Joppe W. Bos is a cryptographic researcher in the Competence Center for Crypto & Security at NXP Semiconductors, Leuven, Belgium. Previously, he was a post-doctoral researcher in the Cryptography Research Group at Microsoft Research and obtained his PhD at EPFL, Lausanne, Switzerland. His research focuses on computational number theory and high-performance arithmetic as used in (post-quantum) public-key cryptography. He is the editor of the book Topics in Computational Number Theory Inspired by Peter L. Montgomery and serves as the Editor of the Cryptology ePrint Archive, and he is elected as the Secretary of the International Association for Cryptologic Research. Martijn Stam is a Chief Research Scientist in cryptology at Simula UiB, Bergen, Norway. Previously, he was a Reader in Cryptology at the University of Bristol, UK and a postdoc at EPFL Switzerland. He obtained his PhD at the TU Eindhoven in the Netherlands. His research ranges a wide area of cryptologic research, with an emphasis on a mathematical and algorithmic perspective. He has won two FSE Best Paper Awards and served as elected Secretary of the IACR from 2011 to 2016.

'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

• 1. Introduction Joppe Bos and Martijn Stam; Part I. Crytanalysis: 2. Lattice attacks on NTRU and LWE – a history of refinements Martin Albrecht and Léo Ducas; 3. History of integer factorization Samuel Wagstaff, Jr.; 4. Lattice-based integer factorization – an introduction to Coppersmith's Method Alexander May; 5. Computing discrete logarithms Rob Granger and Antoine Joux; 6. RSA, DH, and DSA in the wild Nadia Heninger; 7. A survey of chosen-prefix collisions attacks Marc Stevens; Part II. Implementations: 8. Efficient modular arithmetic Joppe Bos, Thorsten Kleinjung and Daniel Page; 9. Arithmetic software libraries Victor Shoup; 10. XTR and Tori Martijn Stam; 11. History of cryptographic key sizes Nigel Smart and Emmanuel Thomé.