Discrete logarithm problem in some families of sandpile groups

TL;DR

This paper investigates the discrete logarithm problem in sandpile groups of specific graphs, revealing cases where the problem is efficiently solvable and analyzing factors affecting its difficulty.

Contribution

It extends prior work by analyzing the discrete logarithm problem in non-cyclic sandpile groups of certain graphs and identifies conditions that simplify solving it.

Findings

Discrete logarithm problem is solvable in non-cyclic sandpile groups under certain conditions.

Knowing generators or the pseudoinverse of the Laplacian simplifies the problem.

In some cases, the sandpile group is cyclic, enabling easier solutions.

Abstract

Biggs proposed the sandpile group of certain modified wheel graphs for cryptosystems relying on the difficulty of the discrete logarithm problem. Blackburn and independently Shokrieh showed that the discrete logarithm problem is efficiently solvable. We study Shokrieh's method in cases of graphs such that the sandpile group is not cyclic, namely the square cycle graphs and the wheel graphs. Knowing generators of the group or the form of the pseudoinverse of the Laplacian matrix makes the problem more vulnerable. We also consider the discrete logarithm problem in case of the so-called subdivided banana graphs. In certain cases the sandpile group is cyclic and a generator is known and one can solve the discrete logarithm problem without computing the pseudoinverse of the Laplacian matrix.