An L (1/3 + epsilon ) Algorithm for the Discrete Logarithm Problem for Low Degree Curves
نویسندگان
چکیده
The discrete logarithm problem in Jacobians of curves of high genus g over finite fields Fq is known to be computable with subexponential complexity Lqg (1/2, O(1)). We present an algorithm for a family of plane curves whose degrees in X and Y are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of Lqg (1/3, O(1)), and a discrete logarithm computation takes subexponential time of Lqg (1/3 + ε, o(1)) for any positive ε. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.
منابع مشابه
An L(1/3 + ε) Algorithm for the Discrete Logarithm Problem for Low Degree Curves
The discrete logarithm problem in Jacobians of curves of high genus g over finite fields Fq is known to be computable with subexponential complexity Lqg (1/2, O(1)). We present an algorithm for a family of plane curves whose degrees in X and Y are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared...
متن کاملGeneralized 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...
متن کاملAn efficient blind signature scheme based on the elliptic curve discrete logarithm problem
Elliptic Curve Cryptosystems (ECC) have recently received significant attention by researchers due to their high performance such as low computational cost and small key size. In this paper a novel untraceable blind signature scheme is presented. Since the security of proposed method is based on difficulty of solving discrete logarithm over an elliptic curve, performance of the proposed scheme ...
متن کاملOn 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...
متن کاملAn L(1/3) algorithm for discrete logarithm computation and principality testing in certain number fields
We analyse the complexity of solving the discrete logarithm problem and of testing the principality of ideals in a certain class of number fields. We achieve the subexponential complexity in O(L(1/3, O(1))) when both the discriminant and the degree of the extension tend to infinity by using techniques due to Enge, Gaudry and Thomé in the context of algebraic curves over finite fields.
متن کامل