Two variable polynomial congruences and capacity theory

TL;DR

This paper applies capacity theory to analyze Coppersmith's method for solving two-variable polynomial congruences, demonstrating its probabilistic success and failure rates and providing bounds on small solutions.

Contribution

It introduces a capacity-theoretic framework to evaluate the effectiveness and limitations of Coppersmith's method for two-variable polynomial congruences.

Findings

Coppersmith's method succeeds in a positive proportion of cases

It also fails in a positive proportion of cases

Bounds on the number of small solutions are established

Abstract

We use capacity theory to analyze Coppersmith's method for finding small solutions of linear two variable polynomial congruences. We show that the method will succeed in a positive proportion of cases and fail in a different positive proportion of cases. We also bound the number of small solutions via capacity theory.