On reducing factorization to the discrete logarithm problem modulo a composite

Abstract

The discrete logarithm problem modulo a composite—abbreviate it as DLPC—is the following: given a (possibly) composite integer n ≥ 1 and elements ${a, b \in \mathbb{Z}_n^*}$, determine an ${x \in \mathbb{N}}$ satisfying a ^x = b if one exists. The question whether integer factoring can be reduced in deterministic polynomial time to the DLPC remains open. In this paper we consider the problem ${{\rm DLPC}_\varepsilon}$ obtained by adding in the DLPC the constraint ${x\le (1-\varepsilon)n}$, where ${\varepsilon}$ is an arbitrary fixed number, ${0 < \varepsilon\le\frac{1}{2}}$. We prove that factoring n reduces in deterministic subexponential time to the ${{\rm DLPC}_\varepsilon}$ with ${O_\varepsilon((\ln n)^2)}$ queries for moduli less or equal to n.