An introduction to mathematical cryptography
By: by Jeffrey Hoffstein, Jill Pipher, J.H. Silverman.
Contributors: Pipher, Jill Catherine, 1955-; Silverman, Joseph H., 1955-
Series: Undergraduate texts in mathematics
Publisher: Springer
Published: 2008
Place of publication: New York; London
Language: eng
Number of pages: 523
ISBN-13: 9780387779935
Subjects: Coding theory; Cryptography -- Mathematics
Description: This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include: * classical cryptographic constructions, such as Diffie-Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures; * fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms; * an in-depth treatment of important recent cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem. This book is an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online.
Notes: Includes bibliographical references (p. 493-500) and index.
Table of contents: title: Preface
pagenum: v; title: Introduction
pagenum: xi; label: Chapter 1
title: An Introduction to Cryptography
pagenum: 1; label: 1.1
title: Simple substitution ciphers
pagenum: 1; label: 1.2
title: Divisibility and greatest common divisors
pagenum: 10; label: 1.3
title: Modular arithmetic
pagenum: 19; label: 1.4
title: Prime numbers, unique factorization, and finite fields
pagenum: 26; label: 1.5
title: Powers and primitive roots in finite fields
pagenum: 29; label: 1.6
title: Cryptography before the computer age
pagenum: 34; label: 1.7
title: Symmetric and asymmetric ciphers
pagenum: 36; title: Exercises
pagenum: 47; label: Chapter 2
title: Discrete Logarithms and Diffie-Hellman
pagenum: 59; label: 2.1
title: The birth of public key cryptography
pagenum: 59; label: 2.2
title: The discrete logarithm problem
pagenum: 62; label: 2.3
title: Diffie-Hellman key exchange
pagenum: 65; label: 2.4
title: The ElGamal public key cryptosystem
pagenum: 68; label: 2.5
title: An overview of the theory of groups
pagenum: 72; label: 2.6
title: How hard is the discrete logarithm problem?
pagenum: 75; label: 2.7
title: A collision algorithm for the DLP
pagenum: 79; label: 2.8
title: The Chinese remainder theorem
pagenum: 81; label: 2.9
title: The Pohlig-Hellman algorithm
pagenum: 86; label: 2.10
title: Rings, quotients, polynomials, and finite fields
pagenum: 92; title: Exercises
pagenum: 105; label: Chapter 3
title: Integer Factorization and RSA
pagenum: 113; label: 3.1
title: Euler's formula and roots modulo pq
pagenum: 113; label: 3.2
title: The RSA public key cryptosystem
pagenum: 119; label: 3.3
title: Implementation and security issues
pagenum: 122; label: 3.4
title: Primality testing
pagenum: 124; label: 3.5
title: Pollard's p-1 factorization algorithm
pagenum: 133; label: 3.6
title: Factorization via difference of squares
pagenum: 137; label: 3.7
title: Smooth numbers and sieves
pagenum: 146; label: 3.8
title: The index calculus and discrete logarithms
pagenum: 162; label: 3.9
title: Quadratic residues and quadratic reciprocity
pagenum: 165; label: 3.10
title: Probabilistic encryption
pagenum: 172; title: Exercises
pagenum: 176; label: Chapter 4
title: Combinatorics, Probability, and Information Theory
pagenum: 189; label: 4.1
title: Basic principles of counting
pagenum: 190; label: 4.2
title: The Vigenère cipher
pagenum: 196; label: 4.3
title: Probability theory
pagenum: 210; label: 4.4
title: Collision algorithms and meet-in-the-middle attacks
pagenum: 227; label: 4.5
title: Pollard's p method
pagenum: 234; label: 4.6
title: Information theory
pagenum: 243; label: 4.7
title: Complexity Theory and P versus NP
pagenum: 258; title: Exercises
pagenum: 262; label: Chapter 5
title: Elliptic Curves and Cryptography
pagenum: 279; label: 5.1
title: Elliptic curves
pagenum: 279; label: 5.2
title: Elliptic curves over finite fields
pagenum: 286; label: 5.3
title: The elliptic curve discrete logarithm problem
pagenum: 290; label: 5.4
title: Elliptic curve cryptography
pagenum: 296; label: 5.5
title: The evolution of public key cryptography
pagenum: 301; label: 5.6
title: Lenstra's elliptic curve factorization algorithm
pagenum: 303; label: 5.7
title: Elliptic curves over \(\mathbb{F}_2\) and over \(\mathbb{F}_{2^k}\)
pagenum: 308; label: 5.8
title: Bilinear pairings on elliptic curves
pagenum: 315; label: 5.9
title: The Weil pairing over fields of prime power order
pagenum: 325; label: 5.10
title: Applications of the Weil pairing
pagenum: 334; title: Exercises
pagenum: 339; label: Chapter 6
title: Lattices and Cryptography
pagenum: 349; label: 6.1
title: A congruential public key cryptosystem
pagenum: 349; label: 6.2
title: Subset-sum problems and knapsack cryptosystems
pagenum: 352; label: 6.3
title: A brief review of vector spaces
pagenum: 359; label: 6.4
title: Lattices: Basic definitions and properties
pagenum: 363; label: 6.5
title: Short vectors in lattices
pagenum: 370; label: 6.6
title: Babai's algorithm
pagenum: 379; label: 6.7
title: Cryptosystems based on hard lattice problems
pagenum: 383; label: 6.8
title: The GGH public key cryptosystem
pagenum: 384; label: 6.9
title: Convolution polynomial rings
pagenum: 387; label: 6.10
title: The NTRU public key cryptosystem
pagenum: 392; label: 6.11
title: NTRU as a lattice cryptosystem
pagenum: 400; label: 6.12
title: Lattice reduction algorithms
pagenum: 403; label: 6.13
title: Applications of LLL to cryptanalysis
pagenum: 418; title: Exercises
pagenum: 422; label: Chapter 7
title: Digital Signatures
pagenum: 437; label: 7.1
title: What is a digital signature?
pagenum: 437; label: 7.2
title: RSA digital signatures
pagenum: 440; label: 7.3
title: ElGamal digital signatures and DSA
pagenum: 442; label: 7.4
title: GGH lattice-based digital signatures
pagenum: 447; label: 7.5
title: NTRU digital signatures
pagenum: 450; title: Exercises
pagenum: 458; label: Chapter 8
title: Additional Topics in Cryptography
pagenum: 465; label: 8.1
title: Hash functions
pagenum: 466; label: 8.2
title: Random numbers and pseudorandom number generators
pagenum: 468; label: 8.3
title: Zero-knowledge proofs
pagenum: 470; label: 8.4
title: Secret sharing schemes
pagenum: 473; label: 8.5
title: Identification schemes
pagenum: 474; label: 8.6
title: Padding schemes and the random oracle model
pagenum: 476; label: 8.7
title: Building protocols from cryptographic primitives
pagenum: 479; label: 8.8
title: Hyperelliptic curve cryptography
pagenum: 480; label: 8.9
title: Quantum computing
pagenum: 483; label: 8.10
title: Modern symmetric cryptosystems: DES and AES
pagenum: 485; title: List of Notation
pagenum: 489; title: References
pagenum: 493; title: Index
pagenum: 501