§16§Cryptography, as done in this century, is heavily mathematical. But it also has roots in what is computationally feasible.§This unique textbook text balances the theorems of mathematics against the feasibility of computation. Cryptography is something one actually "does", not a mathematical game one proves theorems about. There is deep math; there are some theorems that must be proved; and there is a need to recognize the brilliant work done by those who focus on theory. But at the level of an undergraduate course, the emphasis should be first on knowing and understanding the algorithms and how to implement them, and also to be aware that the algorithms must be implemented carefully to avoid the "easy" ways to break the cryptography. This text covers the algorithmic foundations and is complemented by core mathematics and arithmetic. §04§1. Introduction.- 2. Simple Ciphers.- 3. Divisibility, Congruences, and Modular Arithmetic.- 4. Groups, Rings, Fields.- 5. Square Roots and Quadratic Symbols.- 6. Finite Fields of Characteristic 2.- 7. Elliptic Curves.- 8. Mathematics, Computing, and Arithmetic.- 9. Modern Symmetric Ciphers - DES and AES.- 10. Asymmetric Ciphers - RSA and Others.- 11. How to Factor a Number.- 12. How to Factor More Effectively.- 13. Cycles, Randomness, Discrete Logarithms, and KeyExchange.- 14. Elliptic Curve Cryptography.- 15. Quantum Computing and Cryptography.- 16. Lattice-Based Cryptography.- 17. Homomorphic Encryption.- 18. Exercises. §02§§Duncan Buell, professor emeritus in the Dept. of Computer Science and Engineering at University of South Carolina, also has 15 years of experience at a research lab doing high-performance computing research in support of the US National Security Agency.§
