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.

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.