





Internet Engineering Task Force (IETF)                   D. Harkins, Ed.
Request for Comments: 6932                                Aruba Networks
Category: Informational                                         May 2013
ISSN: 2070-1721


                       Brainpool Elliptic Curves
     for the Internet Key Exchange (IKE) Group Description Registry

Abstract

   This memo allocates code points for four new elliptic curve domain
   parameter sets over finite prime fields into a registry that was
   established by the Internet Key Exchange (IKE) but is used by other
   protocols.

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/rfc6932.

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.





Harkins                       Informational                     [Page 1]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Requirements Language . . . . . . . . . . . . . . . . . .   2
   2.  Brainpool Elliptic Curves . . . . . . . . . . . . . . . . . .   2
     2.1.  Domain Parameters for the 224-Bit Curve . . . . . . . . .   4
     2.2.  Domain Parameters for the 256-Bit Curve . . . . . . . . .   4
     2.3.  Domain Parameters for the 384-Bit Curve . . . . . . . . .   5
     2.4.  Domain Parameters for the 512-Bit Curve . . . . . . . . .   5
   3.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .   6
   4.  Security Considerations . . . . . . . . . . . . . . . . . . .   6
   5.  Use of Brainpool Curves . . . . . . . . . . . . . . . . . . .   7
   6.  References  . . . . . . . . . . . . . . . . . . . . . . . . .   7
     6.1.  Normative References  . . . . . . . . . . . . . . . . . .   7
     6.2.  Informative References  . . . . . . . . . . . . . . . . .   7
   Appendix A.  Test Data  . . . . . . . . . . . . . . . . . . . . .   9
     A.1.  Test Vector for brainpoolP224r1 . . . . . . . . . . . . .   9
     A.2.  Test Vector for brainpoolP256r1 . . . . . . . . . . . . .  10
     A.3.  Test Vector for brainpoolP384r1 . . . . . . . . . . . . .  10
     A.4.  Test Vector for brainpoolP512r1 . . . . . . . . . . . . .  11

1.  Introduction

   [RFC5639] defines new elliptic curve domain parameters for curves
   over a number of different prime fields, each with a "twisted"
   variant.  These curves have a number of interesting security
   properties (as described in [EBP]) that make them desirable to use.

   IANA maintains a registry for [RFC2409] to map complete domain
   parameter sets into easily referenced numbers.  While [RFC2409] is
   deprecated, other protocols, for example [IEEE802.11] and [RFC5931],
   refer to this registry for its convenience.  Therefore, this memo
   instructs IANA to allocate new code points for the Brainpool curves
   defined in [RFC5639] to the registry established by [RFC2409] for use
   by other protocols.

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 RFC 2119 [RFC2119].

2.  Brainpool Elliptic Curves

   [RFC5639] defines several elliptic curves over finite prime fields
   (ECP, in the parlance of [RFC2409]).  The domain parameter sets for
   each of the elliptic curves defined in [RFC5639] are copied here for
   convenient reference.



Harkins                       Informational                     [Page 2]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


   The equation for all elliptic curves defined here is:

      y^2 = x^3 + ax + b (mod p)

   Domain parameter sets consist of:

   o  p: the prime

   o  a, b: parameters to the equation of the curve

   o  x, y: the coordinates of the generator for the group, G

   o  q: the order of the group formed by the generator G

   o  h: the co-factor

   o  z: the "twist" (for conversion into twisted curves)

   [RFC5639] defines elliptic curves over seven (7) prime fields with a
   random and a "twisted" variety for each, for a total of fourteen (14)
   distinct curves.  However, some of those curves are not particularly
   useful: the 160-bit curves provide only 80 bits of strength and that
   is too small to be of use in current cryptographic applications, and
   there is no standard hash function to use with the 196-bit and
   320-bit curves -- it would make more sense to use the 224-bit and
   384-bit curves, respectively, instead.  For this reason, the curves
   defined over 160-bit, 192-bit, and 320-bit primes are not being added
   to the registry created by [RFC2409].

   The twisted curves in [RFC5639] are isomorphic to the random curves
   of the same length.  The curve parameter "a" for the twisted curves
   equals -3 mod p, and there are certain arithmetical advantages to
   using such curves.  It is possible to convert a point from a random
   curve (x,y) into a point on the twisted curve (x', y') and back again
   using this equation:

      (x',y') = (x*z^2, y*z^3)

   This would allow an implementation to internally use the twisted
   version of the curve, taking full advantage of the arithmetical
   advantages, while exchanging points on the random versions of the
   curve with peers.

   Therefore, the twisted curves are not being added to the registry
   created by [RFC2409].  Implementations that desire to use the twisted
   curves internally MUST refer to [RFC5639] for the complete domain
   parameter sets, only the "twist" is defined here.




Harkins                       Informational                     [Page 3]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


2.1.  Domain Parameters for the 224-Bit Curve

   Curve-ID: brainpoolP224r1

   p = D7C134AA264366862A18302575D1D787B09F075797DA89F57EC8C0FF

   A = 68A5E62CA9CE6C1C299803A6C1530B514E182AD8B0042A59CAD29F43

   B = 2580F63CCFE44138870713B1A92369E33E2135D266DBB372386C400B

   x = 0D9029AD2C7E5CF4340823B2A87DC68C9E4CE3174C1E6EFDEE12C07D

   y = 58AA56F772C0726F24C6B89E4ECDAC24354B9E99CAA3F6D3761402CD

   q = D7C134AA264366862A18302575D0FB98D116BC4B6DDEBCA3A5A7939F

   z = 2DF271E14427A346910CF7A2E6CFA7B3F484E5C2CCE1C8B730E28B3F

   h = 1

2.2.  Domain Parameters for the 256-Bit Curve

   Curve-ID: brainpoolP256r1

   p = A9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5377

   A = 7D5A0975FC2C3057EEF67530417AFFE7FB8055C126DC5C6CE94A4B44F330B5D9

   B = 26DC5C6CE94A4B44F330B5D9BBD77CBF958416295CF7E1CE6BCCDC18FF8C07B6

   x = 8BD2AEB9CB7E57CB2C4B482FFC81B7AFB9DE27E1E3BD23C23A4453BD9ACE3262

   y = 547EF835C3DAC4FD97F8461A14611DC9C27745132DED8E545C1D54C72F046997

   q = A9FB57DBA1EEA9BC3E660A909D838D718C397AA3B561A6F7901E0E82974856A7

   z = 3E2D4BD9597B58639AE7AA669CAB9837CF5CF20A2C852D10F655668DFC150EF0

   h = 1












Harkins                       Informational                     [Page 4]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


2.3.  Domain Parameters for the 384-Bit Curve

   Curve-ID: brainpoolP384r1

   p = 8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B412B1DA197FB711
       23ACD3A729901D1A71874700133107EC53

   A = 7BC382C63D8C150C3C72080ACE05AFA0C2BEA28E4FB22787139165EFBA91F9
       0F8AA5814A503AD4EB04A8C7DD22CE2826

   B = 04A8C7DD22CE28268B39B55416F0447C2FB77DE107DCD2A62E880EA53EEB62
       D57CB4390295DBC9943AB78696FA504C11

   x = 1D1C64F068CF45FFA2A63A81B7C13F6B8847A3E77EF14FE3DB7FCAFE0CBD10
       E8E826E03436D646AAEF87B2E247D4AF1E

   y = 8ABE1D7520F9C2A45CB1EB8E95CFD55262B70B29FEEC5864E19C054FF99129
       280E4646217791811142820341263C5315

   q = 8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B31F166E6CAC0425
       A7CF3AB6AF6B7FC3103B883202E9046565

   z = 41DFE8DD399331F7166A66076734A89CD0D2BCDB7D068E44E1F378F41ECBAE
       97D2D63DBC87BCCDDCCC5DA39E8589291C

   h = 1

2.4.  Domain Parameters for the 512-Bit Curve

   Curve-ID: brainpoolP512r1

   p = AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308
       717D4D9B009BC66842AECDA12AE6A380E62881FF2F2D82C68528AA6056583A
       48F3

   A = 7830A3318B603B89E2327145AC234CC594CBDD8D3DF91610A83441CAEA9863
       BC2DED5D5AA8253AA10A2EF1C98B9AC8B57F1117A72BF2C7B9E7C1AC4D77FC
       94CA

   B = 3DF91610A83441CAEA9863BC2DED5D5AA8253AA10A2EF1C98B9AC8B57F1117
       A72BF2C7B9E7C1AC4D77FC94CADC083E67984050B75EBAE5DD2809BD638016
       F723

   x = 81AEE4BDD82ED9645A21322E9C4C6A9385ED9F70B5D916C1B43B62EEF4D009
       8EFF3B1F78E2D0D48D50D1687B93B97D5F7C6D5047406A5E688B352209BCB9
       F822





Harkins                       Informational                     [Page 5]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


   y = 7DDE385D566332ECC0EABFA9CF7822FDF209F70024A57B1AA000C55B881F81
       11B2DCDE494A5F485E5BCA4BD88A2763AED1CA2B2FA8F0540678CD1E0F3AD8
       0892

   q = AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308
       70553E5C414CA92619418661197FAC10471DB1D381085DDADDB58796829CA9
       0069

   z = 12EE58E6764838B69782136F0F2D3BA06E27695716054092E60A80BEDB212B
       64E585D90BCE13761F85C3F1D2A64E3BE8FEA2220F01EBA5EEB0F35DBD29D9
       22AB

   h = 1

3.  IANA Considerations

   IANA has assigned four values from the unassigned portion of the
   "Group Description" component of the [IANA-IKE] registry and updated
   the registry by appending Table 1 to the registry table.

   +----------+-----------------------+-------------+------------------+
   |  Value   |   Group Description   | Reference   |       Note       |
   +----------+-----------------------+-------------+------------------+
   |    27    | 224-bit Brainpool     | RFC 6932,   | Not for RFC 2409 |
   |          |     ECP group         | Section 2.1 |                  |
   |          |                       |             |                  |
   |    28    | 256-bit Brainpool     | RFC 6932,   | Not for RFC 2409 |
   |          |     ECP group         | Section 2.2 |                  |
   |          |                       |             |                  |
   |    29    | 384-bit Brainpool     | RFC 6932,   | Not for RFC 2409 |
   |          |     ECP group         | Section 2.3 |                  |
   |          |                       |             |                  |
   |    30    | 512-bit Brainpool     | RFC 6932,   | Not for RFC 2409 |
   |          |     ECP group         | Section 2.4 |                  |
   +----------+-----------------------+-------------+------------------+

                    Table 1: Group Description Updates

4.  Security Considerations

   [EBP] describes the security properties of the curves referenced
   here.  The curves support security levels of 112 (Section 2.1), 128
   (Section 2.2), 192 (Section 2.3), and 256 (Section 2.4).  These
   security levels assume that when these elliptic curves are used with
   discrete logarithm cryptography, for example elliptic curve Diffie-
   Hellman, that the private key used is a uniformly random number in
   the range [1..(q-1)], where q is the order from the curve's domain
   parameter set.  In order to achieve system security commensurate with



Harkins                       Informational                     [Page 6]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


   the security level of a particular elliptic curve, it is incumbent
   upon an implementation to choose key derivation functions, hash
   functions, pseudo-random functions, and ciphers according to the
   recommendations from [SP800-57].

5.  Use of Brainpool Curves

   The notes in Table 1 are an administrative prohibition, not a
   technical one.  The notes are there because, although [RFC2409] has
   been deprecated, it is still widely used.  There is a desire among
   some in the IETF to not do anything that would prolong the use of
   [RFC2409], and the addition of these curves was perceived as doing
   just that.  The registry could not have been updated without
   including notes to indicate that these curves are not for use with
   [RFC2409] and not updating the [RFC2409] registry would have a
   detrimental affect on the other protocols that use it.

6.  References

6.1.  Normative References

   [IANA-IKE]   IANA, "Internet Key Exchange (IKE) Attributes", Registry
                Name: Group Description (Value 4),
                <http://www.iana.org/assignments/ipsec-registry>.

   [RFC2119]    Bradner, S., "Key words for use in RFCs to Indicate
                Requirement Levels", BCP 14, RFC 2119, March 1997.

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

6.2.  Informative References

   [EBP]        The Brainpool Workgroup, "ECC Brainpool Standard Curves
                and Curve Generation", October 2005,
                <http://www.ecc-brainpool.org/download/
                Domain-parameters.pdf>.

   [IEEE802.11] IEEE, "Telecommunications and information exchange
                between systems Local and metropolitan area networks--
                Part 11: Wireless LAN Medium Access Control (MAC) and
                Physical Layer (PHY) Specifications", IEEE Std
                802.11-2012, 2012.

   [RFC2409]    Harkins, D. and D. Carrel, "The Internet Key Exchange
                (IKE)", RFC 2409, November 1998.




Harkins                       Informational                     [Page 7]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


   [RFC5931]    Harkins, D. and G. Zorn, "Extensible Authentication
                Protocol (EAP) Authentication Using Only a Password",
                RFC 5931, August 2010.

   [SP800-57]   National Institute of Standards and Technology,
                "Recommendation for Key Management - Part 1: General
                (Revised)", NIST Special Publication 800-57, March 2007.












































Harkins                       Informational                     [Page 8]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


Appendix A.  Test Data

   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 subsequent sections:

   o  dA: the secret key of party A

   o  x_qA: the x-coordinate of the public key of party A

   o  y_qA: the y-coordinate of the public key of party A

   o  dB: the secret key of party B

   o  x_qB: the x-coordinate of the public key of party B

   o  y_qB: the y-coordinate of the public key of party B

   o  x_Z: the x-coordinate of the shared secret that results from
      completion of the Diffie-Hellman computation

   o  y_Z: the y-coordinate of the shared secret that results from
      completion of the Diffie-Hellman computation

A.1.  Test Vector for brainpoolP224r1

   dA =
   7C4B7A2C 8A4BAD1F BB7D79CC 0955DB7C 6A4660CA 64CC4778
   159B495E

   x_qA =
   B104A67A 6F6E85E1 4EC1825E 1539E8EC DBBF5849 22367DD8
   8C6BDCF2
   y_qA =
   46D782E7 FDB5F60C D8404301 AC5949C5 8EDB26BC 68BA0769
   5B750A94

   dB =
   63976D4A AE6CD0F6 DD18DEFE F55D9656 9D0507C0 3E74D648
   6FFA28FB

   x_qB =
   2A97089A 9296147B 71B21A4B 574E1278 245B536F 14D8C2B9
   D07A874E
   y_qB =
   9B900D7C 77A709A7 97276B8C A1BA61BB 95B546FC 29F862E4
   4D59D25B




Harkins                       Informational                     [Page 9]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


   x_Z =
   312DFD98 783F9FB7 7B970494 5A73BEB6 DCCBE3B6 5D0F967D
   CAB574EB
   y_Z =
   6F800811 D64114B1 C48C621A B3357CF9 3F496E42 38696A2A
   012B3C98

A.2.  Test Vector for brainpoolP256r1

   dA =
   041EB8B1 E2BC681B CE8E3996 3B2E9FC4 15B05283 313DD1A8
   BCC055F1 1AE49699

   x_qA =
   78028496 B5ECAAB3 C8B6C12E 45DB1E02 C9E4D26B 4113BC4F
   015F60C5 CCC0D206
   y_qA =
   A2AE1762 A3831C1D 20F03F8D 1E3C0C39 AFE6F09B 4D44BBE8
   0CD10098 7B05F92B

   dB =
   06F5240E ACDB9837 BC96D482 74C8AA83 4B6C87BA 9CC3EEDD
   81F99A16 B8D804D3

   x_qB =
   8E07E219 BA588916 C5B06AA3 0A2F464C 2F2ACFC1 610A3BE2
   FB240B63 5341F0DB
   y_qB =
   148EA1D7 D1E7E54B 9555B6C9 AC90629C 18B63BEE 5D7AA694
   9EBBF47B 24FDE40D

   x_Z =
   05E94091 5549E9F6 A4A75693 716E3746 6ABA79B4 BF291987
   7A16DD2C C2E23708
   y_Z =
   6BC23B67 02BC5A01 9438CEEA 107DAAD8 B94232FF BBC350F3
   B137628F E6FD134C

A.3.  Test Vector for brainpoolP384r1

   dA =
   014EC075 5B78594B A47FB0A5 6F617304 5B4331E7 4BA1A6F4
   7322E70D 79D828D9 7E095884 CA72B73F DABD5910 DF0FA76A








Harkins                       Informational                    [Page 10]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


   x_qA =
   45CB26E4 384DAF6F B7768853 07B9A38B 7AD1B5C6 92E0C32F
   01253327 78F3B8D3 F50CA358 099B30DE B5EE69A9 5C058B4E
   y_qA =
   8173A1C5 4AFFA7E7 81D0E1E1 D12C0DC2 B74F4DF5 8E4A4E3A
   F7026C5D 32DC530A 2CD89C85 9BB4B4B7 68497F49 AB8CC859

   dB =
   6B461CB7 9BD0EA51 9A87D682 8815D8CE 7CD9B3CA A0B5A826
   2CBCD550 A015C900 95B976F3 52995750 6E1224A8 61711D54

   x_qB =
   01BF92A9 2EE4BE8D ED1A9111 25C209B0 3F99E316 1CFCC986
   DC771138 3FC30AF9 CE28CA33 86D59E2C 8D72CE1E 7B4666E8
   y_qB =
   3289C4A3 A4FEE035 E39BDB88 5D509D22 4A142FF9 FBCC5CFE
   5CCBB302 68EE4748 7ED80448 58D31D84 8F7A95C6 35A347AC

   x_Z =
   04CC4FF3 DCCCB07A F24E0ACC 529955B3 6D7C8077 72B92FCB
   E48F3AFE 9A2F370A 1F98D3FA 73FD0C07 47C632E1 2F1423EC
   y_Z =
   7F465F90 BD69AFB8 F828A214 EB9716D6 6ABC59F1 7AF7C75E
   E7F1DE22 AB5D0508 5F5A01A9 382D05BF 72D96698 FE3FF64E

A.4.  Test Vector for brainpoolP512r1

   dA =
   636B6BE0 482A6C1C 41AA7AE7 B245E983 392DB94C ECEA2660
   A379CFE1 59559E35 75818253 91175FC1 95D28BAC 0CF03A78
   41A383B9 5C262B98 3782874C CE6FE333

   x_qA =
   0562E68B 9AF7CBFD 5565C6B1 6883B777 FF11C199 161ECC42
   7A39D17E C2166499 389571D6 A994977C 56AD8252 658BA8A1
   B72AE42F 4FB75321 51AFC3EF 0971CCDA
   y_qA =
   A7CA2D81 91E21776 A89860AF BC1F582F AA308D55 1C1DC613
   3AF9F9C3 CAD59998 D7007954 8140B90B 1F311AFB 378AA81F
   51B275B2 BE6B7DEE 978EFC73 43EA642E

   dB =
   0AF4E7F6 D52EDD52 907BB8DB AB3992A0 BB696EC1 0DF11892
   FF205B66 D381ECE7 2314E6A6 EA079CEA 06961DBA 5AE6422E
   F2E9EE80 3A1F236F B96A1799 B86E5C8B






Harkins                       Informational                    [Page 11]

RFC 6932          Brainpool ECC for IKE Group Registry          May 2013


   x_qB =
   5A7954E3 2663DFF1 1AE24712 D87419F2 6B708AC2 B92877D6
   BFEE2BFC 43714D89 BBDB6D24 D807BBD3 AEB7F0C3 25F862E8
   BADE4F74 636B97EA ACE739E1 1720D323
   y_qB =
   96D14621 A9283A1B ED84DE8D D64836B2 C0758B11 441179DC
   0C54C0D4 9A47C038 07D171DD 544B72CA AEF7B7CE 01C7753E
   2CAD1A86 1ECA55A7 1954EE1B A35E04BE

   x_Z =
   1EE8321A 4BBF93B9 CF8921AB 209850EC 9B7066D1 984EF08C
   2BB72323 6208AC8F 1A483E79 461A00E0 D5F6921C E9D36050
   2F85C812 BEDEE23A C5B210E5 811B191E
   y_Z =
   2632095B 7B936174 B41FD2FA F369B1D1 8DCADEED 7E410A7E
   251F0831 097C50D0 2CFED026 07B6A2D5 ADB4C000 60085622
   08631875 B58B54EC DA5A4F9F E9EAABA6

Author's Address

   Dan Harkins (editor)
   Aruba Networks
   1322 Crossman avenue
   Sunnyvale , California   94089
   United States of America

   Phone: +1 408 227 4500
   EMail: dharkins@arubanetworks.com























Harkins                       Informational                    [Page 12]

