This book offers an introduction to cryptology, the science that makes secure communications possible, and addresses its two complementary aspects: cryptography---the art of making secure building blocks---and cryptanalysis---the art of breaking them. The text describes some of the most important systems in detail, including AES, RSA, group-based and lattice-based cryptography, signatures, hash functions, random generation, and more, providing detailed underpinnings for most of them. With regard to cryptanalysis, it presents a number of basic tools such as the differential and linear methods and lattice attacks. This text, based on lecture notes from the author's many courses on the art of cryptography, consists of two interlinked parts. The first, modern part explains some of the basic systems used today and some attacks on them. However, a text on cryptology would not be complete without describing its rich and fascinating history. As such, the colorfully illustrated historical part interspersed throughout the text highlights selected inventions and episodes, providing a glimpse into the past of cryptology. The first sections of this book can be used as a textbook for an introductory course to computer science or mathematics students. Other sections are suitable for advanced undergraduate or graduate courses. Many exercises are included. The emphasis is on providing reasonably complete explanation of the background for some selected systems.

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.

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.