A deterministic algorithm for the discrete logarithm problem in a semigroup

Abstract The discrete logarithm problem (DLP) in a finite group is the basis for many protocols cryptography. best general algorithms which solve this have time complexity of O ( N log ) O\left(\sqrt{N}\log N) and space O\left(\sqrt{N}) , where N order group. (If unknown, simple modification would achieve 2 O\left(\sqrt{N}{\left(\log N)}^{2}) .) These require inversion some elements or rely on finding collisions existence inverses, thus do not adapt to work semigroup setting. For semigroups, probabilistic with similar been proposed. main result article deterministic algorithm solving DLP semigroup. Specifically, let x x be an element having {N}_{x} . provides algorithm, which, given any y ∈ ⟨ ⟩ y\in \langle x\rangle all natural numbers m m = {x}^{m}=y has O\left(\sqrt{{N}_{x}}{\left(\log {N}_{x})}^{2}) steps. also gives analysis success rates existing algorithms, were so far only conjectured stated loosely.