Network Working Group                                         T. Kivinen
Request for Comments: 3526                                       M. Kojo
Category: Standards Track                    SSH Communications Security
                                                               May 2003


        More Modular Exponential (MODP) Diffie-Hellman groups
                   for Internet Key Exchange (IKE)

Status of this Memo

  This document specifies an Internet standards track protocol for the
  Internet community, and requests discussion and suggestions for
  improvements.  Please refer to the current edition of the "Internet
  Official Protocol Standards" (STD 1) for the standardization state
  and status of this protocol.  Distribution of this memo is unlimited.

Copyright Notice

  Copyright (C) The Internet Society (2003).  All Rights Reserved.

Abstract

  This document defines new Modular Exponential (MODP) Groups for the
  Internet Key Exchange (IKE) protocol.  It documents the well known
  and used 1536 bit group 5, and also defines new 2048, 3072, 4096,
  6144, and 8192 bit Diffie-Hellman groups numbered starting at 14.
  The selection of the primes for theses groups follows the criteria
  established by Richard Schroeppel.

Table of Contents

  1.   Introduction. . . . . . . . . . . . . . . . . . . . . . .  2
  2.   1536-bit MODP Group . . . . . . . . . . . . . . . . . . .  3
  3.   2048-bit MODP Group . . . . . . . . . . . . . . . . . . .  3
  4.   3072-bit MODP Group . . . . . . . . . . . . . . . . . . .  4
  5.   4096-bit MODP Group . . . . . . . . . . . . . . . . . . .  5
  6.   6144-bit MODP Group . . . . . . . . . . . . . . . . . . .  6
  7.   8192-bit MODP Group . . . . . . . . . . . . . . . . . . .  6
  8.   Security Considerations . . . . . . . . . . . . . . . . .  8
  9.   IANA Considerations . . . . . . . . . . . . . . . . . . .  8
  10.  Normative References. . . . . . . . . . . . . . . . . . .  8
  11.  Non-Normative References. . . . . . . . . . . . . . . . .  8
  12.  Authors' Addresses  . . . . . . . . . . . . . . . . . . .  9
  13.  Full Copyright Statement. . . . . . . . . . . . . . . . . 10






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1.  Introduction

  One of the important protocol parameters negotiated by Internet Key
  Exchange (IKE) [RFC-2409] is the Diffie-Hellman "group" that will be
  used for certain cryptographic operations.  IKE currently defines 4
  groups.  These groups are approximately as strong as a symmetric key
  of 70-80 bits.

  The new Advanced Encryption Standard (AES) cipher [AES], which has
  more strength, needs stronger groups.  For the 128-bit AES we need
  about a 3200-bit group [Orman01].  The 192 and 256-bit keys would
  need groups that are about 8000 and 15400 bits respectively.  Another
  source [RSA13] [Rousseau00] estimates that the security equivalent
  key size for the 192-bit symmetric cipher is 2500 bits instead of
  8000 bits, and the equivalent key size 256-bit symmetric cipher is
  4200 bits instead of 15400 bits.

  Because of this disagreement, we just specify different groups
  without specifying which group should be used with 128, 192 or 256-
  bit AES.  With current hardware groups bigger than 8192-bits being
  too slow for practical use, this document does not provide any groups
  bigger than 8192-bits.

  The exponent size used in the Diffie-Hellman must be selected so that
  it matches other parts of the system.  It should not be the weakest
  link in the security system.  It should have double the entropy of
  the strength of the entire system, i.e., if you use a group whose
  strength is 128 bits, you must use more than 256 bits of randomness
  in the exponent used in the Diffie-Hellman calculation.






















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2.  1536-bit MODP Group

  The 1536 bit MODP group has been used for the implementations for
  quite a long time, but was not defined in RFC 2409 (IKE).
  Implementations have been using group 5 to designate this group, we
  standardize that practice here.

  The prime is: 2^1536 - 2^1472 - 1 + 2^64 * { [2^1406 pi] + 741804 }

  Its hexadecimal value is:

     FFFFFFFF FFFFFFFF C90FDAA2 2168C234 C4C6628B 80DC1CD1
     29024E08 8A67CC74 020BBEA6 3B139B22 514A0879 8E3404DD
     EF9519B3 CD3A431B 302B0A6D F25F1437 4FE1356D 6D51C245
     E485B576 625E7EC6 F44C42E9 A637ED6B 0BFF5CB6 F406B7ED
     EE386BFB 5A899FA5 AE9F2411 7C4B1FE6 49286651 ECE45B3D
     C2007CB8 A163BF05 98DA4836 1C55D39A 69163FA8 FD24CF5F
     83655D23 DCA3AD96 1C62F356 208552BB 9ED52907 7096966D
     670C354E 4ABC9804 F1746C08 CA237327 FFFFFFFF FFFFFFFF

  The generator is: 2.

3.  2048-bit MODP Group

  This group is assigned id 14.

  This prime is: 2^2048 - 2^1984 - 1 + 2^64 * { [2^1918 pi] + 124476 }

  Its hexadecimal value is:

     FFFFFFFF FFFFFFFF C90FDAA2 2168C234 C4C6628B 80DC1CD1
     29024E08 8A67CC74 020BBEA6 3B139B22 514A0879 8E3404DD
     EF9519B3 CD3A431B 302B0A6D F25F1437 4FE1356D 6D51C245
     E485B576 625E7EC6 F44C42E9 A637ED6B 0BFF5CB6 F406B7ED
     EE386BFB 5A899FA5 AE9F2411 7C4B1FE6 49286651 ECE45B3D
     C2007CB8 A163BF05 98DA4836 1C55D39A 69163FA8 FD24CF5F
     83655D23 DCA3AD96 1C62F356 208552BB 9ED52907 7096966D
     670C354E 4ABC9804 F1746C08 CA18217C 32905E46 2E36CE3B
     E39E772C 180E8603 9B2783A2 EC07A28F B5C55DF0 6F4C52C9
     DE2BCBF6 95581718 3995497C EA956AE5 15D22618 98FA0510
     15728E5A 8AACAA68 FFFFFFFF FFFFFFFF

  The generator is: 2.








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4.  3072-bit MODP Group

  This group is assigned id 15.

  This prime is: 2^3072 - 2^3008 - 1 + 2^64 * { [2^2942 pi] + 1690314 }

  Its hexadecimal value is:

     FFFFFFFF FFFFFFFF C90FDAA2 2168C234 C4C6628B 80DC1CD1
     29024E08 8A67CC74 020BBEA6 3B139B22 514A0879 8E3404DD
     EF9519B3 CD3A431B 302B0A6D F25F1437 4FE1356D 6D51C245
     E485B576 625E7EC6 F44C42E9 A637ED6B 0BFF5CB6 F406B7ED
     EE386BFB 5A899FA5 AE9F2411 7C4B1FE6 49286651 ECE45B3D
     C2007CB8 A163BF05 98DA4836 1C55D39A 69163FA8 FD24CF5F
     83655D23 DCA3AD96 1C62F356 208552BB 9ED52907 7096966D
     670C354E 4ABC9804 F1746C08 CA18217C 32905E46 2E36CE3B
     E39E772C 180E8603 9B2783A2 EC07A28F B5C55DF0 6F4C52C9
     DE2BCBF6 95581718 3995497C EA956AE5 15D22618 98FA0510
     15728E5A 8AAAC42D AD33170D 04507A33 A85521AB DF1CBA64
     ECFB8504 58DBEF0A 8AEA7157 5D060C7D B3970F85 A6E1E4C7
     ABF5AE8C DB0933D7 1E8C94E0 4A25619D CEE3D226 1AD2EE6B
     F12FFA06 D98A0864 D8760273 3EC86A64 521F2B18 177B200C
     BBE11757 7A615D6C 770988C0 BAD946E2 08E24FA0 74E5AB31
     43DB5BFC E0FD108E 4B82D120 A93AD2CA FFFFFFFF FFFFFFFF

  The generator is: 2.

























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5.  4096-bit MODP Group

  This group is assigned id 16.

  This prime is: 2^4096 - 2^4032 - 1 + 2^64 * { [2^3966 pi] + 240904 }

  Its hexadecimal value is:

     FFFFFFFF FFFFFFFF C90FDAA2 2168C234 C4C6628B 80DC1CD1
     29024E08 8A67CC74 020BBEA6 3B139B22 514A0879 8E3404DD
     EF9519B3 CD3A431B 302B0A6D F25F1437 4FE1356D 6D51C245
     E485B576 625E7EC6 F44C42E9 A637ED6B 0BFF5CB6 F406B7ED
     EE386BFB 5A899FA5 AE9F2411 7C4B1FE6 49286651 ECE45B3D
     C2007CB8 A163BF05 98DA4836 1C55D39A 69163FA8 FD24CF5F
     83655D23 DCA3AD96 1C62F356 208552BB 9ED52907 7096966D
     670C354E 4ABC9804 F1746C08 CA18217C 32905E46 2E36CE3B
     E39E772C 180E8603 9B2783A2 EC07A28F B5C55DF0 6F4C52C9
     DE2BCBF6 95581718 3995497C EA956AE5 15D22618 98FA0510
     15728E5A 8AAAC42D AD33170D 04507A33 A85521AB DF1CBA64
     ECFB8504 58DBEF0A 8AEA7157 5D060C7D B3970F85 A6E1E4C7
     ABF5AE8C DB0933D7 1E8C94E0 4A25619D CEE3D226 1AD2EE6B
     F12FFA06 D98A0864 D8760273 3EC86A64 521F2B18 177B200C
     BBE11757 7A615D6C 770988C0 BAD946E2 08E24FA0 74E5AB31
     43DB5BFC E0FD108E 4B82D120 A9210801 1A723C12 A787E6D7
     88719A10 BDBA5B26 99C32718 6AF4E23C 1A946834 B6150BDA
     2583E9CA 2AD44CE8 DBBBC2DB 04DE8EF9 2E8EFC14 1FBECAA6
     287C5947 4E6BC05D 99B2964F A090C3A2 233BA186 515BE7ED
     1F612970 CEE2D7AF B81BDD76 2170481C D0069127 D5B05AA9
     93B4EA98 8D8FDDC1 86FFB7DC 90A6C08F 4DF435C9 34063199
     FFFFFFFF FFFFFFFF

  The generator is: 2.



















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6.  6144-bit MODP Group

  This group is assigned id 17.

  This prime is: 2^6144 - 2^6080 - 1 + 2^64 * { [2^6014 pi] + 929484 }

  Its hexadecimal value is:

  FFFFFFFF FFFFFFFF C90FDAA2 2168C234 C4C6628B 80DC1CD1 29024E08
  8A67CC74 020BBEA6 3B139B22 514A0879 8E3404DD EF9519B3 CD3A431B
  302B0A6D F25F1437 4FE1356D 6D51C245 E485B576 625E7EC6 F44C42E9
  A637ED6B 0BFF5CB6 F406B7ED EE386BFB 5A899FA5 AE9F2411 7C4B1FE6
  49286651 ECE45B3D C2007CB8 A163BF05 98DA4836 1C55D39A 69163FA8
  FD24CF5F 83655D23 DCA3AD96 1C62F356 208552BB 9ED52907 7096966D
  670C354E 4ABC9804 F1746C08 CA18217C 32905E46 2E36CE3B E39E772C
  180E8603 9B2783A2 EC07A28F B5C55DF0 6F4C52C9 DE2BCBF6 95581718
  3995497C EA956AE5 15D22618 98FA0510 15728E5A 8AAAC42D AD33170D
  04507A33 A85521AB DF1CBA64 ECFB8504 58DBEF0A 8AEA7157 5D060C7D
  B3970F85 A6E1E4C7 ABF5AE8C DB0933D7 1E8C94E0 4A25619D CEE3D226
  1AD2EE6B F12FFA06 D98A0864 D8760273 3EC86A64 521F2B18 177B200C
  BBE11757 7A615D6C 770988C0 BAD946E2 08E24FA0 74E5AB31 43DB5BFC
  E0FD108E 4B82D120 A9210801 1A723C12 A787E6D7 88719A10 BDBA5B26
  99C32718 6AF4E23C 1A946834 B6150BDA 2583E9CA 2AD44CE8 DBBBC2DB
  04DE8EF9 2E8EFC14 1FBECAA6 287C5947 4E6BC05D 99B2964F A090C3A2
  233BA186 515BE7ED 1F612970 CEE2D7AF B81BDD76 2170481C D0069127
  D5B05AA9 93B4EA98 8D8FDDC1 86FFB7DC 90A6C08F 4DF435C9 34028492
  36C3FAB4 D27C7026 C1D4DCB2 602646DE C9751E76 3DBA37BD F8FF9406
  AD9E530E E5DB382F 413001AE B06A53ED 9027D831 179727B0 865A8918
  DA3EDBEB CF9B14ED 44CE6CBA CED4BB1B DB7F1447 E6CC254B 33205151
  2BD7AF42 6FB8F401 378CD2BF 5983CA01 C64B92EC F032EA15 D1721D03
  F482D7CE 6E74FEF6 D55E702F 46980C82 B5A84031 900B1C9E 59E7C97F
  BEC7E8F3 23A97A7E 36CC88BE 0F1D45B7 FF585AC5 4BD407B2 2B4154AA
  CC8F6D7E BF48E1D8 14CC5ED2 0F8037E0 A79715EE F29BE328 06A1D58B
  B7C5DA76 F550AA3D 8A1FBFF0 EB19CCB1 A313D55C DA56C9EC 2EF29632
  387FE8D7 6E3C0468 043E8F66 3F4860EE 12BF2D5B 0B7474D6 E694F91E
  6DCC4024 FFFFFFFF FFFFFFFF

  The generator is: 2.

7.  8192-bit MODP Group

  This group is assigned id 18.

  This prime is: 2^8192 - 2^8128 - 1 + 2^64 * { [2^8062 pi] + 4743158 }







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  Its hexadecimal value is:

     FFFFFFFF FFFFFFFF C90FDAA2 2168C234 C4C6628B 80DC1CD1
     29024E08 8A67CC74 020BBEA6 3B139B22 514A0879 8E3404DD
     EF9519B3 CD3A431B 302B0A6D F25F1437 4FE1356D 6D51C245
     E485B576 625E7EC6 F44C42E9 A637ED6B 0BFF5CB6 F406B7ED
     EE386BFB 5A899FA5 AE9F2411 7C4B1FE6 49286651 ECE45B3D
     C2007CB8 A163BF05 98DA4836 1C55D39A 69163FA8 FD24CF5F
     83655D23 DCA3AD96 1C62F356 208552BB 9ED52907 7096966D
     670C354E 4ABC9804 F1746C08 CA18217C 32905E46 2E36CE3B
     E39E772C 180E8603 9B2783A2 EC07A28F B5C55DF0 6F4C52C9
     DE2BCBF6 95581718 3995497C EA956AE5 15D22618 98FA0510
     15728E5A 8AAAC42D AD33170D 04507A33 A85521AB DF1CBA64
     ECFB8504 58DBEF0A 8AEA7157 5D060C7D B3970F85 A6E1E4C7
     ABF5AE8C DB0933D7 1E8C94E0 4A25619D CEE3D226 1AD2EE6B
     F12FFA06 D98A0864 D8760273 3EC86A64 521F2B18 177B200C
     BBE11757 7A615D6C 770988C0 BAD946E2 08E24FA0 74E5AB31
     43DB5BFC E0FD108E 4B82D120 A9210801 1A723C12 A787E6D7
     88719A10 BDBA5B26 99C32718 6AF4E23C 1A946834 B6150BDA
     2583E9CA 2AD44CE8 DBBBC2DB 04DE8EF9 2E8EFC14 1FBECAA6
     287C5947 4E6BC05D 99B2964F A090C3A2 233BA186 515BE7ED
     1F612970 CEE2D7AF B81BDD76 2170481C D0069127 D5B05AA9
     93B4EA98 8D8FDDC1 86FFB7DC 90A6C08F 4DF435C9 34028492
     36C3FAB4 D27C7026 C1D4DCB2 602646DE C9751E76 3DBA37BD
     F8FF9406 AD9E530E E5DB382F 413001AE B06A53ED 9027D831
     179727B0 865A8918 DA3EDBEB CF9B14ED 44CE6CBA CED4BB1B
     DB7F1447 E6CC254B 33205151 2BD7AF42 6FB8F401 378CD2BF
     5983CA01 C64B92EC F032EA15 D1721D03 F482D7CE 6E74FEF6
     D55E702F 46980C82 B5A84031 900B1C9E 59E7C97F BEC7E8F3
     23A97A7E 36CC88BE 0F1D45B7 FF585AC5 4BD407B2 2B4154AA
     CC8F6D7E BF48E1D8 14CC5ED2 0F8037E0 A79715EE F29BE328
     06A1D58B B7C5DA76 F550AA3D 8A1FBFF0 EB19CCB1 A313D55C
     DA56C9EC 2EF29632 387FE8D7 6E3C0468 043E8F66 3F4860EE
     12BF2D5B 0B7474D6 E694F91E 6DBE1159 74A3926F 12FEE5E4
     38777CB6 A932DF8C D8BEC4D0 73B931BA 3BC832B6 8D9DD300
     741FA7BF 8AFC47ED 2576F693 6BA42466 3AAB639C 5AE4F568
     3423B474 2BF1C978 238F16CB E39D652D E3FDB8BE FC848AD9
     22222E04 A4037C07 13EB57A8 1A23F0C7 3473FC64 6CEA306B
     4BCBC886 2F8385DD FA9D4B7F A2C087E8 79683303 ED5BDD3A
     062B3CF5 B3A278A6 6D2A13F8 3F44F82D DF310EE0 74AB6A36
     4597E899 A0255DC1 64F31CC5 0846851D F9AB4819 5DED7EA1
     B1D510BD 7EE74D73 FAF36BC3 1ECFA268 359046F4 EB879F92
     4009438B 481C6CD7 889A002E D5EE382B C9190DA6 FC026E47
     9558E447 5677E9AA 9E3050E2 765694DF C81F56E8 80B96E71
     60C980DD 98EDD3DF FFFFFFFF FFFFFFFF

  The generator is: 2.




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8.  Security Considerations

  This document describes new stronger groups to be used in IKE.  The
  strengths of the groups defined here are always estimates and there
  are as many methods to estimate them as there are cryptographers.
  For the strength estimates below we took the both ends of the scale
  so the actual strength estimate is likely between the two numbers
  given here.

  +--------+----------+---------------------+---------------------+
  | Group  | Modulus  | Strength Estimate 1 | Strength Estimate 2 |
  |        |          +----------+----------+----------+----------+
  |        |          |          | exponent |          | exponent |
  |        |          | in bits  | size     | in bits  | size     |
  +--------+----------+----------+----------+----------+----------+
  |   5    | 1536-bit |       90 |     180- |      120 |     240- |
  |  14    | 2048-bit |      110 |     220- |      160 |     320- |
  |  15    | 3072-bit |      130 |     260- |      210 |     420- |
  |  16    | 4096-bit |      150 |     300- |      240 |     480- |
  |  17    | 6144-bit |      170 |     340- |      270 |     540- |
  |  18    | 8192-bit |      190 |     380- |      310 |     620- |
  +--------+----------+---------------------+---------------------+

9.  IANA Considerations

  IKE [RFC-2409] defines 4 Diffie-Hellman Groups, numbered 1 through 4.

  This document defines a new group 5, and new groups from 14 to 18.
  Requests for additional assignment are via "IETF Consensus" as
  defined in RFC 2434 [RFC-2434].  Specifically, new groups are
  expected to be documented in a Standards Track RFC.

10.  Normative References

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

  [RFC-2434]   Narten, T. and H. Alvestrand, "Guidelines for Writing an
               IANA Considerations Section in RFCs", BCP 26, RFC 2434,
               October 1998.

11.  Non-Normative References

  [AES]        NIST, FIPS PUB 197, "Advanced Encryption Standard
               (AES)," November 2001.
               http://csrc.nist.gov/publications/fips/fips197/fips-
               197.{ps,pdf}




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  [RFC-2412]   Orman, H., "The OAKLEY Key Determination Protocol", RFC
               2412, November 1998.

  [Orman01]    Orman, H. and P. Hoffman, "Determining Strengths For
               Public Keys Used For Exchanging Symmetric Keys", Work in
               progress.

  [RSA13]      Silverman, R. "RSA Bulleting #13: A Cost-Based Security
               Analysis of Symmetric and Asymmetric Key Lengths", April
               2000, http://www.rsasecurity.com/rsalabs/bulletins/
               bulletin13.html

  [Rousseau00] Rousseau, F. "New Time and Space Based Key Size
               Equivalents for RSA and Diffie-Hellman", December 2000,
               http://www.sandelman.ottawa.on.ca/ipsec/2000/12/
               msg00045.html

12.  Authors' Addresses

  Tero Kivinen
  SSH Communications Security Corp
  Fredrikinkatu 42
  FIN-00100 HELSINKI
  Finland

  EMail: [email protected]


  Mika Kojo
  HELSINKI
  Finland

  EMail: [email protected]


















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13.  Full Copyright Statement

  Copyright (C) The Internet Society (2003).  All Rights Reserved.

  This document and translations of it may be copied and furnished to
  others, and derivative works that comment on or otherwise explain it
  or assist in its implementation may be prepared, copied, published
  and distributed, in whole or in part, without restriction of any
  kind, provided that the above copyright notice and this paragraph are
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Kivinen & Kojo              Standards Track                    [Page 10]


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