Internet Engineering Task Force (IETF) J. Merkle
Request for Comments: 6954 secunet Security Networks
Category: Informational M. Lochter
ISSN: 2070-1721 BSI
July 2013
Using the Elliptic Curve Cryptography (ECC) Brainpool Curves
for the Internet Key Exchange Protocol Version 2 (IKEv2)
Abstract
This document specifies use of the Elliptic Curve Cryptography (ECC)
Brainpool elliptic curve groups for key exchange in the Internet Key
Exchange Protocol version 2 (IKEv2).
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 Engineering Task Force
(IETF). It represents the consensus of the IETF community. It has
received public review and has been approved for publication by the
Internet Engineering Steering Group (IESG). Not all documents
approved by the IESG are a candidate for any level of Internet
Standard; see Section 2 of RFC 5741.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
http://www.rfc-editor.org/info/rfc6954.
Copyright Notice
Copyright (c) 2013 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
(http://trustee.ietf.org/license-info) in effect on the date of
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described in the Simplified BSD License.
Merkle & Lochter Informational [Page 1]RFC 6954 Brainpool Curves for IKEv2 Key Exchange July 2013
Table of Contents
1. Introduction ....................................................2
1.1. Requirements Language ......................................2
2. IKEv2 Key Exchange Using the ECC Brainpool Curves ...............3
2.1. Diffie-Hellman Group Transform IDs .........................3
2.2. Using the Twisted Brainpool Curves Internally ..............3
2.3. Key Exchange Payload and Shared Secret .....................3
3. Security Considerations .........................................4
4. IANA Considerations .............................................5
5. References ......................................................5
5.1. Normative References .......................................5
5.2. Informative References .....................................6
Appendix A. Test Vectors ...........................................8
A.1. 224-Bit Curve ...............................................8
A.2. 256-Bit Curve ...............................................9
A.3. 384-Bit Curve ...............................................9
A.4. 512-Bit Curve ..............................................10
1. Introduction
[RFC5639] specified a new set of elliptic curve groups over finite
prime fields for use in cryptographic applications. These groups,
denoted as ECC Brainpool curves, were generated in a verifiably
pseudo-random way and comply with the security requirements of
relevant standards from ISO [ISO1] [ISO2], ANSI [ANSI1], NIST [FIPS],
and the Standards for Efficient Cryptography Group [SEC2].
While the ASN.1 object identifiers defined in RFC 5639 allow usage of
the ECC Brainpool curves in certificates and certificate revocation
lists, their utilization for key exchange in IKEv2 [RFC5996] requires
the definition and assignment of additional Diffie-Hellman Group
Transform IDs in the respective IANA registry. This document
specifies transform IDs for four curves from RFC 5639, as well as the
encoding of the key exchange payload and derivation of the shared
secret when using one of these curves.
1.1. Requirements Language
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
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2. IKEv2 Key Exchange Using the ECC Brainpool Curves
2.1. Diffie-Hellman Group Transform IDs
In order to use the ECC Brainpool curves for key exchange within
IKEv2, the Diffie-Hellman Group Transform IDs (Transform Type 4)
listed in the following table have been registered with IANA
[IANA-IKE2]. The parameters associated with these curves are defined
in RFC 5639 [RFC5639].
+-----------------+--------------+
| Curve | Transform ID |
+-----------------+--------------+
| brainpoolP224r1 | 27 |
| brainpoolP256r1 | 28 |
| brainpoolP384r1 | 29 |
| brainpoolP512r1 | 30 |
+-----------------+--------------+
Table 1
Test vectors for the groups defined by the ECC Brainpool curves are
provided in Appendix A.
2.2. Using the Twisted Brainpool Curves Internally
In [RFC5639], for each random curve, a "twisted curve" (defined by a
quadratic twist; see [HMV]) is defined that offers the same level of
security but potentially allows more efficient arithmetic due to the
curve parameter A = -3. The transform IDs listed in Table 1 also
allow using the twisted curve corresponding to the specified random
curve: points (x,y) of any of the listed curves can be efficiently
transformed to the corresponding point (x',y') on the twisted curve
of the same bit length -- and vice versa -- by setting (x',y') =
(x*Z^2, y*Z^3) with the coefficient Z specified for that curve
[RFC5639].
2.3. Key Exchange Payload and Shared Secret
For the encoding of the key exchange payload and the derivation of
the shared secret, the methods specified in [RFC5903] are adopted.
In an Elliptic Curve Group over GF[P] (ECP) key exchange in IKEv2,
the Diffie-Hellman public value passed in a key establishment (KE)
payload consists of two components, x and y, corresponding to the
coordinates of an elliptic curve point. Each component MUST be
computed from the corresponding coordinate using the FieldElement-to-
OctetString conversion method specified in [SEC1] and MUST have a bit
Merkle & Lochter Informational [Page 3]RFC 6954 Brainpool Curves for IKEv2 Key Exchange July 2013
length as indicated in Table 2. This length is enforced by the
FieldElement-to-OctetString conversion method, if necessary, by
prepending the value with zeros.
Note: The FieldElement-to-OctetString conversion method specified in
[SEC1] is equivalent to applying the conversion between integers and
octet strings (as described in Section 6 of [RFC6090]) after
representing the field element as an integer in the interval
[0, p-1].
+---------------------+-----------------------+---------------------+
| Curves | Bit length of each | Bit length of key |
| | component (x or y) | exchange payload |
+---------------------+-----------------------+---------------------+
| brainpoolP224r1 | 224 | 448 |
| brainpoolP256r1 | 256 | 512 |
| brainpoolP384r1 | 384 | 768 |
| brainpoolP512r1 | 512 | 1024 |
+---------------------+-----------------------+---------------------+
Table 2
From these components, the key exchange payload MUST be computed as
the concatenation of the x- and y-coordinates. Hence, the key
exchange payload has the bit length indicated in Table 2.
The Diffie-Hellman shared secret value consists only of the x value.
In particular, the shared secret value MUST be computed from the
x-coordinate of the Diffie-Hellman common value using the
FieldElement-to-OctetString conversion method specified in [SEC1] and
MUST have bit length as indicated in Table 2.
3. Security Considerations
The security considerations of [RFC5996] apply accordingly.
In order to thwart certain active attacks, the validity of the other
peer's public Diffie-Hellman value (x,y) recovered from the received
key exchange payload needs to be verified. In particular, it MUST be
verified that the x- and y-coordinates of the public value satisfy
the curve equation. For additional information, we refer the reader
to [RFC6989].
The confidentiality, authenticity, and integrity of a secure
communication based on IKEv2 are limited by the weakest cryptographic
primitive applied. In order to achieve a maximum security level when
Merkle & Lochter Informational [Page 4]RFC 6954 Brainpool Curves for IKEv2 Key Exchange July 2013
using one of the elliptic curves from Table 1 for key exchange, the
following should be chosen according to the recommendations of
[NIST800-57] and [RFC5639]:
o key derivation function
o algorithms and key lengths of symmetric encryption and message
authentication
o algorithm, bit length, and hash function used for signature
generation
Furthermore, the private Diffie-Hellman keys should be selected with
the same bit length as the order of the group generated by the base
point G and with approximately maximum entropy.
Implementations of elliptic curve cryptography for IKEv2 could be
susceptible to side-channel attacks. Particular care should be taken
for implementations that internally use the corresponding twisted
curve to take advantage of an efficient arithmetic for the special
parameters (A = -3): although the twisted curve itself offers the
same level of security as the corresponding random curve (through
mathematical equivalence), an arithmetic based on small curve
parameters could be harder to protect against side-channel attacks.
General guidance on resistance of elliptic curve cryptography
implementations against side-channel attacks is given in [BSI1] and
[HMV].
4. IANA Considerations
IANA has updated its "Transform Type 4 - Diffie-Hellman Group
Transform IDs" registry in [IANA-IKE2] to include the groups listed
in Table 1.
5. References
5.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC5996] Kaufman, C., Hoffman, P., Nir, Y., and P. Eronen,
"Internet Key Exchange Protocol Version 2 (IKEv2)",
RFC 5996, September 2010.
[RFC5639] Lochter, M. and J. Merkle, "Elliptic Curve Cryptography
(ECC) Brainpool Standard Curves and Curve Generation",
RFC 5639, March 2010.
Merkle & Lochter Informational [Page 5]RFC 6954 Brainpool Curves for IKEv2 Key Exchange July 2013
[RFC6989] Sheffer, Y. and S. Fluhrer, "Additional Diffie-Hellman
Tests for the Internet Key Exchange Protocol Version 2
(IKEv2)", RFC 6989, July 2013.
[IANA-IKE2] Internet Assigned Numbers Authority, "Internet Key
Exchange Version 2 (IKEv2) Parameters",
<http://www.iana.org/assignments/ikev2-parameters>.
[SEC1] Certicom Research, "Elliptic Curve Cryptography",
Standards for Efficient Cryptography (SEC) 1,
September 2000.
5.2. Informative References
[RFC5903] Fu, D. and J. Solinas, "Elliptic Curve Groups modulo a
Prime (ECP Groups) for IKE and IKEv2", RFC 5903,
June 2010.
[RFC6090] McGrew, D., Igoe, K., and M. Salter, "Fundamental
Elliptic Curve Cryptography Algorithms", RFC 6090,
February 2011.
[ANSI1] American National Standards Institute, "Public Key
Cryptography For The Financial Services Industry: The
Elliptic Curve Digital Signature Algorithm (ECDSA)",
ANSI X9.62, 2005.
[BSI1] Bundesamt fuer Sicherheit in der Informationstechnik,
"Minimum Requirements for Evaluating Side-Channel Attack
Resistance of Elliptic Curve Implementations", July
2011.
[FIPS] National Institute of Standards and Technology, "Digital
Signature Standard (DSS)", FIPS PUB 186-2, December
1998.
[HMV] Hankerson, D., Menezes, A., and S. Vanstone, "Guide to
Elliptic Curve Cryptography", Springer-Verlag, 2004.
[ISO1] International Organization for Standardization,
"Information Technology -- Security Techniques --
Digital Signatures with Appendix - Part 3: Discrete
Logarithm Based Mechanisms", ISO/IEC 14888-3, 2006.
[ISO2] International Organization for Standardization,
"Information Technology -- Security Techniques --
Cryptographic Techniques Based on Elliptic Curves -
Part 2: Digital signatures", ISO/IEC 15946-2, 2002.
Merkle & Lochter Informational [Page 6]RFC 6954 Brainpool Curves for IKEv2 Key Exchange July 2013
[NIST800-57] National Institute of Standards and Technology,
"Recommendation for Key Management -- Part 1: General
(Revised)", NIST Special Publication 800-57, March 2007.
[SEC2] Certicom Research, "Recommended Elliptic Curve Domain
Parameters", Standards for Efficient Cryptography (SEC)
2, September 2000.
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Appendix A. Test Vectors
This section provides some test vectors, for example, Diffie-Hellman
key exchanges using each of the curves defined in Section 2. The
following notation is used in the subsequent subsections:
d_A: the secret key of party A
x_qA: the x-coordinate of the public key of party A
y_qA: the y-coordinate of the public key of party A
d_B: the secret key of party B
x_qB: the x-coordinate of the public key of party B
y_qB: the y-coordinate of the public key of party B
x_Z: the x-coordinate of the shared secret that results from
completion of the Diffie-Hellman computation
y_Z: the y-coordinate of the shared secret that results from
completion of the Diffie-Hellman computation
The field elements x_qA, y_qA, x_qB, y_qB, x_Z, and y_Z are
represented as hexadecimal values using the FieldElement-to-
OctetString conversion method specified in [SEC1].
A.1. 224-Bit Curve
Curve brainpoolP224r1
dA = 39F155483CEE191FBECFE9C81D8AB1A03CDA6790E7184ACE44BCA161
x_qA = A9C21A569759DA95E0387041184261440327AFE33141CA04B82DC92E
y_qA = 98A0F75FBBF61D8E58AE5511B2BCDBE8E549B31E37069A2825F590C1
dB = 6060552303899E2140715816C45B57D9B42204FB6A5BF5BEAC10DB00
x_qB = 034A56C550FF88056144E6DD56070F54B0135976B5BF77827313F36B
y_qB = 75165AD99347DC86CAAB1CBB579E198EAF88DC35F927B358AA683681
x_Z = 1A4BFE705445120C8E3E026699054104510D119757B74D5FE2462C66
y_Z = BB6802AC01F8B7E91B1A1ACFB9830A95C079CEC48E52805DFD7D2AFE
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A.2. 256-Bit Curve
Curve brainpoolP256r1
dA =
81DB1EE100150FF2EA338D708271BE38300CB54241D79950F77B063039804F1D
x_qA =
44106E913F92BC02A1705D9953A8414DB95E1AAA49E81D9E85F929A8E3100BE5
y_qA =
8AB4846F11CACCB73CE49CBDD120F5A900A69FD32C272223F789EF10EB089BDC
dB =
55E40BC41E37E3E2AD25C3C6654511FFA8474A91A0032087593852D3E7D76BD3
x_qB =
8D2D688C6CF93E1160AD04CC4429117DC2C41825E1E9FCA0ADDD34E6F1B39F7B
y_qB =
990C57520812BE512641E47034832106BC7D3E8DD0E4C7F1136D7006547CEC6A
x_Z =
89AFC39D41D3B327814B80940B042590F96556EC91E6AE7939BCE31F3A18BF2B
y_Z =
49C27868F4ECA2179BFD7D59B1E3BF34C1DBDE61AE12931648F43E59632504DE
A.3. 384-Bit Curve
Curve brainpoolP384r1
dA = 1E20F5E048A5886F1F157C74E91BDE2B98C8B52D58E5003D57053FC4B0BD6
5D6F15EB5D1EE1610DF870795143627D042
x_qA = 68B665DD91C195800650CDD363C625F4E742E8134667B767B1B47679358
8F885AB698C852D4A6E77A252D6380FCAF068
y_qA = 55BC91A39C9EC01DEE36017B7D673A931236D2F1F5C83942D049E3FA206
07493E0D038FF2FD30C2AB67D15C85F7FAA59
dB = 032640BC6003C59260F7250C3DB58CE647F98E1260ACCE4ACDA3DD869F74E
01F8BA5E0324309DB6A9831497ABAC96670
x_qB = 4D44326F269A597A5B58BBA565DA5556ED7FD9A8A9EB76C25F46DB69D19
DC8CE6AD18E404B15738B2086DF37E71D1EB4
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y_qB = 62D692136DE56CBE93BF5FA3188EF58BC8A3A0EC6C1E151A21038A42E91
85329B5B275903D192F8D4E1F32FE9CC78C48
x_Z = 0BD9D3A7EA0B3D519D09D8E48D0785FB744A6B355E6304BC51C229FBBCE2
39BBADF6403715C35D4FB2A5444F575D4F42
y_Z = 0DF213417EBE4D8E40A5F76F66C56470C489A3478D146DECF6DF0D94BAE9
E598157290F8756066975F1DB34B2324B7BD
A.4. 512-Bit Curve
Curve brainpoolP512r1
dA = 16302FF0DBBB5A8D733DAB7141C1B45ACBC8715939677F6A56850A38BD87B
D59B09E80279609FF333EB9D4C061231FB26F92EEB04982A5F1D1764CAD5766542
2
x_qA = 0A420517E406AAC0ACDCE90FCD71487718D3B953EFD7FBEC5F7F27E28C6
149999397E91E029E06457DB2D3E640668B392C2A7E737A7F0BF04436D11640FD0
9FD
y_qA = 72E6882E8DB28AAD36237CD25D580DB23783961C8DC52DFA2EC138AD472
A0FCEF3887CF62B623B2A87DE5C588301EA3E5FC269B373B60724F5E82A6AD147F
DE7
dB = 230E18E1BCC88A362FA54E4EA3902009292F7F8033624FD471B5D8ACE49D1
2CFABBC19963DAB8E2F1EBA00BFFB29E4D72D13F2224562F405CB80503666B2542
9
x_qB = 9D45F66DE5D67E2E6DB6E93A59CE0BB48106097FF78A081DE781CDB31FC
E8CCBAAEA8DD4320C4119F1E9CD437A2EAB3731FA9668AB268D871DEDA55A54731
99F
y_qB = 2FDC313095BCDD5FB3A91636F07A959C8E86B5636A1E930E8396049CB48
1961D365CC11453A06C719835475B12CB52FC3C383BCE35E27EF194512B7187628
5FA
x_Z = A7927098655F1F9976FA50A9D566865DC530331846381C87256BAF322624
4B76D36403C024D7BBF0AA0803EAFF405D3D24F11A9B5C0BEF679FE1454B21C4CD
1F
y_Z = 7DB71C3DEF63212841C463E881BDCF055523BD368240E6C3143BD8DEF8B3
B3223B95E0F53082FF5E412F4222537A43DF1C6D25729DDB51620A832BE6A26680
A2
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Authors' Addresses
Johannes Merkle
secunet Security Networks
Mergenthaler Allee 77
65760 Eschborn
Germany
Phone: +49 201 5454 3091
EMail: johannes.merkle@secunet.com
Manfred Lochter
Bundesamt fuer Sicherheit in der Informationstechnik (BSI)
Postfach 200363
53133 Bonn
Germany
Phone: +49 228 9582 5643
EMail: manfred.lochter@bsi.bund.de
Merkle & Lochter Informational [Page 11]
