Multiple discrete logarithm problems with auxiliary inputs

Abstract

Let g be an element of prime order p in an abelian group and let (Formula presented.) for a positive integer L. First, we show that, if (Formula presented.) are given for d | p − 1, all the discrete logarithms αi’s can be computed probabilistically in (Formula presented.) group exponentiations with O(L) storage under the condition that L ≪ min (Formula presented.) Let (Formula presented.) be a polynomial of degree d and let ρf be the number of rational points over Fp on the curve determined by f(x) − f(y) = 0. Second, if (Formula presented.) are given for any d ≥ 1, then we propose an algorithm that solves all αi’s in (Formula presented.) group exponentiations with (Formula presented.) storage. In particular, we have explicit choices for a polynomial f when d | p ± 1, that yield a running time of (Formula presented.) whenever (Formula presented.) for some constant c.