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.