Math 485: Mathematical Cryptography

Math 485

Title

Introduction to cryptography.

Prerequisite

Math 371.

Description

A mathematical introduction to some of the high points of modern cryptography.

Desired Learning Outcomes

This is a course in the mathematics and algorithms of modern cryptography. It complements, rather than being equivalent to, the current CS course on Information Security.

Prerequisite

The requirement for Math 371 ensures both an appropriate level of mathematical maturity and a basic knowledge of linear algebra.

Minimal Learning Outcomes

The student should gain a understanding of the following topics. In particular this includes knowing the definitions, being familiar with standard examples, and being able to solve mathematical and algorithmic problems by directly using the material taught in the course. This includes appropriate use of Maple, Mathematica and Matlab.

1. Classical systems, including substitution theory, block ciphers and Enigma.

2. Elementary number theory as follows: Euclid's algorithm. Modular arithmetic and the algorithm for modular exponentiation. Chinese Remainder Theorem. Fermat and Euler Theorems. Primitive roots. Elementary continued fractions. Simple discussion of finite fields.

3. The DES and AES encryption standards.

4.RSA and its weaknesses. Primality testing and factorization. The Quadratic Sieve. Wiener's continued fraction attack on low decryption exponent.

5. Discrete logarithms. Diffie-Hellman key exchange. ElGamal.

6. Lattices and Lattice Algorithms. The LLL algorithm. The NTRU system. Lattice attacks on RSA.

Additional Topics

If time allows: Elliptic curve cryptography.

Courses for which this course is prerequisite

None.