The area of computational cryptography is dedicated to the development of effective methods in algorithmic number theory that improve implementation of cryptosystems or further their cryptanalysis. This book is a tribute to Arjen K. Lenstra, one of the key contributors to the field, on the occasion of his 65th birthday, covering his best-known scientific achievements in the field. Students and security engineers will appreciate this no-nonsense introduction to the hard mathematical problems used in cryptography and on which cybersecurity is built, as well as the overview of recent advances on how to solve these problems from both theoretical and practical applied perspectives. Beginning with polynomials, the book moves on to the celebrated Lenstra–Lenstra–Lovász lattice reduction algorithm, and then progresses to integer factorization and the impact of these methods to the selection of strong cryptographic keys for usage in widely used standards.

‘This volume celebrates the research career of Arjen Lenstra. The volume covers the latest research in many areas of applied cryptography: from algorithms for factoring and discrete log, to fast implementations of computer algebra, to the selection of cryptographic key sizes. Each topic is masterfully covered by a top researcher in the respective area. The information covered in this volume will serve readers for many years to come, and is sure to inspire further research on these topics.’

Dan Boneh - Stanford University

‘This book demonstrates the breathtaking diversity of Arjen Lenstra’s research over the last forty years, and the deep influence his work has had on computational aspects of cryptography. Each chapter is written by a leading domain expert and provides an “in a nutshell” overview of a specific topic. The book is sure to become an important reference for experts and beginners alike.’

Kenneth Paterson - ETH Zurich

‘With highly accessible surveys by leading cryptographers, this book hits all pins with a single strike: framing the important area of "computational cryptography" through its fascinating history, peeking into its (no less prominent) future, and celebrating the impactful research career of one of its principal architects, Arjen Lenstra.’

Ronald Cramer - CWI Amsterdam and Leiden University