On the Computation of Quadratic 2-class Groups
نویسندگان
چکیده
We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer D 0; 1 mod 4 and the factorization of D, computes the structure of the 2-Sylow subgroup of the class group of the quadratic order of discriminant D in random polynomial time in log jDj.
منابع مشابه
A mathematically simple method based on denition for computing eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices
In this paper, a fundamentally new method, based on the denition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. Some examples are provided to show the accuracy and reliability of the proposed method. It is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to th...
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 کاملExperimental Results on Class Groups of Real Quadratic Fields
In an effort to expand the body of numerical data for real quadratic fields, we have computed the class groups and regulators of all real quadratic fields with discriminant ∆ < 10. We implemented a variation of the group structure algorithm for general finite Abelian groups described in [2] in the C++ programming language using built-in types together with a few routines from the LiDIA system [...
متن کاملExplicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields with Class Numbers 7 and 11
Motivated by a constructive realization of dihedral groups of prime degree as Galois group over the field of rational numbers, we give an explicit construction of the Hilbert class fields of some imaginary quadratic fields with class numbers 7 and 11. This was done by explicitly evaluating the elliptic modular j -invariant at each representative of the ideal class of an imaginary quadratic fiel...
متن کاملImprovements in the computation of ideal class groups of imaginary quadratic number fields
We investigate improvements to the algorithm for the computation of ideal class groups described by Jacobson in the imaginary quadratic case. These improvements rely on the large prime strategy and a new method for performing the linear algebra phase. We achieve a significant speed-up and are able to compute ideal class groups with discriminants of 110 decimal digits in less than a week.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011