A Block Lanczos Algorithm for Finding Dependencies over GF(2)

  • Peter L. Montgomery
Conference paper

DOI: 10.1007/3-540-49264-X_9

Volume 921 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Montgomery P.L. (1995) A Block Lanczos Algorithm for Finding Dependencies over GF(2). In: Guillou L.C., Quisquater JJ. (eds) Advances in Cryptology — EUROCRYPT ’95. EUROCRYPT 1995. Lecture Notes in Computer Science, vol 921. Springer, Berlin, Heidelberg


Some integer factorization algorithms require several vectors in the null space of a sparse m × n matrix over the field GF(2). We modify the Lanczos algorithm to produce a sequence of orthogonal subspaces of GF(2)n, each having dimension almost N, where N is the computer word size, by applying the given matrix and its transpose to N binary vectors at once. The resulting algorithm takes about n/(N − 0.76) iterations. It was applied to matrices larger than 106 × 106 during the factorizations of 105-digit and 119-digit numbers via the general number field sieve.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Peter L. Montgomery
    • 1
  1. 1.San RafaelUSA