Topics in Computational Number Theory Inspired by Peter L. Montgomery

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.

1. Introduction Joppe W. Bos, Arjen K. Lenstra, Herman te Riele and Daniel Shumow; 2. Montgomery arithmetic from a software perspective Joppe W. Bos and Peter L. Montgomery; 3. Hardware aspects of Montgomery modular multiplication Colin D. Walter; 4. Montgomery curves and the Montgomery ladder Daniel J. Bernstein and Tanja Lange; 5. General purpose integer factoring Arjen K. Lenstra; 6. Polynomial selection for the number field sieve Thorsten Kleinjung; 7. The block lanczos algorithm Emmanuel Thomé; 8. FFT extension for algebraic-group factorization algorithms Richard P. Brent, Alexander Kruppa and Paul Zimmermann; 9. Cryptographic pairings Kristin Lauter and Michael Naehrig.

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).