The probability distribution of the Diffie-Hellman key

Abstract

The probability distribution of the key generated by the Diffe-Hellman Public Key-Distribution system is derived. For different prime factorizations of p – 1, where p is the prime modulus of the Diffe-Hellman system, the probabilities of the most and the least likely Diffe-Hellman key are found. A lower bound for the entropy of the Diffie-Hellman key is also derived. For the case p – 1 = 2q, with q prime, it is shown that the key distribution is very dose to the uniform distribution and the key entropy is virtually the maximum possible. A tight upper bound on the probability of the most likely key is also derived, from which the form of the prime factorization of p – 1 maximizing the probability of the most likely Diffie-Hellman key is found. The conditions for generating equally likely Diffie-Hellman keys for any prime factorization of p – 1 is given.