이산수학과 유한체이론 - 주별 강의주제 목록

1. Property of Z

    Z is a commutative ring with 1.
    it is an integral domain.
    it is a Euclidean domain.
    it is a Unique Factorization domain.
    Z/(n) is a ring, domain, ED, and UFD.
    G=U(Z/(n)) is a multiplicative group of order phi(n)
              it is cyclic if and only if n=1,2,4,p^k, 2p^k.
    Z/(n) is a field if and only if n=prime
             U(Z/(p)) is cyclic of order p-1

2. Computations over Z

    Linear equation over  Z/(n)
    Chinese Remainer Theorem
    Quadratic equation over Z/(n)
    Quadratic Reciprocity Theorem - Legendre/Jacobi symbols
    Payley construction of Hadamard matrix
    Big Integer arithematic
    Fast Exponentiation

3. Some Public Key Crypto Algorithms

    Primality testing algorithm
    DLP - analysis
    ElGamal Algorithm
    RSA Algorithm
    Secrete Sharing Algorithm
    Coin-flipping over Telephone
    Public key Envelope

4. Permutations and Counting

   Definition, notation, order, cycles
   unique decomposition
   even and odd permutation
   Not-Burnside Theorem on counting

5. Vectors and Matrix

   n-tuple vector space over F
   Basis, Linear Independance
   simultaneous equation and coefficient matrix
   Gauss Elimination,    rank of a matrix,    LU decomposition
   column space, row space, orthogonal complement
        rank, nullity, basic relation
   Vandemond matrix
  
6. Linear Transformation and Matrix

   Definition of Linear Transformation
   Range space and Null space, basic relation
   relation to Matrix
   Multi-linear transformation and Determinant
   Existence of Determinant

7. Some Problem Discussions

8. midterm

9. Polynomial over GF(p) = Fp[x]

    Fp[x] is a commutative ring with 1.
    it is an integral domain.
    it is a Euclidean domain.
    it is a Unique Factorization domain.
    Fp[x]/(f(x)) is a ring, domain, ED, and UFD.
    Fp[x]/(f(x))  is a field if and only if f(x) is irreducible

10. Structure of Finite Field

   Extension and Subfield
   Additive structure
   Multiplicative structure and conjugate class
   Addone table
   Homomorphism and isomorphism

11. Irreducible Polynomial over finite field

   irreducible polynomials over F2
   Property of minimal polynomials
   Conjugates
   Trace function
   m-sequence

12. Irreducible Polynomial and Cyclotomic Polynomial

   x^{q^n}-x = ㅠ V_d(x)
   Number of irreducible polynomials
   Cyclotomic Polynomials
        over C
        over Finite Field
   Some Factoring

13. Error-correcting linear codes

   binary symmetric channel, binary erasure channel
   binary linear code and minimum distance decoding
   Binary Hamming code and decoding
   minimum distance decoding is ML decoding
   BCH code over Z/(p)
   Some nonlinear simultaneous equations for decoding of BCH code over  Z/(p)

14. Cyclic code

   Hamming code and BCH code as cyclic code
   RS codes - encoding and decoding
   GFFT approach

15. Some Problem Discussions

16. final exam