scholarly journals The reversibility of one-dimensional cellular automata

2021 ◽  
Vol 22 (1) ◽  
pp. 7-15
Author(s):  
Alexey E. Zhukov
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
High Performance ◽  
Finite Automata ◽  
Finite Size ◽  
Transition Function ◽  
One Dimensional ◽  
Enumeration Problems ◽  
Reversible Cellular Automata ◽  
Binary Filter

Recently the reversible cellular automata are increasingly used to build high-performance cryptographic algorithms. The paper establishes a connection between the reversibility of homogeneous one-dimensional binary cellular automata of a finite size and the properties of a structure called binary filter with input memory and such finite automata properties as the prohibitions in automata output and loss of information. We show that finding the preimage for an arbitrary configuration of a one-dimensional cellular automaton of length L with a local transition function f is associated with reversibility of a binary filter with input memory. As a fact, the nonlinear filter with an input memory corresponding to our cellular automaton does not depend on the number of memory cells of the cellular automaton. The results obtained make it possible to reduce the complexity of solving massive enumeration problems related to the issues of reversibility of cellular automata. All the results obtained can be transferred to cellular automata with non-binary cell filling and to cellular automata of dimension greater than 1.

scholarly journals SPECTRAL PROPERTIES OF REVERSIBLE ONE-DIMENSIONAL CELLULAR AUTOMATA

2003 ◽  
Vol 14 (03) ◽  
pp. 379-395 ◽  
Author(s):  
JUAN CARLOS SECK TUOH MORA ◽  
SERGIO V. CHAPA VERGARA ◽  
GENARO JUÁREZ MARTÍNEZ ◽  
HAROLD V. McINTOSH
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Spectral Properties ◽  
Neighborhood Size ◽  
Local Behavior ◽  
One Dimensional ◽  
Single Positive ◽  
Reversible Cellular Automata ◽  
Inverse Rule

Reversible cellular automata are invertible dynamical systems characterized by discreteness, determinism and local interaction. This article studies the local behavior of reversible one-dimensional cellular automata by means of the spectral properties of their connectivity matrices. We use the transformation of every one-dimensional cellular automaton to another of neighborhood size 2 to generalize the results exposed in this paper. In particular we prove that the connectivity matrices have a single positive eigenvalue equal to 1; based on this result we also prove the idempotent behavior of these matrices. The significance of this property lies in the implementation of a matrix technique for detecting whether a one-dimensional cellular automaton is reversible or not. In particular, we present a procedure using the eigenvectors of these matrices to find the inverse rule of a given reversible one-dimensional cellular automaton. Finally illustrative examples are provided.

scholarly journals BLOCK CIPHERS ON THE BASIS OF REVERSIBLE CELLULAR AUTOMATA

2020 ◽  
Vol 10 (1) ◽  
pp. 8-11
Author(s):  
Yuliya Tanasyuk ◽  
Petro Burdeinyi
Keyword(s):  
Cellular Automata ◽  
Block Cipher ◽  
Key Generation ◽  
One Dimensional ◽  
C Programming Language ◽  
C Programming ◽  
Shared Secret ◽  
Reversible Cellular Automata ◽  
The Given ◽  
First Time

The given paper is devoted to the software development of block cipher based on reversible one-dimensional cellular automata and the study of its statistical properties. The software implementation of the proposed encryption algorithm is performed in C# programming language in Visual Studio 2017. The paper presents specially designed approach for key generation. To ensure desired cryptographic stability, the shared secret parameters can be adjusted to contain information needed for creating substitution tables, defining reversible rules, and hiding final data. For the first time, it is suggested to create substitution tables based on iterations of a cellular automaton that is initialized by the key data.

scholarly journals Characterization of the Evolution of Nonlinear Uniform Cellular Automata in the Light of Deviant States

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Pabitra Pal Choudhury ◽  
Sudhakar Sahoo ◽  
Mithun Chakraborty
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Boolean Functions ◽  
State Transition ◽  
The State ◽  
Transition Diagram ◽  
One Dimensional ◽  
State Transition Diagram ◽  
Linear Rule

Dynamics of a nonlinear cellular automaton (CA) is, in general asymmetric, irregular, and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable, primarily due to the presence of a matrix handle. In this paper, we present a novel technique of studying the properties of the State Transition Diagram of a nonlinear uniform one-dimensional cellular automaton in terms of its deviation from a suggested linear model. We have considered mainly elementary cellular automata with neighborhood of size three, and, in order to facilitate our analysis, we have classified the Boolean functions of three variables on the basis of number and position(s) of bit mismatch with linear rules. The concept of deviant and nondeviant states is introduced, and hence an algorithm is proposed for deducing the State Transition Diagram of a nonlinear CA rule from that of its nearest linear rule. A parameter called the proportion of deviant states is introduced, and its dependence on the length of the CA is studied for a particular class of nonlinear rules.

ON THE STRUCTURE OF REAL-VALUED ONE-DIMENSIONAL CELLULAR AUTOMATA

2011 ◽  
Vol 21 (05) ◽  
pp. 1265-1279 ◽  
Author(s):  
XU XU ◽  
STEPHEN P. BANKS ◽  
MAHDI MAHFOUF
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Cellular Systems ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
New Type ◽  
Complex Patterns

It is well-known that binary-valued cellular automata, which are defined by simple local rules, have the amazing feature of generating very complex patterns and having complicated dynamical behaviors. In this paper, we present a new type of cellular automaton based on real-valued states which produce an even greater amount of interesting structures such as fractal, chaotic and hypercyclic. We also give proofs to real-valued cellular systems which have fixed points and periodic solutions.

scholarly journals CELLULAR AUTOMATON SUPERCOLLIDERS

2011 ◽  
Vol 22 (04) ◽  
pp. 419-439 ◽  
Author(s):  
GENARO J. MARTÍNEZ ◽  
ANDREW ADAMATZKY ◽  
CHRISTOPHER R. STEPHENS ◽  
ALEJANDRO F. HOEFLICH
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Elastic Collision ◽  
Collective Excitations ◽  
Compact Groups ◽  
Dimensional Torus ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Spatially Extended ◽  
Binary Strings

Gliders in one-dimensional cellular automata are compact groups of non-quiescent and non-ether patterns (ether represents a periodic background) translating along automaton lattice. They are cellular automaton analogous of localizations or quasi-local collective excitations traveling in a spatially extended nonlinear medium. They can be considered as binary strings or symbols traveling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We analyze what types of interaction occur between gliders traveling on a cellular automaton "cyclotron" and build a catalog of the most common reactions. We demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in nonlinear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analyzed via implementation of cyclic tag systems.

scholarly journals Representing Reversible Cellular Automata with Reversible Block Cellular Automata

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Jérôme Durand-Lose
Keyword(s):  
System Theory ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Linear Time ◽  
Fixed Size ◽  
Mathematical System ◽  
Reversible Cellular Automaton ◽  
International Audience ◽  
Infinite Lattices ◽  
Reversible Cellular Automata

International audience Cellular automata are mappings over infinite lattices such that each cell is updated according tothe states around it and a unique local function.Block permutations are mappings that generalize a given permutation of blocks (finite arrays of fixed size) to a given partition of the lattice in blocks.We prove that any d-dimensional reversible cellular automaton can be exp ressed as thecomposition of d+1 block permutations.We built a simulation in linear time of reversible cellular automata by reversible block cellular automata (also known as partitioning CA and CA with the Margolus neighborhood) which is valid for both finite and infinite configurations. This proves a 1990 conjecture by Toffoli and Margolus <i>(Physica D 45)</i> improved by Kari in 1996 <i>(Mathematical System Theory 29)</i>.

Wave Propagation in Filamental Cellular Automata

2012 ◽  
pp. 60-73
Author(s):  
Alan Gibbons ◽  
Martyn Amos
Keyword(s):  
Wave Propagation ◽  
Distributed Computing ◽  
Cellular Automata ◽  
Numerical Simulations ◽  
Finite Automata ◽  
Cooperative Behaviour ◽  
One Dimensional ◽  
Minimum Requirements ◽  
State Changes

Motivated by questions in biology and distributed computing, the authors investigate the behaviour of particular cellular automata, modelled as one-dimensional arrays of identical finite automata. They investigate what kinds of self-stabilising cooperative behaviour may be induced in terms of waves of cellular state changes along a filament of cells. The authors report the minimum requirements, in terms of numbers of states and the range of communication between automata, for this behaviour to be observed in individual filaments. They also discover that populations of growing filaments may have useful features not possessed by individual filaments, and they report the results of numerical simulations.

PARALLEL GENERATION AND PARSING OF ARRAY LANGUAGES USING REVERSIBLE CELLULAR AUTOMATA

1994 ◽  
Vol 08 (02) ◽  
pp. 543-561 ◽  
Author(s):  
KENICHI MORITA ◽  
SATOSHI UENO
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Reversible Cellular Automata ◽  
New System

We propose a new system of generating array languages in parallel, based on a partitioned cellular automaton (PCA), a kind of cellular automaton. This system is called a PCA array generator (PCAAG). The characteristic of PCAAG is that a”reversible” version is easily defined. A reversible PCA (RPCA) is a backward deterministic PCA, and we can construct a deterministic “inverse” PCA that undoes the operations of the RPCA. Thus if an array language is generated by an RPCA, it can be parsed in parallel by a deterministic inverse PCA without backtracking. We also define two subclasses of PCAAG, and give examples of them that generate geometrical figures.

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