Topics in Computational Number Theory Inspired by Peter L. Montgomery

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Joppe W. Bos, Arjen K. Lenstra
Cambridge University Press, Oct 12, 2017 - Mathematics - 276 pages
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Peter 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
Subject Index
261
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About the author (2017)

Joppe W. Bos is a cryptographic researcher at the Innovation Center for Cryptography and Security at NXP Semiconductors, Belgium. He also currently serves as the Secretary of the International Association for Cryptologic Research (IACR). His research focuses on computational number theory and high-performance arithmetic as used in public-key cryptography.

Arjen K. Lenstra is Professor of Computer Science at École Polytechnique Fédérale de Lausanne. His research focuses on cryptography and computational number theory, especially in areas such as integer factorization. He was closely involved in the development of the number field sieve method for integer factorization as well as several other cryptologic results. He is the recipient of the Excellence in the Field of Mathematics RSA Conference 2008 Award and a Fellow of the International Association for Cryptologic Research (IACR).

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