What is included with this book?
Preface | p. v |
Introduction | p. xi |
An Introduction to Cryptography | p. 1 |
Simple substitution ciphers | p. 1 |
Divisibility and greatest common divisors | p. 10 |
Modular arithmetic | p. 19 |
Prime numbers, unique factorization, and finite fields | p. 26 |
Powers and primitive roots in finite fields | p. 29 |
Cryptography before the computer age | p. 34 |
Symmetric and asymmetric ciphers | p. 36 |
Exercises | p. 47 |
Discrete Logarithms and Diffie-Hellman | p. 59 |
The birth of public key cryptography | p. 59 |
The discrete logarithm problem | p. 62 |
Diffie-Hellman key exchange | p. 65 |
The ElGamal public key cryptosystem | p. 68 |
An overview of the theory of groups | p. 72 |
How hard is the discrete logarithm problem? | p. 75 |
A collision algorithm for the DLP | p. 79 |
The Chinese remainder theorem | p. 81 |
The Pohlig-Hellman algorithm | p. 86 |
Rings, quotients, polynomials, and finite fields | p. 92 |
Exercises | p. 105 |
Integer Factorization and RSA | p. 113 |
Euler's formula and roots modulo pq | p. 113 |
The RSA public key cryptosystem | p. 119 |
Implementation and security issues | p. 122 |
Primality testing | p. 124 |
Pollard's p - 1 factorization algorithm | p. 133 |
Factorization via difference of squares | p. 137 |
Smooth numbers and sieves | p. 146 |
The index calculus and discrete logarithms | p. 162 |
Quadratic residues and quadratic reciprocity | p. 165 |
Probabilistic encryption | p. 172 |
Exercises | p. 176 |
Combinatorics, Probability, and Information Theory | p. 189 |
Basic principles of counting | p. 190 |
The Vigenere cipher | p. 196 |
Probability theory | p. 210 |
Collision algorithms and meet-in-the-middle attacks | p. 227 |
Pollard's [rho] method | p. 234 |
Information theory | p. 243 |
Complexity Theory and P versus NP | p. 258 |
Exercises | p. 262 |
Elliptic Curves and Cryptography | p. 279 |
Elliptic curves | p. 279 |
Elliptic curves over finite fields | p. 286 |
The elliptic curve discrete logarithm problem | p. 290 |
Elliptic curve cryptography | p. 296 |
The evolution of public key cryptography | p. 301 |
Lenstra's elliptic curve factorization algorithm | p. 303 |
Elliptic curves over F[subscript 2] and over F[subscript 2 superscript k] | p. 308 |
Bilinear pairings on elliptic curves | p. 315 |
The Weil pairing over fields of prime power order | p. 325 |
Applications of the Weil pairing | p. 334 |
Exercises | p. 339 |
Lattices and Cryptography | p. 349 |
A congruential public key cryptosystem | p. 349 |
Subset-sum problems and knapsack cryptosystems | p. 352 |
A brief review of vector spaces | p. 359 |
Lattices: Basic definitions and properties | p. 363 |
Short vectors in lattices | p. 370 |
Babai's algorithm | p. 379 |
Cryptosystems based on hard lattice problems | p. 383 |
The GGH public key cryptosystem | p. 384 |
Convolution polynomial rings | p. 387 |
The NTRU public key cryptosystem | p. 392 |
NTRU as a lattice cryptosystem | p. 400 |
Lattice reduction algorithms | p. 403 |
Applications of LLL to cryptanalysis | p. 418 |
Exercises | p. 422 |
Digital Signatures | p. 437 |
What is a digital signature? | p. 437 |
RSA digital signatures | p. 440 |
ElGamal digital signatures and DSA | p. 442 |
GGH lattice-based digital signatures | p. 447 |
NTRU digital signatures | p. 450 |
Exercises | p. 458 |
Additional Topics in Cryptography | p. 465 |
Hash functions | p. 466 |
Random numbers and pseudorandom number generators | p. 468 |
Zero-knowledge proofs | p. 470 |
Secret sharing schemes | p. 473 |
Identification schemes | p. 474 |
Padding schemes and the random oracle model | p. 476 |
Building protocols from cryptographic primitives | p. 479 |
Hyperelliptic curve cryptography | p. 480 |
Quantum computing | p. 483 |
Modern symmetric cryptosystems: DES and AES | p. 485 |
List of Notation | p. 489 |
References | p. 493 |
Index | p. 501 |
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