A Fast Search Algorithm for <m,m,m> Triple Product Property Triples and an Application for 5x5 Matrix Multiplication

TL;DR

This paper introduces a significantly faster search algorithm for <m,m,m> TPP triples, enabling the analysis of larger groups and confirming the optimality of Makarov's 5x5 matrix multiplication method.

Contribution

A new parallelized search algorithm for <m,m,m> TPP triples that outperforms previous methods by up to 194 times and applies it to analyze groups for 5x5 matrix multiplication.

Findings

Achieved a 40 to 194 times speed-up over previous algorithms.

Identified groups capable of realizing 5x5 matrix multiplication with under 125 scalar multiplications.

Confirmed no group can improve upon Makarov's 5x5 matrix multiplication algorithm.

Abstract

We present a new fast search algorithm for <m,m,m> Triple Product Property (TPP) triples as defined by Cohn and Umans in 2003. The new algorithm achieves a speed-up factor of 40 up to 194 in comparison to the best known search algorithm. With a parallelized version of the new algorithm we are able to search for TPP triples in groups up to order 55. As an application we identify a list of groups that would realize 5x5 matrix multiplication with under 100 resp. 125 scalar multiplications (the best known upper bound by Makarov 1987 resp. the trivial upper bound) if they contain a <5,5,5> TPP triple. With our new algorithm we show that no group can realize 5x5 matrix multiplication better than Makarov's algorithm.