Statement of Course Outcomes

Course Number: CS 503

Course Name: Discrete Mathematics for Cryptography

Course Coordinator: Susanne Wetzel

Graduate or Undergraduate Equivalent:

 

Catalog Description:  Topics include elements of number theory including the prime number theorem, the Euler phi function, the Euclidean algorithm, the Chinese remainder theorem; elements of abstract algebra and finite fields including basic fundamentals of groups, rings, polynomial rings, vector spaces and finite fields. Carries credit toward the Applied Mathematics degree only when followed by CS 579. Recommended for high-level undergraduate students. Prerequisite: MA 502.

 

Course Outcomes:

All of these course outcomes support the Program Outcome core: math - stat. 

1. Know the definition of divisibility in the integers and the statement of the Division Algorithm.

2. Know the definitions of the algebraic notions semigroup, monoid , group , ring and field; the notion of an ideal in a ring and its application to the definitions of gcd and lcm.

3. Know the definition of the Euler Phi function and be competent in calculating its values.

4. Know the definition of a prime number and the statement of the Prime Number Theorem.

5. Know the definition of a finite cyclic group and the orders of a subgroup and an element in a finite group.

6. Know the definition of congruence mod n, the ring of integers mod n and the characterization of the units in this ring; be competent at mod n arithmetic.

7. Know when the group of units of the integers mod n is cyclic.

8. Know the Chinese Remainder Theorem and Gauss' algorithm for finding a solution guaranteed by the theorem.

9. Know Fermat's Theorem and Euler's Theorem.

10. Know the definition of a quadratic residue mod n ; the definitions and properties of the Legendre and Jacobi symbols and be competent at calculating them.

11. Know how to add and multiply polynomials with coefficients from a field ( specifically Zp ); know and apply the Division Algorithm for polynomials.

12. Know the definition of the gcd of a pair of polynomials and the development of the notion using the notion of an ideal in the ring of polynomials .

13. Know the definition of irreducibility and the factorization of a polynomial into irreducible factors.

14. Know the definitions of addition and multiplication in the ring of polynomials modulo a polynomial and when this ring is a field.

15. Know the definition of the characteristic of a finite field and the linear algebraic meaning of k in the expression p^k giving the number of elements in such a field.

16. Know the definition of the minimal polynomial of an element in a finite field and when such a polynomial is a primitive polynomial;

be able to calculate it for the case of a " small " field.

17.Know the nature of the factorization of x^(p^k) - x into irreducible factors with coefficients from Zp and be able to obtain such a factorization for small p^k

18.Know the application of the factorization alluded to in 17. to proving the existence of GF(p^k) using Mobius inversion.

19.Know the nature of the subfields of GF(p^k).