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Shahab Saquib Sohail#, Musheer Ahmad*
#Department of Computer Science, Aligarh Muslim UniversityAligarh, U.P 202002, India
shahabsaquibsohail@hotmail.com
*Department of Computer Engineering, Faculty of Engineering and Technology,
 Jamia Millia Islamia,  New Delhi 110025
musheer.cse@gmail.com

Abstract High randomness is an intuitive property of the sequence generated by a chaos based system. Pseudorandom properties of binary sequences are widely exploited to build the cryptographic systems known as stream ciphers. In this paper, the randomness performance of binary sequence generated by proposed chaos based keystream generator is evaluated. The proposed generator is based on the three dimensional Lorenz chaotic system. NIST statistical package is used to evaluate the randomness of the sequence generated. Frequency, block-frequency, runs, longest runs,  linear complexity, cusum-forward, cususm-reverse, rank, serial etc tests are performed. The experimental results show that the binary sequence generated by the proposed chaos based generator has randomness similar to a noise sequence.

Keywords  security, randomness, keystream generator, Lorenz chaotic system.
Introduction
Communication is the best mean to convey the message. When we talk about communication, network comes into the picture. For communication we may have wired or wireless networks. The rapidly growing communication media raises the issues related to securities. Security is the major aspect in communication. Pseudorandom sequences are widely used in secure communication. The quality of these sequences is very vital for cryptographic techniques used to make communication secure. Pseudorandom sequences are also applied to various engineering area like computer simulation, aerospace, random sampling and biology system control, etc. in the design of random binary sequence generator chaotic system is regarded as an important pseudorandom source. Progress in chaos theory for recent years and its applications are reported in [1]-[5].

The chaotic systems have certain desirable characteristics from cryptographic viewpoint. They are simple in realization and there is no need to determine the sampling frequency in comparison with the true random sources such as noise of circuit. The chaotic system is better than the LFSR in properties with periodicity, correlation and complexity. 
Chaotic sequence has very good pseudo-random characteristics. It is much sensitive to initial condition. Therefore they fit the requirement of pseudorandom sequence. Thus chaotic systems are widely used in secure communication [7].
Cryptographically desirable binary sequences which are utilized to have a secure communication should have high pseudo-randomness performance. In this paper we propose a pseudo-random binary sequence generator using Lorenz chaotic system. We analyse the pseudo-random binary sequence generated by the proposed generator. Standard NIST statistical package is used to evaluate the performance of the generator under frequency, block-frequency, runs, linear complexity tests etc. The results confirm that the proposed chaos based generator has randomness similar to a noise sequence.
Rest of the paper is organized as follows, in section II background of Lorenz chaotic system is presented. In section III the detail of design of proposed pseudo-random binary sequence generator is given. In section IV the randomness performance of the binary sequence generator is analysed. Finally things are concluded in section V. 
Chaos based Pseudo-random Binary Sequence Generator
The well known Lorenz equation is defined as:


 = �(y-x)
 = rx-xz-y		(1)
 = xy-bz


The values t, x, y, z, �, r, b are real numbers and � ,r, b are positive constants termed as control parameters. � is the              Prandtl	 number, r is the Rayleigh number and b is a fraction. The Lorenz chaotic system makes use of the numerical integration to obtain the real valued chaotic sequence. Runge Kutta-4 method is applied to solve the equation (1).
The following initial values are used in the proposed system.
Control parameters: � = 10, r = 28, b = 8/3
Initial condition:
x(0) = 13.3604,  y(0) = 7.2052,           z(0) = 21.5026	 
Integer stepsize: h = 0.001
Iteration steps: 2500 
The following figure illustrates the process.













         Fig. SEQ "Fig." \*Arabic 1Chaos based PRBS generator


 After each iteration of the Lorenz chaotic system, the preprocessing of real valued chaotic sequence is done. The preprocessing includes the following steps.
Extract the fractional part of the real valued sequence
Transform x using following formula
x = 106(x) - ( floor [106 (x)] )

Now, x lies in (0, 1), Where floor [.] rounds to the nearest smallest integer. After the pre-processing step, the quantization is done with the help of  a quantizer as shown in the Fig.1.In quantization, the coming sequence of  value is made either 0 or 1.Let x be the coming sequence then



	           0 if x<0.5
  	x  =  
                         1 if xe"0.5


So after the quantization binary bit streams are generated. We have taken these bits as bx , by and bz for corresponding x, y and z sequences. Boolean functions f1 and f2 are operated over the bit streams obtained after quantization. These functions have following form:
f1(x,y,z) = y*z*y.z*x.z    
f2(x,y,z) = x*z*x.z*y.z 
where  *  and  .  denote XOR and AND operations respectively. After these functions being operated over the incoming bits we get two bits z1 and z2 as output and these bits are treated as the input stream for obtaining final binary sequence as output. We perform  XOR operation over z1 and z2 to get zout which is the final bit received as output, hence giving a pseudo-random binary sequence.
RANDOMNESS EVALUATION
The randomness performance of the proposed scheme is evaluated using the statistical package by the National Institute of Standards and Technology (NIST) [6]. The statistical tests are performed as it is necessary for determining the randomness attributes. One of the reasons for choosing this test package is that it is a benchmark for the evaluation of possible candidate generator for Advanced Encryption Standards [6]. The statistical testing of binary sequence generators is one of the root requirements for cryptographic applications.
 The proposed generator is analysed under the following randomness tests: 
1. Frequency (Monobit) Test
2. Block Frequency Test
3. Cumulative Sums (Cusum) Test
4. Runs Test
5. Longest Run Test
6. Rank Test
7. FFT Test
8. Linear Complexity Test
9. Approximate Entropy Test
10 Overlapping  Template
11.Serial Test

The purpose of Frequency (monobit) test is to determine whether the number of ones and zeroes in a sequence is approximately the same as would be expected for a truly random sequence. The number of ones and zeroes in a sequence should be about the same. All subsequent tests depend on passing of this test. 
The Block Frequency test is used to determine whether the frequency of one in an M-bit block is exactly M/2. If block size is made to M, subsequently the test degenerates to Frequency test. 
The cumulative sum test which is often termed as cusum test is categorized as cusum-forward and cusum-reverse test. The focus of this test is the maximal excursion of the random walked defined by the cumulative sum of the adjusted digits in the sequence. The purpose of this test is to whether the cumulative sum of the partial sequences occurring in the tested sequence is too large or too small relative to the expected behaviour of that cumulative sum of random sequence. 
The Runs test has its main focus over the total number of runs in the sequence, where a run is an uninterrupted sequence of identical bits. A run of length k consists of exactly k identical bits. In particular this test determines whether the oscillation between zeroes and ones is too fast or too slow.
The focus of the longest run test is the longest run of ones within m-bit blocks. The purpose of this test is to determine whether the length of the longest run of ones within the tested sequence is consistent with the  length of the longest run of ones that would be expected in a random sequence.
Rank test focuses on the rank of disjoint sub-matrices of the entire sequence. The purpose of this test is to check for linear dependence among fixed length substrings of the original sequence.
The linear complexity test uses linear complexity to test for randomness. Its purpose is to determine whether or not a sequence is complex enough to be considered as random.
Focus of the Approximate Entropy Test is the frequency of all possible overlapping m-bit patterns across the entire sequence. The purpose of this test is to compare the frequency of overlapping blocks of two consecutive lengths against the expected result of the random sequence. The result is tabulated below.
The focus of the overlapping Template matching test is the number of occurrences of pre-specified target string. It uses an m-bit window to search for a specific m-bit pattern. If pattern is not found the window slides one bit position.
A significance level (�) can be chosen for the tests. If p-value is greater than or equal to � then the sequence appears to be random. If p-value is less than �, then the sequence appears to be non-random. Typically, � is chosen in the range [0.001, 0.01]. An � of 0.01 indicates that one would expect 1 sequence in 100 sequences to be rejected. A p-value e" 0.01 would mean that the sequence would be considered to be random with a confidence of 99 %. A p-value < 0.01 would mean that the conclusion was that the sequence is non-random with a confidence of 99 %.
 p-value lies between [0,1]. If pe"0.01 then tested sequence is considered to be random. Bigger the p-value better is the pseudo-random property of the tested value [6]. p-values obtained by NIST statistical package is tabulated below:	















TABLE  SEQ "Table" \*ROMAN INIST randomness test results

Test Type
p-values
ResultsFrequency Test0.283131SuccessBlock Frequency Test0.924531SuccessCusum-Forward Test0.401523SuccessCusum-Reverse Test0.433749SuccessRuns Test0.129669SuccessLongest Runs Test0.755207SuccessRank Test0.622138SuccessFFT Test0.222920SuccessLinear Complexity Test0.245109SuccessSerial Test0.812251SuccessApprox Entropy Test0.965153SuccessOverlapping Template0.324241Success
Conclusions
In this paper we proposed a chaos based pseudo-random binary sequence generator. An analysis of the generated binary sequences is also performed using NIST statistical package. The experimental results verify that the sequence generated is highly random and can be used eminently in cryptographic techniques.
References
[1] Kinsner W. Characterizing chaos through lyapunov metrics. IEEE Trans Syst Man Cybern C 2006;36(2):141 51.
[2] Xiang F, Qiu SS. Analysis on stability of binary chaotic pseudorandom sequence. IEEE Commun Lett 2008;12(5):337 9.
[3] Manuel Delgado-Restituto, Angel Rodriguez-Vazquez. Mixed-signal map-configurable integrated chaos generator for chaotic communications. IEEE
Trans Circ Syst I 2001;48(12):1462 74.
[4] Wang L, Wang F, Wang Z. Novel chaos-based pseudo-random number generator. Acta Phys Sin 2006;55(8):3964 8.
[5] Sun FY, Liu ST. Cryptographic pseudo-random sequence from the spatial chaotic map. Chaos Soliton Fract 2009;41(5):2216 9.
[6]Rukhin A, et al. "A Statistical Test Suite for Random and Pseudo-random Number Generators for Cryptographic Applications , NIST Special Publication 800-22, 2001.
[7]Mita R, Palumbo G, Pennisi  S, and Poli M, "A novel pseudo random bit generator for cryptography applications", Electronics, Circuits and Systems, 2002. 9th International Conference on, volume. 2, pp 489-492, 2002.


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Youez - 2016 - github.com/yon3zu
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