The Proceedings contain twenty selected, refereed contributions arising from the International Conference on Public-Key Cryptography and Computational Number Theory held in Warsaw, Poland, on September 11-15, 2000. The conference, attended by eightyfive mathematicians from eleven countries, was organized by the Stefan Banach International Mathematical Center. This volume contains articles from leading experts in the world on cryptography and computational number theory, providing an account of the state of research in a wide variety of topics related to the conference theme. It is dedicated to the memory of the Polish mathematicians Marian Rejewski (1905-1980), Jerzy Rooycki (1909-1942) and Henryk Zygalski (1907-1978), who deciphered the military version of the famous Enigma in December 1932 ? January 1933. A noteworthy feature of the volume is a foreword written by Andrew Odlyzko on the progress in cryptography from Enigma time until now."

The volume is devoted to the interaction of modern scientific computation and classical number theory. The contributions, ranging from effective finiteness results to efficient algorithms in elementary, analytical and algebraic number theory, provide a broad view of the methods and results encountered in the new and rapidly developing area of computational number theory. Topics covered include finite fields, quadratic forms, number fields, modular forms, elliptic curves and diophantine equations. In addition, two new number theoretical software packages, KANT and SIMATH, are described in detail with emphasis on algorithms in algebraic number theory.

Although the Lucas sequences were known to earlier investigators such as Lagrange, Legendre and Genocchi, it is because of the enormous number and variety of results involving them, revealed by Édouard Lucas between 1876 and 1880, that they are now named after him.  Since Lucas’ early work, much more has been discovered concerning these remarkable mathematical objects, and the objective of this book is to provide a much more thorough discussion of them than is available in existing monographs.  In order to do this a large variety of results, currently scattered throughout the literature, are brought together. Various sections are devoted to the intrinsic arithmetic properties of these sequences, primality testing, the Lucasnomials, some associated density problems and Lucas’ problem of finding a suitable generalization of them. Furthermore, their application, not only to primality testing, but also to integer factoring, efficient solution of quadratic and cubic congruences, cryptography and Diophantine equations are briefly discussed.  Also, many historical remarks are sprinkled throughout the book, and a biography of Lucas is included as an appendix.Much of the book is not intended to be overly detailed. Rather, the objective is to provide a good, elementary and clear explanation of the subject matter without too much ancillary material. Most chapters, with the exception of the second and the fourth, will address a particular theme, provide enough information for the reader to get a feel for the subject and supply references to more comprehensive results.  Most of this work should be accessible to anyone with a basic knowledge of elementary number theory and abstract algebra.  The book’s intended audience is number theorists, both professional and amateur, students and enthusiasts.

The objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics. The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi's unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory. The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics.

In the summer of 1981 Allen Gersho organized the first major open co, ifcrcncc c ci devoted to cryptologic research This meeting, Crypto '81, was held at the Universitc nl California campus in Santa Barbara Since then the Crypto' conference has become dii annual event These are the proceedings of the fifth1 of these confercnces, Crypt0 XS Each section of this volume corresponds to a session at the meeting. Thr paperk were accepted by the program committee, sometimes on the basis of an abstract only. and appear here without having been otherwise refereed. The last section contains papers lor some of the impromptu talks given at the traditional rump session. Each of thew pq1ci-k was refereed by a single member of the program committee. An author index;is wcll;I\ 'I keyword index, the entries for which were mainly supplied by the authors. appear at 11ic end of the volume. Unfortunately, two of the papers accepted for presentation at Crypto '85 could noi be included in this book they are: Unique Extrapolation of Polynomial Recurrences J. C. Lagarias and J. A. Reeds (A. T. & T Bell Labs) Some Cryptographic Applications of Permutation I'olynomials and Permutation Functions Rupert Nobarer (Universitat fur Bildungswissenschaftttn, Austria) It is my great pleasure to acknowledge the efforts of all of those who contributed to making these proceedings possible: the authors, program committee, other orgmizrrs ot the meeting, IACR officers and directors. and aU the attendees.