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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

An algorithm for evaluation
of discrete logarithms in some
nonprime finite fields


Author: Igor A. Semaev
Journal: Math. Comp. 67 (1998), 1679-1689
MSC (1991): Primary 11T71, 11Y16, 94A60
MathSciNet review: 1474656
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Abstract: In this paper we propose an algorithm for evaluation of logarithms in the finite fields $F_{p^n}$, where the number $p^n-1$ has a small primitive factor $r$. The heuristic estimate of the complexity of the algorithm is equal to
$\exp((c+o(1))(\log p\,r\log^2r)^{1/3})$, where $n$ grows to $\infty$, and $p$ is limited by a polynomial in $n$. The evaluation of logarithms is founded on a new congruence of the kind of D. Coppersmith, $C(x)^k\equiv D(x)$, which has a great deal of solutions-pairs of polynomials $C(x),D(x)$ of small degrees.


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Additional Information

Igor A. Semaev
Affiliation: 43-2 Profsoyuznaya Street, Apartment #723, 117420 Moscow, Russia

DOI: http://dx.doi.org/10.1090/S0025-5718-98-00969-7
PII: S 0025-5718(98)00969-7
Keywords: Cryptography, discrete logarithms, finite fields
Received by editor(s): March 30, 1993
Received by editor(s) in revised form: August 30, 1995
Article copyright: © Copyright 1998 American Mathematical Society