Probabilistic analysis of Wiedemann's algorithm for minimal polynomial computation

TL;DR

This paper provides probabilistic formulas and bounds for the success of Wiedemann's minimal polynomial algorithm, demonstrating that small block sizes can yield high success probabilities across various matrix sizes and field sizes.

Contribution

It introduces exact probability formulas based on the generalized Jordan form and derives sharp bounds applicable to all finite field sizes.

Findings

Exact formulas for success probability of Wiedemann's algorithm.

Bounds valid for all finite field sizes and matrix dimensions.

Small blocking factors increase success probability across scenarios.

Abstract

Blackbox algorithms for linear algebra problems start with projection of the sequence of powers of a matrix to a sequence of vectors (Lanczos), a sequence of scalars (Wiedemann) or a sequence of smaller matrices (block methods). Such algorithms usually depend on the minimal polynomial of the resulting sequence being that of the given matrix. Here exact formulas are given for the probability that this occurs. They are based on the generalized Jordan normal form (direct sum of companion matrices of the elementary divisors) of the matrix. Sharp bounds follow from this for matrices of unknown elementary divisors. The bounds are valid for all finite field sizes and show that a small blocking factor can give high probability of success for all cardinalities and matrix dimensions.