This thesis consists of three chapters. Chapter 1 is general introduction to information theory, cryptology, coding theory.
Let $F_n^2$ be the n-dimensional vector space over $\text{GF}(2).$ Chapter 2 deals with expansive permutation on $F_2^n$ that is used to be the cryptographic function in S-box of DES. In Section 2 of the Chapter 2, we define a permutation P on $F_2^n$ to be the locally expanding permutation if P satisfies the following condition: For Hamming distance d and positive integers r,s, if the elements x,y in $F_2^n$ satisfies d(x,y)=r, then d(p(x),p(y))≥s.(See Definition 2.2.1.) And we show some simple properties of the locally expanding permutation.
In Section 3 of the Chapter 2, we define a permutation P of $F_2^n$ to be maximally separating permutation if P satisfies the following condition: For Hamming distance d and non-negative integer k, we first define the metric $d_k^P$ on $F_2^n$ as $d_k^P(x,y)=max_{0≤i≤k}{d(P^i(x),P^i(y))}$ for x,y in $F_2^n$, where $P^i$ means i-times composition of P. If there exists integer $k≥0$ such that for all x,y in $F_2^n$ with x≠y, $d_k^P(x,y)=n$, we define P to be maximally separating permutation on $F_2^n.$(See Definition 2.3.1.) In Theorem 2.3.3, we show that the only two kinds of permutations can be maximally separating permutation.
But, in view of Definition, a locally expanding permutation seems to be strong candidate for maximally separating permutation. However, in the last Remark of Section 3 of the Chapter 2, we show that there is no strong relationship between locally expanding permutation and maximally separating permutation.
Chapter 3 deals with the decoding algorithm of a syndrome-distribution decoding of mols $L_p$ codes. Let P be an odd prime number. We introduce simple and useful decoding algorithm for orthogonal Latin square codes of order p. Let H be the parity check matrix of orthogonal Latin square code. For any x∈ GF$(p)^n,$ we call $xH^T$ the syndrome of x. This method is based on the syndrome decoding for linear codes.
이 논문은 세 개의 장으로 구성되어 있다. 제 1 장은 정보이론, 암호론, 부호이론에 대한 일반적인 소개이다. 제 2 장은 DES의 S 상자에 쓰이는 $F_2^n$ 위에서의 팽창 치환을 다룬다. 제 2.2 절에서는 국소 팽창 치환을 정의하고 그 성질을 살펴보고, 제 2.3 절에서는 극대 팽창 치환을 정의하고, 그 성질을 살펴보며, 국소 팽창 치환과 극대 팽창 치환과의 관계를 살펴 본다. 제 3 장에서는 위수 p (소수)인 직교라면 평방코드의 단순하고 유용한 해독 알고리듬을 다룬다.
