Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
On the equivalence between Berlekamp's and Euclid's algorithms
IEEE Transactions on Information Theory
Analysis of euclidean algorithms for polynomials over finite fields
Journal of Symbolic Computation
Processor efficient parallel solution of linear systems over an abstract field
SPAA '91 Proceedings of the third annual ACM symposium on Parallel algorithms and architectures
Matrix-free linear system solving and applications to symbolic computation
Matrix-free linear system solving and applications to symbolic computation
Modern computer algebra
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Early termination in Ben-Or/Tiwari sparse interpolation and a hybrid of Zippel's algorithm
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Computational aspects of discrete logarithms
Computational aspects of discrete logarithms
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
Reliable Krylov-based algorithms for matrix null space and rank
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Signature of symmetric rational matrices and the unitary dual of lie groups
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
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Krylov-based algorithms have recently been used (alone, or in combination with other methods) in order to solve systems of linear equations that arise during integer factorization and discrete logarithm computations. Since these include systems over small finite fields, the behaviour of these algorithms in this setting is of interest.Unfortunately, the application of these methods is complicated by the possibility of several kinds of breakdown. Orthogonal vectors can arise when a variant of the Lanczos algorithm is used to generate a basis, and zero-discrepancies can arise during the computation of minimal polynomials of linearly recurrent sequences when Wiedemann's algorithm is applied.Several years ago, Austin Lobo reported experimental evidence that zero-discrepancies are extremely unlikely when a randomized version of Wiedemann's algorithm is applied to solve systems over large fields. With high probability, results are correct if a computation is terminated as soon as such a sequence is detected. "Early termination" has consequently been included in recent implementations.In this paper, we analyze the probability of long sequences of zero-discrepancies during computations of minimal polynomials of the linearly recurrent sequences that arise when simple Krylov-based algorithms are used to solve systems over very small finite fields. Variations of these algorithms that incorporate early termination are briefly presented and analyzed in the small field case.