## Cryptanalysis of Number Theoretic CiphersAt the heart of modern cryptographic algorithms lies computational number theory. Whether you're encrypting or decrypting ciphers, a solid background in number theory is essential for success. Written by a number theorist and practicing cryptographer, Cryptanalysis of Number Theoretic Ciphers takes you from basic number theory to the inner workings of ciphers and protocols. First, the book provides the mathematical background needed in cryptography as well as definitions and simple examples from cryptography. It includes summaries of elementary number theory and group theory, as well as common methods of finding or constructing large random primes, factoring large integers, and computing discrete logarithms. Next, it describes a selection of cryptographic algorithms, most of which use number theory. Finally, the book presents methods of attack on the cryptographic algorithms and assesses their effectiveness. For each attack method the author lists the systems it applies to and tells how they may be broken with it. Computational number theorists are some of the most successful cryptanalysts against public key systems. Cryptanalysis of Number Theoretic Ciphers builds a solid foundation in number theory and shows you how to apply it not only when breaking ciphers, but also when designing ones that are difficult to break. |

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### Contents

Terminology of Cryptography | 3 |

Probability Theory | 13 |

Divisibility and Arithmetic | 27 |

Primes | 45 |

Congruences | 61 |

Eulers Theorem and Its Consequences | 75 |

Second Degree Congruences | 93 |

Information Theory | 111 |

Random Number Generation | 211 |

Private Key Ciphers | 221 |

Public Key Ciphers | 231 |

Signature Algorithms | 239 |

Key Exchange Algorithms | 245 |

Simple Protocols | 253 |

Complicated Protocols | 263 |

Complete Systems | 275 |

Groups Rings and Fields | 125 |

Exponential Methods of Factoring Integers | 143 |

Finding Large Primes | 155 |

Elliptic Curves | 171 |

Subexponential Factoring Algorithms | 185 |

Computing Discrete Logarithms | 203 |

Direct Attacks | 281 |

Exploiting an Error | 289 |

Active Attacks | 297 |

303 | |

### Common terms and phrases

Alice and Bob Alice's arithmetic attack bank bit operations Bob's byte Chapter Chinese remainder theorem chooses a random ciphertext coefficients coin compute congruence classes congruence x2 Cryptanalysis Cryptanalysis of Number cryptographic deciphering decryption defined discrete logarithm problem divides eavesdropper element elliptic curve enciphered encryption entropy equation Euler pseudoprime example exclusive-or exponent factoring algorithm fast exponentiation Fermat's little theorem function gcd(a inverse irreducible Jacobi symbol large prime Legendre symbol Lemma linear loop method mod q multiplication number field Number Theoretic Ciphers odd prime OxB7 pairs plaintext polynomial positive integer primality test prime divisor prime number primitive root modulo private key probable prime PROOF protocol prove pseudoprime to base public key quadratic nonresidue quadratic residue quadratic residue modulo random number relatively prime Rijndael secret key sends sieve signature signed solution square roots Step stream cipher strong pseudoprime Suppose trial division

### Popular passages

Page xii - the Center for Education and Research in Information Assurance and Security, CERIAS, its sponsors and its director, Professor Eugene Spafford, for support while this book was being written.