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

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

Gluing of graphs and their Jacobians

The Jacobian of a graph is a discrete analogue of the Jacobian of a Riemann surface. In this paper, we explore how Jacobians of graphs change when we glue two graphs along a common subgraph focusing on the case of cycle graphs. Then, we link the computation of Jacobians of graphs with cycle matrices. Finally, we prove that Tutte's rotor construction with his original example produces two graphs with isomorphic Jacobians when all involved graphs are planar. This answers the question posed by Clancy, Leake, and Payne, stating it is affirmative in this case.Comment: v3; After we posted our first version on arXiv, we learned from Matt Baker the work of Chen and Ye by which one can obtain our Theorem B as a special case. Also, we added the work of Alfaro and Villagr{\'a}n which has the same definition in our matrix B_

Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem

Let $\Gamma$ be a compact tropical curve (or metric graph) of genus $g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on $\Gamma$. We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an "integral" version of this result which is of independent interest. As an application, we provide a "geometric proof" of (a dual version of) Kirchhoff's celebrated Matrix-Tree Theorem. Indeed, we show that each weighted graph model $G$ for $\Gamma$ gives rise to a canonical polyhedral decomposition of the $g$-dimensional real torus ${\rm Pic}^g(\Gamma)$ into parallelotopes $C_T$, one for each spanning tree $T$ of $G$, and the dual Kirchhoff theorem becomes the statement that the volume of ${\rm Pic}^g(\Gamma)$ is the sum of the volumes of the cells in the decomposition.Comment: 20 pages -- Final version to appear in Forum of Math, Sigm