- Email: [email protected]
Efficient Cascaded 1-D and 2-D Chaotic Generators
Efficient Cascaded 1-D and 2-D Chaotic Generators Hassan Noura*, Sebastien Henaff**, Ina Taralova**, Safwan El Assad*, *IREENA : Institue de Recherche en Electronique et Electrotechnique de Nantes Atlantique Polytech’Nantes, Rue Christian pauc BP 50609 Nantes Cedex 3, France. Tel: + 33 2 40 68 30 36; Fax: + 33 2 40 68 32 32 (Email: [email protected], [email protected]) **Ecole centrale de Nantes, IRCCyN: Institue de Recherche en Communication et Cybernetique de Nantes (Email:[email protected], [email protected])} Abstract: In order to improve the communication security, two novel chaotic generators are proposed. The first one is a single order chaotic generator using two cascading stages, and the second one is a second order chaotic generator with single and cascaded layer. The goal is to generate binary pseudo random chaotic signal with high degree of randomness. We analyze their global dynamics properties using system and signal processing tools (Lyapunov exponents, bifurcation diagrams, distribution, pseudo phase, autocorrelation, cross correlation, NIST test). A complete analysis is provided. Experimental and theoretical analyses show that the proposed generators have good cryptographic properties.
1. INTRODUCTION Chaos based communication systems have attracted increasing interest from researchers and cryptanalysts. Several chaotic cryptographic schemes have been proposed (Alvarez and Li, 2006), (Gotz et al., 1997), (Burghelea et al., 2008), (Lozi, 2007) where, globally, the original message to encrypt (the plaintext) is mixed up with a chaotic signal to form the encrypted text (the cipher text) which is to be transmitted through an unsecured channel. An inverse scheme is placed at the reception to recover the plaintext.
The obtained results exhibit a random-like behavior and have a uniform probability density. The cascaded technique, or the second order chaotic generator allow to expand the periodicity and double the key space which are essential for high security data encryption and increases robustness against cryptanalysts. This paper is organized as follows: in the next section, the two digital chaotic generators are described. In section 3 a comparative study of their performances is presented. Finally, section 4 concludes this paper.
In all chaos-based cryptographic systems, the chaotic generator takes an important part and it must exhibit appropriate cryptographic properties. Generally, if it generates a signal statistically close to a pseudo-random one with the highest possible period length, then its behaviour can be considered as satisfactory. Many digital chaotic generators have been proposed such as the traditional chaotic maps (Logistic, PWLCM, Frey … etc) (Zhou, 1996), (Frey, 1993), but they don’t exhibit very good statistical properties and their cycle length is not long enough.
Nowadays, most researchers investigate chaotic sequences generated by one-dimensional or two-dimensional discrete time non-linear dynamic systems for secure communication information. Two approaches to generate chaotic signals are presented in detail in the next section.
This paper introduces new chaotic generators, and the generated signals are compared to the random ones. In addition, in our approach, each chaotic generator is studied by analyzing its complex dynamics by both deterministic and statistical tools.
The structure of the first chaotic generator is presented in Figure 1. The generator scheme with N bits finite precision consists in a non-linear function FNL( x ) with three delayed feedback loop for each cascade (El Assad et al., 2008).
The chaotic sequence depends on the parameters and the initial conditions (seed), they form the key. The design of the two chaotic generators is presented. The first one is modeled by a one dimensional map with cascaded network, and the second one is modeled by a two dimensional map. According to the spectral analysis, one of the state components had been selected as being the output.
2. Proposed Chaotic Maps
2.1 Proposed generator: First Order Chaotic Generator
The Equation for the single chaotic generator is defined by the following relation: eu (n) = mod [ku1 (n) + c1 × eu ( n − 1) + c2 × eu ( n − 2) + FNL(c3 × eu ( n − 3) ),2 N ]
(1)
normalized by the maximum absolute value of the quantized levels. Hence, the effective generated signal is: 2.2 Proposed Generator: Second Order Chaotic Generator This proposed generator is given by the following system (Kocarev and Makraduli, 2005), (Hénaff and Taralova, 2008): x ( n + 1) = Λ ( p × x ( n) + p × x (n )2 + m(n )) 1 1 1 2 2 x2 ( n + 1) = Λ ( p3 × x2 ( n) + p4 × x1( n) 2 )
Fig. 1.Structure of the proposed generator
Where Λ : The proposed non-linear function FNL(x) is given by Equation (2): N
FNL( x) = mod [x exp (cos( x)),2 ]
(2)
The system states are eu1(n) and eu 2 (n) , with eu1 (n ) = mod [ ku1 (n ) + c11 × eu1 (n − 1) + c12 × eu1( n − 2) +
c23 × eu 2 (n − 3) exp (cos (c23 × eu 2 (n − 3))), 2 N ]
(3) The u index in Equations (1), (3) denotes an unsigned number, and all additions are modulo 2 N . These operators are assumed to be generally non-linear operations, also all operations inside any loop work in the unsigned number representation, modulo 2 N . The input signal ku (n ) plays in our application the role of an additional key, which does not allow an unauthorized eavesdropper to recover the generated signal. However ku ( n) can be used as an information signal to encrypt.
∑ qn × PTch (t − nTch ) n
−2 N −1 2
N −1
≤ qn ≤
(4)
2 N −1− 1 2 N −1 1
And PTch =
0
⇔ −1 ≤ qn ≤ 1
if 0 ≤ t ≤ Tch
(5) otherwhise
In order to reduce the signal’s mean power, and to make its amplitude independent of the number of levels (in fact, on N ), the generated signal e (n ) is first converted into a u2 signed signal es ( n) (The index s means «signed») in the 2’s complement in the 2 N representation set, C2 , 2 N , and then
mod( z, 2) if mod( z ,2) > 1 Λ( z) = mod( − z ,2) else
(7)
m is the input of the function (the message to encrypt) and the pi are the parameters of the map. Let denote the map T : (8)
We shall study the performances of the two chaotic generators using the tools proposed and defined in the next section.
eu 2 (n ) = mod [eu1 (n ) + c21 × eu 2 (n − 1) + c22 × eu 2 (n − 2) +
with
the triangular map, defined by:
T : ( x1( n + 1); x2 ( n + 1)) = T ( x1 (n ); x2 (n ); m)
c13 × eu1( n − 3)exp (cos (c13 × eu1( n − 3))), 2 N ]
es' =
R → [ 0; 1 ] is
(6)
We are first interested in the dynamics of the chaotic generator itself, so we shall consider the zero input case (i.e. no message to encrypt, the chaotic generator works autonomously). The performances of different chaotic generators based on Equations (3) and (6) are quantified and compared in the frame of secure communication. 2.2.1 Study in the Parameter Plan This study allows us to identify which parameter combinations are the most suitable to generate robust chaos. In the plane drawn in Figure 2-a, each point corresponds to a given parameter combination ( p1 , p3 ) . Each orbit's period is represented by a specific color: red for fixed point; blue for period three, etc. A black region corresponds to very long periodic orbits or chaos. Thus, in this Figure, the map has been studied by varying the parameters p1 and p3 from -3 to 3, for p2 = p4 = 1 . For these parameters, the system is mostly periodic and the non-black regions are not suitable to design a chaotic generator for secure communication applications. As it can be observed in the enlargement of Figure 2-b, neighboring regions of periodic behavior are not uniformly chaotic (there are blue points inside the black region). Furthermore, when chaotic, these regions are characterized by a low level of chaoticity defined by the Lyapunov exponents. For these reasons, the system is analyzed later on for parameter combinations which guarantee a higher level of chaoticity
everywhere in the parameter plane; such combination is given for instance by ( p1, p2 , p3 , p4 ) = (5, 3, 5,−15 )
conditions have been generated randomly and a large number of sampled values has been simulated (10000 samples). 3.1 Random-Like Behavior For the two generators, we found that the autocorrelation and cross correlation functions in Figures 3 and 4 and the spectrum DFT in Figure 5 are clearly noise-like. 3.2 Probability Distribution
Fig. 2-a. Bifurcation: Parameter plane ( p1, p3 ) for
p2 = p4 = 1
Another property of the random sequences is that the generator should have a uniform probability distribution in the whole phase space [0; 1], which indicates that the attractor mapping is well distributed. The histograms in Figure 6) and Figure 7) show the values distribution of the chaotic sequences generated by the first and second order generators. Also Table 1 presents the mean value of the generated signal and the percentage of the bits “zero” and “one” which is close to 0.5. This means that the generated chaotic sequences are close to a random sequence.
Fig. 2-b. Zoom of the parameter plane Figure 2-a 2.2.2 Lyapunov Exponents Lyapunov
exponents,
denoted
{ λ1…λn } quantify the
sensitivity to initial conditions from a system point of view: the bigger these exponents are the more complex the dynamics is. For a chaotic behavior, there must be at least one positive Lyapunov exponent. They are defined by the Equation: λi = lim
1
N →∞ N
Where
× ln eigi (T '( x( N ) )T '( x( N −1) )…T '( x(1) ))
' T ( x( N ) ) is
Fig. 3. Auto & Cross correlation of first chaotic generator
(9)
the jacobian matrix of the system
calculated at the considerate state x( N ) , eigi ( A) , denotes the ith Eigen value of matrix A. For the parameter combinations under consideration: ( p1; p 2 ; p 3 ; p 4 ) = ( 5; 3; 5; − 15 ) , the Lyapunov exponents are respectively: ( λ1; λ2 ) = ( 2.7; 1.8 ). 3. COMPARATIVE SIMULATION RESULTS In order to verify the results and to quantify the cryptographic properties of the two generators, some experiments have been carried out. The finite computing precision is N = 32 , all coefficients c11, c12 , c13 , c21, c22 , c23 are equal one. Both initial
Fig. 4. Auto & Cross correlation of x1, x2 for second order chaotic generator
Table 1. Some comparative results of the proposed generators
Generator
mean
Simple first generator Cascaded first order generator Second order generator, output x1 Second order generator ,output x2
0.5035 0.4997 0.5004
0.5023
% of bit 0,and 1 49.74 50.26 50.019 49.981 50.139 49.861
Max crosscorrelation 0.03231
50.14 49.87
0.0320
0.029825 0.04287
3.3 Pseudo phase space trajectories Fig. 5. Spectrum: DFT of the first order chaotic generator
The most important property of a random sequence is that the dynamical values in the sequence are uncorrelated. One of the ways to check if a sequence is uncorrelated is to generate a pseudo phase space. A pseudo phase space is obtained by plotting in 3-D the sequence xn + 2 = f ( xn , xn +1 ) . For a proper random sequence, the pseudo phase space shows no correlation at all. The pseudo phase spaces of the two proposed generators, shown in Figures 8) and 9) indicate clearly that the generated sequences are random.
Fig. 6. Histogram of the first order chaotic generator
Fig. 8. Pseudo phase space of the first order generator
Fig. 7. Histogram of the second order chaotic generator
Table 2- NIST tests of the first order generator Test Name Mean of pProp Res value’s
Fig. 9. Pseudo phase space of the second order generator 3.4 Statistical test NIST Among the numerous standard tests for pseudo randomness, the most important one is the NIST tests (Rukhin et al., 2001]. NIST consist in a battery of 16 statistical tests to detect deviations from randomness in binary sequences. For each test a probability, called the p -value, is extracted. This value summarizes the strength of the evidence against the perfect randomness hypothesis. A p -value less than α = 0.01 (significance level) indicate that the sequence appears to be non-random. A p -value larger than 0.01 means that the sequence is considered to be random with a confidence of 99%. Table 2 shows the NIST results obtained for the first order proposed generator. The NIST Test has been performed on a trial of 100 binary sequences, each containing 106 bit. As a result, the produced binary sequences exhibit randomness properties, so they can be used in secure communication systems. 6. CONCLUSIONS In a crypto-system, the use of a good chaotic generator with desirable dynamical statistical properties is very important. In this paper, we have developed and implemented under Matlab/Simulink two classes of generators. A comparative analysis of their performances using standard criteria (signal statistics, NIST tests) proves the efficiency of the cascading technique and the multi-component approach. As a result, the proposed generators are suitable to be used in crypto-systems to secure data. As a prospect of this work, we are currently working on a generator structure whose cycle length is very large.
Approximate Entropy Block Frequency
0.03080
0.9900
Success
0.9131
1
Success
Cumulative Sums(Forward) Cumulative Sums(Reverse) Fast Fourier Transform Frequency ( Monobit ) Lempel-Ziv Compression Linear Complexity Longest Runs of ones Non-overlapping Template Matching
0.27570 0.017912 0.2403
1.0000 0.9900 1
Success
0.17186
0.9900
Success
0.45593
1
Success
0.12962
0.9800
Success
0.83430 0.31908
0.9800 0.9900
Success Success
0.779188
0.9800
Success
overlapping Template Matching
0.3489
1
Success
Random excursions Random excursions Variant
0.3523 0.5046
0.9900 0.9800
Success Success
Rank
0.36691
0.9700
Success
Runs Serial
0.83430 0.35048 0.16260
0.9900 1.0000 0.9700
Success Success Success
Success
ACKNOWLEDGMENT The authors wish to thank the “Federation AtlanSTIC-CNRS FR2819” and the “National Council for Scientific Research of Lebanon” for funding this work. REFERENCES Alvarez, G., Li, S.,(2006). "Some basic cryptographic requirements for chaos-based cryptosystems", International Journal of Bifurcation and Chaos, vol. 16, no. 8, pp. 2129-2151. Burghelea, R., Mangeot, J. P., Launay, Coirault, P. and El Assad, S. (2008). "Coherent synchronization of a chaotic communication system based on a QAM-4 transmitter", Lecture Notes in Computer Science, Book Image and Signal Processing, vol. 5099, pp. 552-561, SpringerVerlag Berlin. Carroll T. and Pecora L. (1991). "Synchronizing chaotic circuits," IEEE Transactions on Circuits and Systems, vol. 38, no. 4, pp. 453-456.
El Assad, S., Noura, H., Taralova, I. (2008), «Design and analyses of efficient chaotic generators for cryptosystems», Lecture Notes in IAENG Transactions on Electrical and Electronics Engineering, vol. I, 2008, 10 Pages (to appear). Frey, D. (1993). "Chaotic digital encoding: an approach to secure communication", IEEE Transactions on Circuits and Systems II, vol. 40, no. 10, pp. 660-666. Gotz, M., Kelber K. and Schwarz, W. (1997). "Discrete-time chaotic encryption systems, i. statistical design approach", IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 44, no. 10, pp. 963-970. Hénaff, S. and Taralova, I. (2008). "Joint signal-system approach for chaotic generators selection", ECIT (European conference on iteration theory), Yalta, Ukraine. Kocarev, J. A. P., Makraduli, L. (2005). "Public-key encryption based on cheby-shev polynomials", IEEE Circuits Systems and Signal Processing, vol. 24, no. 5, pp. 497-517. Lozi, R. (2007). "New enhanced chaotic number generators", Indian journal of industrial and applied mathematics, vol. 1, no. 1, pp. 1-23. Rukhin, A., Soto, J.,Nechvatal, J., Smid, M., (2001). “A Statistical Test Suite For Random and Pseudorandom Number Generators FOR CRYPTOGRAPHIIC APPLIICATIIONS”. NIST Special Publication 800-22. Zhou, H. “A design methodology of chaotic stream ciphers and the realization problems in finite precision,” Ph.D. thesis, Department of Electrical Engineering, Fudan University, Shanghai, China 1996.
1. INTRODUCTION Chaos based communication systems have attracted increasing interest from researchers and cryptanalysts. Several chaotic cryptographic schemes have been proposed (Alvarez and Li, 2006), (Gotz et al., 1997), (Burghelea et al., 2008), (Lozi, 2007) where, globally, the original message to encrypt (the plaintext) is mixed up with a chaotic signal to form the encrypted text (the cipher text) which is to be transmitted through an unsecured channel. An inverse scheme is placed at the reception to recover the plaintext.
The obtained results exhibit a random-like behavior and have a uniform probability density. The cascaded technique, or the second order chaotic generator allow to expand the periodicity and double the key space which are essential for high security data encryption and increases robustness against cryptanalysts. This paper is organized as follows: in the next section, the two digital chaotic generators are described. In section 3 a comparative study of their performances is presented. Finally, section 4 concludes this paper.
In all chaos-based cryptographic systems, the chaotic generator takes an important part and it must exhibit appropriate cryptographic properties. Generally, if it generates a signal statistically close to a pseudo-random one with the highest possible period length, then its behaviour can be considered as satisfactory. Many digital chaotic generators have been proposed such as the traditional chaotic maps (Logistic, PWLCM, Frey … etc) (Zhou, 1996), (Frey, 1993), but they don’t exhibit very good statistical properties and their cycle length is not long enough.
Nowadays, most researchers investigate chaotic sequences generated by one-dimensional or two-dimensional discrete time non-linear dynamic systems for secure communication information. Two approaches to generate chaotic signals are presented in detail in the next section.
This paper introduces new chaotic generators, and the generated signals are compared to the random ones. In addition, in our approach, each chaotic generator is studied by analyzing its complex dynamics by both deterministic and statistical tools.
The structure of the first chaotic generator is presented in Figure 1. The generator scheme with N bits finite precision consists in a non-linear function FNL( x ) with three delayed feedback loop for each cascade (El Assad et al., 2008).
The chaotic sequence depends on the parameters and the initial conditions (seed), they form the key. The design of the two chaotic generators is presented. The first one is modeled by a one dimensional map with cascaded network, and the second one is modeled by a two dimensional map. According to the spectral analysis, one of the state components had been selected as being the output.
2. Proposed Chaotic Maps
2.1 Proposed generator: First Order Chaotic Generator
The Equation for the single chaotic generator is defined by the following relation: eu (n) = mod [ku1 (n) + c1 × eu ( n − 1) + c2 × eu ( n − 2) + FNL(c3 × eu ( n − 3) ),2 N ]
(1)
normalized by the maximum absolute value of the quantized levels. Hence, the effective generated signal is: 2.2 Proposed Generator: Second Order Chaotic Generator This proposed generator is given by the following system (Kocarev and Makraduli, 2005), (Hénaff and Taralova, 2008): x ( n + 1) = Λ ( p × x ( n) + p × x (n )2 + m(n )) 1 1 1 2 2 x2 ( n + 1) = Λ ( p3 × x2 ( n) + p4 × x1( n) 2 )
Fig. 1.Structure of the proposed generator
Where Λ : The proposed non-linear function FNL(x) is given by Equation (2): N
FNL( x) = mod [x exp (cos( x)),2 ]
(2)
The system states are eu1(n) and eu 2 (n) , with eu1 (n ) = mod [ ku1 (n ) + c11 × eu1 (n − 1) + c12 × eu1( n − 2) +
c23 × eu 2 (n − 3) exp (cos (c23 × eu 2 (n − 3))), 2 N ]
(3) The u index in Equations (1), (3) denotes an unsigned number, and all additions are modulo 2 N . These operators are assumed to be generally non-linear operations, also all operations inside any loop work in the unsigned number representation, modulo 2 N . The input signal ku (n ) plays in our application the role of an additional key, which does not allow an unauthorized eavesdropper to recover the generated signal. However ku ( n) can be used as an information signal to encrypt.
∑ qn × PTch (t − nTch ) n
−2 N −1 2
N −1
≤ qn ≤
(4)
2 N −1− 1 2 N −1 1
And PTch =
0
⇔ −1 ≤ qn ≤ 1
if 0 ≤ t ≤ Tch
(5) otherwhise
In order to reduce the signal’s mean power, and to make its amplitude independent of the number of levels (in fact, on N ), the generated signal e (n ) is first converted into a u2 signed signal es ( n) (The index s means «signed») in the 2’s complement in the 2 N representation set, C2 , 2 N , and then
mod( z, 2) if mod( z ,2) > 1 Λ( z) = mod( − z ,2) else
(7)
m is the input of the function (the message to encrypt) and the pi are the parameters of the map. Let denote the map T : (8)
We shall study the performances of the two chaotic generators using the tools proposed and defined in the next section.
eu 2 (n ) = mod [eu1 (n ) + c21 × eu 2 (n − 1) + c22 × eu 2 (n − 2) +
with
the triangular map, defined by:
T : ( x1( n + 1); x2 ( n + 1)) = T ( x1 (n ); x2 (n ); m)
c13 × eu1( n − 3)exp (cos (c13 × eu1( n − 3))), 2 N ]
es' =
R → [ 0; 1 ] is
(6)
We are first interested in the dynamics of the chaotic generator itself, so we shall consider the zero input case (i.e. no message to encrypt, the chaotic generator works autonomously). The performances of different chaotic generators based on Equations (3) and (6) are quantified and compared in the frame of secure communication. 2.2.1 Study in the Parameter Plan This study allows us to identify which parameter combinations are the most suitable to generate robust chaos. In the plane drawn in Figure 2-a, each point corresponds to a given parameter combination ( p1 , p3 ) . Each orbit's period is represented by a specific color: red for fixed point; blue for period three, etc. A black region corresponds to very long periodic orbits or chaos. Thus, in this Figure, the map has been studied by varying the parameters p1 and p3 from -3 to 3, for p2 = p4 = 1 . For these parameters, the system is mostly periodic and the non-black regions are not suitable to design a chaotic generator for secure communication applications. As it can be observed in the enlargement of Figure 2-b, neighboring regions of periodic behavior are not uniformly chaotic (there are blue points inside the black region). Furthermore, when chaotic, these regions are characterized by a low level of chaoticity defined by the Lyapunov exponents. For these reasons, the system is analyzed later on for parameter combinations which guarantee a higher level of chaoticity
everywhere in the parameter plane; such combination is given for instance by ( p1, p2 , p3 , p4 ) = (5, 3, 5,−15 )
conditions have been generated randomly and a large number of sampled values has been simulated (10000 samples). 3.1 Random-Like Behavior For the two generators, we found that the autocorrelation and cross correlation functions in Figures 3 and 4 and the spectrum DFT in Figure 5 are clearly noise-like. 3.2 Probability Distribution
Fig. 2-a. Bifurcation: Parameter plane ( p1, p3 ) for
p2 = p4 = 1
Another property of the random sequences is that the generator should have a uniform probability distribution in the whole phase space [0; 1], which indicates that the attractor mapping is well distributed. The histograms in Figure 6) and Figure 7) show the values distribution of the chaotic sequences generated by the first and second order generators. Also Table 1 presents the mean value of the generated signal and the percentage of the bits “zero” and “one” which is close to 0.5. This means that the generated chaotic sequences are close to a random sequence.
Fig. 2-b. Zoom of the parameter plane Figure 2-a 2.2.2 Lyapunov Exponents Lyapunov
exponents,
denoted
{ λ1…λn } quantify the
sensitivity to initial conditions from a system point of view: the bigger these exponents are the more complex the dynamics is. For a chaotic behavior, there must be at least one positive Lyapunov exponent. They are defined by the Equation: λi = lim
1
N →∞ N
Where
× ln eigi (T '( x( N ) )T '( x( N −1) )…T '( x(1) ))
' T ( x( N ) ) is
Fig. 3. Auto & Cross correlation of first chaotic generator
(9)
the jacobian matrix of the system
calculated at the considerate state x( N ) , eigi ( A) , denotes the ith Eigen value of matrix A. For the parameter combinations under consideration: ( p1; p 2 ; p 3 ; p 4 ) = ( 5; 3; 5; − 15 ) , the Lyapunov exponents are respectively: ( λ1; λ2 ) = ( 2.7; 1.8 ). 3. COMPARATIVE SIMULATION RESULTS In order to verify the results and to quantify the cryptographic properties of the two generators, some experiments have been carried out. The finite computing precision is N = 32 , all coefficients c11, c12 , c13 , c21, c22 , c23 are equal one. Both initial
Fig. 4. Auto & Cross correlation of x1, x2 for second order chaotic generator
Table 1. Some comparative results of the proposed generators
Generator
mean
Simple first generator Cascaded first order generator Second order generator, output x1 Second order generator ,output x2
0.5035 0.4997 0.5004
0.5023
% of bit 0,and 1 49.74 50.26 50.019 49.981 50.139 49.861
Max crosscorrelation 0.03231
50.14 49.87
0.0320
0.029825 0.04287
3.3 Pseudo phase space trajectories Fig. 5. Spectrum: DFT of the first order chaotic generator
The most important property of a random sequence is that the dynamical values in the sequence are uncorrelated. One of the ways to check if a sequence is uncorrelated is to generate a pseudo phase space. A pseudo phase space is obtained by plotting in 3-D the sequence xn + 2 = f ( xn , xn +1 ) . For a proper random sequence, the pseudo phase space shows no correlation at all. The pseudo phase spaces of the two proposed generators, shown in Figures 8) and 9) indicate clearly that the generated sequences are random.
Fig. 6. Histogram of the first order chaotic generator
Fig. 8. Pseudo phase space of the first order generator
Fig. 7. Histogram of the second order chaotic generator
Table 2- NIST tests of the first order generator Test Name Mean of pProp Res value’s
Fig. 9. Pseudo phase space of the second order generator 3.4 Statistical test NIST Among the numerous standard tests for pseudo randomness, the most important one is the NIST tests (Rukhin et al., 2001]. NIST consist in a battery of 16 statistical tests to detect deviations from randomness in binary sequences. For each test a probability, called the p -value, is extracted. This value summarizes the strength of the evidence against the perfect randomness hypothesis. A p -value less than α = 0.01 (significance level) indicate that the sequence appears to be non-random. A p -value larger than 0.01 means that the sequence is considered to be random with a confidence of 99%. Table 2 shows the NIST results obtained for the first order proposed generator. The NIST Test has been performed on a trial of 100 binary sequences, each containing 106 bit. As a result, the produced binary sequences exhibit randomness properties, so they can be used in secure communication systems. 6. CONCLUSIONS In a crypto-system, the use of a good chaotic generator with desirable dynamical statistical properties is very important. In this paper, we have developed and implemented under Matlab/Simulink two classes of generators. A comparative analysis of their performances using standard criteria (signal statistics, NIST tests) proves the efficiency of the cascading technique and the multi-component approach. As a result, the proposed generators are suitable to be used in crypto-systems to secure data. As a prospect of this work, we are currently working on a generator structure whose cycle length is very large.
Approximate Entropy Block Frequency
0.03080
0.9900
Success
0.9131
1
Success
Cumulative Sums(Forward) Cumulative Sums(Reverse) Fast Fourier Transform Frequency ( Monobit ) Lempel-Ziv Compression Linear Complexity Longest Runs of ones Non-overlapping Template Matching
0.27570 0.017912 0.2403
1.0000 0.9900 1
Success
0.17186
0.9900
Success
0.45593
1
Success
0.12962
0.9800
Success
0.83430 0.31908
0.9800 0.9900
Success Success
0.779188
0.9800
Success
overlapping Template Matching
0.3489
1
Success
Random excursions Random excursions Variant
0.3523 0.5046
0.9900 0.9800
Success Success
Rank
0.36691
0.9700
Success
Runs Serial
0.83430 0.35048 0.16260
0.9900 1.0000 0.9700
Success Success Success
Success
ACKNOWLEDGMENT The authors wish to thank the “Federation AtlanSTIC-CNRS FR2819” and the “National Council for Scientific Research of Lebanon” for funding this work. REFERENCES Alvarez, G., Li, S.,(2006). "Some basic cryptographic requirements for chaos-based cryptosystems", International Journal of Bifurcation and Chaos, vol. 16, no. 8, pp. 2129-2151. Burghelea, R., Mangeot, J. P., Launay, Coirault, P. and El Assad, S. (2008). "Coherent synchronization of a chaotic communication system based on a QAM-4 transmitter", Lecture Notes in Computer Science, Book Image and Signal Processing, vol. 5099, pp. 552-561, SpringerVerlag Berlin. Carroll T. and Pecora L. (1991). "Synchronizing chaotic circuits," IEEE Transactions on Circuits and Systems, vol. 38, no. 4, pp. 453-456.
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