# Improved method for finding optimal formulae for bilinear maps in a finite field

## TL;DR

This paper introduces a new pruning criterion that enhances the search for optimal bilinear map formulae over finite fields, leading to new optimal solutions and insights into matrix product decompositions.

## Contribution

It presents a novel pruning criterion for exhaustive search, enabling discovery of new optimal formulae and proving uniqueness of certain matrix product decompositions.

## Key findings

- New optimal formulae for short product modulo X^5 and circulant product modulo (X^5 - 1)
- Proof of uniqueness of the optimal decomposition for 3x2 by 2x3 matrix products
- Enhanced search efficiency for bilinear map formulae

## Abstract

In 2012, Barbulescu, Detrey, Estibals and Zimmermann proposed a new framework to exhaustively search for optimal formulae for evaluating bilinear maps, such as Strassen or Karatsuba formulae. The main contribution of this work is a new criterion to aggressively prune useless branches in the exhaustive search, thus leading to the computation of new optimal formulae, in particular for the short product modulo X 5 and the circulant product modulo (X 5 -- 1). Moreover , we are able to prove that there is essentially only one optimal decomposition of the product of 3 x 2 by 2 x 3 matrices up to the action of some group of automorphisms.