Fast Algorithms for Linear Algebra Modulo N

Abstract

Many linear algebra problems over the ring ℤ _N of integers modulo N can be solved by transforming via elementary row operations an n × m input matrix A to Howell form H. The nonzero rows of H give a canonical set of generators for the submodule of (ℤ _N )^m generated by the rows of A. In this paper we present an algorithm to recover H together with an invertible transformation matrix P which satisfies PA = H. The cost of the algorithm is O(nm ^ω−1) operations with integers bounded in magnitude by N. This leads directly to fast algorithms for tasks involving ℤ _N -modules, including an O(nm ^ω−1) algorithm for computing the general solution over ℤ _N of the system of linear equations xA = b, where b ∈ (ℤ _N )^m.

This work has been supported by grants from the Swiss Federal Office for Education and Science in conjunction with partial support by ESPRIT LTR Project no. 20244 — ALCOM-IT.