Computing Asymptotic Bounds for Small Roots in Coppersmith's Method via Sumset Theory

Coppersmith's method is a well-known and practical method for solving polynomial modular equations involved in some cryptosystems such as RSA. An important and tedious task in this method consists in computing the asymptotic bounds. In this work, we address the challenge of computing such asymptotic bounds by introducing the Sumsets theory from Additive Combinatorics as a new analytical tool, which significantly streamlines manual calculations. More precisely, we develop the first provable algorithm for determining these asymptotic bounds, whereas the recent methods based on simple Lagrange interpolation are heuristic. Moreover, the experiments showed that our method is much more efficient than the previous method in practice. We also employ our method to improve the cryptanalytic results for the Commutative Isogeny Hidden Number Problem. Our approach may deepen the understanding of Coppersmith's method and inspire more security analysis methodologies.