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AuthorKoblitz, Neal. author
TitleAlgebraic Aspects of Cryptography [electronic resource] / by Neal Koblitz
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1998
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Descript IX, 206 p. online resource


This is a textbook for a course (or self-instruction) in cryptography with emphasis on algebraic methods. The first half of the book is a self-contained informal introduction to areas of algebra, number theory, and computer science that are used in cryptography. Most of the material in the second half - "hidden monomial" systems, combinatorial-algebraic systems, and hyperelliptic systems - has not previously appeared in monograph form. The Appendix by Menezes, Wu, and Zuccherato gives an elementary treatment of hyperelliptic curves. This book is intended for graduate students, advanced undergraduates, and scientists working in various fields of data security. From the reviews: "... 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 Patarin'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 their 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." P. Hajnal, Acta Scientiarum Mathematicarum 64.1998 "... Overall, the book is highly recommended to everyone who has the requisite mathematical sophistication." E.Leiss, Computing Reviews 1998


1. Cryptography -- ยง1. Early History -- ยง2. The Idea of Public Key Cryptography -- ยง3. The RSA Cryptosystem -- ยง4. Diffie-Hellman and the Digital Signature Algorithm -- ยง5. Secret Sharing, Coin Flipping, and Time Spent on Homework -- ยง6. Passwords, Signatures, and Ciphers -- ยง7. Practical Cryptosystems and Useful Impractical Ones -- 2. Complexity of Computations -- ยง1. The Big-O Notation -- ยง2. Length of Numbers -- ยง3. Time Estimates -- ยง4. P, NP, and NP-Completeness -- ยง5. Promise Problems -- ยง6. Randomized Algorithms and Complexity Classes -- ยง7. Some Other Complexity Classes -- 3. Algebra -- ยง1. Fields -- ยง2. Finite Fields -- ยง3. The Euclidean Algorithm for Polynomials -- ยง4. Polynomial Rings -- ยง5. Grรถbner Bases -- 4. Hidden Monomial Cryptosystems -- ยง 1. The Imai-Matsumoto System -- ยง2. Patarinโ{128}{153}s Little Dragon -- ยง3. Systems That Might Be More Secure -- 5. Combinatorial-Algebraic Cryptosystems -- ยง1. History -- ยง2. Irrelevance of Brassardโ{128}{153}s Theorem -- ยง3. Concrete Combinatorial-Algebraic Systems -- ยง4. The Basic Computational Algebra Problem -- ยง5. Cryptographic Version of Ideal Membership -- ยง6. Linear Algebra Attacks -- ยง7. Designing a Secure System -- 6. Elliptic and Hyperelliptic Cryptosystems -- ยง 1. Elliptic Curves -- ยง2. Elliptic Curve Cryptosystems -- ยง3. Elliptic Curve Analogues of Classical Number Theory Problems -- ยง4. Cultural Background: Conjectures on Elliptic Curves and Surprising Relations with Other Problems -- ยง5. Hyperelliptic Curves -- ยง6. Hyperelliptic Cryptosystems -- ยง1. Basic Definitions and Properties -- ยง2. Polynomial and Rational Functions -- ยง3. Zeros and Poles -- ยง4. Divisors -- ยง5. Representing Semi-Reduced Divisors -- ยง6. Reduced Divisors -- ยง7. Adding Reduced Divisors -- Exercises -- Answers to Exercises.ย 

Computer science Data structures (Computer science) Data encryption (Computer science) Computers Number theory Combinatorics Computer Science Theory of Computation Number Theory Data Structures Cryptology and Information Theory Combinatorics Data Encryption


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