An L(1/3) algorithm for discrete logarithm computation and principality testing in certain number fields

TL;DR

This paper presents a subexponential algorithm with complexity O(L(1/3)) for solving discrete logarithms and testing principality in specific number fields, advancing computational number theory methods.

Contribution

It introduces an L(1/3) algorithm for discrete logarithm and principality testing in certain number fields, extending techniques from algebraic curves to number field problems.

Findings

Achieves subexponential complexity O(L(1/3))

Applicable to number fields with large discriminant and degree

Extends algebraic curve techniques to number field computations

Abstract

We analyse the complexity of solving the discrete logarithm problem and of testing the principality of ideals in a certain class of number fields. We achieve the subexponential complexity in $O(L(1/3,O(1)))$ when both the discriminant and the degree of the extension tend to infinity by using techniques due to Enge, Gaudry and Thom\'{e} in the context of algebraic curves over finite fields.