Fast Algorithms for Linear Algebra Modulo N
Many linear algebra problems over the ring ℤN 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 (ℤN)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(nmω−1) operations with integers bounded in magnitude by N. This leads directly to fast algorithms for tasks involving ℤN-modules, including an O(nmω−1) algorithm for computing the general solution over ℤN of the system of linear equations xA = b, where b ∈ (ℤN)m.
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- 1.A. V. Aho, J. E. Hopcroft, and J. D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, 1974.Google Scholar
- 2.E. Bach. Linear algebra modulo N. Unpublished manuscript., December 1992.Google Scholar
- 5.M. Newman. Integral Matrices. Academic Press, 1972.Google Scholar
- 7.A. Storjohann and G. Labahn. Asymptotically fast computation of Hermite normal forms of integer matrices. In Y. N. Lakshman, editor, Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC’ 96, pages 259–266. ACM Press, 1996.Google Scholar