A Deterministic Algorithm for the Discrete Logarithm Problem in a Semigroup

The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N})$, where $N$ is the order of the group. These algorithms require the inversion of some some group elements or rely on finding collisions, and thus do not adapt to work in the general semigroup setting. For semigroups, probabilistic algorithms with similar time complexity have been proposed. The main result of this paper is a deterministic algorithm for solving the discrete logarithm problem in a semigroup. Specifically, let $x$ be an element in a semigroup having finite order $N_x$. If $y\in \langle x \rangle $ is given the paper provides an algorithm having time complexity $O(\sqrt{N_x}\log N_x)$ to find all natural numbers m with $x^m=y$. The paper also give an analysis of the success rates of the existing probabilistic algorithms, which were so far only conjectured or stated loosely.