This monograph describes and implements partially homomorphic encryption functions using a unified notation. After introducing the appropriate mathematical background, the authors offer a systematic examination of the following known algorithms: Rivest-Shamir-Adleman; Goldwasser-Micali; ElGamal; Benaloh; Naccache-Stern; Okamoto-Uchiyama; Paillier; Damgaard-Jurik; Boneh-Goh-Nissim; and Sander-Young-Yung. Over recent years partially and fully homomorphic encryption algorithms have been proposed and researchers have addressed issues related to their formulation, arithmetic, efficiency and security. Formidable efficiency barriers remain, but we now have a variety of algorithms that can be applied to various private computation problems in healthcare, finance and national security, and studying these functions may help us to understand the difficulties ahead. The book is valuable for researchers and graduate students in Computer Science, Engineering, and Mathematics who are engaged with Cryptology.

The chapters in this book present the work of researchers, scientists, engineers, and teachers engaged with developing unified foundations, principles, and technologies for cyber-physical security. They adopt a multidisciplinary approach to solving related problems in next-generation systems, representing views from academia, government bodies, and industrial partners, and their contributions discuss current work on modeling, analyzing, and understanding cyber-physical systems.

Software-based cryptography can be used for security applications where data traffic is not too large and low encryption rate is tolerable. But hardware methods are more suitable where speed and real-time encryption are needed. Until now, there has been no book explaining how cryptographic algorithms can be implemented on reconfigurable hardware devices. This book covers computational methods, computer arithmetic algorithms, and design improvement techniques needed to implement efficient cryptographic algorithms in FPGA reconfigurable hardware platforms. The author emphasizes the practical aspects of reconfigurable hardware design, explaining the basic mathematics involved, and giving a comprehensive description of state-of-the-art implementation techniques.

Cryptographic Engineering is the first book that discusses the design techniques and methods. The material of this book is scattered in journal and conference articles, and authors lecture notes. This is a first attempt by top cryptographic engineers to bring this material in a book form and make it available to electrical engineering and computer science students and engineers working for the industry.

This book is intended for a graduate-level course in Cryptographic Engineering to be taught in Electrical Engineering, Computer Engineering, and Computer Science departments. Students will have to have the knowledge of basic cryptographic algorithms before taking this course which will teach them how to design cryptographic hardware (FPGA, ASIC, custom) and embedded software to be used in secure systems.

Additionally, engineers working in the industry will be interested in this book to learn how to design cryptographic chips and embedded software. Engineers working on the design of cellular phones, mobile computing and sensor systems, web and enterprise security systems which rely upon cryptographic hardware and software will be interested in this book. Essential and advanced design techniques for cryptography will be covered by this book."

Cryptographic solutions using software methods can be used for those security applications where data traffic is not too large and low encryption rate is tolerable. On the other hand, hardware methods offer high-speed solutions making them highly suitable for applications where data traffic is fast and large data is required to be encrypted in real time. VLSI (also known as ASIC), and FPGAs (Field Programmable Gate Arrays) are two alternatives for implementing cryptographic algorithms in hardware. FPGAs offer several benefits for cryptographic algorithm implementations over VLSI as they offer high flexibility. Due to its reconfigurable property, keys can be changed rapidly. Moreover, basic primitives in most cryptographic algorithms can efficiently be implemented in FPGAs.

Since the invention of the Data Encryption Standard (DES), some 40 years ago, a considerable amount of cryptographic algorithm implementation literature has been produced both, for software and hardware platforms. Unfortunately, virtually there exists no book explaining how the main cryptographic algorithms can be implemented on reconfigurable hardware devices.

This book will cover the study of computational methods, computer arithmetic algorithms, and design improvement techniques needed to implement efficient cryptographic algorithms in FPGA reconfigurable hardware platforms. The concepts and techniques to be reviewed in this book will make special emphasis on the practical aspects of reconfigurable hardware design, explaining the basic mathematics related and giving a comprehensive description of state-of-the-art implementation techniques. Thus, the main goal of this monograph is to show how high-speed cryptographic algorithms implementations can be achieved on reconfigurable hardware devices without posing prohibited high requirements for hardware resources.

Cryptographic Engineering is the first book that discusses the design techniques and methods. The material of this book is scattered in journal and conference articles, and authors lecture notes. This is a first attempt by top cryptographic engineers to bring this material in a book form and make it available to electrical engineering and computer science students and engineers working for the industry.

This book is intended for a graduate-level course in Cryptographic Engineering to be taught in Electrical Engineering, Computer Engineering, and Computer Science departments. Students will have to have the knowledge of basic cryptographic algorithms before taking this course which will teach them how to design cryptographic hardware (FPGA, ASIC, custom) and embedded software to be used in secure systems.

Additionally, engineers working in the industry will be interested in this book to learn how to design cryptographic chips and embedded software. Engineers working on the design of cellular phones, mobile computing and sensor systems, web and enterprise security systems which rely upon cryptographic hardware and software will be interested in this book. Essential and advanced design techniques for cryptography will be covered by this book."

These are the proceedings of WAIFI 2008, the second workshop on the Ari- metic of Finite Fields, that was held in Siena, Italy, July 6-9, 2008. The ?rst workshop, WAIFI 2007, which was held in Madrid (Spain), was received quite enthusiasticallybymathematicians, computerscientists, engineersandphysicists who are performing research on ?nite ?eld arithmetic. We believe that there is a need for a workshop series bridging the gap between the mathematical theory of ?nite ?elds and their hardware/software implementations and technical - plications. We hope that the WAIFI workshopseries, which from now on will be held on even years, will help to ?ll this gap. Therewere 34 submissionsto WAIFI 2008, of which the ProgramCommittee selected 16 for presentation. Each submission was reviewed by at least three reviewers. Our thanks go to the Program Committee members for their many contributionsandhardwork.Wearealsogratefultotheexternalreviewerslisted below for their expertise and assistance in the deliberations. In addition to the contributions appearing in these proceedings, the workshop program included an invited lecture given by Amin Shokrollahi. Special compliments go out to Enrico Martinelli, General Co-chair, and to Roberto Giorgi and Sandro Bartolini, local organizers of WAIFI 2008, who broughtthe workshopto Siena, oneofthe mostbeautiful citiesofTuscany, Italy. WAIFI 2008 was organized by the Dipartimento di Ingegneria dell'Informazione of the University of Siena, Italy. The submission and selection of papers were done using the iChair software, developed at EPFL by Thomas Baign eres and Matthieu Finiasz. We also thank Deniz Karakoyunlu for his help in this matter.

This monograph describes and implements partially homomorphic encryption functions using a unified notation. After introducing the appropriate mathematical background, the authors offer a systematic examination of the following known algorithms: Rivest-Shamir-Adleman; Goldwasser-Micali; ElGamal; Benaloh; Naccache-Stern; Okamoto-Uchiyama; Paillier; Damgaard-Jurik; Boneh-Goh-Nissim; and Sander-Young-Yung. Over recent years partially and fully homomorphic encryption algorithms have been proposed and researchers have addressed issues related to their formulation, arithmetic, efficiency and security. Formidable efficiency barriers remain, but we now have a variety of algorithms that can be applied to various private computation problems in healthcare, finance and national security, and studying these functions may help us to understand the difficulties ahead. The book is valuable for researchers and graduate students in Computer Science, Engineering, and Mathematics who are engaged with Cryptology.

This book constitutes the refereed proceedings of the 5th International Workshop on the Arithmetic of Finite Field, WAIFI 2014, held in Gebze, Turkey, in September 2014. The 9 revised full papers and 43 invited talks presented were carefully reviewed and selected from 27 submissions. This workshop is a forum of mathematicians, computer scientists, engineers and physicists performing research on finite field arithmetic, interested in communicating the advances in the theory, applications, and implementations of finite fields. The workshop will help to bridge the gap between the mathematical theory of finite fields and their hardware/software implementations and technical applications.