Hard Instances of Discrete Logarithm Problem and Cryptographic Applications

TL;DR

This paper constructs finitely generated groups where the discrete logarithm problem's difficulty can be arbitrarily set, offering new insights into cryptographic hardness and potential protocol designs.

Contribution

It introduces a method to create finitely generated groups with customizable discrete log problem complexity, including NP-complete instances.

Findings

Existence of finitely generated groups with arbitrarily hard discrete log problems

Construction of two-generated groups with polynomial-time word problem and NP-complete discrete log problem

Proposal of a generic cryptographic protocol scheme

Abstract

Let f be an arbitrary positive integer valued function. The goal of this note is to show that one can construct a finitely generated group in which the discrete log problem is polynomially equivalent to computing the function f. In particular, we provide infinite, but finitely generated groups, in which the discrete logarithm problem is arbitrarily hard. As another application, we construct a family of two-generated groups that have polynomial time word problem and NP-complete discrete log problem. Additionally, using our framework, we propose a generic scheme of cryptographic protocols, which might be of independent interest.