Practical Improvements to Class Group and Regulator Computation of Real Quadratic Fields
نویسندگان
چکیده
We present improvements to the index-calculus algorithm for the computation of the ideal class group and regulator of a real quadratic field. Our improvements consist of applying the double large prime strategy, an improved structured Gaussian elimination strategy, and the use of Bernstein’s batch smoothness algorithm. We achieve a significant speed-up and are able to compute the ideal class group structure and the regulator corresponding to a number field with a 110decimal digit discriminant.
منابع مشابه
Subexponential Class Group Computation in Quadratic Orders (abstract)
In 1989, the first subexponential algorithm for computing the class group of an imaginary quadratic order was introduced by Hafner and McCurley. Their algorithm is based on an integer factorization algorithm due to Seysen, and is conditional on the truth of the Extended Riemann Hypothesis. Not long after, their result was generalized to arbitrary algebraic number fields by Buchmann. Efficient v...
متن کاملOn the real quadratic fields with certain continued fraction expansions and fundamental units
The purpose of this paper is to investigate the real quadratic number fields $Q(sqrt{d})$ which contain the specific form of the continued fractions expansions of integral basis element where $dequiv 2,3( mod 4)$ is a square free positive integer. Besides, the present paper deals with determining the fundamental unit$$epsilon _{d}=left(t_d+u_dsqrt{d}right) 2left.right > 1$$and $n_d$ and $m_d...
متن کاملAsymptotically Fast Discrete Logarithms in Quadratic Number Fields
This article presents algorithms for computing discrete logarithms in class groups of quadratic number elds. In the case of imaginary quadratic elds, the algorithm is based on methods applied by Hafner and McCurley HM89] to determine the structure of the class group of imaginary quadratic elds. In the case of real quadratic elds, the algorithm of Buchmann Buc89] for computation of class group a...
متن کاملApproximating Euler Products and Class Number Computation in Algebraic Function Fields
We provide a number of results that can be used to derive approximations for the Euler product representation of the zeta function of an arbitrary algebraic function field. Three such approximations are given here. Our results have two main applications. They lead to a computationally suitable algorithm for computing the class number of an arbitrary function field. The ideas underlying the clas...
متن کاملAn Accelerated Buchmann Algorithm for Regulator Computation in Real Quadratic Fields
We present a probabilistic algorithm for computing the regulator R of a real quadratic order of discriminant ∆ running in time L( 1 2 , 3/ √ 8 + o(1)).
متن کامل