| Preface for the Second Edition Preface 1 Integers 1.1 Basics 1.2 Divisibility 1.3 Representation of Integers 1.4 O-and Ω-Notation 1.5 Cost of Addition, Multiplication, and Division with Remainder 1.6 Polynomial Time 1.7 Greatest Common Divisor 1.8 Euclidean Algorithm 1.9 Extended Euclidean Algorithm 1.10 Analysis of the Extended Euclidean Algorithm 1.11 Factoring into Primes 1.12 Exercises 2 Congruences and Residue Class Rings 2.1 Congruences 2.2 Semigroups 2.3 Groups 2.4 Residue Class Ring 2.5 Fields 2.6 Division in the Residue Class Ring 2.7 Analysis of the Operations in the Residue Class Ring 2.8 Multiplicative Group of Residues mod rn 2.9 Order of Group Elements 2.10 Subgroups 2.11 Fermat's Little Theorem 2.12 Fast Exponentiation 2.13 Fast Evaluation of Power Products 2.14 Computation of Element Orders 2.15 The Chinese Remainder Theorem 2.16 Decomposition of the Residue Class Ring 2.17 A Formula for the Euler фo-Function 2.18 Polynomials 2.19 Polynomials over Fields 2.20 Construction of Finite Fields 2.21 The Structure of the Unit Group of Finite Fields 2.22 Structure of the Multiplicative Group of Residues Modulo a Prime Number 2.23 Exercises 3 Encryption 3.1 Encryption Schemes 3.2 Symmetric and Asymmetric Cryptosystems 3.3 Cryptanalysis 3.4 Alphabets and Words 3.5 Permutations 3.6 Block Ciphers 3.7 Multiple Encryption 3.8 The Use of Block Ciphers 3.9 Stream Ciphers 3.10 The Affine Cipher 3.11 Matrices and Linear Maps 3.12 Affine Linear Block Ciphers 3.13 Vigenere, Hill, and Permutation Ciphers …… 4 Probability and Perfect Secrecy 5 DES 6 AES 7 Prime Number Generation 8 Public-key Encryption 9 Factoring 10 Discrete Logarithms 11 Cryptographic Hash Functions 12 Digital Signatures 13 Other Systems 14 Identification 15 Secret Sharing 16 Public-key Infrastructures Solutions of the exercises References Index |
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