This Article 
 Bibliographic References 
 Add to: 
Analysis of One-Dimensional Linear Hybrid Cellular Automata over GF(q)
July 1996 (vol. 45 no. 7)
pp. 782-792

Abstract—This paper studies theoretical aspects of one-dimensional linear hybrid cellular automata over a finite (Galois) field. General results concerning the characteristic polynomials of such automata are presented. A probabilistic synthesis algorithm for determining such a linear hybrid cellular automaton with a specific characteristic polynomial is given, along with empirical results and a theoretical analysis. Cyclic-boundary cellular automata are defined and related to the more common null-boundary cellular automata. An explicit similarity transform between a cellular automaton and its corresponding linear feedback shift register is derived.

[1] P.H. Bardell, Analysis of Cellular Automata Used as a Pseudo-Random Pattern Generators Proc. Int'l Test Conf., pp. 762-768, 1990.
[2] D. Roy Chowdhury, S. Basu, I. Sen Gupta, and P. Pal Chaudhuri, "Design of CAECC—Cellular Automata Based Error Correcting code," IEEE Trans. Computers, vol. 43, no. 6, pp. 759-764, June 1994.
[3] A. Das and P.P. Chaudhuri, “Pseudo-Exhaustive Test Pattern Generation Using Cellular Automata,” IEEE Trans. Computers, Vol. 42, No. 3, Mar. 1993, pp. 340-352.
[4] G. Edirisooriya and J. Robinson, "Aliasing in Multiple-Valued Test Data Compaction," Proc. 22nd Int'l Symp. Multiple-Valued Logic, pp. 43-50, 1992.
[5] D.M. Miller, J.C. Muzio, M. Serra, X. Sun, S. Zhang, and R.D. McLeod, "Cellular Automata Techniques for Compaction Based BIST," Proc. IEEE Int'l Symp. Circuits and Systems, pp. 1,893-1,896, 1991.
[6] M. Serra et al., "The Analysis of One-Dimensional Linear Cellular Automata and Their Aliasing Properties," IEEE Trans. Computer-Aided Design, vol. 9, no. 7, pp. 767-778, July 1990.
[7] Ph. Tsalides, T.A. York, and A. Thanailakis, "Pseudorandom Number Generators for Systems Based on Linear Cellular Automata," IEE Proc. Part E: Computers and Digital Techniques, vol. 138, pp. 241-249, 1991.
[8] S. Zhang, R. Byrne, J.C. Muzio, and D.M. Miller, "Why Cellular Automata Are Better than LFSRs as Built-In Self-Test Generators for Sequential-Type Faults," Proc. IEEE Int'l Symp. Circuits and Systems, vol. 1, pp. 69-72, 1994.
[9] P. Guan and Y. He, "Exact Rules for Deterministic Cellular Automata with Additive Rules," J. Statistical Physics, vol. 43, pp. 463-478, 1986.
[10] E. Jen, "Global Properties of Cellular Automata," J. Statistical Physics, vol. 43, pp. 219-242, 1986.
[11] H.Y. Lee and Y. Kawahara, "On Dynamical Behaviours of Cellular Automata ca-60," Bulletin Informatics and Cybernetics, vol. 25, pp. 21-25, 1992.
[12] M. Nohmi, "On a Polynomial Representation of Finite Linear Cellular Automata," Bulletin Informatics and Cybernetics, vol. 24, pp. 137-145, 1991.
[13] S. Wolfram, "Statistical Mechanics of Cellular Automata," Review Modern Physics, vol. 55, pp. 601-644, 1983.
[14] S. Wolfram, "Universality and Complexity in Cellular Automata," Physica, vol. 10D, pp. 1-35, 1984.
[15] Z. Zilic and Z. Vranesic,“Current mode CMOS Galois field circuits,” Proc. 23rd Int’l Symp. Multiple-Valued Logic, pp. 245-250, May 1993.
[16] K. Cattell and J.C. Muzio, "Synthesis of One-Dimensional Linear Hybrid Cellular Automata," IEEE Trans. Computer-Aided Design, vol. 15, pp. 325-335, 1996.
[17] J.M. Mesirov and M.M. Sweet, "Continued Fraction Expansions of Rational Expressions with Irreducible Denominators in Characteristic 2," J. Number Theory, vol. 27, pp. 144-148, 1987.
[18] R. Lidl and H. Niederreiter,An Introduction to Finite Fields and Their Applications.Cambridge: Cambridge Univ. Press, 1986.
[19] R.J. McEliece, Finite Fields for Computer Scientists and Engineers. Kluwer Academic, 1987.
[20] T. Hansen and G.K. Mullen, "Primitive Polynomials Over Finite Fields," Math. Comp., vol. 59, pp. 639-643, 1992.
[21] K. Cattell and M. Serra, "The Analysis of One Dimensional Multiple-Valued Linear Cellular Automata," Proc. 20th Int'l Symp. Multiple-Valued Logic, pp. 402-409, 1990.
[22] H.S. Stone, Discrete Mathematical Structures and Their Applications. Science Research Associates Inc., 1973.

Index Terms:
Cellular automata, characteristic polynomial, finite field, LFSM.
Kevin Cattell, Jon C. Muzio, "Analysis of One-Dimensional Linear Hybrid Cellular Automata over GF(q)," IEEE Transactions on Computers, vol. 45, no. 7, pp. 782-792, July 1996, doi:10.1109/12.508317
Usage of this product signifies your acceptance of the Terms of Use.