(Short Paper) How to Solve DLOG Problem with Auxiliary Input

Abstract

Let G be a group of prime order p with a generator g. We first show that if p- 1 = d1… dt and g, gx, gx(p-1)/d1,…,gx(p-1)/(d1…dt-1) are given, then x can be computed in time O(d1+…+dt) exponentiations. Further suppose that p-1=d1e1…dtet, where d1, …, dt are relatively prime. We then show that x can be computed in time O(e1d1+…+etdt) exponentiations if g and gx(p-1)/di,…,gx(p-1)/diei-1 are given for i= 1, …, t.