The asymptotic complexity of matrix reduction over finite fields

TL;DR

This paper proves that the striped elimination algorithm for matrix reduction over finite fields is asymptotically optimal, requiring nearly the minimum possible number of row operations for almost all invertible matrices.

Contribution

It establishes the asymptotic optimality of the striped elimination algorithm for matrix reduction over finite fields, matching the lower bound on required operations.

Findings

Striped elimination algorithm has complexity rac{n^2}{ ext{log}_q n}.

Almost all matrices in GL(n;q) require rac{n^2}{ ext{log}_q n} operations asymptotically.

The algorithm is proven to be asymptotically optimal.

Abstract

Consider an invertible n \times n matrix over some field. The Gauss-Jordan elimination reduces this matrix to the identity matrix using at most n^2 row operations and in general that many operations might be needed. In [1] the authors considered matrices in GL(n;q), the set of n \times n invertible matrices in the finite field of q elements, and provided an algorithm using only row operations which performs asymptotically better than the Gauss-Jordan elimination. More specifically their `striped elimination algorithm' has asymptotic complexity \frac{n^2}{\log_q{n}}. Furthermore they proved that up to a constant factor this algorithm is best possible as almost all matrices in GL(n;g) need asymptotically at least \frac{n^2}{2\log_q{n}} operations. In this short note we show that the `striped elimination algorithm' is asymptotically optimal by proving that almost all matrices in GL(n;q) need asymptotically at least frac{n^2}{\log_q{n}} operations.