Biffy crypto
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The item type is invalid. Please select an image. Alice computes ga mod p and sends it to Bob. Bob computes gb mod p and sends it to Carol. Carol computes gc mod p and sends it to Alice. An eavesdropper has been able to see ga mod p, gb mod p, gc mod p, gab mod p, gac mod p, and gbc mod p, but cannot use any combination of these to efficiently reproduce gabc mod p.
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 the 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 final 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 every 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. 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.
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 be large, particularly if the same group is used for large amounts of traffic.
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. 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 it 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.
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