On improvements of $r$-adding walks to solve the Discrete Logarithm Problem

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 does just that. It reduces 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 analyse it and we compare it with the original $r$-adding walk. In the final chapter, we discuss an improvement of original $r$-adding walk on elliptic curve over $\mathbb{F}_p$.