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Internet Engineering Task Force (IETF)
Request for Comments: 7027
Updates: 4492
Category: Informational
ISSN: 2070-1721
J. Merkle
secunet Security Networks
M. Lochter
BSI
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 © 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 publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License.

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].

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- 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.
   
   [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.

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].

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

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

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

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