Mathematics of public key cryptography /

"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 st...

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Bibliographic Details
Main Author: Galbraith, Steven D. (Author)
Format: eBook
Published: Cambridge : Cambridge University Press, 2012.
Online Access:CONNECT
Table of Contents:
  • 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.