Abstract
The problem of computing logarithms over finite fields has proved to be of interest in different fields [4]. Subexponential time algorithms for computing logarithms over the special cases GF(p), GF(p 2) and GF(p m) for a fixed p and m → ∞ have been obtained. In this paper, we present some results for obtaining a subexponential time algorithms for the remaining cases GF(p m) for p → ∞ and fixed m ≠ 1, 2. The algorithm depends on mapping the field GF(p m) into a suitable cyclotomic extension of the integers (or rationals). Once an isomorphism between GF(p m) and a subset of the cyclotomic field Q(θ q) is obtained, the algorithms becomes similar to the previous algorithms for m = 1.2.
A rigorous proof for subexponential time is not yet available, but using some heuristic arguments we can show how it could be proved. If a proof would be obtained, it would use results on the distribution of certain classes of integers and results on the distribution of some ideal classes in cyclotomic fields.
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References
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© 1986 Springer-Verlag Berlin Heidelberg
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ElGamal, T. (1986). On Computing Logarithms Over Finite Fields. In: Williams, H.C. (eds) Advances in Cryptology — CRYPTO ’85 Proceedings. CRYPTO 1985. Lecture Notes in Computer Science, vol 218. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-39799-X_28
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DOI: https://doi.org/10.1007/3-540-39799-X_28
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