Mathematics of Public Key Cryptography

This advanced graduate textbook gives an authoritative and insightful description of the major ideas and techniques of public key cryptography.

Steven D. Galbraith (Author)

9781107013926, Cambridge University Press

Hardback, published 15 March 2012

630 pages, 11 b/w illus. 750 exercises
25.4 x 17.8 x 3.5 cm, 1.28 kg

'… the book gathers the main mathematical topics related to public key cryptography and provides an excellent source of information for both students and researchers interested in the field.' Juan Tena Ayuso, Zentralblatt MATH

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.

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.

Subject Areas: Algorithms & data structures [UMB], Discrete mathematics [PBD], Coding theory & cryptology [GPJ]