Diffie-Hellman key exchange

Diffie-Hellman (D-H) key exchange is a cryptographic protocol which allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure communications channel. This key can then be used to encrypt subsequent communications using a symmetric key cipher.

Synonyms of Diffie-Hellman key exchange include:

Diffie-Hellman key agreement
Diffie-Hellman key establishment
Diffie-Hellman key negotiation
exponential key exchange

The scheme was first published publicly by Whitfield Diffie and Martin Hellman in 1976, although it later emerged that it had been discovered a few years earlier within GCHQ, the British signals intelligence agency, by Malcolm J. Williamson but was kept classified. In 2002, Hellman suggested the algorithm be called Diffie-Hellman-Merkle key exchange in recognition of Ralph Merkle's contribution to the invention of public-key cryptography (Hellman, 2002).

Although Diffie-Hellman key agreement itself is an anonymous (non-authenticated) key agreement protocol, it provides the basis for a variety of authenticated protocols, and is used to provide perfect forward secrecy in TLS's ephemeral modes.

History of the protocol

Diffie-Hellman key agreement was invented in 1976 during a collaboration between Whitfield Diffie and Martin Hellman and was the first practical method for establishing a shared secret over an unprotected communications channel. Ralph Merkle's work on public key distribution was an influence. John Gill suggested application of the discrete logarithm problem. It had been discovered by Malcolm Williamson of GCHQ in the UK some years previously, but GCHQ chose not make it public until 1997, by which time it had no influence on research in academia.

The method was followed shortly afterwards by RSA, another implementation of public key cryptography using asymmetric algorithms.

In 2002, Martin Hellman wrote:

"The system...has since become known as Diffie-Hellman key exchange. While that system was first described in a paper by Diffie and me, it is a public key distribution system, a concept developed by Merkle, and hence should be called 'Diffie-Hellman-Merkle key exchange' if names are to be associated with it. I hope this small pulpit might help in that endeavor to recognize Merkle's equal contribution to the invention of public key cryptography." ^*

, now expired, describes the algorithm and credits Hellman, Diffie, and Merkle as inventors.

Description

The simplest, and original, implementation of the protocol uses the multiplicative group of integers modulo p, where p is prime and g is primitive mod p. Modulo (or mod) means that the integers between 0 and p − 1 are used with normal addition, subtraction, multiplication, and exponentiation, except that after each operation the result keeps only the remainder after dividing by p. Here is an example of the protocol:

Alice
Sec		Calc
	p, g
a
		g^a mod p
	…
	(g^b mod p)^a mod p

$\rightarrow$

$\leftarrow$

Bob
Calc		Sec
	p, g
		b
	…
g^b mod p
	(g^a mod p)^b mod p

Alice and Bob agree to use a prime number p=23 and base g=5.
Alice chooses a secret integer a=6, then sends Bob (g^a mod p)
- 5⁶ mod 23 = 8.
Bob chooses a secret integer b=15, then sends Alice (g^b mod p)
- 5¹⁵ mod 23 = 19.
Alice computes (g^b mod p)^a mod p
- 19⁶ mod 23 = 2.
Bob computes (g^a mod p)^b mod p
- 8¹⁵ mod 23 = 2.

Both Alice and Bob have arrived at the same value, because g^ab and g^ba are equal. Note that only a, b and g^ab = g^ba are kept secret. All the other values are sent in the clear. Once Alice and Bob compute the shared secret they can use it as an encryption key, known only to them, for sending messages across the same open communications channel. Of course, much larger values of a,b, and p would be needed to make this example secure, since it is easy to try all the possible values of g^ab mod 23 (there will be, at most, 22 such values, even if a and b are large). If p was a prime of more than 300 digits, and a and b were at least 100 digits long, then even the best known algorithms for finding a given only g, p, and g^a mod p (known as the discrete logarithm problem) would take longer than the lifetime of the universe to run. g need not be large at all, and in practice is usually either 2 or 5.

Here's a more general description of the protocol:

Alice and Bob agree on a finite cyclic group G and a generating element g in G. (This is usually done long before the rest of the protocol; g is assumed to be known by all attackers.) We will write the group G multiplicatively.
Alice picks a random natural number a and sends g^a to Bob.
Bob picks a random natural number b and sends g^b to Alice.
Alice computes (g^b)^a.
Bob computes (g^a)^b.

Both Alice and Bob are now in possession of the group element g^ab which can serve as the shared secret key. The values of (g^b)^a and (g^a)^b are the same because groups are power associative. (See also exponentiation.)

Chart

Here is a chart to help simplify who knows what. (Eve is an eavesdropper - she watches what is sent between Alice and Bob, but she does not alter the contents of their communications.)

Let s = shared secret key. s = 2

let a = Alice's private key. a = 6

let b = Bob's private key. b = 15

let g = public base. g=5

let p = public (prime) number. p = 23

Alice
knows	doesn't know
p = 23	b = 15
base g = 5
a = 6
5⁶ mod 23 = 8
5^b mod 23 = 19
19⁶ mod 23 = 2
8^b mod 23 = 2
19⁶ mod 23 = 8^b mod 23
s = 2

Bob
knows	doesn't know
p = 23	a = 6
base g = 5
b = 15
5¹⁵ mod 23 = 19
5^a mod 23 = 8
8¹⁵ mod 23 = 2
19^a mod 23 = 2
8¹⁵ mod 23 = 19^a mod 23
s = 2

Eve
knows	doesn't know
p = 23	a = 6
base g = 5	b = 15
	s = 2
5^a mod 23 = 8
5^b mod 23 = 19
19^a mod 23 = s
8^b mod 23 = s
19^a mod 23 = 8^b mod 23

Note: It should be difficult for Alice to solve for Bob's private key or for Bob to solve for Alice's private key. If it isn't difficult for Alice to solve for Bob's private key (or vice versa), Eve may simply substitute her own private / public key pair, plug Bob's public key into her private key, produce a fake shared secret key, and solve for Bob's private key (and use that to solve for the shared secret key. Eve may attempt to choose a public / private key pair that will make it easy for her to solve for Bob's private key).

Security

The protocol is considered secure against eavesdroppers if G and g are chosen properly. The eavesdropper ("Eve") must solve the Diffie-Hellman problem to obtain g^ab. This is currently considered difficult. An efficient algorithm to solve the discrete logarithm problem would make it easy to compute a or b and solve the Diffie-Hellman problem, making this and many other public key cryptosystems insecure.

The order of G should be prime or have a large prime factor to prevent use of the Pohlig-Hellman algorithm to obtain a or b. For this reason, a Sophie Germain prime q is sometimes used to calculate p=2q+1, called a safe prime, since the order of G is then only divisible by 2 and q. g is then sometimes chosen to generate the order q subgroup of G, rather than G, so that the Legendre symbol of g^a never reveals the low order bit of a.

If Alice and Bob use random number generators whose outputs are not completely random but can be predicted to some extent, then Eve's task is much easier.

The secret integers a and b are discarded at the end of the session. Therefore, Diffie-Hellman key exchange by itself trivially achieves perfect forward secrecy because no long-term private keying material exists to be disclosed.

Authentication

In the original description, the Diffie-Hellman exchange by itself does not provide authentication of the parties, and is thus vulnerable to man in the middle attack. The man-in-the-middle may establish two distinct Diffie-Hellman keys, one with Alice and the other with Bob, and then try to masquerade as Alice to Bob and/or vice-versa, perhaps by decrypting and re-encrypting messages passed between them. Some method to authenticate these parties to each other is generally needed.

A variety of cryptographic authentication solutions incorporate a Diffie-Hellman exchange. When Alice and Bob have a public key infrastructure they may digitally sign the agreed key, or g^a and g^b, as in MQV, STS and the IKE component of the IPsec protocol suite for securing Internet Protocol communications. When Alice and Bob share a password, they may use a password-authenticated key agreement form of Diffie-Hellman.

References

Non-Secret Encryption Using a Finite Field MJ Williamson, January 21, 1974.
Thoughts on Cheaper Non-Secret Encryption MJ Williamson, August 10, 1976.
New Directions in Cryptography W. Diffie and M. E. Hellman, IEEE Transactions on Information Theory, vol. IT-22, Nov. 1976, pp: 644-654.
Cryptographic apparatus and method Martin E. Hellman, Bailey W. Diffie, and Ralph C. Merkle, U.S. Patent #4,200,770, 29 April 1980
The History of Non-Secret Encryption JH Ellis 1987 (28K PDF file) (HTML version)
The First Ten Years of Public-Key Cryptography Whitfield Diffie, Proceedings of the IEEE, vol. 76, no. 5, May 1988, pp: 560-577 (1.9MB PDF file)
Menezes, Alfred; van Oorschot, Paul; Vanstone, Scott (1997). Handbook of Applied Cryptography Boca Raton, Florida: CRC Press. ISBN 0-8493-8523-7. (Available online)
Singh, Simon (1999) The Code Book: the evolution of secrecy from Mary Queen of Scots to quantum cryptography New York: Doubleday ISBN 0-385-49531-5
An Overview of Public Key Cryptography Martin E. Hellman, IEEE Communications Magazine, May 2002, pp:42-49. (123kB PDF file)

External links

RFC 2631, Diffie-Hellman Key Agreement Method E. Rescorla June 1999. Full text of RFC 2631
Summary of ANSI X9.42: Agreement of Symmetric Keys Using Discrete Logarithm Cryptography (64K PDF file) (Description of ANSI 9 Standards)
Diffie-Hellman Key Exchange - A Non-Mathematician's Explanation — by Keith Palmgren
Diffie-Hellman explained visually

Cryptographic protocols | Asymmetric-key cryptosystems

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