Asymptotically Fast Discrete Logarithms in Quadratic Number Fields

Abstract

This article presents algorithms for computing discrete logarithms in class groups of quadratic number fields. In the case of imaginary quadratic fields, the algorithm is based on methods applied by Hafner and McCurley [HM89] to determine the structure of the class group of imaginary quadratic fields. In the case of real quadratic fields, the algorithm of Buchmann [Buc89] for computation of class group and regulator forms the basis. We employ the rigorous elliptic curve factorization algorithm of Pomerance [Pom87], and an algorithm for solving systems of linear Diophantine equations proposed and analysed by Mulders and Storjohann [MS99]. Under the assumption of the Generalized Riemann Hypothesis, we obtain for fields with discriminant d a rigorously proven time bound of \(L_{|d|} [\frac{1}{2}, \frac{3}{4}\sqrt{2}]\).