, Volume 1, Issue 1, pp 115
Discrete logarithms inGF(p)
 Don CoppersmithAffiliated withIBM Research
 , Andrew M. OdlzykoAffiliated withAT & T Bell Laboratories
 , Richard SchroeppelAffiliated withInference Corporation
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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))\(\sqrt {\log p \log \log p} \)) 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.
Key words
Cryptography Number theory Discrete logarithms Title
 Discrete logarithms inGF(p)
 Journal

Algorithmica
Volume 1, Issue 14 , pp 115
 Cover Date
 198611
 DOI
 10.1007/BF01840433
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Cryptography
 Number theory
 Discrete logarithms
 Industry Sectors
 Authors

 Don Coppersmith ^{(1)}
 Andrew M. Odlzyko ^{(2)}
 Richard Schroeppel ^{(3)}
 Author Affiliations

 1. IBM Research, 10598, Yorktown Heights, NY, USA
 2. AT & T Bell Laboratories, 07974, Murray Hill, NJ, USA
 3. Inference Corporation, 90045, Los Angeles, CA, USA