On the discrete logarithm problem in class groups of curves

On the discrete logarithm problem in class groups of curves

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 hyperell...

Generalized Jacobian and Discrete Logarithm Problem on Elliptic Curves

Let E be an elliptic curve over the finite field F_{q}, P a point in E(F_{q}) of order n, and Q a point in the group generated by P. The discrete logarithm problem on E is to find the number k such that Q = kP. In this paper we reduce the discrete logarithm problem on E[n] to the discrete logarithm on the group F*_{q} , the multiplicative group of nonzero elements of Fq, in the case where n | q...

On arithmetic and the discrete logarithm problem in class groups of curves

On the discrete logarithm problem in elliptic curves II

We continue our study on the elliptic curve discrete logarithm problem over finite extension fields. We show, among other results, the following two results: For sequences of prime powers (qi)i∈N and natural numbers (ni)i∈N with ni −→ ∞ and ni log(qi) −→ 0 for i −→ ∞, the discrete logarithm problem in the groups of rational points of elliptic curves over the fields Fqi i can be solved in subexp...

On the discrete logarithm problem in elliptic curves

We study the elliptic curve discrete logarithm problem over finite extension fields. We show that for any sequences of prime powers (qi)i∈N and natural numbers (ni)i∈N with ni −→ ∞ and ni log(qi) −→ 0 for i −→ ∞, the elliptic curve discrete logarithm problem restricted to curves over the fields Fqi i can be solved in subexponential expected time (qi i ) . We also show that there exists a sequen...