On the Complexity of Generalized Discrete Logarithm Problem

TL;DR

This paper proves that the Generalized Discrete Logarithm Problem (GDLP) is NP-hard for symmetric groups and even when base elements are permutations of at most three elements, highlighting its computational difficulty.

Contribution

The paper establishes NP-hardness of GDLP in symmetric groups and for permutations of limited size, advancing understanding of its complexity in algebraic structures.

Findings

GDLP is NP-hard for symmetric groups

GDLP remains NP-hard with permutations of up to 3 elements

Implications for classical and quantum complexity theory discussed

Abstract

Generalized Discrete Logarithm Problem (GDLP) is an extension of the Discrete Logarithm Problem where the goal is to find $x\in\mathbb{Z}_s$ such $g^x\mod s=y$ for a given $g,y\in\mathbb{Z}_s$. Generalized discrete logarithm is similar but instead of a single base element, uses a number of base elements which does not necessarily commute with each other. In this paper, we prove that GDLP is NP-hard for symmetric groups. Furthermore, we prove that GDLP remains NP-hard even when the base elements are permutations of at most 3 elements. Lastly, we discuss the implications and possible implications of our proofs in classical and quantum complexity theory.