Journal of KIISE:Computer Systems and Theory (한국정보과학회논문지:시스템및이론)

Korean Institute of Information Scientists and Engineers (한국정보과학회)
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Modular Multiplier based on Cellular Automata Over $GF(2^m)$

셀룰라 오토마타를 이용한 $GF(2^m)$ 상의 곱셈기

  • Published : 2004.02.01
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Abstract

In this paper, we propose a suitable multiplication architecture for cellular automata in a finite field $GF(2^m)$. Proposed least significant bit first multiplier is based on irreducible all one Polynomial, and has a latency of (m+1) and a critical path of $ 1-D_{AND}＋1-D{XOR}$.Specially it is efficient for implementing VLSI architecture and has potential for use as a basic architecture for division, exponentiation and inverses since it is a parallel structure with regularity and modularity. Moreover our architecture can be used as a basic architecture for well-known public-key information service in $GF(2^m)$ such as Diffie-Hellman key exchange protocol, Digital Signature Algorithm and ElGamal cryptosystem.

본 논문에서는 유한 체 $GF(2^m)$상에서 셀룰라 오토마타 (Cellular Automata)의 구조에 적합한 곱셈기 구조를 제안한다. 제안된 LSB 우선 곱셈 구조는 AOP(All One Polynomial)를 기약 다항식으로 사용하며, m＋1의 지연시간과 $ 1-D_{AND}＋1-D{XOR}$의 임계경로를 갖는다. 특히 정규성, 모듈성, 병렬성을 가지기 때문에 VLSI구현에 효율적이고 나눗셈기, 지수기 및 역원기를 설계하는 데 기본 구조로 사용될 수 있다 또한, 이 구조는 유한 체 상에서 Diffie-Hellman 키 교환 프로토콜, 디지털 서명 알고리즘, 및 ElGamal 암호화와 같이 잘 알려진 공개키 정보 보호 서비스를 위한 기본 구조로 사용될 수 있다.

Keywords

References

  1. D. E. R. Denning, Cryptography and data security Reading, MA: Addison-Wesley, 1983
  2. R. L. Rivest, A. Shamir, and L. Adleman, 'A Method for Obtaining Digital Signatures and Public-key Cryptosystems,' Comm AMC Vol. 21, pp.120-126, 1978 https://doi.org/10.1145/359340.359342
  3. E. R. Berlekamp, Algebraic Coding Theory, New York: McGraw-Hill, 1986
  4. R. J. McEliece, Finite fields for Computer Scientists and Engineers, New York: Kluwer-Academic, 1987
  5. C. S. Yeh. S. Reed, and T. K. Truong, 'Systolic multipliers for finite fields (2$^m$),' IEEE Trans. on Computers. Vol. 33, pp.357-360, Apr. 1984 https://doi.org/10.1109/TC.1984.1676441
  6. S. K. Jain and L. Song, 'Efficient Semi systolic Architectures for finite field Arithmetic,' IEEE Trans. on VLSI Systems, Vol. 6, No.1, Mar. 1998 https://doi.org/10.1109/92.661252
  7. J. L. Massey and J. K. Omura, Computational method and apparatus for finite field arithmetic, U. S. Patent application, submitted 1981
  8. S. W. Wei, 'A systolic power-sum circuit for GF($(2^m)$),' IEEE Trans. Comput., Vol. 43, pp.226-229, Feb. 1994 https://doi.org/10.1109/12.262128
  9. T. Itoh and S. Tsujii, 'Structure of parallel multipliers for a class of finite fields GF(2m),' Info. Camp. Vol. 83, pp.21-40, 1989 https://doi.org/10.1016/0890-5401(89)90045-X
  10. M. A. Hasan, M. Z. Wang and V. K. Bhargava, 'Modular Construction of low complexity parallel multipliers for a class of finite fields GF(2m),' IEEE Trans. on Computers. Vo1.8. pp.962-971, Aug. 1992 https://doi.org/10.1109/12.156539
  11. J. V. Newmann, The theory of self-reproducing automata, Univ. of lllinois Press, Urbana (London, 1966
  12. P. P. Choudhury, 'Cellular Automata Based VLSI Architecture for Computing Multiplication And Inverses In GF($GF(2^m)$),' IEEE 7th International Conference on VLSI Design, pp.279-282. Jan. 1994 https://doi.org/10.1109/ICVD.1994.282702
  13. S. T. J. Fenn et ai, 'Bit-serial Multiplication in GF(2$^m$) using irreducible all one polynomials,' lEE. Proc. Comput. Digit. Tech, Vol. 144. ·No. 6. Nov. 1997