Saturday, May 12, 2007

Complexity and Cryptography, An Introduction

book cover

This book originated in a well-established yet constantly evolving course on Complexity and Cryptography which we have both given to final year Mathematics undergraduates at Oxford for many years. It has also formed part of an M.Sc. course on Mathematics and the Foundations of Computer Science, and has been the basis for a more recent course on Randomness and Complexity for the same groups of students.

One of the main motivations for setting up the course was to give mathematicians, who traditionally meet little in the way of algorithms, a taste for the beauty and importance of the subject. Early on in the book the reader will have
gained sufficient background to understand what is now regarded as one of the top ten major open questions of this century, namely the P = NP question. At the same time the student is exposed to the mathematics underlying the security
of cryptosystems which are now an integral part of the modern ‘email age’. Although this book provides an introduction to many of the key topics in complexity theory and cryptography, we have not attempted to write a comprehensive text. Obvious omissions include cryptanalysis, elliptic curve cryptography, quantum cryptography and quantum computing. These omissions have allowed us to keep the mathematical prerequisites to a minimum. Throughout the text the emphasis is on explaining the main ideas and proving the mathematical results rigorously. Thus we have not given every result in complete generality.

The exercises at the end of many sections of the book are in general meant to be routine and are to be used as a check on the understanding of the preceding principle; the problems at the end of each chapter are often harder.

Chapter 01: Basics of cryptography
Chapter 02: Complexity theory
Chapter 03: Non-deterministiccomputation
Chapter 04: Probabilistic computation
Chapter 05: Symmetric cryptosystems
Chapter 06: One wayfunctions
Chapter 07: Public key cryptography
Chapter 08: Digital signatures
Chapter 09: Key establishment protocols
Chapter 10: Secure encryption
Chapter 11: Identification schemes
Appendix 1: Basic mathematical background
Appendix 2: Graph theory definitions
Appendix 3: Algebra and number theory
Appendix 4: Probability theory
Appendix 5: Hints to selected exercises and problems
Appendix 6: Answers to selected exercises and problems

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