Universitext

Elementary Number Theory, Cryptography and Codes

Authors: Baldoni, M. Welleda, Ciliberto, Ciro, Piacentini Cattaneo, G.M.

  • Nice, elementary introduction to number theory on two of its real-world applications

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About this Textbook

In this volume one finds basic techniques from algebra and number theory (e.g. congruences, unique factorization domains, finite fields, quadratic residues, primality tests, continued fractions, etc.) which in recent years have proven to be extremely useful for applications to cryptography and coding theory. Both cryptography and codes have crucial applications in our daily lives, and they are described here, while the complexity problems that arise in implementing the related numerical algorithms are also taken into due account. Cryptography has been developed in great detail, both in its classical and more recent aspects. In particular public key cryptography is extensively discussed, the use of algebraic geometry, specifically of elliptic curves over finite fields, is illustrated, and a final chapter is devoted to quantum cryptography, which is the new frontier of the field. Coding theory is not discussed in full; however a chapter, sufficient for a good introduction to the subject, has been devoted to linear codes. Each chapter ends with several complements and with an extensive list of exercises, the solutions to most of which are included in the last chapter.

Though the book contains advanced material, such as cryptography on elliptic curves, Goppa codes using algebraic curves over finite fields, and the recent AKS polynomial primality test, the authors' objective has been to keep the exposition as self-contained and elementary as possible. Therefore the book will be useful to students and researchers, both in theoretical (e.g. mathematicians) and in applied sciences (e.g. physicists, engineers, computer scientists, etc.) seeking a friendly introduction to the important subjects treated here. The book will also be useful for teachers who intend to give courses on these topics.

Reviews

From the reviews:

"The aim of the book is to introduce the basic concepts on the two topics of Cryptography and Error Correcting Codes, presented as applications of Number Theory and Finite Fields … . The book is addressed to undergraduate students and is as self-contained as possible. It is well written and rigorous and each chapter is complemented with three lists of exercises … . it is good news to have the translation into English of this work that will allow a much wider public access." (Juan Tena Ayuso, Zentralblatt MATH, Vol. 1162, 2009)


Table of contents (9 chapters)

  • A round-up on numbers

    Pages 1-85

  • Computational complexity

    Pages 87-113

  • From infinite to finite

    Pages 115-147

  • Finite is not enough: factoring integers

    Pages 149-212

  • Finite fields and polynomial congruences

    Pages 213-260

Buy this book

eBook n/a
  • ISBN 978-3-540-69200-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
Softcover n/a
  • ISBN 978-3-540-69199-0
  • Free shipping for individuals worldwide
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Bibliographic Information

Bibliographic Information
Book Title
Elementary Number Theory, Cryptography and Codes
Authors
Translated by
Gewurz, D.
Series Title
Universitext
Copyright
2009
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
Distribution Rights
Distribution rights for India: CBS Publishers, New Delhi, India
eBook ISBN
978-3-540-69200-3
DOI
10.1007/978-3-540-69200-3
Softcover ISBN
978-3-540-69199-0
Series ISSN
0172-5939
Edition Number
1
Number of Pages
XVI, 522
Number of Illustrations and Tables
10 b/w illustrations
Additional Information
Translated from the Italian by Daniele A. Gewurz
Topics