Characteristic polynomials of one-dimensional linear hybrid cellular automata

Date

2018-06-12

Authors

Cattell, Kevin Michael

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Abstract

A one-dimensional linear hybrid cellular automaton (CA) is a specialised form of linear finite state machine. These machines are of interest, both for their theoretical properties and for their applications in VLSI built-in-self-test, random number generation, cryptography, coding theory, and other areas. This work is a study of the algebraic properties of the characteristic polynomials of CA, primarily for machines defined over GF(2). Several problems, previously open, are solved: the efficient synthesis of a CA from an irreducible polynomial, the existence and uniqueness of CA for irreducible polynomials, the reducibility of the characteristic polynomial of a cyclic-boundary CA, and the form of a similarity transform between CA and linear feedback shift registers. A probabilistic algorithm for the synthesis of CA over finite fields other than GF(2) is presented. Various other results concerning the characteristic polynomial of CA are derived, and possible directions for future research are discussed.

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Keywords

Cellular automata

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