Internet Engineering Task Force (IETF) J. Merkle

Internet Engineering Task Force (IETF) J. Merkle
Request for Comments: 7027 secunet Security Networks
Updates: 4492 M. Lochter
Category: Informational BSI
ISSN: 2070-1721 October 2013

Elliptic Curve Cryptography (ECC) Brainpool Curves
for Transport Layer Security (TLS)

Abstract

This document specifies the use of several Elliptic Curve
Cryptography (ECC) Brainpool curves for authentication and key
exchange in the Transport Layer Security (TLS) protocol.

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/rfc7027.

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
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the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.

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Table of Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Brainpool NamedCurve Types . . . . . . . . . . . . . . . . . . 2
3. IANA Considerations . . . . . . . . . . . . . . . . . . . . . . 3
4. Security Considerations . . . . . . . . . . . . . . . . . . . . 3
5. References . . . . . . . . . . . . . . . . . . . . . . . . . . 4
5.1. Normative References . . . . . . . . . . . . . . . . . . . 4
5.2. Informative References . . . . . . . . . . . . . . . . . . 4
Appendix A. Test Vectors . . . . . . . . . . . . . . . . . . . . . 6
A.1. 256-Bit Curve . . . . . . . . . . . . . . . . . . . . . . . 7
A.2. 384-Bit Curve . . . . . . . . . . . . . . . . . . . . . . . 8
A.3. 512-Bit Curve . . . . . . . . . . . . . . . . . . . . . . . 9

1. Introduction

[RFC5639] specifies 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 SecG [SEC2].

[RFC4492] defines the usage of elliptic curves for authentication and
key agreement in TLS 1.0 and TLS 1.1; these mechanisms may also be
used with TLS 1.2 [RFC5246]. While the ASN.1 object identifiers
defined in [RFC5639] already allow usage of the ECC Brainpool curves
for TLS (client or server) authentication through reference in X.509
certificates according to [RFC3279] and [RFC5480], their negotiation
for key exchange according to [RFC4492] requires the definition and
assignment of additional NamedCurve IDs. This document specifies
such values for three curves from [RFC5639].

2. Brainpool NamedCurve Types

According to [RFC4492], the name space NamedCurve is used for the
negotiation of elliptic curve groups for key exchange during a
handshake starting a new TLS session. This document adds new
NamedCurve types to three elliptic curves defined in [RFC5639] as
follows:

enum {
brainpoolP256r1(26),
brainpoolP384r1(27),
brainpoolP512r1(28)
} NamedCurve;

These curves are suitable for use with Datagram TLS [RFC6347].

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Test vectors for a Diffie-Hellman key exchange using these elliptic
curves are provided in Appendix A.

3. IANA Considerations

IANA has assigned numbers for the ECC Brainpool curves listed in
Section 2 in the "EC Named Curve" [IANA-TLS] registry of the
"Transport Layer Security (TLS) Parameters" registry as follows:

+-------+-----------------+---------+-----------+
| Value | Description | DTLS-OK | Reference |
+-------+-----------------+---------+-----------+
| 26 | brainpoolP256r1 | Y | RFC 7027 |
| 27 | brainpoolP384r1 | Y | RFC 7027 |
| 28 | brainpoolP512r1 | Y | RFC 7027 |
+-------+-----------------+---------+-----------+

Table 1

4. Security Considerations

The security considerations of [RFC5246] apply to the ECC Brainpool
curves described in this document.

The confidentiality, authenticity, and integrity of the TLS
communication is limited by the weakest cryptographic primitive
applied. In order to achieve a maximum security level when using one
of the elliptic curves from Table 1 for authentication and/or key
exchange in TLS, the key derivation function; the algorithms and key
lengths of symmetric encryption; and message authentication (as well
as the algorithm, bit length, and hash function used for signature
generation) should be chosen according to the recommendations of
[NIST800-57] and [RFC5639]. 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 TLS may be
susceptible to side-channel attacks. Particular care should be taken
for implementations that internally transform curve points to points
on the corresponding "twisted curve", using the map (x',y') = (x*Z^2,
y*Z^3) with the coefficient Z specified for that curve in [RFC5639],
in order to take advantage of an efficient arithmetic based on the
twisted curve's 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 may be harder to protect against side-

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channel attacks. General guidance on resistance of elliptic curve
cryptography implementations against side-channel-attacks is given in
[BSI1] and [HMV].

5. References

5.1. Normative References

[IANA-TLS] Internet Assigned Numbers Authority, "Transport Layer
Security (TLS) Parameters",
<http://www.iana.org/assignments/tls-parameters>.

[RFC4492] Blake-Wilson, S., Bolyard, N., Gupta, V., Hawk, C., and
B. Moeller, "Elliptic Curve Cryptography (ECC) Cipher
Suites for Transport Layer Security (TLS)", RFC 4492,
May 2006.

[RFC5246] Dierks, T. and E. Rescorla, "The Transport Layer
Security (TLS) Protocol Version 1.2", RFC 5246,
August 2008.

[RFC5639] Lochter, M. and J. Merkle, "Elliptic Curve Cryptography
(ECC) Brainpool Standard Curves and Curve Generation",
RFC 5639, March 2010.

[RFC6347] Rescorla, E. and N. Modadugu, "Datagram Transport Layer
Security Version 1.2", RFC 6347, January 2012.

5.2. Informative References

[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.

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[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.

[NIST800-57] National Institute of Standards and Technology,
"Recommendation for Key Management - Part 1: General
(Revised)", NIST Special Publication 800-57,
March 2007.

[RFC3279] Bassham, L., Polk, W., and R. Housley, "Algorithms and
Identifiers for the Internet X.509 Public Key
Infrastructure Certificate and Certificate Revocation
List (CRL) Profile", RFC 3279, April 2002.

[RFC5480] Turner, S., Brown, D., Yiu, K., Housley, R., and T.
Polk, "Elliptic Curve Cryptography Subject Public Key
Information", RFC 5480, March 2009.

[SEC1] Certicom Research, "Elliptic Curve Cryptography",
Standards for Efficient Cryptography (SEC) 1,
September 2000.

[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 Table 1. The
following notation is used in the subsequent sections:

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, i.e., the hex
representation of the pre-master secret

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].

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A.1. 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

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A.2. 384-Bit Curve

Curve brainpoolP384r1

dA = 1E20F5E048A5886F1F157C74E91BDE2B98C8B52D58E5003D57053FC4B0BD6
5D6F15EB5D1EE1610DF870795143627D042

x_qA = 68B665DD91C195800650CDD363C625F4E742E8134667B767B1B47679358
8F885AB698C852D4A6E77A252D6380FCAF068

y_qA = 55BC91A39C9EC01DEE36017B7D673A931236D2F1F5C83942D049E3FA206
07493E0D038FF2FD30C2AB67D15C85F7FAA59

dB = 032640BC6003C59260F7250C3DB58CE647F98E1260ACCE4ACDA3DD869F74E
01F8BA5E0324309DB6A9831497ABAC96670

x_qB = 4D44326F269A597A5B58BBA565DA5556ED7FD9A8A9EB76C25F46DB69D19
DC8CE6AD18E404B15738B2086DF37E71D1EB4

y_qB = 62D692136DE56CBE93BF5FA3188EF58BC8A3A0EC6C1E151A21038A42E91
85329B5B275903D192F8D4E1F32FE9CC78C48

x_Z = 0BD9D3A7EA0B3D519D09D8E48D0785FB744A6B355E6304BC51C229FBBCE2
39BBADF6403715C35D4FB2A5444F575D4F42

y_Z = 0DF213417EBE4D8E40A5F76F66C56470C489A3478D146DECF6DF0D94BAE9
E598157290F8756066975F1DB34B2324B7BD

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A.3. 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: [email protected]

Manfred Lochter
Bundesamt fuer Sicherheit in der Informationstechnik (BSI)
Postfach 200363
53133 Bonn
Germany

Phone: +49 228 9582 5643
EMail: [email protected]

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