the projection and the analysis of the cellular automata ...orar.upit.ro/docmanagerpub/file/7 -...

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THE PROJECTION AND THE ANALYSIS OF THE CELLULAR AUTOMATA FOR PROCESSING OF INFORMATION

Petre ANGHELESCU, Emil SOFRON, Silviu IONIŢĂ

Department of Electronics and Computers, University of Pitesti Targul din Vale, 0300, Pitesti, Romania

[email protected]

Keywords: cellular automata, programmable cellular automata, dynamic systems, chaotic dynamic systems, cryptography, artificial life, reconfigurable hardware structures, evolutionary hardware, VLSI, FPGA, optimization.

Abstract: The major objective of this research is to develop techniques for realizing a software and hardware encryption system with low cost, high speed and good security. This has been achieved by using a simple structure called cellular automata. In present, the necessity of a circuits and communication systems with reduced power consumption, ample transmission capacity, flexibility and reliability at a low fabrication cost are essential technical-economic challenge.

INTRODUCTION The cellular automata (CA) domain is very

dynamic, actuality and great perspective. Cellular automata are massive parallel computational systems used with success to modeling complex natural systems, replacing the traditional methods that use for modeling complicated equations systems. CA are very suitable to be implemented both in software and hardware in FPGA (Field Programmable Gate Array) circuits because they has a regular structure and are composed from simple components with local interconnections.

I have to mention that there is not a method of ideal encryption at present and this is an additional reason regarding the suitability of the elaboration of new methods and encryption systems. The research involves a series of stages, combining the fundamental research, the simulation of the proposed model, the development of a simulator with facilities specific of evaluation of the performances, the implementation in FPGA using the VHDL (VHSIC Hardware Description Language, VHSIC Very High Speed Integrated Circuits) program as a support of description, simulation, test and optimization.

The actual stage in the CA theory and technology is marked by the evolution of software and hardware technology.

Notable is the fact that the strong relation between the cellular automata and cryptography can be found in fundamental Shanon’s paper [Shan49]: “Good mixing transformations are often formed by repeated products of two simple non-commuting operations. Hopf has shown, for example, that pastry dough can be mixed by such a sequence of operations. The dough is first rolled out into a thin slab, then folded over, then rolled, and the folded again, etc.” As we can observe, Shanon write about a system composed from simple components that interaction between them – with a transparent local comportment – but the global comportment of the entire system unsuspected, things that are well know in the cellular automata theory. In fact is a careless thing to use one or more nonlinear functions (in the case of cellular automat theory, the functions are represented by evolution rules) for projection of modern cipher, where these functions can be considerate as variant in continuous or discrete time of the cellular automaton.

From the documentary research effectuated I observed that also from the 90’s years many researches have apprise an

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interesting association between cellular automata and cryptography, many properties of the cellular automata having some correspondence in classical encryption systems that are based on the computational methods [Fran03]. For example, the ergodicity property of the cellular automata is identical with confusion because the output signal will have the same distribution as for any text used as input signal (constrained to encryption). Also, the sensibility to the initial conditions and control parameters, specifically to cellular automata, is identically with the diffusion property at a low change of text or secret key in case of classical encryption methods, because a small modification of initial condition or control parameters will determine a substantial modification of output signal. In addition, the deterministic chaotically dynamic, is similar with pseudorandom systems used in classical methods and the complexity of a dynamical systems that determine the efficiency of the entire information protection process is equivalent with the algorithm complexity from classical encryption.

In the domain of researches having as subject the association between the cellular automata and cryptography was reported more encryption systems based on the cellular automata theory.

So that, a very simple variant used for codification using cellular automata is reported by Stephen Wolfram in [Wolf85] [Wolf02] and is based on the fact that the cellular automata from class III, conform the Wolfram classification [Wolf94], are dynamical chaotic systems. In this case, the evolution of the cellular automaton depends considerable of the initial state, but we can say that after some time the state is forgotten in sense in which cannot be found from current configuration analyses. Anyway, if we repeat the initial state, the evolution will be the same. The encryption system proposed by Wolfram can be included in category “Chaotic stream ciphers based on the pseudorandom number generator” (PRNG). The based principle of these ciphers is to obtain the encryption text by mixing the output of these pseudorandom number generators with the message [Lee03]. Another variant of encryption system based on the cellular automata, that consider also the inverse iteration, is presented in [Adam94] and [Mart04]. Here is used a bi-dimensional cellular

automaton and the dates are the initial state of the cellular automaton. Using a reversible evolution rule the initial message is modified progressive. The message is decrypted rolling the inverse rule the same number of iterations as to encryption. This encryption system can be included in category of “Stream ciphers based on inverse iteration-with reaction”. These systems can be also based on a series of evolution rules that served as chaotic system, rules used for encryption and decryption.

Other cryptosystem realized with the help of cellular automata combine the direct and inverse iteration [Guto94], “Block ciphers based on the direct and inverse iteration” [Masu02] – these ciphers was as a general rule proposed for image encryption. Here is used a bi-dimensional cellular automaton, the message being the initial state of this. The codification implied the inverse iteration of a rule, the key is a rule. This rule is not necessary to be reversible: for inverse iteration is chosen randomly one of the possible states of the cellular automaton. For decryption we must know the rule that is direct iterated and use the same number of steps.

The cryptographic systems anterior presented use one or more evolution rule for realizing of the cryptosystem, and the initial condition and/or the control parameter specific to this rule are the key. Beside the assurance of the confidentiality, the encryption systems based on the cellular automata theory can be used to insurance some other security services, as: attestation and integrity of the dates [Wolf02].

Also a relative small number of cryptosystems based on the theory of the cellular automata was proposed, some of them are vulnerable. The majority of articles that present this subject were published in diverse journals and less in the specialty journals. This explains why we find so few comments regarding to this subject and of course why are so few amendments to this systems. Most of all is difficult to evaluate the performance of them in a systematic mode, because at the presentations are omitted essential things, as: implementations details, the key management and the methods for analysis of the proposed systems. The most variants proposed ignore or treats superficially this kind of problems.

P. ANGHELESCU, E. SOFRON, S. IONIŢĂ – The Projection and the Analysis of the Cellular Automata for Processing of Information 45

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The organization of this paper was done with respect to the principles of an applicative research.

In the first chapter we present in the first instance the cellular automata domain, presented the inedited theory and the cause that determined the development: the continuous increase of complexity of software and hardware systems. Here is presented the cellular automata concept. Cellular automata are a particular class of dynamical systems discrete in time and space, composed from a matrix of nodes (cells) by different dimensions, in which the nodes (cells) interact with each other to produce complex forms of comportment [Sung04] (figure 1).

Figure 1. Usual geometries of cellular automata

(a) Unidimensional cellular automata (b), (c), (d) bidimensional cellular automata (e) tridimensional cellular automata

Diversified in variants more and more sophisticated, cellular automaton was the base for neural computer, genetics algorithms, and finally at a complex and controversial couple named artificial life. Cellular automata represent a “converse pole” as computing architecture in comparison with sequential model: are parallel systems without central processing unit in which the computation powerful of his elements are very reduced [Dasc03]. Cellular automaton can be look like an artificial universe that can be particularized choosing the structure and laws and than it respect those laws.

Any cellular automaton is composed by the following elements:

1. A lattice of cells – a spatial system composed from cells – each cell can be in a finite

number of states and is restricted to local neighborhood interactions only, and as a result it is incapable of immediate global communication. The lattice of cells can have any dimension and can be expand infinite. Simple most cellular automaton is one-dimensional A(i), Figure 1, in which every cell had only two neighbors and the ensemble of N cells that getting out the automaton can or not compose a closed form like an ring.

2. The width and topology of the neighborhood of the cell, id est. the layout and the number of cells that are used in establishment of the next state of a cell. We can define the neighborhood of a cell as the mass cells from the row that can have some influence in establishment of the next state of a cell (because from that cells only two of them are neighbor with the goal cell – speaking about one-dimensional cellular automata – in other case we use the nomenclature pseudo-neighbors in case in which the neighbors including also other cells beside the adjacent cells). The width of each side of the array wj is the width of the jth side of the array, where j=1, 2, 3 … n. The width of the neighborhood of the cell j is the width of the neighborhood at the jth side of the array. Classical examples for cell neighborhood are presented in figure 2 (Von Neumann Neighborhood) with 3 cells for one-dimensional cellular automata respective 5 for bi-dimensional cellular automata, case in which are considerate the direct neighbors for establishment of the next state of a one cell and figure 3 (Moore Neighborhood) with 3 cells for one-dimensional cellular automata respective 9 cells for bi-dimensional cellular automata, case in which are considerate the direct and the diagonal neighbor

The cellular automata’s local rules (or diagram of finite automata), rules that implies, as inputs, also the state of the neighbor, prominence the automat dynamics. The next-state of a cell is assumed to depend on itself and on its neighbors. As arguments, this update function takes the cell's present state and the states of the cells in its interaction neighborhood as shown in figure 4.


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Figure 2. von Neumann Neighborhood Figure 3. Moore Neighborhood

If the next-state function of a cell is expressed in the form of a truth table, then the decimal equivalent of the output is conventionally called the rule number for the cell. As the cellular automaton evolves, the update function will determine how microscopic (or local) interactions influence the overall macroscopic (or global) behavior of the complete system. Also, we must explicit specify which cells will represent the neighbors of a central cell.

Figure 4. The cell state modification

Also, in this chapter are presented some interconnections between CA and complexity theory, explained the fashion of construction and study of this model and fundamental aspects regarding to the practical applications. Here are presented the CA comportment, the CA classification, the computational capability, the structure and the dynamic behavior and the problem of CA synthesis. This chapter justifies the motivation for the research subject and prominence the aspects regarding to the CA.

The chapter 2 is dedicated for the analyses of the cellular automata as pseudo-random number generators (CA-PRNG) and programmable cellular automata (PCA). From the effectuated analyses results the regular and chaotic comportment of the CA, thing very important in the process of information

protection where the principal goal of the realized actions is to assure the recuperation of information at the reception. For this reason, in this chapter were proposed methods to analyze the CA comportment and also methods for synchronization and control of chaotic comportment. Also in this chapter was analyzed the hardware structures that are suitable for implementation of the CA encryption system. The analyze of a CA-PRNG used for encryption

systems

The CA-PRNG used for the proposed encryption system is a null-boundary CA configured with rules 90 and 150 having primitive characteristic polynomials that will run through the maximum length of (2N-1) distinct nonzero status [Hort89]. The two rules are presented in table 1.

Table 1 Rules used for the next state of a CA cell

Regula

7 111

6 110

5 101

4 100

3 011

2 010

1 001

0 000

90 0 1 0 1 1 0 1 0 150 1 0 0 1 0 1 1 0

27 26 25 24 23 22 21 20

The corresponding combinational logic of rules 90 and 150 for CA can be expressed as follows:

Rule 90:

)(1)(1)1( tiatiatia +⊕−=+

Rule 150: )(1)()(1)1( tiatiatiatia +⊕⊕−=+


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Such a maximum length CA will generate

high-quality pseudo-random patterns. In figure 5 we present a state transition graph of a 4-bit hybrid CA with rules <90, 150, 90 and 150>. It generates a 4-bit pseudo-random pattern.

Figure 5 The state transition diagram of a maximum

length CA

In continuance we analyzed with the help of NIST (The National Institute of Standards and Technology) statistical tests [NIST05] the statistical properties of the sequences generated by the CA. Also, we compared the results withthe other pseudo-random generators as classical LFSR (Linear Feedback Shift Register). We find that this CA has better results than LFSR [Petr06] and also could be implemented very easy in hardware, in FPGA circuits (CA are parallel systems without central processing unit).

The conception and the analyses of a PCA used

for encryption systems

As succession of a lot of simulation with various structural and behavioral parameters we concept and realized an original PCA that combine in some way the rules 51, 60 and 102 to generate sequences that are repeated periodically, after a number of evolution steps. The three rules are presented in table 2.

Table 2 Rules used for the next state of a PCA cell

Regula

7 111

6 110

5 101

4 100

3 011

2 010

1 001

0 000

51 0 0 1 1 0 0 1 1 60 0 0 1 1 1 1 0 0

102 0 1 1 0 0 1 1 0 27 26 25 24 23 22 21 20

The corresponding combinational logic of

rules 51, 60 and 102 for PCA can be expressed as follows:

Rule 51: )()1( tata ii =+

Rule 60: )()()1( 1 tatata iii −⊕=+

Rule 102: )()()1( 1 tatata iii +⊕=+

In figure 6 we present a state transition graph of a 8-bit hybrid PCA with rules 51, 60 (or 102).

a)

b) Figure 6 The state transition diagram of a PCA with

8 cells with rules 51, 60 or 102 a) initial state 00000000 b) initial state 01010101

Table 3 shows the number of 8-cell CA

configurations, each generates cycles of length 2, 4, 8 or 16.

Table 3 CA having even length cycles of length 2, 4,

8, 16 or combination of them Number of CA configurations generating

even length cycles

Rules applied to cells

8-cell CA

having 2

length cycles

8-cell CA

having 4

length cycles

8-cell CA

having 8

length cycles

8-cell CA

having 16

length cycles

8-cell CA

having 2, 4, 8 or 16

length cycles

51, 60 (or

102) 7 327 156 2 20

An 8-cell CA configured with rules 51, 60

(or 102) has 512 kinds of configurations, but only 156 of them have cycle length 8. The others have cycles of 2, 4, 16 or combination of them.

In the third chapter we present the modality of projection of an encryption system that work on the basis of the CA theory. The encryption system is composed from the two CA presented in chapter 2. The encryption method proposed in this chapter is based on the fact that

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the CAs from class III are chaotic dynamical systems and CAs from class II exhibit periodic behavior. In these cases, their evolution depends essentially of the initial state, but we can say that after a while the initial state is “forgotten”, in sense that the initial state cannot be retrievable through analyses of the current configuration.

Here are presented two encryption schemes: one uses a CA-PRNG and a PCA and the other use a CA-PRNG and 5 PCA in pipeline. The second scheme is used to increase the security of the system.

In figure 7 and 8 we present the bloc scheme of the encryption system and the 5 PCA pipeline.

Figure 7. Bloc scheme of the encryption system

In the block cipher scheme, one 8-bit

message block is enciphered by one enciphered function. For the sake of simplicity, the enciphering function has 5 PCA to operate on 8-bit data. This PCA are constructed using the 8-bit null boundary CA with rules 51, 60 or 102. Each CA configuration generates cycles of length 8. The block cipher (decipher) procedure can be defined as follows:

• Load the PCA with one byte plaintext (ciphertext) from I/O.

• Load a rule configuration control word from the CA-PRNG or memory into the PCA.

• Run the PCA for 1…7 cycles.

• Repeat steps 2 and 3 for 5 times.

• Send one byte ciphertext (plaintext) to I/O.

• If not end of plaintext (ciphertext) go to step 1. Otherwise, stop the process.

The rules employed in encryption are

shown in Table 4. The system designer is free to take any numbers in the 156 combinations to enhance security.

Table 4 Rules used in the encryption process

Rule No.

Rules applied to

PCAs

Binary reprezentation

Hexadecimal reprezentation

0 51 60 60 60 51 51 51 51 000001110 0e

1 51 51 60 60 60 51 51 51 000011100 1c

2 60 51 60 60 60 51 51 51 000011101 1d

3 51 102 102 102 102 51

51 102 110011110 19e

4

102 102 102 51 102

102 102 102

111110111 1f7

…… …… …… ……

Figure 8. The PCA pipeline architecture


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Figure 9. Control logic – detail scheme

In concordance with the CA theory, a single basic programmable CA cell was designed (see figure 10). It consists of a D flip-flop and a circuit combinational logic (CLC).

Figure 10 The structure of a PCA cell

8 kind of these cells are connected together to form a PCA on 8 bits. The control logic is the “heart” of the encryption system. It is a finite state machine automaton that controls all the operation realized by the CA-PRNG and PCA and also enables the control signals (see figure 9 and 11).

Figure 11 Control logic – bloc scheme

The last part of this chapter is dedicated to

the analysis of the security of the encryption system. The available key space for the scheme can be evaluated as follows. The number of selections of q fundamental transformations (PCA) is 5

156C (where C is the number of combinations, 156 is the number of rules that has 8 cycles length). These q PCA could be arranged in q! different ways. So, the enciphering

functions can be stored in ∏−

=

−1

0

)!(p

i

iq ways.

Hence, the key space is:


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∏−

=

−•1

0

5156 )!(

p

i

iqC

For example, for the designed encryption

system the key space is:

2071156

0

5156 108275.4)!5( •=−•∏

−

=i

iC .

This is an extremely large value. The designed blocks presented in this

chapter remain valid for both software and hardware implementation.

In the fourth chapter we present the

implementation of the proposed encryption system presented in the precedent chapters. Also, here we analyzed the obtained results. In software the encryption system was implemented in the C++ and C# languages and in hardware was described in VHDL language and implemented on FPGA Spartan 3 XC3S400fg-456 (figure 12).

Figure 12 FPGA platform used to implement the

encryption system

The process of project verification was performed with two major tools: the simulator and the analyzer. The simulator was used to verify the function of the encryption and control logic and compared with theoretical results. The timing analyzer was used to determine the critical path and the maximum operating frequency.

In figure 13 we present the evolution of a PCA configured dynamic with rules 51, 60 or 102. Here the yellow cells correspond to logic “0” and blue cells correspond to logic “1”.

Figure 13 The evolution of a PCA with 8 cells with rules 51, 60 or 102

In figure 14 we present the evolution of

the proposed encryption system with one PCA.

In figure 15 we present the evolution of the proposed encryption system with five PCA.

The distribution of the encrypted text is uniform in all intervals. For example, for a text file, the encrypted text will be distributed in all ASCII intervals and not only in zone of alphanumeric intervals (see figure 16).


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Figure 14 Encryption systems with one PCA

Figure 15 Encryption systems with 5 PCA in pipeline


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Figure 16 The distribution of the encrypted text

The cryptogram was tested and verified

using an illustrative example in area of e-health applications and messenger conversations. An encouraging result was the perfect concordance between encryption system implemented by hardware on a FPGA and the software simulation.

Due to the complexity of cryptosystems, the theory to support current research is not sufficient for practical use. Several questions are still difficult to answer, e.g. the security of the system under different types of attack, the optimized number of PCA for the algorithm, etc. The answers to these questions require more detailed mathematical analysis.

The general conclusion of this work is to show that it is possible to build evolutionary encryption systems based on a simple mathematical model specific of CA by introducing the local interaction between cells, local rules, and their robustness to variations of the model. This implies a large possibility to implement very efficient encryption systems in software as well in hardware.

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