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Mathematics of Public Key Cryptography

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.

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