A fast implementation of the Monster group

Abstract

Let $\mathbb{M}$ be the Monster group, which is the largest sporadic finite simple group, and has first been constructed in 1982 by Griess. In 1985 Conway has constructed a 196884-dimensional rational epresentation $\rho$ of $\mathbb{M}$ with matrix entries in $\mathbb{Z}[\frac{1}{2}]$. We describe a new and very fast algorithm for performing the group operation in $\mathbb{M}$. For an odd integer $p > 1$ let $\rho_p$ be the representation $\rho$ with matrix entries taken modulo $p$. We use a generating set $\Gamma$ of $\mathbb{M}$, such that the operation of a generator in $\Gamma$ on an element of $\rho_p$ can easily be computed. We construct a triple $(v_1, v^+, v^-)$ of elements of the module $\rho_{15}$, such that an unknown $g \in \mathbb{M}$ can be effectively computed as a word in $\Gamma$ from the images $(v_1 g, v^+ g, v^- g)$. Our new algorithm based on this idea multiplies two random elements of $\mathbb{M}$ in less than 30~milliseconds on a standard PC with an Intel i7-8750H CPU at 4 GHz. This is more than 100000 times faster than estimated by Wilson in 2013.