Paper 2023/1953

Efficient quantum algorithms for some instances of the semidirect discrete logarithm problem

Abstract

The semidirect discrete logarithm problem (SDLP) is the following analogue of the standard discrete logarithm problem in the semidirect product semigroup $G\rtimes \mathrm{End}(G)$ for a finite semigroup $G$. Given $g\in G, \sigma\in \mathrm{End}(G)$, and $h=\prod_{i=0}^{t-1}\sigma^i(g)$ for some integer $t$, the SDLP$(G,\sigma)$, for $g$ and $h$, asks to determine $t$. As Shor's algorithm crucially depends on commutativity, it is believed not to be applicable to the SDLP. Previously, the best known algorithm for the SDLP was based on Kuperberg's subexponential time quantum algorithm. Still, the problem plays a central role in the security of certain proposed cryptosystems in the family of $\textit{semidirect product key exchange}$. This includes a recently proposed signature protocol called SPDH-Sign. In this paper, we show that the SDLP is even easier in some important special cases. Specifically, for a finite group $G$, we describe quantum algorithms for the SDLP in $G\rtimes \mathrm{Aut}(G)$ for the following two classes of instances: the first one is when $G$ is solvable and the second is when $G$ is a matrix group and a power of $\sigma$ with a polynomially small exponent is an inner automorphism of $G$. We further extend the results to groups composed of factors from these classes. A consequence is that SPDH-Sign and similar cryptosystems whose security assumption is based on the presumed hardness of the SDLP in the cases described above are insecure against quantum attacks. The quantum ingredients we rely on are not new: these are Shor's factoring and discrete logarithm algorithms and well-known generalizations.