Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory. This edition now includes over 150 new exercises, ranging from the routine to the challenging, that flesh out the material presented in the body of the text, and which further develop the theory and present new applications. The material has also been reorganized to improve clarity of exposition and presentation. Ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.

These are the proceedings of Crypto 2005, the 25th Annual International Cr- tology Conference. The conference was sponsored by the International Assoc- tion for Cryptologic Research (IACR) in cooperation with the IEEE Computer ScienceTechnicalCommitteeonSecurityandPrivacyandtheComputerScience DepartmentoftheUniversityofCaliforniaatSantaBarbara.Theconferencewas held in Santa Barbara, California, August 14-18, 2005. Theconferencereceived178submissions,outofwhichtheprogramcommittee selected33forpresentation.Theselectionprocesswascarriedoutbytheprogram committee via an "online" meeting. The authors of selected papers had a few weeks to prepare ?nal versions of their papers, aided by comments from the reviewers. However, most of these revisions were not subject to any editorial review. This year, a "Best Paper Award" was given to Xiaoyun Wang, Yiqun Lisa Yin, and Hongbo Yu, for their paper "Finding Collisions in the Full SHA-1." The conference program included two invited lectures. Ralph Merkle del- ered an IACR Distinguished Lecture, entitled "The Development of Public Key Cryptography:a PersonalView; and Thoughts on Nanotechnology." Dan Boneh gave an invited talk, entitled "Bilinear Maps in Cryptography." We continued the tradition of a "rump session," featuring short, informal presentations (usually serious, sometimes entertaining, and occasionally both). " The rump session was chaired this year by Phong Q. Nguy en.