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• Preface
• Contents
• Introduction
• An Introduction to Cryptography
• Simple substitution ciphers
• Divisibility and greatest common divisors
• Modular arithmetic
• Prime numbers, unique factorization, and finite fields
• Powers and primitive roots in finite fields
• Cryptography before the computer age
• Symmetric and asymmetric ciphers
• Exercises
• Discrete Logarithms and Diffie–Hellman
• The birth of public key cryptography
• The discrete logarithm problem
• Diffie–Hellman key exchange
• The ElGamal public key cryptosystem
• An overview of the theory of groups
• How hard is the discrete logarithm problem?
• A collision algorithm for the DLP
• The Chinese remainder theorem
• The Pohlig–Hellman algorithm
• Rings, quotients, polynomials, and finite fields
• Exercises
• Integer Factorization and RSA
• Euler’s formula and roots modulo pq
• The RSA public key cryptosystem
• Implementation and security issues
• Primality testing
• Pollard’s bold0mu mumu ppunitspppp-1 factorization algorithm
• Factorization via difference of squares
• Smooth numbers and sieves
• The index calculus and discrete logarithms
• Quadratic residues and quadratic reciprocity
• Probabilistic encryption
• Exercises
• Combinatorics, Probability, and Information Theory
• Basic principles of counting
• The Vigenère cipher
• Probability theory
• Collision algorithms and meet-in-the-middle attacks
• Pollard’s bold0mu mumu units method
• Information theory
• Complexity Theory and P versus NP
• Exercises
• Elliptic Curves and Cryptography
• Elliptic curves
• Elliptic curves over finite fields
• The elliptic curve discrete logarithm problem
• Elliptic curve cryptography
• The evolution of public key cryptography
• Lenstra’s elliptic curve factorization algorithm
• Elliptic curves over F2k and over F2k
• Bilinear pairings on elliptic curves
• The Weil pairing over fields of prime power order
• Applications of the Weil pairing
• Exercises
• Lattices and Cryptography
• A congruential public key cryptosystem
• Subset-sum problems and knapsack cryptosystems
• A brief review of vector spaces
• Lattices: Basic definitions and properties
• Short vectors in lattices
• Babai’s algorithm
• Cryptosystems based on hard lattice problems
• The GGH public key cryptosystem
• Convolution polynomial rings
• The NTRU public key cryptosystem
• NTRU as a lattice cryptosystem
• Lattice reduction algorithms
• Applications of LLL to cryptanalysis
• Exercises
• Digital Signatures
• What is a digital signature?
• RSA digital signatures
• ElGamal digital signatures and DSA
• GGH lattice-based digital signatures
• NTRU digital signatures
• Exercises
• Additional Topics in Cryptography
• Hash functions
• Random numbers and pseudorandom number generators
• Zero-knowledge proofs
• Secret sharing schemes
• Identification schemes
• Padding schemes and the random oracle model
• Building protocols from cryptographic primitives
• Hyperelliptic curve cryptography
• Quantum computing
• Modern symmetric cryptosystems: DES and AES
• List of Notation
• References
• Index