Exponentiation Speed Up For Diffie-Hellman Key Agreement Protocol

The Diffie-Hellman key agreement protocol is a well known method in which two parties agree on a secret key by mean of public communication. From the computational point of view, the protocol requires, from both paries, to execute two exponentiations in some group, where the secret exponent is chosen, independently, by each user. In environments with limited resources, such as smartcards, these computations are considered heavy. In this paper we propose a method for decreasing the compu- tational work involved with this exponentiation. If the required entropy of the secret exponent is K, the straightforward approach is to select this key as a string of K random bits. Instead, we choose here longer keys with imposed limitations on their Hamming weight, in a way that the entropy remains K. We show how this can reduce the exponentiation time by » 4% to » 9%, depending on the group. The relative performance gain is shown to be independent of K. Finally, we show how our method can be combined with more sophisticated exponentiation algorithms, yielding smaller, but sometimes still significant extra- gain in performance.