The Journal of the Korea institute of electronic communication sciences (한국전자통신학회논문지)

Korea Institute of Electronic Communication Science (한국전자통신학회)
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Characteristic Polynomials of 90/150 CA <10 ⋯ 0>

90/150 CA <10 ⋯ 0>의 특성다항식

  • 김진경 (부경대학교 응용수학과) ;
  • 조성진 (부경대학교 응용수학과) ;
  • 최언숙 (동명대학교 정보통신공학과) ;
  • 김한두 (인제대학교 응용수학과) ;
  • 강성원 (부경대학교 응용수학과)
  • Received : 2018.09.07
  • Accepted : 2018.12.15
  • Published : 2018.12.31
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Abstract

90/150 CA which are used as key generators of the cipher system have more randomness than LFSRs, but synthesis methods of 90/150 CA are difficult. Therefore, 90/150 CA synthesis methods have been studied by many researchers. In order to synthesize a suitable CA, the analysis of the characteristic polynomial of 90/150 CA should be preceded. In general, the characteristic of polynomial ${\Delta}_n$ of n cell 90/150 CA is obtained by using ${\Delta}_{n-1}$ and ${\Delta}_{n-2}$. Choi et al. analyzed $H_{2^n}(x)$ and $H_{2^n-1}(x)$, where $H_k(x)$ is the characteristic polynomial of k cell 90/150 CA with state transition rule <$10{\cdots}0$>. In this paper, we propose an efficient method to obtain $H_n(x)$ from $H_{n-1}(x)$ and an efficient algorithm to obtain $H_{2^n+i}(x)$ and $H_{2^n-i}(x)$ ($1{\leq}i{\leq}2^{n-1}$) from $H_{2^n}(x)$ by using this method.

암호 시스템의 키 생성기로 응용되는 90/150 CA는 LFSR보다 난수성이 뛰어나지만 합성법이 어렵기 때문에 CA 합성법에 대한 연구가 많은 연구자에 의해 진행되어 왔다. 적합한 CA를 합성하기 위해 90/150 CA의 특성다항식에 대한 분석이 선행되어야 한다. 일반적으로 n셀 90/150 CA의 특성다항식 ${\Delta}_n$는 ${\Delta}_{n-1}$와 ${\Delta}_{n-2}$을 이용하여 구한다. 본 논문에서는 n셀 90/150 CA <$10{\cdots}0$>의 특성다항식 $H_n(x)$을 (n-1)셀 90/150 CA <$10{\cdots}0$>의 특성다항식 $H_{n-1}(x)$로부터 구하는 방법과 이 방법을 이용하여 $H_{2^n}(x)$로부터 $H_{2^n+i}(x)$와 $H_{2^n+i}(x)$ ($1{\leq}i{\leq}2^{n-1}$)을 효과적으로 구하는 알고리즘을 제안한다.

Keywords

KCTSAD_2018_v13n6_1301_f0001.png 이미지

그림 1. Wolfram의 표기법으로 나타낸 전이규칙 90과 전이규칙 150 Fig. 1 The transition rules 90 and 150 represented by Wolfram’s notation

KCTSAD_2018_v13n6_1301_f0002.png 이미지

그림 2. 90 UCA의 상태전이에 대응하는 시어핀스키 삼각형 형태의 프랙탈 구조 Fig. 2 Fractal structure in the form of Sierpinski’s triangle corresponding to state transition of 90 UCA

KCTSAD_2018_v13n6_1301_f0003.png 이미지

그림 3. Hn(χ)을 구하는 알고리즘 Fig. 3 Algorithm for finding Hn(χ)

KCTSAD_2018_v13n6_1301_f0004.png 이미지

그림 4. H0(χ) ∼ H230(χ)의 계수들로 이루어진 시어핀스키 삼각형 형태의 프랙탈 구조 Fig. 4 Fractal structure in the form of Sierpinski’s triangle composed of coefficients from H0(χ) to H230(χ)

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