Fast multiquadratic S-unit computation and application to the calculation of class groups

Abstract

√ d n) be a real multiquadratic field and S be a set of prime ideals of L. In this paper, we present a heuristic algorithm for the computation of the S-class group and the S-unit group that runs in time Poly(log(), Size(S))e O(√ ln d) where d = i≤n d i and is the discriminant of L. We use this method to compute the ideal class group of the maximal order O L of L in time Poly(log())e O(√ log d). When log(d) ≤ log(log()) c for some constant c < 2, these methods run in polynomial time. We implemented our algorithm using Sage 7.5.1.

Norm relations and computational problems in number fields

“…Indeed, every nontrivial element of G has order p or q, there is a unique subgroup of order p, there are p subgroups of order q, and every nontrivial element is contained in exactly one of them. This is the relation used by Wada [41], Bauch, Bernstein, de Valence, Lange and van Vredendaal [8] as well as by Biasse and van Vredendaal [10].…”

“…These generalize those classical relations between norms where the coefficient of the trivial group is nonzero, which are exactly the relations one needs when determining invariants of K using subfields. Although the terminology is new, these norm relations have been used in an ad-hoc way implicitly for G = C 2 × C 2 by Wada [41], [8] and Biasse-van Vredendaal [10] and for G = C 3 × C 3 by Parry [35] and Lesavourey, Plantard and Susilo [1]. We give a systematic study of these relations and proof the following existence result (see Theorem 2.11).…”

“…In recent work of Bauch, Bernstein, de Valence, Lange and van Vredendaal [8], it is described how to reduce the computation of principal ideal generators in multiquadratic fields (that is, G = C n 2 ) to quadratic subfields, thus for the first time improving (both in theory and practice) upon classical algorithms by exploiting subfields. This was then generalized to the computation of S-units by Biasse and van Vredendaal [10] and to multicubic fields (that is, G = C n 3 ) by Lesavourey, Plantard and Susilo [1].…”

An algorithm for computing the Stickelberger elements for imaginary multiquadratic fields

“…Ввиду большой группы Галуа интересными полями являются мультиквадратичные. Недавние работы по эффективному вычислению короткого вектора в мультиквадратичных полях [2] и по вычислению группы классов [3] подводят к вопросу нахождения коротких векторов в произвольных идеалах. Следуя примеру круговых полей, логично обратить внимание на структуру идеала Штикельбергера для мультиквадратичных расширений.…”

A proof of the conjectured run time of the Hafner-McCurley class group algorithm

“…The methods of [10] can be specialized to the case of cyclotomic fields for a better asymptotic complexity [9]. Based on [2], Biasse and van Vredendaal [16] described an algorithm for computing the class group of…”