# Probabilistic computation of the Smith normal form of a sparse integer matrix

## Abstract

We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix *A ∈* ℤ^{m×n}. The algorithm treats *A* as a “black-box”; *A* is only used to compute matrix-vector products and we don't access individual entries in *A* directly. The algorithm requires about *O*(*m*^{2} log ∥*A*∥) such black-box evaluations reduced modulo word-sized primes *p* on vectors in ℤ _{p} ^{n×1} , plus *O*(*m*^{2}*n* log ∥*A*∥) additional bit operations. For sparse matrices this represents a substantial improvement over previously known algorithms. For example, on an n×n integer matrix *A* with O(*n* log *n*) non-zero entries, only about *O*(*n*^{3} log^{2} ∥*A*∥) bit operations are required to find the Smith form using standard integer arithmetic. The new algorithm suffers from no “fill-in” or intermediate value explosion, and uses very little additional space. The algorithm is probabilistic of the Monte Carlo type — on any input it returns the correct answer with a controllable, exponentially small probability of error.

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