Network Working Group                                      Markus Friedl
INTERNET-DRAFT                                              Niels Provos
Expires in six months                                 William A. Simpson
                                                            January 2001

   Diffie-Hellman Group Exchange for the SSH Transport Layer Protocol

1.  Status of this Memo

     This document is an Internet-Draft and is in full conformance with
     all provisions of Section 10 of RFC2026.

     Internet-Drafts are working documents of the Internet Engineering
     Task Force (IETF), its areas, and its working groups.  Note that
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     Internet-Drafts are draft documents valid for a maximum of six
     months and may be updated, replaced, or obsoleted by other docu-
     ments at any time.  It is inappropriate to use Internet- Drafts as
     reference material or to cite them other than as "work in

     The list of current Internet-Drafts can be accessed at

     The list of Internet-Draft Shadow Directories can be accessed at

2.  Copyright Notice

     Copyright (C) 2000 by Markus Friedl, Niels Provos and William A.

3.  Abstract

     This memo describes a new key exchange method for the SSH protocol.
     It allows the SSH server to propose to the client new groups on
     which to perform the Diffie-Hellman key exchange.  The proposed
     groups need not be fixed and can change with time.

4.  Overview and Rational

     SSH [4,5,6,7] is a a very common protocol for secure remote login
     on the Internet.  Currently, SSH performs the initial key exchange

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     using the "diffie-hellman-group1-sha1" method.  This method pre-
     scribes a fixed group on which all operations are performed.

     The Diffie-Hellman key exchange provides a shared secret that can
     not be determined by either party alone.  In SSH, the key exchange
     is signed with the host key to provide host authentication.

     The security of the Diffie-Hellman key exchange is based on the
     difficulty of solving the Discrete Logarithm Problem (DLP).  Since
     we expect that the SSH protocol will be in use for many years in
     the future, we fear that extensive precomputation and more effi-
     cient algorithms to compute the discrete logarithm over a fixed
     group might pose a security threat to the SSH protocol.

     The ability to propose new groups will reduce the incentive to use
     precomputation for more efficient calculation of the discrete loga-
     rithm.  The server can constantly compute new groups in the back-

5.  Diffie-Hellman Group and Key Exchange

     The server keeps a list of safe primes and corresponding generators
     that it can select from.  A prime p is safe, if p = 2q + 1, and q
     is prime.  New primes can be generated in the background.

     The generator g should be chosen such that the order of the gener-
     ated subgroup does not factor into small primes, i.e., with p = 2q
     + 1, the order has to be either q or p - 1.  If the order is p - 1,
     then the exponents generate all possible public-values, evenly dis-
     tributed throughout the range of the modulus p, without cycling
     through a smaller subset. Such a generator is called a "primitive
     root" (which is trivial to find when p is "safe").

     Implementation Notes:

          One useful technique is to select the generator, and then
          limit the modulus selection sieve to primes with that genera-

            2   when p (mod 24) = 11.
            5   when p (mod 10) = 3 or 7.

          It is recommended to use 2 as generator, because it improves
          efficiency in multiplication performance.  It is usable even
          when it is not a primitive root, as it still covers half of
          the space of possible residues.

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     Servers and clients SHOULD support groups with a modulus length of
     k bits, where 1024 <= k <= 8192.

     The client requests a minimum modulus size from the server.  In the
     following description (C is the client, S is the server; n is the
     minimal number of bits in the subgroup; p is a large safe prime and
     g is a generator for a subgroup of GF(p); V_S is S's version
     string; V_C is C's version string; K_S is S's public host key; I_C
     is C's KEXINIT message and I_S S's KEXINIT message which have been
     exchanged before this part begins):

     1.   C sends "n", the minimal number of bits in the subgroup that
          the server should reply with.

     2.   S finds a group that best matches the client's request, and
          sends "p || g" to C.

     3.   C generates a random number x (1 < x < (p-1)/2). It computes e
          = g^x mod p, and sends "e" to S.

     4.   S generates a random number y (0 < y < (p-1)/2) and computes f
          = g^y mod p. S receives "e".  It computes K = e^y mod p, H =
          hash(V_C || V_S || I_C || I_S || K_S || n || p || g || e || f
          || K) (these elements are encoded according to their types;
          see below), and signature s on H with its private host key.  S
          sends "K_S || f || s" to C.  The signing operation may involve
          a second hashing operation.

     5.   C verifies that K_S really is the host key for S (e.g. using
          certificates or a local database).  C is also allowed to
          accept the key without verification; however, doing so will
          render the protocol insecure against active attacks (but may
          be desirable for practical reasons in the short term in many
          environments).  C then computes K = f^x mod p, H = hash(V_C ||
          V_S || I_C || I_S || K_S || n || p || g || e || f || K), and
          verifies the signature s on H.

          Either side MUST NOT send or accept e or f values that are not
          in the range [1, p-1]. If this condition is violated, the key
          exchange fails.  To prevent confinement attacks, they MUST
          accept the shared secret K only , if 1 < K < p - 1.

     The server should return the smallest group it knows about that is
     larger than the size the client requested.  If the server does not
     know a group that is larger than the client request, then it has to
     return the largest group it knows.  In all cases, the size of the
     returned group SHOULD be at least 1024 bits.

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     This is implemented with the following messages.  The hash algo-
     rithm for computing the exchange hash is defined by the method
     name, and is called HASH.  The public key algorithm for signing is
     negotiated with the KEXINIT messages.

     First, the client sends:
       byte      SSH_MSG_KEY_DH_GEX_REQUEST
       uint32    n, number of bits the subgroup should have at least

     The server responds with
       byte      SSH_MSG_KEX_DH_GEX_GROUP
       mpint     p, safe prime
       mpint     g, generator for subgroup in GF(p)

     The client responds with:
       byte      SSH_MSG_KEX_DH_GEX_INIT
       mpint     e

     The server responds with:
       byte      SSH_MSG_KEX_DH_GEX_REPLY
       string    server public host key and certificates (K_S)
       mpint     f
       string    signature of H

     The hash H is computed as the HASH hash of the concatenation of the
       string    V_C, the client's version string (CR and NL excluded)
       string    V_S, the server's version string (CR and NL excluded)
       string    I_C, the payload of the client's SSH_MSG_KEXINIT
       string    I_S, the payload of the server's SSH_MSG_KEXINIT
       string    K_S, the host key
       uint32    n, number of bits the client requested
       mpint     p, safe prime
       mpint     g, generator for subgroup
       mpint     e, exchange value sent by the client
       mpint     f, exchange value sent by the server
       mpint     K, the shared secret

     This value is called the exchange hash, and it is used to authenti-
     cate the key exchange.

6.  diffie-hellman-group-exchange-sha1

     The "diffie-hellman-group-exchange-sha1" method specifies Diffie-
     Hellman Group and Key Exchange with SHA-1 as HASH.

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7.  Summary of Message numbers

     The following message numbers have been defined in this document.

       #define SSH_MSG_KEX_DH_GEX_REQUEST      30
       #define SSH_MSG_KEX_DH_GEX_GROUP        31
       #define SSH_MSG_KEX_DH_GEX_INIT         32
       #define SSH_MSG_KEX_DH_GEX_REPLY        33

     The numbers 30-49 are key exchange specific and may be redefined by
     other kex methods.

8.  Security Considerations

     This protocol aims to be simple and uses only well understood prim-
     itives.  This encourages acceptance by the community and allows for
     ease of implementation, which hopefully leads to a more secure sys-

     The use of multiple moduli inhibits a determined attacker from pre-
     calculating moduli exchange values, and discourages dedication of
     resources for analysis of any particular modulus.

     It is important to only employ safe primes as moduli.  Oorshot and
     Wiener note that using short private exponents with a random prime
     modulus p makes the computation of the discrete logarithm easy [1].
     However, they also state that this problem does not apply to safe

     The least significant bit of the private exponent can be recovered,
     when the modulus is a safe prime [2].  However, this is not a prob-
     lem, if the size of the private exponent is big enough.  Related to
     this, Waldvogel and Massey note: When private exponents are chosen
     independently and uniformly at random from {0,...,p-2}, the key
     entropy is less than 2 bits away from the maximum, lg(p-1) [3].

9.  Acknowledgments

     The document is derived in part from "SSH Transport Layer Protocol"
     by T. Ylonen, T. Kivinen, M. Saarinen, T. Rinne and S. Lehtinen.

     Markku-Juhani Saarinen pointed out that the least significant bit
     of the private exponent can be recovered efficiently when using
     safe primes and a subgroup with an order divisible by two.

     Bodo Moeller suggested that the server send only one group, reduc-
     ing the complexity of the implementation and the amount of data
     that needs to be exchanged between client and server.

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10.  Bibliography

     [1]  P. C. van Oorschot and M. J. Wiener, On Diffie-Hellman key
          agreement with short exponents, In Advances in Cryptology -
          EUROCRYPT'96, LNCS 1070, Springer-Verlag, 1996, pp.332-343.

     [2]  Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Van-
          stone.  Handbook of Applied Cryptography. CRC Press, 1996.

     [3]  C. P. Waldvogel and J. L. Massey, The probability distribution
          of the Diffie-Hellman key, in Proceedings of AUSCRYPT 92, LNCS
          718, Springer- Verlag, 1993, pp. 492-504.

     [4]  Ylonen, T., et al: "SSH Protocol Architecture", Internet-
          Draft, draft-secsh-architecture-07.txt

     [5]  Ylonen, T., et al: "SSH Transport Layer Protocol", Internet-
          Draft, draft-ietf-secsh-transport-09.txt

     [6]  Ylonen, T., et al: "SSH Authentication Protocol", Internet-
          Draft, draft-ietf-secsh-userauth-09.txt

     [7]  Ylonen, T., et al: "SSH Connection Protocol", Internet-Draft,

11.  Appendix A:  Generation of safe primes

     The Handbook of Applied Cryptography [2] lists the following algo-
     rithm to generate a k-bit safe prime p.  It has been modified so
     that 2 is a generator for the multiplicative group mod p.

      1. Do the following:
        1.1 Select a random (k-1)-bit prime q, so that q mod 12 = 5.
        1.2 Compute p := 2q + 1, and test whether p is prime, (using, e.g.
            trial division and the Rabin-Miller test.)
        Repeat until p is prime.

   If an implementation uses the OpenSSL libraries, a group consisting
   of a 1024-bit safe prime and 2 as generator can be created as fol-

      DH *d = NULL;
      d = DH_generate_parameters(1024, DH_GENERATOR_2, NULL, NULL);
      BN_print_fp(stdout, d->p);

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      The order of the subgroup generated by 2 is q = p - 1.

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12.  Author's Address

     Markus Friedl
     Ganghoferstr. 7
     80339 Munich


     Niels Provos
     Center for Information Technology Integration
     519 W. William Street
     Ann Arbor, MI, 48103

     Phone: (734) 764-5207

     William Allen Simpson
     Computer Systems Consulting Services
     1384 Fontaine
     Madion Heights, Michigan 48071


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