This textbook introduces elliptic curves and modular forms, focusing on their connections to number theory, particularly the congruent number problem. It aims to make the subject accessible to those less familiar with advanced mathematics, featuring practical examples and exercises. It's suitable for graduate students and advanced undergraduates in related fields.
This book is a comprehensive introduction to p-adic number theory, now enhanced with a detailed exploration of p-adic functions, including the Iwasawa logarithm and p-adic gamma-function. It features new exercises and an appendix with solutions. Neal Koblitz, an expert in the field, provides valuable insights from his extensive research and teaching experience.
This text serves as a course resource on cryptography, focusing on algebraic methods and designed for graduate or advanced undergraduate students, as well as scientists from other fields. The initial three chapters provide a self-contained introduction to fundamental concepts and techniques, using an intuitive and informal approach. For instance, Chapter 2 discusses computational complexity with a focus on its relevance to cryptography, despite lacking formal rigor. Chapters 4-6 and the Appendix introduce material that has not been widely available in textbooks, featuring three innovative types of cryptography: "hidden monomial" systems, combinatorial-algebraic systems, and hyperelliptic systems, which are still in early development. The practical utility of these cryptosystems remains uncertain, but exploring new one-way constructions from various mathematical areas is essential in the evolving field of cryptography. The book encourages readers to engage in ongoing research. It is not intended as a comprehensive reference but as a selective textbook, complete with numerous exercises (answers provided) to support math or computer science courses and independent study programs.
Neal Koblitz is a co-inventor of one of the two most popular forms of encryption and digital signature, and his autobiographical memoirs are collected in this volume. Besides his own personal career in mathematics and cryptography, Koblitz details his travels to the Soviet Union, Latin America, Vietnam and elsewhere;
Focusing on p-adic analysis and number theory, this work presents recent advancements made by mathematicians over the past five years. It begins with foundational concepts, including the construction of p-adic numbers and power series, before delving into the properties of p-adic Dirichlet L-series through p-adic measures. The text introduces p-adic gamma functions and examines their connections to L-series, emphasizing parallels with complex analysis. Additionally, it proves a formula for Gauss sums related to p-adic gamma functions, making it valuable for graduate students and researchers in relevant fields.
The book explores the evolving relevance of number theory, particularly in the realms of cryptography and information transmission, contrasting G. H. Hardy's views on its remoteness from practical applications. It highlights the rise of computational number theory, driven by advancements in computer technology. Aimed at readers with little background in algebra or number theory, it introduces key arithmetic topics—both historical and contemporary—using an algorithmic approach to emphasize the efficiency of various techniques.
Crypto '96, the Sixteenth Annual Crypto Conference, is sponsored by the International Association for Cryptologic Research (IACR), in cooperation with the IEEE Computer Society Technical Committee on Security and P- vacy and the Computer Science Department of the University of California at Santa Barbara (UCSB). It takes place at UCSB from August 18 to 22, 1996. The General Chair, Richard Graveman, is responsible for local organization and registration. The scientific program was organized by the 16-member Program C- mittee. We considered 115 papers. (An additional 15 submissions had to be summarily rejected because of lateness or major noncompliance with the c- ditions in the Call for Papers.) Of these, 30 were accepted for presentation. In addition, there will be five invited talks by Ernest Brickell. Andrew Clark, Whitfield Diffie, Ronald Rivest, and Cliff Stoll. A Rump Session will be chaired by Stuart Haber. These proceedings contain the revised versions of the 30 contributed talks. least three com- The submitted version of each paper was examined by at mittee members and/or outside experts, and their comments were taken into account in the revisions. However, the authors (and not the committee) bear full responsibility for the content of their papers.
Książka ta jest poświęcona związkom teorii liczb z kryptografią. Do zrozumienia omawianych w niej problemów nie jest właściwie potrzebna znajomość algebry i teorii liczb. Można tu bowiem znaleźć definicje wszystkich niezbędnych pojęć i dowody wielu prostych twierdzeń. Wykład jest prowadzony w przystępny i interesujący sposób, z wieloma trafnie dobranymi przykładami. Na końcu każdego rozdziału znajdują się ćwiczenia umożliwiające sprawdzenie nabytych wiadomości, a na końcu książki - poprawne do nich odpowiedzi.