Discrete logarithms inGF(p) (original) (raw)

Abstract

Several related algorithms are presented for computing logarithms in fields_GF_(p),p a prime. Heuristic arguments predict a running time of exp((1+o(1))\(\sqrt {\log p \log \log p} \)) for the initial precomputation phase that is needed for each_p_, 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 about_p_. 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.

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Authors and Affiliations

  1. IBM Research, 10598, Yorktown Heights, NY, USA
    Don Coppersmith
  2. AT & T Bell Laboratories, 07974, Murray Hill, NJ, USA
    Andrew M. Odlzyko
  3. Inference Corporation, 90045, Los Angeles, CA, USA
    Richard Schroeppel

Authors

  1. Don Coppersmith
  2. Andrew M. Odlzyko
  3. Richard Schroeppel

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Communicated by C. K. Wong.

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Coppersmith, D., Odlzyko, A.M. & Schroeppel, R. Discrete logarithms in_GF_(p).Algorithmica 1, 1–15 (1986). https://doi.org/10.1007/BF01840433

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