged a length-shortening attack
that enabled them to remove the cloaking elements and then solve a
conjugacy search problem to derive the private keys. Their attack
requires both knowledge of the permutation being cloaked and also
that the cloaking elements themselves are conjugates. By adding
additional concealed cloaking elements, the attack requires an N!
search for each cloaking element. By inserting k concealed cloaking
elements, this requires the attacker to perform (N!)^k work. This
allows k to be set to meet the desired security level.
Finally, Merz and Petit discovered that using a Garside Normal Form
of a WalnutDSA signature enabled them to find commonalities with the
Garside Normal Form of the encoded message. Using those
commonalities, they were able to splice into a signature and create
forgeries. Increasing the number of cloaking elements, specifically
within the encoded message, sufficiently obscures the commonalities
and blocks this attack.
In summary, most of these attacks are exponential in runtime and it
can be shown that current parameters put the runtime beyond the
desired security level. The final two attacks are also sufficiently
blocked to the desired security level.
6. IANA Considerations
IANA has added entries for WalnutDSA signatures in the "COSE
Algorithms" registry and WalnutDSA public keys in the "COSE Key
Types" and "COSE Key Type Parameters" registries.
6.1. COSE Algorithms Registry Entry
The following new entry has been registered in the "COSE Algorithms"
registry:
Name: WalnutDSA
Value: -260
Description: WalnutDSA signature
Reference: RFC 9021
Recommended: No
6.2. COSE Key Types Registry Entry
The following new entry has been registered in the "COSE Key Types"
registry:
Name: WalnutDSA
Value: 6
Description: WalnutDSA public key
Reference: RFC 9021
6.3. COSE Key Type Parameters Registry Entries
The following sections detail the additions to the "COSE Key Type
Parameters" registry.
6.3.1. WalnutDSA Parameter: N
The new entry, N, has been registered in the "COSE Key Type
Parameters" registry as follows:
Key Type: 6
Name: N
Label: -1
CBOR Type: uint
Description: Group and Matrix (NxN) size
Reference: RFC 9021
6.3.2. WalnutDSA Parameter: q
The new entry, q, has been registered in the "COSE Key Type
Parameters" registry as follows:
Key Type: 6
Name: q
Label: -2
CBOR Type: uint
Description: Finite field F_q
Reference: RFC 9021
6.3.3. WalnutDSA Parameter: t-values
The new entry, t-values, has been registered in the "COSE Key Type
Parameters" registry as follows:
Key Type: 6
Name: t-values
Label: -3
CBOR Type: array (of uint)
Description: List of T-values, entries in F_q
Reference: RFC 9021
6.3.4. WalnutDSA Parameter: matrix 1
The new entry, matrix 1, has been registered in the "COSE Key Type
Parameters" registry as follows:
Key Type: 6
Name: matrix 1
Label: -4
CBOR Type: array (of array of uint)
Description: NxN Matrix of entries in F_q in column-major form
Reference: RFC 9021
6.3.5. WalnutDSA Parameter: permutation 1
The new entry, permutation 1, has been registered in the "COSE Key
Type Parameters" registry as follows:
Key Type: 6
Name: permutation 1
Label: -5
CBOR Type: array (of uint)
Description: Permutation associated with matrix 1
Reference: RFC 9021
6.3.6. WalnutDSA Parameter: matrix 2
The new entry, matrix 2, has been registered in the "COSE Key Type
Parameters" registry as follows:
Key Type: 6
Name: matrix 2
Label: -6
CBOR Type: array (of array of uint)
Description: NxN Matrix of entries in F_q in column-major form
Reference: RFC 9021
7. References
7.1. Normative References
[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>.
[RFC8152] Schaad, J., "CBOR Object Signing and Encryption (COSE)",
RFC 8152, DOI 10.17487/RFC8152, July 2017,
<https://www.rfc-editor.org/info/rfc8152>.
[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>.
[SHA2] National Institute of Standards and Technology (NIST),
"Secure Hash Standard (SHS)", DOI 10.6028/NIST.FIPS.180-4,
August 2015, <https://doi.org/10.6028/NIST.FIPS.180-4>.
[WALNUTDSA]
Anshel, I., Atkins, D., Goldfeld, D., and P. Gunnells,
"WalnutDSA(TM): A group theoretic digital signature
algorithm", DOI 10.1080/23799927.2020.1831613, November
2020, <https://doi.org/10.1080/23799927.2020.1831613>.
7.2. Informative References
[BH2013] Ptacek, T., Ritter, J., Samuel, J., and A. Stamos, "The
Factoring Dead: Preparing for the Cryptopocalypse", August
2013, <https://www.slideshare.net/astamos/bh-slides>.
[GTC] Vasco, M. and R. Steinwandt, "Group Theoretic
Cryptography", ISBN 9781584888369, April 2015,
<https://www.crcpress.com/Group-Theoretic-Cryptography/
Vasco-Steinwandt/p/book/9781584888369>.
[NAS2019] National Academies of Sciences, Engineering, and Medicine,
"Quantum Computing: Progress and Prospects",
DOI 10.17226/25196, 2019,
<https://doi.org/10.17226/25196>.
[PQC] Bernstein, D., "Introduction to post-quantum
cryptography", DOI 10.1007/978-3-540-88702-7, 2009,
<https://doi.org/10.1007/978-3-540-88702-7>.
[RFC4086] Eastlake 3rd, D., Schiller, J., and S. Crocker,
"Randomness Requirements for Security", BCP 106, RFC 4086,
DOI 10.17487/RFC4086, June 2005,
<https://www.rfc-editor.org/info/rfc4086>.
[WalnutDSAAnalysis]
Anshel, I., Atkins, D., Goldfeld, D., and P. Gunnells,
"Defeating the Hart et al, Beullens-Blackburn, Kotov-
Menshov-Ushakov, and Merz-Petit Attacks on WalnutDSA(TM)",
May 2019, <https://eprint.iacr.org/2019/472>.
[WALNUTSPEC]
Anshel, I., Atkins, D., Goldfeld, D., and P. Gunnells,
"The Walnut Digital Signature Algorithm Specification",
Post-Quantum Cryptography, November 2018,
<https://csrc.nist.gov/projects/post-quantum-cryptography/
round-1-submissions>.
Acknowledgments
A big thank you to Russ Housley for his input on the concepts and
text of this document.
Author's Address
Derek Atkins
Veridify Security
100 Beard Sawmill Rd, Suite 350
Shelton, CT 06484
United States of America
Phone: +1 617 623 3745
Email: datkins@veridify.com
