Abstract
In this paper, we propose a universal algorithm designed for solving large sparse linear systems over finite fields with a large prime number of elements. Such systems arise in the solution of the discrete logarithm problem modulo a prime number. The algorithm has been developed for parallel computing systems with various parallel architectures and properties. The new method inherits the structural properties of the Lanczos method. However, it provides flexible control over the complexity of parallel computations and the intensity of exchanges.
References
M. A. Cherepnev, “A block algorithm of Lanczos type for solving sparse systems of linear equations,” Diskr. Mat., 20, No. 1, 145–150 (2008).
M. A. Cherepnev, “Version of block Lanczos-type algorithm for solving sparse linear systems.,” Bull. Math. Soc. Sci. Math. Roumanie, 53 (101), No. 3, 225–230 (2010).
D. Coppersmith, “Solving linear equations over GF(2): Block Lanczos algorithm,” Linear Algebra Appl., 193, 33–60 (1993).
A. Ya. Dorofeev, “Computation of logarithms in a finite simple field by the linear sieve method,” Tr. Diskr. Mat., 5, 29–50 (2002).
A. Ya. Dorofeev, “Solving systems of linear equations arising in the computation of logarithms in a finite prime field,” Mat. Vopr. Kriptogr., 3, No. 1, 5–51 (2012)
M. H. Gutknecht. “A completed theory of the unsymmetric Lanczos process and related algorithms. I,” SIAM J. Matrix Anal. Appl., 13, No. 2, 594–639 (1992).
M. H. Gutknecht. “A completed theory of the unsymmetric Lanczos process and related algorithms. II,” SIAM J. Matrix Anal. Appl., 15, No. 1, 15–58 (1994).
C. Lanczos, “An iteration method for the solution of the eigenvalue problem of linear differential and integral operators,” J. Res. Nat. Bur. Standards, 45, 255–282 (1950).
P. L. Montgomery, “A block Lanczos algorithm for finding dependencies over GF(2),” in: Advances in Cryptology — EUROCRYPT ’95, Lect. Notes Comp. Sci., Vol. 921, Springer, Berlin (1995), pp. 106–120.
M. Peterson and C. Monico, “\( {\mathbb{F}}_2 \) Lanczos revisited,” Linear Algebra Appl., 428, 1135–1150 (2008).
I. A. Popovyan, Yu. V. Nesterenko, and E. A. Grechnikov, Complex Computations in Number Theory. Tutorial [in Russian], Izd. Mosk. Univ., Moscow (2012).
N. L. Zamarashkin, Algorithms for Sparse Linear Systems over GF(2). Tutorial [in Russian], Izd. Mosk. Univ., Moscow (2013).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 6, pp. 225–249, 2014.
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Cherepniov, M.A., Zamarashkin, N.L. The Universal Block Lanczos–Padé Method for Linear Systems Over Large Prime Fields. J Math Sci 221, 461–478 (2017). https://doi.org/10.1007/s10958-017-3238-2
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DOI: https://doi.org/10.1007/s10958-017-3238-2