A fast implementation of the Monster group

Abstract

Let $M$ 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 representation ρ of $M$ with matrix entries in $Z\left[\frac{1}{2}\right]$. We describe a new and very fast algorithm for performing the group operation in $M$.

For an odd integer $p>1$ let ${\rho }_p$ be the representation ρ with matrix entries taken modulo p. We use a generating set Γ of $M$, such that the operation of a generator in Γ 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 $g\in M$ can be effectively computed as a word in Γ from the images $(v_{1}g,v^{+}g,v^{-}g)$.

Our new algorithm based on this idea multiplies two random elements of $M$ 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.