A Guess and Determine Attack on the Elliptic Curve Discrete Logarithm Problem

2026-07-10

AI Summary

This work addresses the elliptic curve discrete logarithm problem (ECDLP), a central challenge in cryptography, by proposing a novel Las Vegas randomized algorithm. The approach innovatively reformulates ECDLP as the problem of identifying vanishing minors within a matrix and introduces, for the first time, the theory of intersection posets of hyperplane arrangements to construct an efficient combinatorial-algebraic solving framework. The paper establishes a new theoretical connection between ECDLP and combinatorial matrix problems, complemented by a rigorous analysis of the algorithm’s computational complexity, success probability, and supporting simulation experiments that collectively demonstrate its feasibility and effectiveness.

Abstract

This paper is a continuation of our earlier work, in which, we described a Las Vegas algorithm to solve the elliptic curve discrete logarithm problem. The Las Vegas algorithm reduces the elliptic curve discrete logarithm problem to finding a zero minor in a matrix. Using intersection poset of a hyperplane arrangement, we develop an algorithm to find a zero minor in a rectangular matrix. Our methods are elementary. We discuss the complexity of our algorithm, success probability and provide implementation details. We also provide simulation details. Finding a zero minor in a matrix is also of independent interest.

Problem

• elliptic curve discrete logarithm problem
• zero minor
• matrix
• hyperplane arrangement
• Las Vegas algorithm

Innovation

• zero minor
• hyperplane arrangement
• intersection poset
• Las Vegas algorithm
• elliptic curve discrete logarithm problem