Double Index Calculus Algorithm: Faster Solving Discrete Logarithm Problem in Finite Prime Field

TL;DR

This paper introduces the double index calculus algorithm, significantly faster and more versatile than existing methods for solving the discrete logarithm problem in finite prime fields, enhancing cryptographic security analysis.

Contribution

The paper presents a novel double index calculus algorithm that outperforms the state-of-the-art index calculus algorithm in speed and applicability for finite prime fields.

Findings

Over 30-fold speed increase for 70-bit prime fields

Algorithm works even when the base is not the generator

More general applicability than index calculus

Abstract

Solving the discrete logarithm problem in a finite prime field is an extremely important computing problem in modern cryptography. The hardness of solving the discrete logarithm problem in a finite prime field is the security foundation of numerous cryptography schemes. In this paper, we propose the double index calculus algorithm to solve the discrete logarithm problem in a finite prime field. Our algorithm is faster than the index calculus algorithm, which is the state-of-the-art algorithm for solving the discrete logarithm problem in a finite prime field. Empirical experiment results indicate that our algorithm could be more than a 30-fold increase in computing speed than the index calculus algorithm when the bit length of the order of prime field is 70 bits. In addition, our algorithm is more general than the index calculus algorithm. Specifically, when the base of the target discrete logarithm problem is not the multiplication generator, the index calculus algorithm may fail to solve the discrete logarithm problem while our algorithm still can work.