On the complexity of the discrete logarithm and Diffie-Hellman problems
Journal of Complexity - Special issue on coding and cryptography
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Let g be a primitive root modulo a (n+1)-bit prime p. In this paper we prove the uniformity of distribution of the Diffie--Hellman triples (gx, gy, gxy) as the exponents x and y run through the set of n-bit integers with precisely k nonzero bits in their bit representation provided that $k \ge 0.35 n$. Such "sparse" exponents are of interest because for these the computation of gx, gy, gxy is faster than for arbitrary x and y. In the latter case, that is, for arbitrary exponents, similar (albeit stronger) uniformity of distribution results have recently been obtained by R. Canetti, M. Larsen, D. Lieman, S. Konyagin [ Israel J. Math, 120 (2000), pp. 23--46], and the authors.