Mathematics of Public Key Cryptography

by Steven D. Galbraith

Overview

Public key cryptography is a major interdisciplinary subject with many real-world applications, such as digital signatures. A strong background in the mathematics underlying public key cryptography is essential for a deep understanding of the subject, and this book provides exactly that for students and researchers in mathematics, computer science and electrical engineering. Carefully written to communicate the major ideas and techniques of public key cryptography to a wide readership, this text is enlivened throughout with historical remarks and insightful perspectives on the development of the subject. Numerous examples, proofs and exercises make it suitable as a textbook for an advanced course, as well as for self-study. For more experienced researchers it serves as a convenient reference for many important topics: the Pollard algorithms, Maurer reduction, isogenies, algebraic tori, hyperelliptic curves and many more.

Product Details

ISBN-13:	9781107013926
Publisher:	Cambridge University Press
Publication date:	03/15/2012
Pages:	630
Product dimensions:	7.01(w) x 10.00(h) x 1.38(d)

About the Author

Steven D. Galbraith is a leading international authority on the mathematics of public key cryptography. He is an Associate Professor in the Department of Mathematics at the University of Auckland.

Table of Contents

Preface; Acknowledgements; 1. Introduction; Part I. Background: 2. Basic algorithmic number theory; 3. Hash functions and MACs; Part II. Algebraic Groups: 4. Preliminary remarks on algebraic groups; 5. Varieties; 6. Tori, LUC and XTR; 7. Curves and divisor class groups; 8. Rational maps on curves and divisors; 9. Elliptic curves; 10. Hyperelliptic curves; Part III. Exponentiation, Factoring and Discrete Logarithms: 11. Basic algorithms for algebraic groups; 12. Primality testing and integer factorisation using algebraic groups; 13. Basic discrete logarithm algorithms; 14. Factoring and discrete logarithms using pseudorandom walks; 15. Factoring and discrete logarithms in subexponential time; Part IV. Lattices: 16. Lattices; 17. Lattice basis reduction; 18. Algorithms for the closest and shortest vector problems; 19. Coppersmith's method and related applications; Part V. Cryptography Related to Discrete Logarithms: 20. The Diffie–Hellman problem and cryptographic applications; 21. The Diffie–Hellman problem; 22. Digital signatures based on discrete logarithms; 23. Public key encryption based on discrete logarithms; Part VI. Cryptography Related to Integer Factorisation: 24. The RSA and Rabin cryptosystems; Part VII. Advanced Topics in Elliptic and Hyperelliptic Curves: 25. Isogenies of elliptic curves; 26. Pairings on elliptic curves; Appendix A. Background mathematics; References; Author index; Subject index.

Editorial Reviews

".. the reader is assumed to have a minimum background in group theory, algorithms and complexity, together with a basic knowledge of abstract algebra that includes ring and field theory. The book is suitable in principle for PhD. students in mathematics and related areas."
-Mathematical Reviews

From the Publisher