On the Semidirect Discrete Logarithm Problem in Finite Groups

Abstract

We present an efficient quantum algorithm for solving the semidirect discrete logarithm problem (SDLP) in any finite group. The believed hardness of the semidirect discrete logarithm problem under- lies more than a decade of works constructing candidate post-quantum cryptographic algorithms from nonabelian groups. We use a series of reduction results to show that it suffices to consider SDLP in finite sim- ple groups. We then apply the celebrated Classification of Finite Sim- ple Groups to consider each family. The infinite families of finite simple groups admit, in a fairly general setting, linear algebraic attacks pro- viding a reduction to the classical discrete logarithm problem. For the sporadic simple groups, we show that their inherent properties render them unsuitable for cryptographically hard SDLP instances, which we illustrate via a Baby-Step Giant-Step style attack against SDLP in the Monster Group. Our quantum SDLP algorithm is fully constructive for all but three re- maining cases that appear to be gaps in the literature on constructive recognition of groups; for these cases SDLP is no harder than finding a linear representation. We conclude that SDLP is not a suitable post- quantum hardness assumption for any choice of finite group.