Showcasing straight-line programs with memory via matrix Bruhat decomposition

TL;DR

This paper emphasizes the importance of analyzing both computational complexity and memory usage in algebraic straight-line programs, demonstrating an efficient approach for matrix Bruhat decomposition over finite fields.

Contribution

It introduces a formal framework for complexity analysis of straight-line programs and constructs memory-efficient programs for matrix Bruhat decomposition.

Findings

Programs have length O(d^2 log(q))

Memory requirement is only O(log(q)) matrices

Efficient algebraic computation over finite fields

Abstract

We suggest that straight-line programs designed for algebraic computations should be accompanied by a comprehensive complexity analysis that takes into account both the number of fundamental algebraic operations needed, as well as memory requirements arising during evaluation. We introduce an approach for formalising this idea and, as illustration, construct and analyse straight-line programs for the Bruhat decomposition of $d\times d$ matrices with determinant $1$ over a finite field of order $q$ that have length $O(d^2\log(q))$ and require storing only $O(\log(q))$ matrices during evaluation.