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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 solutionsβ€”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

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