The monodromy pairing and discrete logarithm on the Jacobian of finite graphs

TL;DR

This paper explores a bilinear pairing on the Jacobian group of finite graphs, demonstrating how to compute it and use it to efficiently solve discrete logarithms, thereby compromising cryptographic security.

Contribution

It introduces a method to compute a bilinear pairing on graph Jacobians and applies it to efficiently solve discrete logarithms, revealing vulnerabilities in proposed cryptographic schemes.

Findings

The pairing can be computed efficiently.

Discrete logarithms are solvable using this pairing.

Cryptographic schemes based on graph Jacobians are insecure.

Abstract

Every graph has a canonical finite abelian group attached to it. This group has appeared in the literature under a variety of names including the sandpile group, critical group, Jacobian group, and Picard group. The construction of this group closely mirrors the construction of the Jacobian variety of an algebraic curve. Motivated by this analogy, it was recently suggested by Norman Biggs that the critical group of a finite graph is a good candidate for doing discrete logarithm based cryptography. In this paper, we study a bilinear pairing on this group and show how to compute it. Then we use this pairing to find the discrete logarithm efficiently, thus showing that the associated cryptographic schemes are not secure. Our approach resembles the MOV attack on elliptic curves.