Mathematics of Public Key Cryptography

Steven Galbraith


Cambridge University Press



Full Extended Text

NOTE: Most of these chapters are "extended versions" of chapters in the book and so have additional material. Chapter 19a is an additional chapter. Section/Theorem/Lemma/page numberings do not necessarily match those in the published version of the book.

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Table of contents

Table of notation
1. Introduction

Part I: Background
2. Basic Algorithms
3. Hash Functions

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 Reduction
18. Algorithms for the Closest and Shortest Vector Problem
19. Coppersmith's Method and Other Applications
19a. Cryptosystems Based on Lattices (does not appear in published version of book)

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

A. Background Mathematics
B. Hints and Solutions to Exercises