Network Working Group
Request for Comments: 3526
Category: Standards Track

T. Kivinen
M. Kojo
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

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.

   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

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.

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:

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:

The generator is: 2.

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:

The generator is: 2.

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:

The generator is: 2.

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 }

Its hexadecimal value is:

The generator is: 2.

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}
   
   [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:

          kivinen@ssh.fi

Mika Kojo
HELSINKI
Finland

   EMail: mika.kojo@helsinki.fi

13. Full Copyright Statement

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 included on all such copies and derivative works. However, this document itself may not be modified in any way, such as by removing the copyright notice or references to the Internet Society or other Internet organizations, except as needed for the purpose of developing Internet standards in which case the procedures for copyrights defined in the Internet Standards process must be followed, or as required to translate it into languages other than English.

The limited permissions granted above are perpetual and will not be revoked by the Internet Society or its successors or assigns.

This document and the information contained herein is provided on an "AS IS" basis and THE INTERNET SOCIETY AND THE INTERNET ENGINEERING TASK FORCE DISCLAIMS 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.

Acknowledgement

Funding for the RFC Editor function is currently provided by the Internet Society.

More Modular Exponential (MODP) Diffie-Hellman groups

for Internet Key Exchange (IKE)

Status of this Memo

Copyright Notice

Abstract

Table of Contents

1. Introduction

2. 1536-bit MODP Group

3. 2048-bit MODP Group

4. 3072-bit MODP Group

5. 4096-bit MODP Group

6. 6144-bit MODP Group

7. 8192-bit MODP Group

8. Security Considerations

9. IANA Considerations

10. Normative References

11. Non-Normative References

12. Authors' Addresses

13. Full Copyright Statement

Acknowledgement