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Note: It should be difficult for Alice to solve for Bob's private key or for Bob to solve for Alice's private key.

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Another demonstration of Diffie—Hellman also using numbers too small for practical use is given here. This is usually done long before the rest of the protocol; g is assumed to be known by all attackers.

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The group G is written multiplicatively. Alice computes gb a. Bob computes ga b.

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Both Alice and Bob are now in possession of the group element gab, which can serve as the shared secret key. The group G satisfies the requisite condition for secure communication if there is not an efficient algorithm for determining gab given g, ga, and gb.

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For example, the elliptic curve Diffie—Hellman protocol is a variant that uses elliptic curves instead of the multiplicative group of integers modulo p. Variants using hyperelliptic curves have also been proposed.

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The supersingular isogeny key exchange is a Diffie—Hellman variant that has been designed to be secure against quantum computers.

Operation with more than two parties[ edit ] Diffie—Hellman key agreement is not limited to negotiating a key shared by only two participants.

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Any number of users can take part in an agreement by performing iterations of the agreement protocol and exchanging intermediate data which does not itself need to be kept secret. For example, Alice, Bob, and Carol could participate in a Diffie—Hellman agreement as follows, with all operations taken to be modulo p: The parties agree on the algorithm parameters p and g. The parties generate their private keys, named a, b, and c.

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Alice computes ga and sends it to Bob. Bob computes gb and sends it to Carol. Carol computes gc and sends it to Alice.

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An eavesdropper has been able to primele 5 schimburi criptografice ga, gb, gc, gab, gac, and gbc, but cannot use any combination of these to efficiently reproduce gabc. To extend this mechanism to larger groups, two basic principles must be followed: Starting with an "empty" key consisting only of g, the secret is made by raising the current value to every participant's private exponent once, in any order primele 5 schimburi criptografice first such exponentiation yields the participant's own public key.

Any intermediate value having up to N-1 exponents applied, where N is the number of participants in the group may be revealed publicly, but the primele 5 schimburi criptografice value having had all N exponents applied constitutes the shared secret and hence must never be revealed publicly. Thus, each user must obtain their copy of the secret by applying their own private key last otherwise there would be no way for the last contributor to communicate the final key to its recipient, as that last contributor would have turned the key into the very secret the group wished to protect.

These principles leave open various options for choosing in which order participants contribute to keys. The simplest and most obvious solution is to arrange the N participants in a circle and have N keys rotate around the circle, until eventually primele 5 schimburi criptografice key has been contributed to by all N participants ending with its owner and each participant has contributed to N keys ending with their own.

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However, this requires that every participant perform N modular exponentiations. In return, participants A, B, C, and D receive gefgh.

Participants A and B each perform one exponentiation, yielding gefghab, which they send to C and D, while C and D do the same, yielding gefghcd, which they send to A and B.

Participant A performs an exponentiation, yielding gefghcda, which it sends to B; similarly, B sends gefghcdb to A. C and D do similarly. Participants E through H simultaneously perform the same operations using gabcd as their starting point.

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Once this operation has been completed all participants will possess the secret gabcdefgh, but each participant will have performed only four modular exponentiations, rather than the eight implied by a simple circular arrangement. Security[ edit ] The protocol is considered secure against eavesdroppers if G and g are chosen properly. In particular, the order of the group G must primele 5 schimburi criptografice large, particularly if the same group is used for large amounts of traffic.

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The eavesdropper has to solve the Diffie—Hellman problem to obtain gab. This is currently considered difficult for groups whose order is large enough.

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. Fields of small characteristic may be less secure.

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A protocol using such a choice is for example IKEv2. Because of the random self-reducibility of the discrete logarithm problem a small g is equally secure as any other generator of the same group. If Alice and Bob use random number generators whose outputs are not completely random and can be predicted to some extent, then primele 5 schimburi criptografice is much easier to eavesdrop.

In the original description, the Diffie—Hellman exchange by itself does not provide authentication of the communicating parties and is thus vulnerable to a man-in-the-middle attack. Mallory an active attacker executing the man-in-the-middle attack may establish two distinct key exchanges, one with Alice and the other with Bob, effectively masquerading as Alice to Bob, and vice versa, allowing her to decrypt, then re-encrypt, the messages passed between them.

Note that Mallory must continue to be in the middle, actively decrypting and re-encrypting messages every time Alice and Bob communicate. If she is ever absent, her previous presence is then revealed to Alice and Bob. They will know that all of their private conversations had been intercepted and decoded by someone in the channel.

In most cases it will not help them get Mallory's private key, even if she used the same key for both exchanges. A method to authenticate the communicating parties to each other is generally needed to prevent this type of attack. Variants of Diffie—Hellman, such as STS protocolmay be used instead to avoid these types of attacks.

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Practical attacks on Internet traffic[ edit ] The number field sieve algorithm, which is generally the most effective in solving the discrete logarithm problemconsists of four computational steps. The first three steps only depend on the order of the group G, not on the specific number whose finite log is desired. The Logjam attack used this vulnerability to compromise a variety of Internet services that allowed the use of groups whose order was a bit prime number, so called export grade.

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The authors needed several thousand CPU cores for a week to precompute data for a single bit prime. Once that was done, individual logarithms could be solved in about a minute using two core Intel Xeon CPUs.

The Logjam authors speculate that precomputation against widely reused bit DH primes is behind claims in leaked NSA documents that NSA is able to break much of current cryptography.

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Failing that, they recommend that the order, p, of the Diffie—Hellman group should be at least  bits. They estimate that the pre-computation required for a bit prime is times more difficult than for bit primes.

The first such scheme is the ElGamal encryption.

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A more modern variant is the Integrated Encryption Scheme. Forward secrecy[ edit ] Protocols that achieve forward secrecy generate new key pairs for each session and discard them at the end of the session. The Diffie—Hellman key exchange is a frequent choice for such protocols, because of its fast key generation.

Password-authenticated key agreement[ edit ] When Alice and Bob share a password, they may use a password-authenticated key agreement PK form of Diffie—Hellman to prevent man-in-the-middle attacks.

One simple scheme is to compare the hash of s concatenated with the password calculated independently on both ends of channel.

Diffie–Hellman key exchange - Wikipedia

A feature of these schemes is that an attacker can only test one specific password on each iteration with the other party, and so the system provides good security with relatively weak passwords.

Public key[ edit ] It is also possible to use Diffie—Hellman as part of a public key infrastructureallowing Bob to encrypt a message so that only Alice will be able to decrypt it, with no prior communication between them other than Bob having trusted knowledge of Alice's public key.

Alice's public key is.