On improvements of the $r$-adding walk in a finite field of characteristic 2

TL;DR

This paper critically analyzes and compares a modified $r$-adding walk algorithm, which aims to reduce the work per iteration in solving discrete logarithm problems over finite fields of characteristic 2.

Contribution

It provides a detailed critique and comparison of the modified $r$-adding walk against the original, assessing its effectiveness in reducing iteration work.

Findings

The modified walk potentially reduces work per iteration.

Comparison shows trade-offs between original and modified methods.

Analysis highlights conditions where the modification is beneficial.

Abstract

It is currently known from the work of Shoup and Nechaev that a generic algorithm to solve the discrete logarithm problem in a group of prime order must have complexity at least $k\sqrt{N}$ where $N$ is the order of the group. In many collision search algorithms this complexity is achieved. So with generic algorithms one can only hope to make the $k$ smaller. This $k$ depends on the complexity of the iterative step in the generic algorithms. The $\sqrt{N}$ comes from the fact there is about $\sqrt{N}$ iterations before a collision. So if we can find ways that can reduce the amount of work in one iteration then that is of great interest and probably the only possible modification of a generic algorithm. The modified $r$-adding walk allegedly does just that. It claims to reduce the amount of work done in one iteration of the original $r$-adding walk. In this paper we study this modified $r$-adding walk, we critically analyze it and we compare it with the original $r$-adding walk.