A Deterministic Algorithm for the Discrete Logarithm Problem in a Semigroup

TL;DR

This paper introduces a deterministic algorithm for solving the discrete logarithm problem in semigroups, achieving similar efficiency to probabilistic methods in groups, and analyzes the success of existing probabilistic algorithms.

Contribution

It presents the first deterministic algorithm for discrete logs in semigroups with comparable complexity to probabilistic group algorithms.

Findings

Deterministic algorithm with $O( oot{N_x}(\log N_x)^2)$ complexity

Algorithm finds all solutions $m$ for $x^m=y$ in a semigroup

Analysis of success rates of prior probabilistic algorithms

Abstract

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 $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of $\mathcal{O}(\sqrt{N})$ where $N$ is the order of the group. (If $N$ is unknown, a simple modification would achieve a time complexity of $\mathcal{O}(\sqrt{N}(\log N)^2)$.) These algorithms require the inversion of some group elements or rely on finding collisions and the existence of inverses, 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 $N_x$. The paper provides an algorithm, which, given any element $y\in \langle x \rangle $, provides all natural numbers $m$ with $x^m=y$, and has time complexity $O(\sqrt{N_x}(\log N_x)^2)$ steps. The paper also gives an analysis of the success rates of the existing probabilistic algorithms, which were so far only conjectured or stated loosely.