Codes, Cryptology and Curves with Computer Algebra

This well-balanced text touches on theoretical and applied aspects of protecting digital data. The reader is provided with the basic theory and is then shown deeper fascinating detail, including the current state of the art. Readers will soon become familiar with methods of protecting digital data while it is transmitted, as well as while the data is being stored. Both basic and advanced error-correcting codes are introduced together with numerous results on their parameters and properties. The authors explain how to apply these codes to symmetric and public key cryptosystems and secret sharing. Interesting approaches based on polynomial systems solving are applied to cryptography and decoding codes. Computer algebra systems are also used to provide an understanding of how objects introduced in the book are constructed, and how their properties can be examined. This book is designed for Masters-level students studying mathematics, computer science, electrical engineering or physics.

Table of Contents:
Preface Ruud Pellikaan; 1. Error-correcting codes Ruud Pellikaan and Xin-Wen Wu; 2. Code constructions and bounds on codes Ruud Pellikaan and Xin-Wen Wu; 3. Weight enumeration Relinde Jurrius, Ruud Pellikaan and Xin-Wen Wu; 4. Cyclic codes Ruud Pellikaan; 5. Polynomial codes Ruud Pellikaan; 6. Algebraic decoding Ruud Pellikaan and Xin-Wen Wu; 7. Complexity and decoding Stanislav Bulygin, Ruud Pellikaan and Xin-Wen Wu; 8. Codes and related structures Relinde Jurrius and Ruud Pellikaan; 9. Cryptology Stanislav Bulygin; 10. Gröbner bases for coding and cryptology Stanislav Bulygin; 11. Codes on curves Ruud Pellikaan; 12. Coding and cryptology with computer algebra Stanislav Bulygin; References; Index.

About the Author :
Ruud Pellikaan has tenure at the Technische Universiteit Eindhoven, The Netherlands where his research has shifted from a devotion to coding theory, particularly algebraic geometry codes and their decoding, to code-based cryptography. He previously served as an associate editor of the IEEE Transactions of Information Theory and has organised several conferences. Xin-Wen Wu is a Senior Lecturer at the School of Information and Communication Technology, Griffith University, Queensland. His research interests include coding theory and information theory, cyber and data security, applied cryptography communications and networks. He has published extensively in these areas and is a senior member of Institute of Electrical and Electronics Engineers (IEEE). Stanislav Bulygin works as a technology specialist and product manager in the field of IT security and banking services. He previously worked as a researcher focusing on cryptology and IT security at the Technical University of Darmstadt, Germany. His main research activities were connected to the theory of error-correcting codes and their use in cryptography, quantum resistant cryptosystems and algebraic methods in cryptology. Relinde Jurrius is an Assistant Professor at the Université de Neuchatel, Switzerland. Her research interests are in coding theory, network coding and its connection with other branches of mathematics such as matroid theory, algebraic and finite geometry, and combinatorics. Apart from research and teaching, she is active in organising outreach activities, including a math camp for high school students, a public open day for the Faculty of Science and extra-curricular activities for elementary school children.

Review :
'The book under review is intended as an introduction to the field for beginning graduate students. The authors do a good job of covering a wide range of topics and keeping the discussion detailed while still as elementary as one can hope to make it.' Darren Glass, MAA Reviews 'While 'coding' may commonly connote confidential communication and security for sensitive data, coding also enters the engineering of information transmission and retrieval, simply for efficient resilience against mechanical error and corrupting noise. From these two purposes rise the two distinct subjects of cryptology and error-correction, receiving here an unusual, unified treatment. Good codes spring from diverse directions, since so many branches of mathematics inform their development: combinatorics, linear algebra, finite fields, ring theory, algebraic geometry, and computer algebra. The girth of this volume reflects the reasonably detailed exposition of all this background material, most of it likely new to engineering students (but students of pure mathematics should also read this book for practical applications of seemingly abstract material they have likely studied). The authors maintain a high level of rigor, keeping all proofs short by astute organization without ever stinting on detail.' D. V. Feldman, Choice 'This book provides a fine exposition of the topics to those students who are novices to the field. At the same time it will also be of interest to readers who are already familiar with some of the concepts discussed in the book. It provides a valuable schematic summary and consolidated overview of the field.' S. V. Nagaraj, SIGACT News I was impressed by the scope of the book: many topics in algebraic coding theory are addressed and now collected in one book. Someone reading the entire book, will obtain a very good overview of algebraic coding theory. Peter Beelen, Nieuw Archief voor Weskunde