Number Theory and Cryptography

Abstract

The treatment of number theory is elementary, in the technical sense. Number systems, factorization, the Euclidean algorithm, and greatest common divisors are covered, as is the reversal of the Euclidean algorithm to express a greatest common divisor (GCD) as a linear combination. Modular arithmetic and congruence are discussed. Simultaneous congruences are solved, and this leads to the Chinese remainder theorem.

Cryptography is introduced through classical ciphers, the scytale and the Caesar cipher. Additive ciphers, the Vigenè method, and substitution ciphers are discussed.

The treatment of modern cryptography starts with the Rivest, Shamir, and Adleman (RSA) system and public key systems in general. The security of the RSA and similar systems is discussed, together with attacks on RSA and related factorization problems. Signature systems, key exchange, and simulated random acts (such as coin tossing) also appear in this chapter.