Traps to the BGJT-Algorithm for Discrete Logarithms

TL;DR

This paper critically examines the BGJT algorithm for discrete logarithms, identifies issues with its heuristics in non-Kummer fields, and proposes modifications to maintain its quasi-polynomial efficiency, though further validation is needed.

Contribution

It highlights problems in the original heuristics of the BGJT algorithm and proposes a corrected version applicable to non-Kummer fields without losing efficiency.

Findings

Original heuristics are problematic outside Kummer extensions.

Proposed fix maintains quasi-polynomial complexity.

Further research needed to validate the modified algorithm.

Abstract

In the recent breakthrough paper by Barbulescu, Gaudry, Joux and Thom{\'e}, a quasi-polynomial time algorithm (QPA) is proposed for the discrete logarithm problem over finite fields of small characteristic. The time complexity analysis of the algorithm is based on several heuristics presented in their paper. We show that some of the heuristics are problematic in their original forms, in particular, when the field is not a Kummer extension. We believe that the basic idea behind the new approach should still work, and propose a fix to the algorithm in non-Kummer cases, without altering the quasi-polynomial time complexity. The modified algorithm is also heuristic. Further study is required in order to fully understand the effectiveness of the new approach.