HomeDocsIETF standardsRFCRFC 6900 — 6999
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Internet Engineering Task Force (IETF)
Request for Comments: 6954
Category: Informational
ISSN: 2070-1721
J. Merkle
secunet Security Networks
M. Lochter
BSI
July 2013

Using the Elliptic Curve Cryptography (ECC) Brainpool Curves

for the Internet Key Exchange Protocol Version 2 (IKEv2)

Abstract

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 http://www.rfc-editor.org/info/rfc6954.

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
      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", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this 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 [RFC5639].

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]:

  • key derivation function
  • algorithms and key lengths of symmetric encryption and message authentication
  • algorithm, bit length, and hash function used for signature generation

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 [HMV].

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",
                <http://www.iana.org/assignments/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
                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. 

   [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 = 81DB1EE100150FF2EA338D708271BE38300CB54241D79950F77B063039804F1D

x_qA = 44106E913F92BC02A1705D9953A8414DB95E1AAA49E81D9E85F929A8E3100BE5

y_qA = 8AB4846F11CACCB73CE49CBDD120F5A900A69FD32C272223F789EF10EB089BDC

dB = 55E40BC41E37E3E2AD25C3C6654511FFA8474A91A0032087593852D3E7D76BD3

x_qB = 8D2D688C6CF93E1160AD04CC4429117DC2C41825E1E9FCA0ADDD34E6F1B39F7B

y_qB = 990C57520812BE512641E47034832106BC7D3E8DD0E4C7F1136D7006547CEC6A

x_Z = 89AFC39D41D3B327814B80940B042590F96556EC91E6AE7939BCE31F3A18BF2B

y_Z = 49C27868F4ECA2179BFD7D59B1E3BF34C1DBDE61AE12931648F43E59632504DE

A.3. 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.4. 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