Network Working Group Markus Friedl
INTERNET-DRAFT Niels Provos
Expires in six months William A. Simpson
May 2000
Diffie-Hellman Group Exchange for the SSH Transport Layer Protocol
draft-provos-secsh-dh-group-exchange-00.txt
Status of this Memo
This document is an Internet-Draft and is in full conformance with
all provisions of Section 10 of RFC2026, except that the right to
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Copyright Notice
Copyright (C) 2000 by Markus Friedl, Niels Provos and William A.
Simpson. All Rights Reserved.
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.
Overview and Rational
SSH is a de-facto standard for secure remote login on the Internet.
Currently, SSH performs the initial key exchange using the "diffie-
hellman-group1-sha1" method. This method prescribes a fixed group on
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which all operations are performed. 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 efficient alogorithms to compute
the Discrete Logarithm might pose a security threat to the SSH
protocol.
The ability to propose new moduli will reduce the possibility to use
precomputation for more efficient calculation of the DL. The server
can constantly compute new moduli in the background.
Diffie-Hellman Group and Key Exchange
The Diffie-Hellman key exchange provides a shared secret that can not
be determined by either party alone. The key exchange is combined
with a signature with the host key to provide host authentication.
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 server
SHOULD know at least one safe prime that has 1024 or more bits.
The generator g should be chosen such that the order of the generated
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 distributed
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 generator:
2 when p (mod 24) = 11.
5 when p (mod 10) = 3 or 7.
It is recommened to 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.
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 the subgroup the server replies with should
have; p is a large safe prime and g is a generator for a subgroup of
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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 the subgroup the server
replies with should have.
2. S finds a group that matches the clients request the closest
and sends p and 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.
This is implemented with the following messages. The hash algorithm
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)
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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
following:
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
authenticate the key exchange.
diffie-hellman-group-exchange-sha1
The "diffie-hellman-group-exchange-sha1" method specifies Diffie-
Hellman Group and Key Exchange with SHA-1 as HASH.
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.
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Security Considerations
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
primes.
The least significant bit of the private exponent can be recovered,
when the modulus is a safe prime [2]. However, this is not a
problem, 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].
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 sends only one group reducing
the complexity of the implementation and the amount of data that
needs to be exchanged between client and server.
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. Vanstone.
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.
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Appendix A: Generation of safe primes
The Handbook of Applied Cryptography [2] lists the following
algorithm 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
follows:
DH *d = NULL;
d = DH_generate_parameters(1024, DH_GENERATOR_2, NULL, NULL);
BN_print_fp(stdout, d->p);
The order of the subgroup generated by 2 is q = p - 1.
Author's Address
Niels Provos
CITI
519 W. William Street
Ann Arbor, MI, 48103
Phone: (734) 764-5207
EMail: provos@citi.umich.edu
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