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# 교재준비작업(1)

2004.10.03 21:27

송홍엽 조회 수:8514

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
Multiplicative structure and conjugate class
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

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