## Topics in Computational Number Theory Inspired by Peter L. MontgomeryPeter L. Montgomery has made significant contributions to computational number theory, introducing many basic tools such as Montgomery multiplication, Montgomery simultaneous inversion, Montgomery curves, and the Montgomery ladder. This book features state-of-the-art research in computational number theory related to Montgomery's work and its impact on computational efficiency and cryptography. Topics cover a wide range of topics such as Montgomery multiplication for both hardware and software implementations; Montgomery curves and twisted Edwards curves as proposed in the latest standards for elliptic curve cryptography; and cryptographic pairings. This book provides a comprehensive overview of integer factorization techniques, including dedicated chapters on polynomial selection, the block Lanczos method, and the FFT extension for algebraic-group factorization algorithms. Graduate students and researchers in applied number theory and cryptography will benefit from this survey of Montgomery's work. |

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### Contents

Overview | 5 |

Using Primes of a Special Form | 22 |

page xi | 33 |

Hardware Aspects of Montgomery Modular | 40 |

Montgomery Curves and the Montgomery Ladder | 82 |

General Purpose Integer Factoring | 116 |

Polynomial Selection for the Number Field Sieve | 161 |

The Block Lanczos Algorithm | 175 |

FFT Extension for AlgebraicGroup Factorization | 189 |

Cryptographic Pairings | 206 |

Bibliography | 235 |

261 | |

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addition chain affine coordinates Algorithm 2.2 approach arithmetic array assume bits block Lanczos bound chapter Cited on pages coefficients columns Computer Science conditional subtraction coprime cost cryptography defined degree described differential addition discrete logarithms efficient effort elements elliptic curve elliptic curve cryptography Equation example extension finite field formulas hardware implementation input integer factorization inversion iteration L-notation Lanczos algorithm large prime Lecture Notes linear algebra matrix Miller’s algorithm modular multiplication modular reduction modulo Montgomery curves Montgomery ladder Montgomery multiplication Montgomery reduction Montgomery’s algorithm multiplication algorithm ninv Notes in Computer number field sieve operations optimal output parallel perform Peter Pollard’s polynomial f(X polynomial pairs precomputed prime ideal processing quadratic sieve quotient radix reduced relations representation result root scalar multiplication Section sequence shift SIMD smoothness square systolic array Tate pairing technique Theorem tion vector Weierstrass curve zero