Fast Algorithms for Linear Algebra Modulo N

Many linear algebra problems over the ring ZZN 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 (ZZN ) 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 ) operations with integers bounded in magnitude by N . This leads directly to fast algorithms for tasks involving ZZN–modules, including an O(nm ! ) algorithm for computing the general solution over ZZN of the system of linear equations xA = b, where b 2 (ZZN ).