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COURSE SYLLABUS 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 applications. 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 TENTATIVE COURSE OUTLINE BY WEEK 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 algorithm. 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 logarithms. 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. Comment: TI calculators as well as MATLAB and MAPLE computer algebra systems will be used.