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. Featuring contributions by the leading experts in the field, this text covers the various hard mathematical problems used in cryptography and the asymptotically best algorithms to solve these problems Presents 20 algorithms with in-depth descriptions on their utility in solving hard problems Teaches students what problems are actually used as the foundation to secure all our digital assets, and about the security assessment used in cryptography and how it relates to selecting key-sizes in practice Advises security engineers how to avoid common pitfalls when deploying security solutions Written in tribute to the scientific research career of Professor Arjen K. Lenstra, on the occasion of his 65th birthday