Kangaroo Methods for Solving the Interval Discrete Logarithm Problem

TL;DR

This paper introduces new multi-kangaroo algorithms for the interval discrete logarithm problem, significantly improving expected running times for more than four kangaroos and providing insights into the efficiency of five versus three kangaroos.

Contribution

It presents the first seven kangaroo method with near-optimal efficiency and compares the performance of five kangaroo algorithms to existing methods.

Findings

Seven kangaroo method with expected time ~1.7195√N group operations.

Five kangaroo algorithm with expected time ~1.737√N group operations.

Improved understanding of the efficiency trade-offs among different numbers of kangaroos.

Abstract

The interval discrete logarithm problem is defined as follows: Given some $g,h$ in a group $G$, and some $N \in \mathbb{N}$ such that $g^z=h$ for some $z$ where $0 \leq z < N$, find $z$. At the moment, kangaroo methods are the best low memory algorithm to solve the interval discrete logarithm problem. The fastest non parallelised kangaroo methods to solve this problem are the three kangaroo method, and the four kangaroo method. These respectively have expected average running times of $\big(1.818+o(1)\big)\sqrt{N}$, and $\big(1.714 + o(1)\big)\sqrt{N}$ group operations. It is currently an open question as to whether it is possible to improve kangaroo methods by using more than four kangaroos. Before this dissertation, the fastest kangaroo method that used more than four kangaroos required at least $2\sqrt{N}$ group operations to solve the interval discrete logarithm problem. In this thesis, I improve the running time of methods that use more than four kangaroos significantly, and almost beat the fastest kangaroo algorithm, by presenting a seven kangaroo method with an expected average running time of $\big(1.7195 + o(1)\big)\sqrt{N} \pm O(1)$ group operations. The question, 'Are five kangaroos worse than three?' is also answered in this thesis, as I propose a five kangaroo algorithm that requires on average $\big(1.737+o(1)\big)\sqrt{N}$ group operations to solve the interval discrete logarithm problem.