Secrecy, authentication, and public key systems.
Secrecy, authentication, and public key systems.
A subexponential-time algorithm for computing discrete logarithms over
IEEE Transactions on Information Theory
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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(p2) and GF(pm) for a fixed p and m 驴 驴 have been obtained. In this paper, we present some results for obtaining a sub exponential time algorithms for the remaining cases GF(pm) for p 驴 驴 and fixed m 驴 1, 2. The algorithm depends on mapping the fieLd GF(pm) into a suitable cyclotomic extension of the integers (or rationals). Once an isomorphism between GF(pm) and a subset of the cyclotomic field Q(驴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.