On the discrete logarithm problem in class groups of curves
نویسنده
چکیده
We study the discrete logarithm problem in degree 0 class groups of curves over finite fields, with particular emphasis on curves of small genus. We prove that for every fixed g ≥ 2, the discrete logarithm problem in degree 0 class groups of curves of genus g can be solved in an expected time of Õ(q 2 g ), where Fq is the ground field. This result generalizes a corresponding result for hyperelliptic curves given in imaginary quadratic representation with cyclic degree 0 class group, and just as this previous result, it is obtained via an index calculus algorithm with double large prime variation. Generalizing this result, we prove that for fixed g0 ≥ 2 the discrete logarithm problem in class groups of all curves C/Fq of genus g ≥ g0 can be solved in an expected time of Õ((qg) 2 g0 (1− 1 g0 ) and in an expected time of Õ(#Cl(C) 2 g0 (1− 1 g0 ). As a complementary result we prove that for any fixed n ∈ N with n ≥ 2 the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields Fqn , q a prime power, can be solved in an expected time of Õ(q2− 2 n ). Furthermore, we give an algorithm for the efficient construction of a uniformly randomly distributed effective divisor of a specific degree, given the curve and its L-polynomial.
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عنوان ژورنال:
- Math. Comput.
دوره 80 شماره
صفحات -
تاریخ انتشار 2011