Global duality, signature calculus and the discrete logarithm problem

TL;DR

This paper introduces a new signature calculus approach using global duality to analyze the discrete logarithm problem over finite fields, establishing equivalence between signature computation and the problem itself.

Contribution

It develops signature calculus based on global duality, providing a unified framework for the discrete logarithm problem on multiplicative groups and elliptic curves.

Findings

Signature calculus generalizes index calculus methods.

Proves polynomial-time equivalence between signature computation and discrete logarithm.

Provides new insights into the structure of discrete logarithm problems.

Abstract

We study the discrete logarithm problem for the multiplicative group and for elliptic curves over a finite field by using a lifting of the corresponding object to an algebraic number field and global duality. We introduce the \textit{signature} of a Dirichlet character (in the multiplicative group case) or principal homogeneous space (in the elliptic curve case), which is a measure of the ramification at certain places. We then develop \textit{signature calculus}, which generalizes and refines the index calculus method. Finally, we show the random polynomial time equivalence for these two cases between the problem of computing signatures and the discrete logarithm problem.