Number Theory Cryptography

3 Credits · Level: Undergraduate

This course begins with a survey of cryptography from Roman times up to today's high tech world. Students will learn how to encrypt messages and how to attempt to break codes. We will discuss the efficiency and security level of different encryption methods. To make the discussion rigorous, we will need tools/concepts/results from mathematics. Mathematical content: mappings and inverse mappings, modular arithmetic, the additive group Z/n, the multiplicative group Z*/n, Euler's phi function, Fermat's Little theorem and Euler's generalization, primitive roots, discrete logarithms, Diophantine equations, the Chinese Remainder Theorem, Wilson’s theorem, multiplicative functions, quadratic residues and the law of quadratic reciprocity. Cryptographic content: classical ciphers and their decryption (shift, affine, and Vigenere ciphers), key exchange protocols (eg, Diffie-Hellman), public key ciphers (eg, RSA).