A fast implementation of the Monster group: The Monster has been tamed

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[12]. We describe a new and very fast algorithm for performing the group operation in M.For an odd integer p>1 let ρp 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 ρp can easily be computed.We construct a triple (v1,v+,v−) of elements of the module ρ15, such that an unknown g∈M can be effectively computed as a word in Γ from the images (v1g,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.