research-article

Discrete logarithms inGF(p)

Authors:

Don Coppersmith,

Andrew M. Odlzyko,

Richard SchroeppelAuthors Info & Claims

Algorithmica, Volume 1, Issue 1-4

Pages 1 - 15

https://doi.org/10.1007/BF01840433

Published: 01 January 1986 Publication History

Abstract

Several related algorithms are presented for computing logarithms in fieldsGF(p),p a prime. Heuristic arguments predict a running time of exp((1+o(1))) for the initial precomputation phase that is needed for eachp, and much shorter running times for computing individual logarithms once the precomputation is done. The running time of the precomputation is roughly the same as that of the fastest known algorithms for factoring integers of size aboutp. The algorithms use the well known basic scheme of obtaining linear equations for logarithms of small primes and then solving them to obtain a database to be used for the computation of individual logarithms. The novel ingredients are new ways of obtaining linear equations and new methods of solving these linear equations by adaptations of sparse matrix methods from numerical analysis to the case of finite rings. While some of the new logarithm algorithms are adaptations of known integer factorization algorithms, others are new and can be adapted to yield integer factorization algorithms.

References

[1]

L. M. Adleman, “A subexponential algorithm for the discrete logarithm problem with applications to cryptography,”Proc. 20th IEEE Found. Comp. Sci. Symp. (1979), 55–60.

[2]

Canfield E. R., Erdös P., and Pomerance C. On a problem of Oppenheim concerning ‘Factorisatio Numerorum’ J. Number Theory 1983 17 1-28

[3]

Coppersmith D. Fast evaluation of logarithms in fields of characteristic two IEEE Trans. Inform. Theory IT 1984 -30 587-594

[4]

Coppersmith D. and Winograd S. On the asymptotic complexity of matrix multiplication SIAM J. Comput. 1982 11 472-492

[5]

T. ElGamal, “A subexponential-time algorithm for computing discrete logarithms overGF(p²),”IEEE Trans. Inform. Theory, to appear.

[6]

Hestenes M. R. and Stiefel E. Method of conjugate gradients for solving linear systems J. Res. Nat. Bur. Standards, Sect. B 1952 49 409-436

[7]

Lanczos C. An iterative method for the solution of the eigenvalue problem of linear differential and integral operators J. Res. Nat. Bur. Standards, Sect. B 1950 45 255-282

[8]

H. W. Lenstra, Jr., paper in preparation.

[9]

A. M. Odlyzko, “Discrete logarithms in finite fields and their cryptographic significance,” to appear,Proceedings of Eurocrypt '84, Springer Lecture Notes in Computer Science.

[10]

Pollard J. M. A Monte Carlo method for factorization BIT 1975 15 331-334

[11]

C. Pomerance, “Analysis and comparison of some integer factoring algorithms,” pp. 89–139 inComputational Methods in Number Theory: Part I, H. W. Lenstra, Jr., and R. Tijdeman, eds., Math. Centre Tract 154, Math. Centre Amsterdam, 1982.

[12]

J. M. Reyneri, unpublished manuscript.

[13]

Strassen V. Gaussian elimination is not optimal Numer. Math. 1969 13 354-356

[14]

D. Wiedemann, “Solving sparse linear equations over finite fields,”IEEE Trans. Inform. Theory, to appear.

[15]

A. E. Western and J. C. P. Miller,Tables of Indices and Primitive Roots, Royal Society Mathematical Tables, vol. 9, Cambridge Univ. Press, 1968.

Cited By

Villard G(2023)Elimination ideal and bivariate resultant over finite fieldsProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597100(526-534)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597100
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Discrete logarithms inGF(p)
1. Computing methodologies

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Reviews

Reviewer: Thomas Rainer Michael Fischer

Several related algorithms are presented for computing logarithms in fields GF( p); that is, given integers a, b and a prime p, computing an integer x such that a x :3W.9T b mod p if such an x exists. The algorithms differ from previous approaches by making use of new methods of generating linear equations over finite rings and also by utilizing new techniques of solving them by adaptations of sparse matrix methods to the case of finite rings. Heuristic arguments predict a running time of exp((1 + o(1)) :.PC12 :3Wz:Hlog :Cp:A log log :Cp:A:9U) for the initial precomputation phase required for each p, and much shorter time for computing individual logarithms once the precomputation is done. All algorithms are described informally. While some of the methods used are modifications of known algorithms, others seem to be new and can perhaps be adapted to integer factorization problems. The achieved running times substantially improve upon previously published algorithms.

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Published In

cover image Algorithmica

Algorithmica Volume 1, Issue 1-4

Nov 1986

517 pages

ISSN:0178-4617

Issue’s Table of Contents

© Springer-Verlag New York Inc. 1986.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 January 1986

Revision received: 01 June 1985

Received: 02 February 1985

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Sakkas GEndres MGuo PWeimer WJhala R(2022)Seq2Parse: neurosymbolic parse error repairProceedings of the ACM on Programming Languages10.1145/35633306:OOPSLA2(1180-1206)Online publication date: 31-Oct-2022
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