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.
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… (More)
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… (More)
It is shown how the use of a certain integral basis for cyclotomic fields enables one to perform the basic operations in their ring of integers efficiently. In particular, from the representation with respect to this basis, one obtains immediately the smallest possible cyclotomic field in which a given sum of roots of unity lies. This is of particular… (More)
We review the well-known relation between Lucas sequences and exponentiation. This leads to the observation that certain public-key cryptosystems that are based on the use of Lucas sequences have some elementary properties their re-inventors were apparently not aware of. In particular, we present a chosen-message forgery for 'LUC' (cf. [21; 25]), and we… (More)
XTR is a general method that can be applied to 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 posed in , concerning the size of XTR-like representations of elements… (More)
For each integer n, let Sn be the set of all class number quotients h(K)/h(K) for number fields K and K of degree n with the same zeta-function. In this note we will give some explicit results on the finite sets Sn, for small n. For example, for every x ∈ Sn with n ≤ 15, x or x −1 is an integer that is a prime power dividing 2 14 · 3 6 · 5 3 .
Brauer and Kuroda showed in the fifties how in a Galois extension of number fields, relations between permutation characters of subgroups provide relations between in-variants, such as the discriminant, class number and regulator, of the corresponding intermediate fields. In this paper we investigate various computational aspects of these relations, we… (More)