Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm
Mathematics of Computation
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
On randomized Lanczos algorithms
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Black box Frobenius decompositions over small fields
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
AAECC-10 Proceedings of the 10th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Solving Large Sparse Linear Systems over Finite Fields
CRYPTO '90 Proceedings of the 10th Annual International Cryptology Conference on Advances in Cryptology
Early termination over small fields
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
A block Lanczos algorithm for finding dependencies over GF(2)
EUROCRYPT'95 Proceedings of the 14th annual international conference on Theory and application of cryptographic techniques
Efficient computation of the characteristic polynomial
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
A reliable block Lanczos algorithm over small finite fields
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
A block Wiedemann rank algorithm
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Fast computation of Smith forms of sparse matrices over local rings
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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Krylov-based algorithms have recently been used, in combination with other methods, to solve systems of linear equations and to perform related matrix computations over finite fields. For example, large and sparse systems of linear equations ;2; are formed during the use of the number field sieve for integer factorization, and elements of the null space of these systems are sampled.Two rather different kinds of block algorithms have recently been considered. Block Wiedemann algorithms have now been presented and fully analyzed. Block Lanczos algorithms were proposed earlier but are not yet as well understood. In particular, proofs of reliability of block Lanczos algorithms are not yet available. Nevertheless, an examination of the computational number theory literature suggests that block Lanczos algorithms continue to be preferred.This report presents a block Lanczos algorithm that is somewhat simpler than block algorithms that are presently in use and provably reliable for computations over large fields. To my knowledge, this is the first block Lanczos algorithm for which a proof of reliability is available.A different Krylov-based approach is considered for computations over small fields: It is shown that if Wiedemann's sparse matrix preconditioner is applied to an arbitrary matrix then the number of nontrivial invariant factors of the result is, with high probability, quite small. A Krylov-based algorithm to compute a partial Frobenius decomposition can then be used to sample from the null space of the original matrix or to compute its rank. This yields a randomized (Monte Carlo) black box algorithm for matrix rank that is asymptotically faster, in the small field case, than any other that is presently known.