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RFC 6954 - Using the Elliptic Curve Cryptography (ECC) Brainpool

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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)


   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

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.

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",
   document are to be interpreted as described in [RFC2119].

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

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

   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

   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

   o  algorithm, bit length, and hash function used for signature

   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

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.

   [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",

   [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

   [FIPS]       National Institute of Standards and Technology, "Digital
                Signature Standard (DSS)", FIPS PUB 186-2, December

   [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.

   [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 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

A.2.  256-Bit Curve

   Curve brainpoolP256r1

      dA =

      x_qA =

      y_qA =

      dB =

      x_qB =

      y_qB =

      x_Z =

      y_Z =

A.3.  384-Bit Curve

   Curve brainpoolP384r1

      dA = 1E20F5E048A5886F1F157C74E91BDE2B98C8B52D58E5003D57053FC4B0BD6

      x_qA = 68B665DD91C195800650CDD363C625F4E742E8134667B767B1B47679358

      y_qA = 55BC91A39C9EC01DEE36017B7D673A931236D2F1F5C83942D049E3FA206

      dB = 032640BC6003C59260F7250C3DB58CE647F98E1260ACCE4ACDA3DD869F74E

      x_qB = 4D44326F269A597A5B58BBA565DA5556ED7FD9A8A9EB76C25F46DB69D19

      y_qB = 62D692136DE56CBE93BF5FA3188EF58BC8A3A0EC6C1E151A21038A42E91

      x_Z = 0BD9D3A7EA0B3D519D09D8E48D0785FB744A6B355E6304BC51C229FBBCE2

      y_Z = 0DF213417EBE4D8E40A5F76F66C56470C489A3478D146DECF6DF0D94BAE9

A.4.  512-Bit Curve

   Curve brainpoolP512r1

      dA = 16302FF0DBBB5A8D733DAB7141C1B45ACBC8715939677F6A56850A38BD87B

      x_qA = 0A420517E406AAC0ACDCE90FCD71487718D3B953EFD7FBEC5F7F27E28C6

      y_qA = 72E6882E8DB28AAD36237CD25D580DB23783961C8DC52DFA2EC138AD472

      dB = 230E18E1BCC88A362FA54E4EA3902009292F7F8033624FD471B5D8ACE49D1

      x_qB = 9D45F66DE5D67E2E6DB6E93A59CE0BB48106097FF78A081DE781CDB31FC

      y_qB = 2FDC313095BCDD5FB3A91636F07A959C8E86B5636A1E930E8396049CB48

      x_Z = A7927098655F1F9976FA50A9D566865DC530331846381C87256BAF322624

      y_Z = 7DB71C3DEF63212841C463E881BDCF055523BD368240E6C3143BD8DEF8B3

Authors' Addresses

   Johannes Merkle
   secunet Security Networks
   Mergenthaler Allee 77
   65760 Eschborn

   Phone: +49 201 5454 3091
   EMail: johannes.merkle@secunet.com

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

   Phone: +49 228 9582 5643
   EMail: manfred.lochter@bsi.bund.de


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