Mathematics of public key cryptography / Stephen D. Galbraith.

Cover; MATHEMATICS OF PUBLIC KEY CRYPTOGRAPHY; Title; Copyright; Contents; Preface; Acknowledgements; 1: Introduction; 1.1 Public key cryptography; 1.2 The textbook RSA cryptosystem; 1.3 Formal definition of public key cryptography; 1.3.1 Security of encryption; 1.3.2 Security of signatures; PART I: BACKGROUND; 2: Basic algorithmic number theory; 2.1 Algorithms and complexity; 2.1.1 Randomised algorithms; 2.1.2 Success probability of a randomised algorithm; 2.1.3 Reductions; 2.1.4 Random self-reducibility; 2.2 Integer operations; 2.2.1 Faster integer multiplication; 2.3 Euclid's algorithm.

2.4 Computing Legendre and Jacobi symbols2.5 Modular arithmetic; 2.6 Chinese remainder theorem; 2.7 Linear algebra; 2.8 Modular exponentiation; 2.9 Square roots modulo p; 2.10 Polynomial arithmetic; 2.11 Arithmetic in finite fields; 2.12 Factoring polynomials over finite fields; 2.13 Hensel lifting; 2.14 Algorithms in finite fields; 2.14.1 Constructing finite fields; 2.14.2 Solving quadratic equations in finite fields; 2.14.3 Isomorphisms between finite fields; 2.15 Computing orders of elements and primitive roots; 2.15.1 Sets of exponentials of products.

2.15.2 Computing the order of a group element2.15.3 Computing primitive roots; 2.16 Fast evaluation of polynomials at multiple points; 2.17 Pseudorandom generation; 2.18 Summary; 3: Hash functions and MACs; 3.1 Security properties of hash functions; 3.2 Birthday attack; 3.3 Message authentication codes; 3.4 Constructions of hash functions; 3.5 Number-theoretic hash functions; 3.6 Full domain hash; 3.7 Random oracle model; PART II: ALGEBRAIC GROUPS; 4: Preliminary remarks on algebraic groups; 4.1 Informal definition of an algebraic group; 4.2 Examples of algebraic groups.

4.3 Algebraic group quotients4.4 Algebraic groups over rings; 5: Varieties; 5.1 Affine algebraic sets; 5.2 Projective algebraic sets; 5.3 Irreducibility; 5.4 Function fields; 5.5 Rational maps and morphisms; 5.6 Dimension; 5.7 Weil restriction of scalars; 6: Tori, LUC and XTR; 6.1 Cyclotomic subgroups of finite fields; 6.2 Algebraic tori; 6.3 The group Gq,2; 6.3.1 The torus T2; 6.3.2 Lucas sequences; 6.4 The group Gq,6; 6.4.1 The torus T6; 6.4.2 XTR; 6.5 Further remarks; 6.6 Algebraic tori over rings; 7: Curves and divisor class groups; 7.1 Non-singular varieties; 7.2 Weierstrass equations.

7.3 Uniformisers on curves7.4 Valuation at a point on a curve; 7.5 Valuations and points on curves; 7.6 Divisors; 7.7 Principal divisors; 7.8 Divisor class group; 7.9 Elliptic curves; 8: Rational maps on curves and divisors; 8.1 Rational maps of curves and the degree; 8.2 Extensions of valuations; 8.3 Maps on divisor classes; 8.4 Riemann-Roch spaces; 8.5 Derivations and differentials; 8.6 Genus zero curves; 8.7 Riemann-Roch theorem and Hurwitz genus formula; 9: Elliptic curves; 9.1 Group law; 9.2 Morphisms between elliptic curves; 9.3 Isomorphisms of elliptic curves; 9.4 Automorphisms.

9.5 Twists.

"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."--Provided by publisher.

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Includes bibliographical references and index.

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