Abstract. The standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of Z"' in a suitable box. However, it suffices to consider those x e Z'" which lie in a suitable ellipsoid having a much smaller volume than the box. We show in this paper that searching through that ellipsoid is in many cases much more efficient. If combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions. Several randomly constructed numerical examples illustrate the superiority of our new method over the known ones.
The problem of determining shortest vectors and reduced bases or successive minima of lattices often occurs in algebra and number theory. Nevertheless, computational methods for the solution hardly exist in the literature. It is the aim of this paper to develop efficient algorithms for this purpose.
The software package KANT V4 for computations in algebraic number fields is now available in version 4. In addition a new user interface has been released. We will outline the features of this new software package.
Now in paperback, this classic book is addressed to all lovers of number theory. On the one hand, it gives a comprehensive introduction to constructive algebraic number theory, and is therefore especially suited as a textbook for a course on that subject. On the other hand many parts go beyond an introduction and make the user familiar with recent research in the field. New methods which have been developed for experimental number theoreticians are included along with new and important results. Both computer scientists interested in higher arithmetic and those teaching algebraic number theory will find the book of value.
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