On the invertible cellular automata 150 over Fp

doi:10.1016/j.amc.2012.11.036

Applied Mathematics and Computation

Volume 219, Issue 10, 15 January 2013, Pages 5427-5432

https://doi.org/10.1016/j.amc.2012.11.036 Get rights and content

Abstract

In this work the reversibility problem for cellular automata with rule number 150 over the finite field $F_{p}$ is tackled. It is shown that when null boundary conditions are stated, the reversibility is independent of the state set $F_{p}$ and it appears when the number of cells of the cellular space satisfies some conditions. Furthermore, the explicit expressions of the inverse cellular automata are computed.

Introduction

Cellular automata are a particular type of finite state machines with a great capacity to simulate physical, biological or environmental complex phenomena ([17]). Moreover, they are also widely used in Cryptography and Information Technologies (see, for example, [11]).

A cellular automaton consists of a discrete spatial lattice of memory units called cells which are endowed with a state from a finite state set at every step of time (see [7]). These states are updated in discrete time steps according to a local transition function which depends on the states of the cells in some neighborhood around it. As the lattice is finite, some type of boundary conditions must be taken into account.

Reversibility implies that information can be neither created or destroyed ([16]). A cellular automaton is said to be reversible when there exists another cellular automaton, its inverse, which makes possible the evolution backwards ([10]). The reversibility of cellular automata has been extensively studied not only from a theoretical point of view (see, for example, [1], [3], [4], [9], [10], [12], [14], [15]), but also from an applicable point of view; in this sense as reversible cellular automata preserve the information given by the initial states throughout the evolution, the majority of their applications concern with information preserving computing processes and cryptography (see, for example, [13]).

Of special interest are those cellular automata for which the state set is $F_{2}$ and the local transition function is a 3-variable boolean function whose variables are the states of the main cell and its two nearest neighbors. They are called elementary cellular automata [17]. One of the most important elementary cellular automata is that one given by the local transition function defined by the XOR sum of the three variables; it is called elementary cellular automata with rule number 150. Its importance is due to its applications to cryptographic protocols (see, for example, [6], [11]). The reversibility problem for this cellular automata was solved in [8] considering the usual finite field $F_{2}$ . In this work, the extended reversibility problem for an arbitrary finite field $F_{p}$ (where p is prime) is tackled and solved.

The rest of the paper is organized as follows: in Section 2 the mathematical background is presented (the DETGTRI algorithm and the basic mathematical theory of cellular automata is introduced); the problem of the reversibility of the cellular automaton considered is solved in Section 3; finally, some illustrative examples are shown in Section 4.

Section snippets

The DETGTRI algorithm

In [5] an efficient algorithm to compute the determinant of a tri-diagonal matrix was proposed. It is as follows: Let $T = (\begin{matrix} d_{1} & a_{1} & 0 & \dots & \dots & 0 \\ b_{2} & d_{2} & a_{2} & ⋱ & ⋮ \\ 0 & b_{3} & d_{3} & ⋱ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & ⋱ & 0 \\ ⋮ & ⋱ & ⋱ & d_{n - 1} & a_{n - 1} \\ 0 & \dots & \dots & 0 & b_{n} & d_{n} \end{matrix}),$ be an nth order tridiagonal matrix and set: $c_{i} = \{\begin{matrix} d_{1}, & if i = 1, \\ d_{i} - \frac{a_{i - 1} b_{i}}{c_{i} - 1}, & if 2 ⩽ i ⩽ n . \end{matrix}$ Then, the determinant of T is computed following the next two steps:

(1)
Compute the parameters $c_{1}, \dots, c_{n}$ given in (2) in such a way that if $c_{i} = 0$ for any i, then set $c_{i} = x$ and continue to compute $c_{i + 1}, c_{i + 2}, \dots$ in terms of the variable x.
(2)
Compute the following

The reversibility

Let $A_{n} (F_{p})$ be the cellular automaton with rule number 150, endowed with null boundary conditions and defined over $F_{p}$ with a cellular space of n cells. The following results hold:

Proposition 1

$A_{n} (F_{p})$ is reversible if and only if $n ≢ 2 (\mod 3)$ .

Proof

Let $M_{n}$ be the transition matrix of $A_{n} (F_{p})$ . Then, using DETGTRI algorithm, we obtain the following expression for the coefficients $c_{i}$ , $c_{i} = \{\begin{matrix} 1, & si i = 1, \\ 1 - \frac{1}{c_{i - 1}}, & si 2 ⩽ i ⩽ n . \end{matrix}$ Consequently, as a simple calculus shows: $\begin{matrix} c_{1} = 1, \\ c_{3 k - 1} = x, k \in Z^{+}, \end{matrix}$ $\begin{matrix} c_{3 k} = \frac{x - 1}{x}, k \in Z^{+}, \\ c_{3 k + 1} = \frac{- 1}{x - 1}, k \in Z^{+} \end{matrix}$ and then $p_{M_{n}} (x) = \prod_{i = 1}^{n} c_{i} = \{\begin{matrix} (- 1)^{k}, & si n = 3 k \end{matrix}$

Conclusions

In this work the reversibility problem for cellular automata with rule number 150 and state set $F_{p}$ is completely solve in the case of null boundary conditions. It is shown that the cellular automata is reversible if and only if the number of cells of the cellular space, n, is of the form $n = 3 k$ or $n = 3 k + 1$ where $k \in Z^{+}$ . Using simple matrix operations the characteristic matrices of the inverse cellular automata are also explicitly computed.

Future work aimed to study the reversibility problem in the

Acknowledgments

We would like to thank the anonymous referees for his/her valuable comments. This work has been supported by Ministry of de Economy and Competitiveness (Spain) and European FEDER Fund under project TUERI (TIN2011-25452).

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On the invertible cellular automata 150 over Fp

Abstract

Introduction

Section snippets

The DETGTRI algorithm

The reversibility

Conclusions

Acknowledgments

Theor. Comput. Sci.

Theor. Comput. Sci.

Appl. Math. Comput.

Theor. Comput. Sci.

Appl. Math. Comput.

J. Franklin

Reversibility of 1D cellular automata with periodic boundary over finite fields

J. Stat. Phys.

On the invertible cellular automata 150 over $F_{p}$