IPSec Working Group J. Solinas, NSA INTERNET-DRAFT Expires November 27, 2005 May 27, 2005 ECP Groups For IKE <draft-ietf-ipsec-ike-ecp-groups-01.txt> Status of this Memo By submitting this Internet-Draft, each author represents that any applicable patent or other IPR claims of which he or she is aware have been or will be disclosed, and any of which he or she becomes aware will be disclosed, in accordance with Section 6 of BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Internet-Drafts. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." The list of current Internet-Drafts can be accessed at http://www.ietf.org/1id-abstracts.html The list of Internet-Draft Shadow Directories can be accessed at http://www.ietf.org/shadow.html Abstract This document describes new ECC groups for use in the Internet Key Exchange (IKE) protocol in addition to previously defined groups. Specifically, the new curve groups are based on modular arithmetic rather than binary arithmetic. These new groups are defined to align IKE with other ECC implementations and standards, particularly NIST standards. In addition, the curves defined here can provide more efficient implementation than previously defined ECC groups. Solinas [Page 1]

INTERNET-DRAFT ECP Groups For IKE May 2005 1. Introduction This document describes default groups for use in elliptic curve Diffie-Hellman in IKE in addition to the Oakley groups included in [IKE] and the groups defined in [RFC-3526] and [BBPS]. The document assumes that the reader is familiar with the IKE protocol and the concept of Oakley Groups, as defined in RFC 2409 [IKE]. RFC 2409 [IKE] defines five standard Oakley Groups - three modular exponentiation groups and two elliptic curve groups over GF[2^N]. One modular exponentiation group (768 bits - Oakley Group 1) is mandatory for all implementations to support, while the other four are optional. Thirteen additional groups subsequently have been defined and assigned values by IANA. All of these additional groups are optional. Of the eighteen groups defined so far, eight are modular exponentiation groups and ten are elliptic curve groups over GF[2^N]. The purpose of this document is to expand the options available to implementers of elliptic curve groups by adding three new elliptic curve groups. Unlike the previous elliptic curve groups, the three groups proposed in this document are defined over GF[p] with p prime. The reasons for adding these new groups include the following. - The groups proposed afford efficiency advantages in software applications since the underlying arithmetic is integer arithmetic modulo a prime rather than binary field arithmetic. (Additional computational advantages for these groups are presented in [GMN].) - The groups proposed encourage alignment with other elliptic curve standards. The proposed groups are among those standardized by NIST, by the SECG, by ISO, and by ANSI. (See section 3 for details.) - The groups proposed are capable of providing security consistent with the new Advanced Encryption Standard. These groups could also be defined using the New Group Mode but including them in this RFC will encourage interoperability of IKE implementations based upon elliptic curve groups. In addition, the availability of standardized groups will result in optimizations for a particular curve and field size as well as allowing precomputation that could result in faster implementations. It is anticipated that the groups proposed here will be assigned identifiers by IANA [IANA]. In that case the full list of assigned values for the Group Description class within IKE will be the following. (The groups defined in this document are listed as 19, 20, and 21.) Solinas [Page 2]

INTERNET-DRAFT ECP Groups For IKE May 2005 Group Description Value ----------------- ----- Default 768-bit MODP group [IKE] 1 Alternate 1024-bit MODP group [IKE] 2 EC2N group over GF[2^155] [IKE] 3 EC2N group over GF[2^185] [IKE] 4 1536-bit MODP group [RFC-3526] 5 EC2N group over GF[2^163] [BBPS] 6 EC2N group over GF[2^163] [BBPS] 7 EC2N group over GF[2^283] [BBPS] 8 EC2N group over GF[2^283] [BBPS] 9 EC2N group over GF[2^409] [BBPS] 10 EC2N group over GF[2^409] [BBPS] 11 EC2N group over GF[2^571] [BBPS] 12 EC2N group over GF[2^571] [BBPS] 13 2048-bit MODP group [RFC-3526] 14 3072-bit MODP group [RFC-3526] 15 4096-bit MODP group [RFC-3526] 16 6144-bit MODP group [RFC-3526] 17 8192-bit MODP group [RFC-3526] 18 256-bit ECP group (EC group modulo a 256-bit prime) 19 384-bit ECP group (EC group modulo a 384-bit prime) 20 521-bit ECP group (EC group modulo a 521-bit prime) 21 The IANA group type [IANA] of the three new groups is 2 (ECP - elliptic curve group over GF(P)). The previous eighteen groups all have group types 1 or 3. In summary, due to the performance advantages of elliptic curve groups in IKE implementations and the need for further alignment with other standards, this document defines three elliptic curve groups based on modular arithmetic. Solinas [Page 3]

INTERNET-DRAFT ECP Groups For IKE May 2005 2. Additional ECC Groups The notation adopted in RFC2409 [IKE] is used below to describe the new groups proposed. 2.1 Nineteenth Group IKE implementations SHOULD support an ECP group with the following characteristics. This group is assigned id 19 (nineteen). The curve is based on the integers modulo the generalized Mersenne prime p given by p = 2^(256)-2^(224)+2^(192)+2^(96)-1 . The equation for the elliptic curve is: y^2 = x^3 - 3 x + b. Field size: 256 Group Prime/Irreducible Polynomial: FFFFFFFF 00000001 00000000 00000000 00000000 FFFFFFFF FFFFFFFF FFFFFFFF Group Curve b: 5AC635D8 AA3A93E7 B3EBBD55 769886BC 651D06B0 CC53B0F6 3BCE3C3E 27D2604B Group Generator point P (x coordinate): 6B17D1F2 E12C4247 F8BCE6E5 63A440F2 77037D81 2DEB33A0 F4A13945 D898C296 Group Generator point P (y coordinate): 4FE342E2 FE1A7F9B 8EE7EB4A 7C0F9E16 2BCE3357 6B315ECE CBB64068 37BF51F5 Group order: FFFFFFFF 00000000 FFFFFFFF FFFFFFFF BCE6FAAD A7179E84 F3B9CAC2 FC632551 The group was chosen verifiably at random using SHA-1 as specified in [IEEE-1363] from the seed: C49D3608 86E70493 6A6678E1 139D26B7 819F7E90 The data in the KE payload when using this group represents the point on the curve obtained by taking the scalar multiple Ka*P, where Ka is the randomly chosen secret. Solinas [Page 4]

INTERNET-DRAFT ECP Groups For IKE May 2005 2.2 Twentieth Group IKE implementations SHOULD support an ECP group with the following characteristics. This group is assigned id 20 (twenty). The curve is based on the integers modulo the generalized Mersenne prime p given by p = 2^(384)-2^(128)-2^(96)+2^(32)-1 . The equation for the elliptic curve is: y^2 = x^3 - 3 x + b. Field size: 384 Group Prime/Irreducible Polynomial: FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF 00000000 00000000 FFFFFFFF Group Curve b: B3312FA7 E23EE7E4 988E056B E3F82D19 181D9C6E FE814112 0314088F 5013875A C656398D 8A2ED19D 2A85C8ED D3EC2AEF Group Generator point P (x coordinate): AA87CA22 BE8B0537 8EB1C71E F320AD74 6E1D3B62 8BA79B98 59F741E0 82542A38 5502F25D BF55296C 3A545E38 72760AB7 Group Generator point P (y coordinate): 3617DE4A 96262C6F 5D9E98BF 9292DC29 F8F41DBD 289A147C E9DA3113 B5F0B8C0 0A60B1CE 1D7E819D 7A431D7C 90EA0E5F Group order: FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF C7634D81 F4372DDF 581A0DB2 48B0A77A ECEC196A CCC52973 The group was chosen verifiably at random using SHA-1 as specified in [IEEE-1363] from the seed: A335926A A319A27A 1D00896A 6773A482 7ACDAC73 The data in the KE payload when using this group represents the point on the curve obtained by taking the scalar multiple Ka*P, where Ka is the randomly chosen secret. Solinas [Page 5]

INTERNET-DRAFT ECP Groups For IKE May 2005 2.3 Twenty-First Group IKE implementations SHOULD support an ECP group with the following characteristics. This group is assigned id 21 (twenty-one). The curve is based on the integers modulo the Mersenne prime p given by p = 2^(521)-1 . The equation for the elliptic curve is: y^2 = x^3 - 3 x + b. Field size: 521 Group Prime/Irreducible Polynomial: 000001FF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF Group Curve b: 00000051 953EB961 8E1C9A1F 929A21A0 B68540EE A2DA725B 99B315F3 B8B48991 8EF109E1 56193951 EC7E937B 1652C0BD 3BB1BF07 3573DF88 3D2C34F1 EF451FD4 6B503F00 Group Generator point P (x coordinate): 000000C6 858E06B7 0404E9CD 9E3ECB66 2395B442 9C648139 053FB521 F828AF60 6B4D3DBA A14B5E77 EFE75928 FE1DC127 A2FFA8DE 3348B3C1 856A429B F97E7E31 C2E5BD66 Group Generator point P (y coordinate): 00000118 39296A78 9A3BC004 5C8A5FB4 2C7D1BD9 98F54449 579B4468 17AFBD17 273E662C 97EE7299 5EF42640 C550B901 3FAD0761 353C7086 A272C240 88BE9476 9FD16650 Group order: 000001FF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFA 51868783 BF2F966B 7FCC0148 F709A5D0 3BB5C9B8 899C47AE BB6FB71E 91386409 The group was chosen verifiably at random using SHA-1 as specified in [IEEE-1363] from the seed: D09E8800 291CB853 96CC6717 393284AA A0DA64BA The data in the KE payload when using this group represents the point on the curve obtained by taking the scalar multiple Ka*P, where Ka is the randomly chosen secret. Solinas [Page 6]

INTERNET-DRAFT ECP Groups For IKE May 2005 3. Alignment with Other Standards The following table summarizes the appearance of these three elliptic curve groups in other standards. Standard Group 19 Group 20 Group 21 NIST [DSS] P-256 P-384 P-521 ISO/IEC [ISO-15946-1] P-256 ISO/IEC [ISO-18031] P-256 P-384 P-521 ANSI [X9.62-1998] Sect. J.5.3, Example 1 ANSI [X9.62-2003] Sect. J.6.5.3 Sect. J.6.6 Sect. J.6.7 ANSI [X9.63] Sect. J.5.4, Sect. J.5.5 Sect. J.5.6 Example 2 SECG [SEC2] secp256r1 secp384r1 secp521r1 See also [NIST], [ISO-14888-3], [ISO-15946-2], [ISO-15946-3], and [ISO-15946-4]. 4. Security Considerations Since this document proposes new groups for use within IKE, many of the security considerations contained within RFC 2409 apply here as well. The groups proposed in this document correspond to the symmetric key sizes 128 bits, 192 bits, and 256 bits. This allows the IKE key exchange to offer security comparable with the AES algorithms [AES]. 5. IANA Considerations Before this document can become an RFC, it is required that IANA update its registry of Diffie-Hellman groups for IKE in [IANA] to include the three groups defined above. Solinas [Page 7]

INTERNET-DRAFT ECP Groups For IKE May 2005 6. References 6.1 Normative [IKE] D. Harkins and D. Carrel, The Internet Key Exchange, RFC 2409, November 1998. 6.2 Informative [AES] U.S. Department of Commerce/National Institute of Standards and Technology, Advanced Encryption Standard (AES), FIPS PUB 197, November 2001. (http://csrc.nist.gov/publications/fips/index.html) [BBPS] S. Blake-Wilson, D. Brown, Y. Poeluev, M. Salter, Additional ECC Groups for IKE, draft-ietf-ipsec-ike-ecc-groups-04.txt, July 2002. [DSS] U.S. Department of Commerce/National Institute of Standards and Technology, Digital Signature Standard (DSS), FIPS PUB 186-2, January 2000. (http://csrc.nist.gov/publications/fips/index.html) [GMN] J. Solinas, Generalized Mersenne Numbers, Combinatorics and Optimization Research Report 99-39, 1999. (http://www.cacr.math.uwaterloo.ca/) [IANA] Internet Assigned Numbers Authority, Internet Key Exchange (IKE) Attributes. (http://www.iana.org/assignments/ipsec-registry) [IEEE-1363] Institute of Electrical and Electronics Engineers. IEEE 1363-2000, Standard for Public Key Cryptography. (http://grouper.ieee.org/groups/1363/index.html) [ISO-14888-3] International Organization for Standardization and International Electrotechnical Commission, ISO/IEC First Committee Draft 14888-3 (2nd ed.), Information Technology: Security Techniques: Digital Signatures with Appendix: Part 3 - Discrete Logarithm Based Mechanisms. [ISO-15946-1] International Organization for Standardization and International Electrotechnical Commission, ISO/IEC 15946-1: 2002-12-01, Information Technology: Security Techniques: Cryptographic Techniques based on Elliptic Curves: Part 1 - General. [ISO-15946-2] International Organization for Standardization and International Electrotechnical Commission, ISO/IEC 15946-2: 2002-12-01, Information Technology: Security Techniques: Cryptographic Techniques based on Elliptic Curves: Part 2 - Digital Signatures. Solinas [Page 8]

INTERNET-DRAFT ECP Groups For IKE May 2005 [ISO-15946-3] International Organization for Standardization and International Electrotechnical Commission, ISO/IEC 15946-3: 2002-12-01, Information Technology: Security Techniques: Cryptographic Techniques based on Elliptic Curves: Part 3 - Key Establishment. [ISO-15946-4] International Organization for Standardization and International Electrotechnical Commission, ISO/IEC 15946-4: 2004-10-01, Information Technology: Security Techniques: Cryptographic Techniques based on Elliptic Curves: Part 4 - Digital Signatures giving Message Recovery. [ISO-18031] International Organization for Standardization and International Electrotechnical Commission, ISO/IEC Final Committee Draft 18031, Information Technology: Security Techniques: Random Bit Generation, October 2004. [NIST] U.S. Department of Commerce/National Institute of Standards and Technology. Recommendation for Key Establishment Schemes Using Discrete Logarithm Cryptography, NIST Special Publication 800-56. (http://csrc.nist.gov/CryptoToolkit/KeyMgmt.html) [RFC-3526] T. Kivinen and M. Kojo, More Modular Exponential (MODP) Diffie-Hellman groups for Internet Key Exchange (IKE), RFC 3526, May 2003. [SEC2] Standards for Efficient Cryptography Group. SEC 2 - Recommended Elliptic Curve Domain Parameters, v. 1.0, 2000. (http://www.secg.org) [X9.62-1998] American National Standards Institute, X9.62-1998: Public Key Cryptography for the Financial Services Industry: The Elliptic Curve Digital Signature Algorithm. January 1999. [X9.62-2003] American National Standards Institute, X9.62-1998: Public Key Cryptography for the Financial Services Industry: The Elliptic Curve Digital Signature Algorithm, Revised-Draft-2003-02-26, February 2003. [X9.63] American National Standards Institute. X9.63-2001, Public Key Cryptography for the Financial Services Industry: Key Agreement and Key Transport using Elliptic Curve Cryptography. November 2001. Solinas [Page 9]

INTERNET-DRAFT ECP Groups For IKE May 2005 7. Author's Address Jerome A. Solinas National Security Agency jasolin@orion.ncsc.mil Comments are solicited and should be addressed to the author. Copyright (C) The Internet Society (2005). This document is subject to the rights, licenses and restrictions contained in BCP 78, and except as set forth therein, the authors retain all their rights. This document and the information contained herein are provided on an "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY AND THE INTERNET ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Expires November 27, 2005 Solinas [Page 10]