Wieb Bosma

The MAGMA algebra system I: the user language

Abstract In the first of two papers on MAGMA , a new system for computational algebra, we present the MAGMA language, outline the design principles and theoretical background, and indicate its scope and use. Particular attention is given to the constructors for structures, maps, and sets.

Some metrical observations on the approximation by continued fractions

Abstract The following conjecture of H.W. Lenstra is proved. Denote by p n / q n , n = 1,2,… the sequence of continued fraction convergents of the irrational number x and define θ n ( x ): = q n | q n x - p n |. Then for every z , 0≤ z ≤1, one has for almost all x Similar results are proved for other functions connected with the regular continued fraction expansion, such as the quotient of |x− p n−1 q n−1 | and |x− p n q n |, as well as for other type of expansions, such as the nearest integer and singular continued fractions. The main tool is the natural extension of the operator x ↦ (1/ x ) − [(1/ x )], recently studied by Hitoshi Nakada.

Programming with algebraic structures: design of the MAGMA language

MAGMA is a new software system for computational algebra, number theory and geometry whose design is centred on the concept of algebraic structure (magma). The use of algebraic structure as a design paradigm provides a natural strong typing mechanism. Further, structures and their morphisms appear in the language as first class objects. Standard mathematical notions are used for the basic data types. The result is a powerful, clean language which deals with objects in a mathematically rigorous manner. The conceptual and implementation ideas behind MAGMA will be examined in this paper. This conceptual base differs significantly from those underlying other computer algebra systems.

Looking beyond XTR

XTR is a general methodthat can be appliedto discrete logarithm based cryptosystems in extension fields of degree six, providing a compact representation of the elements involved. In this paper we present a precise formulation of the Brouwer-Pellikaan-Verheul conjecture, originally posedin [4], concerning the size of XTR-like representations of elements in extension fields of arbitrary degree. If true this conjecture wouldpro vide even more compact representations of elements than XTR in extension fields of degree thirty. We test the conjecture by experiment, showing that in fact it is unlikely that such a compact representation of elements can be achieved in extension fields of degree thirty.

Optimal continued fractions

A new continued fraction algorithm is given and analyzed. It yields approximations for an irrational real number by generating a subsequence of its regular continued fraction convergents that is optimal in several respects.

Faster Primality Testing

Several major improvements to the Jacobi sum primality testing algorithm will speed it up in such a way that proving primality of primes of up to 500 digits will be a matter of routine. Primes of about 800 digits will take at most one night on a Cray.

Computations with finitely generated modules over Dedekind rings

In computer algebra the use of normal forms for matrices is of eminent importance. Especially, Hermite and Smith normal form techniques are frequently used for various computational problems over Euclidean rings. In this paper we discuss a generalization of these concepts to Dedekind rings. We consider the problem of normal forms for matrices in the context of basis transformations for finitely generated modules. If R is a Euclidean ring and A is in Rm” then the columns ai of A can be viewed as generators of an R-module 114 of Rm x 1. Hence, the matrix A describes the presentation of the generators ai by the canonical basis el, . . . . e ~ of Rrn x 1. Since any submodule A4 of Rm x 1 is a free module of rank k < m, there is a matrix T of GL(n, R) such that the first k columns of AT form a basis of A4 and the remaining n – k columns are O. Moreover, AT can be assumed to be in Herm.ite normal form, i.e., in column j (for 1 < j < k) the nonzero element of maximal row index i is contained in a representative set of R modulo its unit group U(R) and all elements in the same row i with column indices 1< j are zero. Similarly, if we allow regular transformations of A from both sides we obtain the Smith normal form of A. If R is not Euclidean but just a principal entire ring these concepts still hold. But usually the computation of generators of ideals becomes impracticable without a Euclidean algorithm. In this paper we discuss a generfllzation to Dedekind rings. We make use of the fact that every ideal a of a Dedekind ring R can be generated by two elements one of which can be chosen quite arbitrarily in

Metrical theory for optimal continued fractions

Received November 12, 1987; revised March 13, 1989 The ergodic system underlying the optimal continued fraction algorithm is introduced and studied. In particular the distribution of the sequence B”(x)” z r, which measures how well a number x is approximated by its convergent% is derived for almost all irrational numbers.

Lattices of compatibly embedded finite fields

Abstract The design of a computational facility for finite fields that allows complete freedom in the manner in which fields are constructed, is complicated by the fact that a field of fixed isomorphism type K may be constructed in many different ways. It is desirable that the user be able to perform simultaneous computations in different versions of K in such a way that isomorphisms identifying elements in the different versions are applied automatically whenever necessary. This paper presents a coherent scheme for solving this problem based on an efficient method for compatibly embedding one field within another. This scheme forms a central component of the MAGMA module for finite fields. The paper also outlines the different representations of finite fields employed in the package and comments briefly on some of the major algorithms.

Approximation by mediants

The distribution is determined of some sequences that measure how well a number is approximated by its mediants (or intermediate continued fraction convergents). The connection with a theorem of Fatou, as well as a new proof of this, is given