Math 4411 - Ohio Northern University

Ohio Northern University
College of Arts and Sciences
Department of Mathematics and Statistics
Date: Fall 2011
Course Math 4411
Name: Number Theory with Applications in Cryptography and
Coding Theory
Credit hours: 3
Lecture hours/week: 3
Lab hours/week: 0
Instructor: Staff
Usual Student Level: Junior
Course required of students in:
Course frequency per semester/year: Offered alternate years; spring semester
Average enrollment per year: 15
This course has a prerequisite:
Math 1631 Calculus 1,
This course is a prerequisite for:
Catalogue Description:
An introduction to number theory, modular arithmetic and finite fields, with applications to pseudorandom sequences, cryptography and coding theory. Offered alternate years.
Course Objectives:
To provide students with an introduction to the main ideas and concepts of number theory and its
Textbook: Typical: Elementary Number Theory – an algebraic approach, by E. D. Bolker (Dover 2007)
Finite Fields and Applications by G.L. Mullen and C. Mummert (AMS, 2007)
Introduction to Cryptography by J. A. Buchanan (Springer Verlag, 2001)
Outline of content follows:
(see attached)
Course Outline
MATH 4411
Number Theory with Applications in Cryptography and Coding Theory
1. Integers and division – basic concepts. The fundamental theorem of Arithmetic. The greatest
common divisor and the least common multiple. The Euclid algorithm and the extended Euclid
2. Number theoretic functions. Sums extended over the divisors.
3. Congruences and modular arithmetic. Linear congruences. The theorems of Euler and Fermat.
4. Residue class rings, modular inverses, fast exponentiation. The Chinese remainder theorem.
5. Structure of the multiplicative group of residues modulo a prime. Primitive roots. Discrete
6. Quadratic residues and quadratic reciprocity. The Legendre symbol. The Jacobi symbol.
7. Affine encryption and its generalizations (n-graph encryption). Hill’s cryptosystem.
8. RSA cryptosystem
9. Diffie-Hellman key exchange. Elgamal cryptosystems.
10. Primality testing. Trial division. Fermat test. Carmichael numbers. Miller-Rabin Test.
11. Rabin encryption. Goldwasser-Micali encryption
12. Factoring. Pollard’s Rho test. Fermat’s method. p – 1 method. Quadratic Sieve.
13. Introduction to Coding Theory. Linear codes. Hamming distance.
14. Bounds for codes. “Good” codes. Examples.
15. Three more classes for tests.
TI calculators as well as MATLAB and MAPLE computer algebra systems will be used.