Quantum computation of discrete logarithms in semigroups

Abstract

We describe an efficient quantum algorithm for computing discrete logarithms in semigroups using Shor's algorithms for period finding and the discrete logarithm problem as subroutines. Thus proposed cryptosystems based on the presumed hardness of discrete logarithms in semigroups are insecure against quantum attacks. In contrast, we show that some generalizations of the discrete logarithm problem are hard in semigroups despite being easy in groups. We relate a shifted version of the discrete logarithm problem in semigroups to the dihedral hidden subgroup problem, and we show that the constructive membership problem with respect to k ≥ 2 generators in a black-box abelian semigroup of order N requires Θ˜(N12-12k)$\tilde{\Theta }(N^{\frac{1}{2}-\frac{1}{2k}})$ quantum queries.

Keywords

• quantum algorithms
• discrete logarithm
• semigroups
• hidden shift problem
• constructive membership
• semigroup actions
• 68q12
• 68q17
• 68w30
• 81p68
• 94a60