Skip to main content
SpringerLink
Log in

The Universal Block Lanczos–Padé Method for Linear Systems Over Large Prime Fields

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. A. Cherepnev, “A block algorithm of Lanczos type for solving sparse systems of linear equations,” Diskr. Mat., 20, No. 1, 145–150 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  2. 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).

  3. D. Coppersmith, “Solving linear equations over GF(2): Block Lanczos algorithm,” Linear Algebra Appl., 193, 33–60 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Ya. Dorofeev, “Computation of logarithms in a finite simple field by the linear sieve method,” Tr. Diskr. Mat., 5, 29–50 (2002).

    Google Scholar 

  5. 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)

  6. 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).

    Article  MathSciNet  MATH  Google Scholar 

  7. 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).

    Article  MathSciNet  MATH  Google Scholar 

  8. 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).

    Article  MathSciNet  Google Scholar 

  9. 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.

  10. M. Peterson and C. Monico, “\( {\mathbb{F}}_2 \) Lanczos revisited,” Linear Algebra Appl., 428, 1135–1150 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  11. I. A. Popovyan, Yu. V. Nesterenko, and E. A. Grechnikov, Complex Computations in Number Theory. Tutorial [in Russian], Izd. Mosk. Univ., Moscow (2012).

  12. N. L. Zamarashkin, Algorithms for Sparse Linear Systems over GF(2). Tutorial [in Russian], Izd. Mosk. Univ., Moscow (2013).

Download references

Author information

Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    M. A. Cherepniov

  2. Institute of Numerical Mathematics, Moscow, Russia

    N. L. Zamarashkin

Authors
  1. M. A. Cherepniov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. L. Zamarashkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. Cherepniov.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 6, pp. 225–249, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-017-3238-2

Search

Navigation