Subexponential Algorithms for Class Group and Unit Computations

Abstract

We describe in detail the implementation of an algorithm which computes the class group and the unit group of a general number field, and solves the principal ideal problem. The basic ideas of this algorithm are due to J. Buchmann. New ideas are the use of LLL-reduction of an ideal in a given direction which replaces the notion of neighbour, and the use of complex logarithmic embeddings of elements which plays a crucial role. Heuristically the algorithm performs in sub-exponential time with respect to the discriminant for fixed degree, and performs well in practice.

Subexponential class group and unit group computation in large degree number fields

“…The p-adic method and the compact representation algorithm were inspired by [7] while the relation search significantly differs from [7]. The results of [7] and this paper are not comparable because [7] consists of practical improvements on subexponential methods for fixed degree classes of number fields [9,12] while here we consider classes of number fields of degree going to infinity.…”

“…This complexity is valid for fixed degree n and ∆ tending to infinity. Practical improvements to Buchmann's algorithm were presented in [12] by Cohen, Diaz Y Diaz and Olivier. More recently, Biasse described an algorithm for computing the ideal class group and the unit group of O = Z[θ] in heuristic complexity bounded by L ∆ ( 1 3 , c) for some c > 0 valid in certain classes of number fields.…”

Topics in computational algebraic number theory

“…LLL pseudo-reduction. This notion was introduced by Buchmann [7], and Cohen et al [12]. Let A an integral ideal, and α ∈ A be the first element of a reduced basis of the lattice (A, T 2 ).…”

Solving Thue equations without the full unit group

“…It is well-known [6,7] that computing a system of fundamental units of a given number field is a hard problem; indeed, it seems to be the major, if not the sole, bottleneck of the method.…”

“…Computing a maximal system of units Computing a system of fundamental units is often a surprisingly difficult task. The currently most popular method, which is due to Hafner and McCurley [9] for quadratic fields and to Buchmann, Cohen, Diaz y Diaz and Olivier [5,7] for general fields, produces a system of units which is always of maximal rank. Under the assumption of the generalized Riemann hypothesis, it can be proved to be fundamental in decent time.…”