A group action on ℤpx and the generalized DLP with auxiliary inputs

Abstract

The Discrete Logarithm Problem with Auxiliary Inputs (DLPwAI) is an important cryptographic hard problem to compute α ∈ ℤp for given g, gα, ⋯, gαd where g is a generator of a group of order p. In this paper, we introduce a generalized version of this problem, so called the generalized DLPwAI (GDLPwAI) problem which is asked to compute α for given g, gαe1, ⋯, g αed, and propose an efficient algorithm when K := {e1, ⋯, ed} is a multiplicative subgroup of ℤ p-1x. Although the previous algorithms can only compute α when p ± 1 has a small divisor d, our algorithm resolves the problem when neither p + 1 or p - 1 has an appropriate small divisor. Our method exploits a group action of K on ℤpx to partition ℤpx efficiently. © 2014 Springer-Verlag.