Fast Algorithms for Linear Algebra Modulo N

  • Arne Storjohann
  • Thom Mulders
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1461)


Many linear algebra problems over the ring 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 (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(nmω−1) operations with integers bounded in magnitude by N. This leads directly to fast algorithms for tasks involving N-modules, including an O(nmω−1) algorithm for computing the general solution over N of the system of linear equations xA = b, where b ∈ (N)m.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Arne Storjohann
    • 1
  • Thom Mulders
    • 1
  1. 1.Institute of Scientific ComputingETH ZurichSwitzerland

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