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RFC 7027 - Elliptic Curve Cryptography (ECC) Brainpool Curves fo


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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
   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

 

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