Algebraic Aspects of Cryptography

• 1. Cryptography.- §1. Early History.- §2. The Idea of Public Key Cryptography.- §3. The RSA Cryptosystem.- §4. Diffie-Hellman and the Digital Signature Algorithm.- §5. Secret Sharing, Coin Flipping, and Time Spent on Homework.- §6. Passwords, Signatures, and Ciphers.- §7. Practical Cryptosystems and Useful Impractical Ones.- 2. Complexity of Computations.- §1. The Big-O Notation.- §2. Length of Numbers.- §3. Time Estimates.- §4. P, NP, and NP-Completeness.- §5. Promise Problems.- §6. Randomized Algorithms and Complexity Classes.- §7. Some Other Complexity Classes.- 3. Algebra.- §1. Fields.- §2. Finite Fields.- §3. The Euclidean Algorithm for Polynomials.- §4. Polynomial Rings.- §5. Gröbner Bases.- 4. Hidden Monomial Cryptosystems.- § 1. The Imai-Matsumoto System.- §2. Patarin’s Little Dragon.- §3. Systems That Might Be More Secure.- 5. Combinatorial-Algebraic Cryptosystems.- §1. History.- §2. Irrelevance of Brassard’s Theorem.- §3. Concrete Combinatorial-Algebraic Systems.- §4. The Basic Computational Algebra Problem.- §5. Cryptographic Version of Ideal Membership.- §6. Linear Algebra Attacks.- §7. Designing a Secure System.- 6. Elliptic and Hyperelliptic Cryptosystems.- § 1. Elliptic Curves.- §2. Elliptic Curve Cryptosystems.- §3. Elliptic Curve Analogues of Classical Number Theory Problems.- §4. Cultural Background: Conjectures on Elliptic Curves and Surprising Relations with Other Problems.- §5. Hyperelliptic Curves.- §6. Hyperelliptic Cryptosystems.- §1. Basic Definitions and Properties.- §2. Polynomial and Rational Functions.- §3. Zeros and Poles.- §4. Divisors.- §5. Representing Semi-Reduced Divisors.- §6. Reduced Divisors.- §7. Adding Reduced Divisors.- Exercises.- Answers to Exercises.

"... This is a textbook in cryptography with emphasis on algebraic methods. It is supported by many exercises (with answers) making it appropriate for a course in mathematics or computer science. ... Overall, this is an excellent expository text, and will be very useful to both the student and researcher." M.V.D.Burmester, Mathematical Reviews 2002 "... I think this book is a very inspiring book on cryptography. It goes beyond the traditional topics (most of the cryptosystems presented here are first time in a textbook, some of Paturi's work is not published yet). This way the reader has the feeling how easy to suggest a cryptosystem, how easy to break a safe looking system and hence how hard to trust one. The interested readers are forced to think together with ther researchers and feel the joy of discovering new ideas. At the same time the importance of "hardcore" mathematics is emphasized and hopefully some application driven students will be motivated to study theory." Peter Hajnal, Acta Scientiarum Mathematicarum, 64.1998, p. 750 "... Overall, the book is highly recommended to everyone who has the requisite mathematical sophistication." E.Leiss, Computing Reviews 1998, p. 506 "... Der Autor, der ... vielen Lesern dieses Rundbriefs gut bekannt sein wird, hat hier ein kleines Werk vorgelegt, das man wohl am Besten als "Lesebuch zu algebraischen Aspekten der Kryptographie mit offentlichem Schlussel" charakterisieren kann. ... Mit zunehmender Schwierigkeit des Material werden die Ausfuhrungen dabei skizzenhafter und beschranken sich immer starker auf den Hinweis auf entsprechende Quellen, was den Charakter eines guten "Lesebuchs", wie ich es oben bezeichnet habe, ausmachen sollte. Das Buch eignet sich damit selbst fur "advanced undergraduates", wie es im Klappentext heisst, als Einstieg und erster Uberblick uber ein Gebiet, in dem sich in den letzten Jahren auf uberraschende Weise praktische Anwendungsmoglichkeiten fur tief innermathematische Themen ergeben haben." Hans-Gert Grabe, Computeralgebra Rundbrief 1999, Issue 25 "... Es gelingt Koblitz, anschaulich und mit elementaren Mitteln auch Dinge zu erlautern, die in vergleichbaren Texten kaum zu finden sind: z.B. den Hilbertschen Basis- und Nullstellensatz, sowie Grobnerbasen. ..." Franz Lemmermeyer, Mathematische Semesterberichte 1999, 46/1