Advances in Cryptology — EUROCRYPT ’95

Volume 921 of the series Lecture Notes in Computer Science pp 106-120


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

  • Peter L. MontgomeryAffiliated with


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.