Algebraic Aspects of Cryptography

This book is intended as a text for a course on cryptography with emphasis on algebraic methods. It is written so as to be accessible to graduate or advanced undergraduate students, as well as to scientists in other fields. The first three chapters form a self-contained introduction to basic concepts and techniques. Here my approach is intuitive and informal. For example, the treatment of computational complexity in Chapter 2, while lacking formalistic rigor, emphasizes the aspects of the subject that are most important in cryptography. Chapters 4-6 and the Appendix contain material that for the most part has not previously appeared in textbook form. A novel feature is the inclusion of three types of cryptography - "hidden monomial" systems, combinatorial-algebraic sys­ tems, and hyperelliptic systems - that are at an early stage of development. It is too soon to know which, if any, of these cryptosystems will ultimately be of practical use. But in the rapidly growing field of cryptography it is worthwhile to continually explore new one-way constructions coming from different areas of mathematics. Perhaps some of the readers will contribute to the research that still needs to be done. This book is designed not as a comprehensive reference work, but rather as a selective textbook. The many exercises (with answers at the back of the book) make it suitable for use in a math or computer science course or in a program of independent study.

Table of Contents:
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.