RFC 4753

RFC 4753: ECP Groups for IKE and IKEv2 🎀

okay bestie, so RFC 4753 is titled "ECP Groups for IKE and IKEv2" 🦋 It was written by D. Fu and J. Solinas from the NSA, it's filed as Informational, and it's dated January 2007. The header doesn't say it obsoletes or updates anything, so it stands on its own 💅

Status of This Memo ✨

This memo provides information for the Internet community. It does not specify an Internet standard of any kind. Distribution of this memo is unlimited.

Copyright Notice 🌸

Copyright (C) The IETF Trust (2007).

Abstract 💜

okay so this doc adds new Elliptic Curve Cryptography (ECC) groups for Internet Key Exchange (IKE) and IKEv2, on top of the ones that already exist 🔥 The new curves are based on modular arithmetic instead of binary arithmetic, which is the big twist. The goal is to line IKE and IKEv2 up with other ECC implementations and standards, especially the NIST ones 🫶 Bonus: these curves can be implemented more efficiently than the older ECC groups, so it's a whole glow-up ✨

Table of Contents

   1. Introduction ....................................................2
   2. Requirements Terminology ........................................3
   3. Additional ECC Groups ...........................................3
      3.1. 256-bit Random ECP Group ...................................3
      3.2. 384-bit Random ECP Group ...................................4
      3.3. 521-bit Random ECP Group ...................................5
   4. Security Considerations .........................................6
   5. Alignment with Other Standards ..................................6
   6. IANA Considerations .............................................6
   7. ECP Key Exchange Data Formats ...................................7
   8. Test Vectors ....................................................7
      8.1. 256-bit Random ECP Group ...................................8
      8.2. 384-bit Random ECP Group ...................................9
      8.3. 521-bit Random ECP Group ..................................10
   9. References .....................................................12

1. Introduction 🧚

okay so this doc is all about default Diffie-Hellman groups for IKE and IKEv2 🌈 They sit alongside the Oakley groups from [IKE] and the extra groups that got added since [IANA-IKE]. You're expected to already know IKE and the whole Oakley groups idea, which is laid out 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 everyone to support, and the other four are optional. Thirteen more groups were defined later and got values from IANA, and all of those are optional too. So of the eighteen groups defined so far, eight are MODP groups (exponentiation groups modulo a prime) and ten are EC2N groups (elliptic curve groups over GF[2^N]) 💖 If you want the MODP details, check out [RFC3526].

so the point of this doc is to give implementers more elliptic curve options by adding three ECP groups (elliptic curve groups modulo a prime) 🎀 Why? Let me list the reasons:

• 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, the Standards for Efficient Cryptography Group (SECG), ISO, and ANSI. (See Section 5 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 putting them in this RFC is way better for interoperability between IKE implementations that use elliptic curve groups 🫶 Having standardized groups also means people can optimize for a specific curve and field size and precompute things, which can make implementations faster 🔥

in summary, elliptic curve groups are great for performance in IKE implementations, and the rest of the standards world needs them to line up 💅 So this document defines three elliptic curve groups based on modular arithmetic 👑

2. Requirements Terminology 💅

The keywords "MUST" and "SHOULD" that appear in this document are to be interpreted as described in [RFC2119]. Those are the official requirement words, so pay attention to them 👀

3. Additional ECC Groups 🦋

the notation from RFC 2409 [IKE] is what we use below to describe the new groups 🌸

3.1. 256-bit Random ECP Group ✨

IKE and IKEv2 implementations SHOULD support an ECP group with the following characteristics. 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 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 generator for this group is given by g=(gx,gy) where:

gx:
 6B17D1F2 E12C4247 F8BCE6E5 63A440F2 77037D81 2DEB33A0 F4A13945 D898C296

gy:
 4FE342E2 FE1A7F9B 8EE7EB4A 7C0F9E16 2BCE3357 6B315ECE CBB64068 37BF51F5

3.2. 384-bit Random ECP Group 🌈

IKE and IKEv2 implementations SHOULD support an ECP group with the following characteristics. 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 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 generator for this group is given by g=(gx,gy) where:

gx:
 AA87CA22 BE8B0537 8EB1C71E F320AD74 6E1D3B62 8BA79B98 59F741E0 82542A38
 5502F25D BF55296C 3A545E38 72760AB7

gy:
 3617DE4A 96262C6F 5D9E98BF 9292DC29 F8F41DBD 289A147C E9DA3113 B5F0B8C0
 0A60B1CE 1D7E819D 7A431D7C 90EA0E5F

3.3. 521-bit Random ECP Group 💎

IKE and IKEv2 implementations SHOULD support an ECP group with the following characteristics. 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:
 01FFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF
 FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF
 FFFF

Group Curve b:
 0051953E B9618E1C 9A1F929A 21A0B685 40EEA2DA 725B99B3 15F3B8B4 89918EF1
 09E15619 3951EC7E 937B1652 C0BD3BB1 BF073573 DF883D2C 34F1EF45 1FD46B50
 3F00

Group Order:
 01FFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF
 FFFA5186 8783BF2F 966B7FCC 0148F709 A5D03BB5 C9B8899C 47AEBB6F B71E9138
 6409

The group was chosen verifiably at random using SHA-1 as specified in [IEEE-1363] from the seed:

D09E8800 291CB853 96CC6717 393284AA A0DA64BA

The generator for this group is given by g=(gx,gy) where:

gx:
 00C6858E 06B70404 E9CD9E3E CB662395 B4429C64 8139053F B521F828 AF606B4D
 3DBAA14B 5E77EFE7 5928FE1D C127A2FF A8DE3348 B3C1856A 429BF97E 7E31C2E5
 BD66

gy:
 01183929 6A789A3B C0045C8A 5FB42C7D 1BD998F5 4449579B 446817AF BD17273E
 662C97EE 72995EF4 2640C550 B9013FAD 0761353C 7086A272 C24088BE 94769FD1
 6650

8. Test Vectors ✨

okay bestie, this is the part where we get to see the actual numbers 🧮💖 these are example IKEv2 key exchange payloads for each of the three groups defined in this document, so you can check your work against them.

quick notation check 🦋 we write g^n for the scalar multiple of the point g by the integer n. it's just another point on the curve. in the wider literature people usually write it as ng, but this doc sticks with g^n so it matches the notation in [IKE] and [IKEv2]. no cap, it's the same idea, just dressed differently 💅

8.1. 256-bit Random ECP Group

IANA assigned the ID value 19 to this Diffie-Hellman group 🌸

let's say the initiator's Diffie-Hellman private key is:

i:
 C88F01F5 10D9AC3F 70A292DA A2316DE5 44E9AAB8 AFE84049 C62A9C57 862D1433

then the public key is g^i=(gix,giy), where:

gix:
 DAD0B653 94221CF9 B051E1FE CA5787D0 98DFE637 FC90B9EF 945D0C37 72581180

giy:
 5271A046 1CDB8252 D61F1C45 6FA3E59A B1F45B33 ACCF5F58 389E0577 B8990BB3

the KEi payload looks like this 🎀

 00000048 00130000 DAD0B653 94221CF9 B051E1FE CA5787D0 98DFE637 FC90B9EF
 945D0C37 72581180 5271A046 1CDB8252 D61F1C45 6FA3E59A B1F45B33 ACCF5F58
 389E0577 B8990BB3

now the responder's turn. let's say their Diffie-Hellman private key is:

r:
 C6EF9C5D 78AE012A 011164AC B397CE20 88685D8F 06BF9BE0 B283AB46 476BEE53

then the public key is g^r=(grx,gry), where:

grx:
 D12DFB52 89C8D4F8 1208B702 70398C34 2296970A 0BCCB74C 736FC755 4494BF63

gry:
 56FBF3CA 366CC23E 8157854C 13C58D6A AC23F046 ADA30F83 53E74F33 039872AB

and the KEr payload is 🫶

 00000048 00130000 D12DFB52 89C8D4F8 1208B702 70398C34 2296970A 0BCCB74C
 736FC755 4494BF63 56FBF3CA 366CC23E 8157854C 13C58D6A AC23F046 ADA30F83
 53E74F33 039872AB

the shared secret value g^ir=(girx,giry) is:

girx:
 D6840F6B 42F6EDAF D13116E0 E1256520 2FEF8E9E CE7DCE03 812464D0 4B9442DE

giry:
 522BDE0A F0D8585B 8DEF9C18 3B5AE38F 50235206 A8674ECB 5D98EDB2 0EB153A2

concatenated, that gives:

g^ir:
 D6840F6B 42F6EDAF D13116E0 E1256520 2FEF8E9E CE7DCE03 812464D0 4B9442DE
 522BDE0A F0D8585B 8DEF9C18 3B5AE38F 50235206 A8674ECB 5D98EDB2 0EB153A2

and this is the value that gets used to form SKEYSEED 🔥

8.2. 384-bit Random ECP Group

IANA assigned the ID value 20 to this Diffie-Hellman group 🦋

the initiator's Diffie-Hellman private key:

i:
 099F3C70 34D4A2C6 99884D73 A375A67F 7624EF7C 6B3C0F16 0647B674 14DCE655
 E35B5380 41E649EE 3FAEF896 783AB194

then the public key is g^i=(gix,giy), where:

gix:
 667842D7 D180AC2C DE6F74F3 7551F557 55C7645C 20EF73E3 1634FE72 B4C55EE6
 DE3AC808 ACB4BDB4 C88732AE E95F41AA

giy:
 9482ED1F C0EEB9CA FC498462 5CCFC23F 65032149 E0E144AD A0241815 35A0F38E
 EB9FCFF3 C2C947DA E69B4C63 4573A81C

the KEi payload:

 00000068 00140000 667842D7 D180AC2C DE6F74F3 7551F557 55C7645C 20EF73E3
 1634FE72 B4C55EE6 DE3AC808 ACB4BDB4 C88732AE E95F41AA 9482ED1F C0EEB9CA
 FC498462 5CCFC23F 65032149 E0E144AD A0241815 35A0F38E EB9FCFF3 C2C947DA
 E69B4C63 4573A81C

the responder's private key:

r:
 41CB0779 B4BDB85D 47846725 FBEC3C94 30FAB46C C8DC5060 855CC9BD A0AA2942
 E0308312 916B8ED2 960E4BD5 5A7448FC

then the public key is g^r=(grx,gry), where:

grx:
 E558DBEF 53EECDE3 D3FCCFC1 AEA08A89 A987475D 12FD950D 83CFA417 32BC509D
 0D1AC43A 0336DEF9 6FDA41D0 774A3571

gry:
 DCFBEC7A ACF31964 72169E83 8430367F 66EEBE3C 6E70C416 DD5F0C68 759DD1FF
 F83FA401 42209DFF 5EAAD96D B9E6386C

and the KEr payload 💅

 00000068 00140000 E558DBEF 53EECDE3 D3FCCFC1 AEA08A89 A987475D 12FD950D
 83CFA417 32BC509D 0D1AC43A 0336DEF9 6FDA41D0 774A3571 DCFBEC7A ACF31964
 72169E83 8430367F 66EEBE3C 6E70C416 DD5F0C68 759DD1FF F83FA401 42209DFF
 5EAAD96D B9E6386C

the shared secret value g^ir=(girx,giry):

girx:
 11187331 C279962D 93D60424 3FD592CB 9D0A926F 422E4718 7521287E 7156C5C4
 D6031355 69B9E9D0 9CF5D4A2 70F59746

giry:
 A2A9F38E F5CAFBE2 347CF7EC 24BDD5E6 24BC93BF A82771F4 0D1B65D0 6256A852
 C983135D 4669F879 2F2C1D55 718AFBB4

concatenated:

g^ir:
 11187331 C279962D 93D60424 3FD592CB 9D0A926F 422E4718 7521287E 7156C5C4
 D6031355 69B9E9D0 9CF5D4A2 70F59746 A2A9F38E F5CAFBE2 347CF7EC 24BDD5E6
 24BC93BF A82771F4 0D1B65D0 6256A852 C983135D 4669F879 2F2C1D55 718AFBB4

this is the value used to form SKEYSEED 👑

8.3. 521-bit Random ECP Group

IANA assigned the ID value 21 to this Diffie-Hellman group 🌈 the biggest group, so the numbers are longer, but the vibe is the same ✨

the initiator's Diffie-Hellman private key:

i:
 0037ADE9 319A89F4 DABDB3EF 411AACCC A5123C61 ACAB57B5 393DCE47 608172A0
 95AA85A3 0FE1C295 2C6771D9 37BA9777 F5957B26 39BAB072 462F68C2 7A57382D
 4A52

then the public key is g^i=(gix,giy), where:

gix:
 0015417E 84DBF28C 0AD3C278 713349DC 7DF153C8 97A1891B D98BAB43 57C9ECBE
 E1E3BF42 E00B8E38 0AEAE57C 2D107564 94188594 2AF5A7F4 601723C4 195D176C
 ED3E

giy:
 017CAE20 B6641D2E EB695786 D8C94614 6239D099 E18E1D5A 514C739D 7CB4A10A
 D8A78801 5AC405D7 799DC75E 7B7D5B6C F2261A6A 7F150743 8BF01BEB 6CA3926F
 9582

the KEi payload:

 0000008C 00150000 0015417E 84DBF28C 0AD3C278 713349DC 7DF153C8 97A1891B
 D98BAB43 57C9ECBE E1E3BF42 E00B8E38 0AEAE57C 2D107564 94188594 2AF5A7F4
 601723C4 195D176C ED3E017C AE20B664 1D2EEB69 5786D8C9 46146239 D099E18E
 1D5A514C 739D7CB4 A10AD8A7 88015AC4 05D7799D C75E7B7D 5B6CF226 1A6A7F15
 07438BF0 1BEB6CA3 926F9582

the responder's private key:

r:
 0145BA99 A847AF43 793FDD0E 872E7CDF A16BE30F DC780F97 BCCC3F07 8380201E
 9C677D60 0B343757 A3BDBF2A 3163E4C2 F869CCA7 458AA4A4 EFFC311F 5CB15168
 5EB9

then the public key is g^r=(grx,gry), where:

grx:
 00D0B397 5AC4B799 F5BEA16D 5E13E9AF 971D5E9B 984C9F39 728B5E57 39735A21
 9B97C356 436ADC6E 95BB0352 F6BE64A6 C2912D4E F2D0433C ED2B6171 640012D9
 460F

gry:
 015C6822 6383956E 3BD066E7 97B623C2 7CE0EAC2 F551A10C 2C724D98 52077B87
 220B6536 C5C408A1 D2AEBB8E 86D678AE 49CB5709 1F473229 6579AB44 FCD17F0F
 C56A

and the KEr payload:

 0000008c 00150000 00D0B397 5AC4B799 F5BEA16D 5E13E9AF 971D5E9B 984C9F39
 728B5E57 39735A21 9B97C356 436ADC6E 95BB0352 F6BE64A6 C2912D4E F2D0433C
 ED2B6171 640012D9 460F015C 68226383 956E3BD0 66E797B6 23C27CE0 EAC2F551
 A10C2C72 4D985207 7B87220B 6536C5C4 08A1D2AE BB8E86D6 78AE49CB 57091F47
 32296579 AB44FCD1 7F0FC56A

the shared secret value g^ir=(girx,giry):

girx:
 01144C7D 79AE6956 BC8EDB8E 7C787C45 21CB086F A64407F9 7894E5E6 B2D79B04
 D1427E73 CA4BAA24 0A347868 59810C06 B3C715A3 A8CC3151 F2BEE417 996D19F3
 DDEA

giry:
 01B901E6 B17DB294 7AC017D8 53EF1C16 74E5CFE5 9CDA18D0 78E05D1B 5242ADAA
 9FFC3C63 EA05EDB1 E13CE5B3 A8E50C3E B622E8DA 1B38E0BD D1F88569 D6C99BAF
 FA43

concatenated:

g^ir:
 01144C7D 79AE6956 BC8EDB8E 7C787C45 21CB086F A64407F9 7894E5E6 B2D79B04
 D1427E73 CA4BAA24 0A347868 59810C06 B3C715A3 A8CC3151 F2BEE417 996D19F3
 DDEA01B9 01E6B17D B2947AC0 17D853EF 1C1674E5 CFE59CDA 18D078E0 5D1B5242
 ADAA9FFC 3C63EA05 EDB1E13C E5B3A8E5 0C3EB622 E8DA1B38 E0BDD1F8 8569D6C9
 9BAFFA43

and this is the value used to form SKEYSEED 🔥

9. References

9.1. Normative References

[IANA-IKE] Internet Assigned Numbers Authority, Internet Key Exchange (IKE) Attributes. (http://www.iana.org/assignments/ipsec-registry)

[IANA-IKEv2] IKEv2 Parameters. (http://www.iana.org/assignments/ikev2-parameters)

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

[IKEv2] Kaufman, C., "Internet Key Exchange (IKEv2) Protocol", RFC 4306, December 2005.

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

9.2. Informative References 📚✨

okay so these are the extra-credit reads, bestie 🤓 you don't need them to implement the spec, but this is where all the background lives 🧚

• [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)
• [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/)
• [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 14888-3:2006, 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.
• [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 18031:2005, Information Technology: Security Techniques: Random Bit Generation.
• [NIST] U.S. Department of Commerce/National Institute of Standards and Technology. Recommendation for Pair-Wise Key Establishment Schemes Using Discrete Logarithm Cryptography, NIST Special Publication Publication 800-56A, March 2006. (http://csrc.nist.gov/CryptoToolkit/KeyMgmt.html)
• [RFC3526] Kivinen, T. 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.
• [X9.62-2005] American National Standards Institute, X9.62:2005: Public Key Cryptography for the Financial Services Industry: The Elliptic Curve Digital Signature Algorithm (ECDSA).
• [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.

Authors' Addresses 💌

David E. Fu 🌸 National Information Assurance Research Laboratory National Security Agency

EMail: defu@orion.ncsc.mil

Jerome A. Solinas 🎀 National Information Assurance Research Laboratory National Security Agency

EMail: jasolin@orion.ncsc.mil

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