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J. Software Engineering & Applications, 2010, 3, 1141-1147 doi:10.4236/jsea.2010.312133 Published Online December 2010 (http://www.scirp.org/journal/jsea) Copyright © 2010 SciRes. JSEA 1141 An Efficient PRBG Based on Chaotic Map and Engel Continued Fractions Atef Masmoudi1,2, William Puech2, Mohamed Salim Bouhlel1 1Research Unit: Sciences and Technologies of Image and Telecommunications, Higher Institute of Biotechnology, University of Sfax, Sfax, Tunisia; 2Laboratory LIRMM, University of Montpellier II, Montpellier, France E-mail: atef.masmoudi@lirmm.fr, william.puech@lirmm.fr, medsalim.bouhlel@enis.rnu.tn Received March 3rd, 2010; revised April 12th, 2010; accepted April 16th, 2010. ABSTRACT In recent years, a variety of chaos-based cryptosystems have been proposed. Some of these systems are used in design- ing a pseudo random bit genera tor (PRBG) for stream cipher application s. Most of the chaotic systems used in crypto- graphy have good chaotic properties like ergodicity, sensitivity to initial values and sensitivity to control parameters. However, some of them are not very suitable for use in cryptography because of their non-uniform density function, and their relatively small key space. To be used in cryptography, a PRBG may need to meet stronger requirements than for other applications. In particular, various statistical tests can be applied to the outputs of such generators to conclude whether the generator produ ces a truly random sequen ce or not. In this pap er, we propose a PRBG based on the use of the standard chaotic map with large key space and the Engle Continued Fractions (ECF) map. The outputs of the stan- dard map are used as the inputs of ECF-map. The chaotic nature of the standard map and the good statistical proper- ties of the ECF map motivate us to design a new PRBG for stream cipher applications. The numerical simulation anal- ysis indicates that our PRBG produces bit sequences possessing excellent statistical and cryptographic properties. Keywords: Cry ptography , Continued Fraction, Statistical Tests, PRBG, Chaotic Map 1. Introduction Recently, a variety of crypto-systems have been pro- po sed. Many of them are based on chaotic systems [1-5] which possess good cryptographic characteristics. Chaos systems have many important features like ergodicity, sensitivity to initial condition, sensitivity to control pa- rameters and randomness. These features are very im- portant in cryptography as they form the basis of some new and efficient ways to develop encryption algorithms for secure digital image transmission over the Internet and through public networks. In addition, most of the chaos-based image cryptosystems are based on a single chaotic system. And it is possible according to the chaos theory to extract some useful information about the chaotic system from its orbit, which makes chaotic sys- tems in sec ur e [6-8 ]. To over co me th ese dr awb ack s, some techniques such as multiple chaotic systems [9-12], high-dimensional chaotic systems [13-15], multiple ite- rations of chaotic systems, and many other techniques have been proposed to improve chaos-based ciphers [6,16]. To be used in cryptographic applications, a cha- otic system must sat isfy two important characteristics; its large key space, to resist bruit force attacks, and its ability to generate a sequence that has a uniform inva- riant density function to make it resistant to statistical attacks. The problem is that not all chaotic systems sa- tisfy these characteristics. For example, the chaotic logi- stic map is widely used to design cryptosystems. The known one-Dimensional logistic map is defined as 11 nnn x xx where 0,4 , 0, 1,2,nand 1,0 n x. Mi et al. [17] proposed a chaotic encryption scheme based on randomized arithmetic coding using the logistic map as the PRBG. In [18], Kanso and Smaoui proposed two Pseudo Random Bit Gener ator (PRBG) for stream cipher applications. The first is based on a single 1-D logistic map and the second is based on a combina- tion of two logistic maps. The logistic map is weak in security because it neither satisfies the uniform distribu- tion proper ty nor d o es its k ey space [1 9,20]. Fo r this map, the key size is determined by the initial value 0 x and the control parameter ; 50 bits with a precision of 10-14. In [11], Patidar and Sud proposed a PRBG for stream cipher applications based on two chaotic standard maps running side-by-side and starting from randomly inde An Efficient PRBG Based on Chaotic Map and Engel Continued Fractions Copyright © 2010 SciRes. JSEA 1142 pendent initial conditions. The pseudo random bit se- quence is generated by comparing the outputs of both the chaotic standard maps. They presented the detailed re- sults of the statistical testing on generated bit sequences, using two statistical test suites: the NIST [21] and the DIEHARD [22]. Thus, the need to find a secure and effi- cient chaotic-based cryptosystem motivates us to propose a new scheme which consists of using the standard chao- tic map and the Engle Continued Fractions (ECF) map. The use of ECF-map increases the complexity of a cryp- tosystem based only on standard chaotic system and thus makes difficulties in extraction of information about it. In addition, ECF-map conserves the cryptography prop- erties of the chaotic system; like sensitivity to initial values/control parameters, non periodicity and random- ness; and adds interesting statistical properties such uni- form distribution density function and zero co-corre- lation. The rest of this paper is organized as follows. In Sec- tion 2, we discuss the CF theory and we present a brief description of the ECF-map along with some important features. Section 3 details our proposed PRBG for stream cipher applications. In Section 4, we analyze the security of the proposed PRBG and discuss experimental results based on statistical testing. The concluding remarks are given in Section 5. 2. Continued Fraction Continued fractions (CF) [23-26] refer to all expressions of the form: 1 0 1 2 2 3 4 3... a xb a ba ba b (1) Where i a(i > 0) are the partial numerators, i b are the partial denominators, 0 b is the integer part of the con- tinued fraction and x is a real numb er. Hartono et al. [27] introduce a new contin ued fraction expansion, called Engel continued fraction (ECF) expan- sion. The Engel continued fraction (ECF) map :[0,1)0,1 E T is given by : 11 10 1 00 if x xx Tx Ex if x (2) For any 1,0x, the ECF-map generates a new and unique continued fraction expansion of x of the form: 1 1 12 21 3 1,, nnn n n x bINbb b bb bb bb (3) Let 1,0 x, and define: 11 1 bbx x 11 1,2, 0, nn nn EE bbxbT xnT x (4) Where xxTE 0and xTTxTn EE n E1 for 1n. From definition of E Tit follows that: 11 1 12 21 3 1 1, E nn nnE xbbTx b bb bb bbbTx (5) We describe the method for generating the ECF-con- tinued fraction expansion of x as follows. Algorithm 1 ECF expansion Initialize 0, 0 ixx while 0 1 x do 11 1 1 11 1 1 1 1 ii i i i i xx x x x b ii end while From the theorem presented in [27], if we let 1,0 x, then x has a finite ECF-expansion (i.e., 0xT n E for some 1n) if and only if x Q. In this paper, we pay most attention to the following sequence: ,1. n nnE Zx bxTxn (6) The sequence 1 n ii Zx is in 0,1 and uniformly distributed for almost all points x (for a proof see [27]). So, the ECF-map generates a random and unpredictable sequence 1 n ii Zx with a uniform distribution. These An Efficient PRBG Based on Chaotic Map and Engel Continued Fractions Copyright © 2010 SciRes. JSEA 1143 properties, which are very useful in cryptography, moti- vate us to propose a new PRBG for stream cipher appli- cations based on ECF-ma p. 3. The Proposed Pseudo Random Bit Generator Algorithm In this section, we describe the process of the proposed PRBG. The first step in designing the proposed PRBG is to choose an n-Dimensional chaotic map [10,28]. Choo- sing maps for encryption algorithms is the most impor- tant task. The use of chaotic maps can make the output very sensitive to the input and in our PRBG, the outputs of the chosen chaotic map are used as the input to the ECF-map for generating sequences with desirable chao- tic and statistical properties. In the proposed PRBG, we suggest to use the standard map due to their good chaotic properties like sensitivity to the initial values, sensitivity to the control parameter and its large key space. The 2-D map function known as the standard map is defined by: 10 1 1 sin , jj j jj j xx py yy x (7) Where j x and j y are taken modulo 2 . The secret key in the proposed PRBG is a set of three floating point numbers and one integer 000 0 ,,, x ypN , where 00 ,[0,2)xy is the initial values, 0 p is the control parameter value which can have any real value greater than 0.18 and 0 N is the number of initial iterations of the standard chaotic map. The standard map has a good chaotic properties and a large key space of the order 157 bits [10] with an accuracy of 14 10. This key space is sufficient enough to resist the brute-force attack. In the following paragraph, we give the detailed procedure to generate pseudo random binary sequences using the standard and ECF maps. We define a funct i on :[0,1)0,1G such that: , ijiji jj Gx ZxZx (8) Where j Z is the set calculated according to (6) using ECF-map. In addition, assuming that we have defined a function 1,01,0:F that converts the real number i x to a discrete bit symbol as follows: 00.5 , 1 i i ifx Fx otherwise (9) We propose to use the 2-D standard map, with parame- ters )2,0[, 00 yx are the initial values and 0 p is the control parameter of the chaotic map. Firstly, we propose to iterate the chaotic map0 ptimes. And the operation procedures of the proposed PRBG are described as fol- lows: 1) Step 1: The standard map is iterated continuously. For the jth iteration, the output of the standard map is a new 2-tuplet ,0,2 jj xy . 2) Step 2: Now, we propose to calculate the set 1 N j j a using the relation: jjj jj axyxy (10) 3) Step 3: Finally, the sequence 1 N Njj Kk repre- sents the random binary sequence and it is generated by: jj kFGa (11) The standard and ECF maps are iterated until the gen- eration of a key stream with length N. In order to gener- ate the random binary sequence 1 N j j k, an initial se- quence 1 N j j a has to be created using the standard map. From a cryptographic point of view, the sequence 1 N j j a is not good enough for designing a PRBG be- cause it is not random. Therefore, we propose to use the ECF-map to convert the gener ated sequence 1 N j j a to a binary sequence 1 N j j k of the same length by applying (11). 4. Statistical Tests In this section, we apply the statistical test suite d esigned by NIST [21] to verify the randomness of the binary se- quences generated by our PRBG. The NIST test suite includes 16 independent and computationally intensive statistical tests. These tests are useful in detecting devia- tions of a binary sequence from randomness. The 16 tests can be classified into two categories: 1) Non-parameterized tests: Frequency (monobit) test (FT), Runs test (RT), Test for longest run of ones in a block (LROT), Lempel-Ziv compression test (LZT), Binary matrix rank test (MRT), Cumulative sums test (CST), Discrete Fourier transform (spectral) test (SPT), Random excursions test (RET) and Random excursions variant test (REVT). 2) Parameterized tests: Frequency test within a block (BFT), Approximate entropy test (AET), Linear com- plexity test (LCT), Maurer’s universal statistical test (MUST), Serial test (ST), Overlapping template match- ing test (OTMT) and Non-overlapping template match- ing test (NOTMT). The detailed description of all 16 tests of NIST suite can be found in [21]. 4.1. Statistical Analysis of the Proposed PRBG First, the generator should produce a binary sequence of 0s and 1s of a given length N. Next, we invoke the NIST An Efficient PRBG Based on Chaotic Map and Engel Continued Fractions Copyright © 2010 SciRes. JSEA 1144 Statistical Test Suite using the generated sequence. Fi- nally, a set of p_values for each statistical test will be generated by the test suite. Based on these p_values, a conclusion regarding the quality of the sequences can be drawn. The significance level for all tests in NIST Suite is set to 0.01. A p_value less than would mean that the sequence is non-random. If a p_value is greater than , we accept the sequence as random. In Table 1, we list the results of the statistical tests (the p_values) applied on the sequences 1 N j j k and 1 N j j b of length 1.000.000N using random initial values and parame- ters produced by our PRBG and by a pseudo random bit generator based only on the standard map, respectively. The sequence 1 N j j bis generated as follows: j b j F afor 1jN . The exact parameters values used in these examples have been included in parentheses, besides the name of th e statistical test. Table 1 sh ow s th e p_values of each test of the sequences with and without applying the ECF-map. It is clear that some tests failed if the sequence is simply generated from the standard map. However, a noticeable improvement is observed if we use standard map with the ECF-map, as all the tests are passed. 4.2. The Interpretation of Empirical Results The interpretation of empirical results can be conducted in a number of ways. Two approaches have been adopted: 1) The examination of the proportion of passing se- quences: in the final analysis report file generated by the NITS statistical suite, a value called the proportion was listed for each test. The proportion is the number of se- quences having a p_value greater than the significance level , divided by the total number of bit sequences tested, i.e the p ercentage of passed tests. NIST specifies a, range of acceptable proportions. The range is determined by using the confidence interval defined as mppp / ˆ 1 ˆ 3 ˆ, where ˆ1p , and m is the sample size. If the proportion falls outside this interval then it is sufficient to deduce that the data is non-random. For this experiment, we generate 100m binary se- quences, each containing 000.000.1 random bits. The Table 1. Statistical tests on the sequences 1 N j j k and 1 N j j b with different initial states. Test No. Test Name 0 0 0 3.59587469543 0.8512974635 120.9625487136 x y p 0250N 0 0 5.02548745491 2.9654128766 x y 0100.6p 0 250N 1 N jj k 1 N jj b 1 N jj k 1 N jj b 1 FT 0.95056 0.00000 0.57139 0.00000 2 BFT(m = 128) 0.48770 0.00499 0.60654 0.02557 3 RT 0.85244 0.00000 0.58803 0.00000 4 LROT 0.90989 0.21701 0.67662 0.41932 5 MRT 0.93152 0.40617 0.10481 0.76072 6 SPT 0.76038 0.41730 0.06727 0.01983 7 NOTMT (m = 9, B = 000000001) 0.97615 0.00407 0.28535 0.00040 8 OTMT(m = 9, B = 111 111111) 0.52804 0.00034 0.50918 0.19895 9 MUST(L = 7,Q=128 0 ) 0.18980 0.02664 0.08763 0.29615 10 LZT 0.53715 0.23431 0.06145 0.00234 11 LCT(M = 500) 0.48293 0.27597 0.68564 0.82922 12 ST(m = 16) 0.44260 0.11511 0.25245 0. 952 71 13 AET 0.18228 0.00000 0.78445 0.00000 14 CST(Forward) CST(Reverse) 0.83761 0.80126 0.00000 0.00000 0.60651 0.22321 0.00000 0.00000 15 RET( 1x ) 0.93862 0.00000 0.40331 0.00000 16 REVT( 1x ) 0.24142 0.00000 0.76430 0.00000 An Efficient PRBG Based on Chaotic Map and Engel Continued Fractions Copyright © 2010 SciRes. JSEA 1145 confidence interval is 0.99 0.029849 and then the range of acceptable proportion is from 960150.0 to 019849.1 . Table 2 and 3 present the proportions of all tests. Figure 1 shows the proportion for all tests. Since the proportion for each test is within the range, so we are confident to accept the sequence as random bit sequence. 2) The examination of p value’s uniformity: The dis- tribution of p_values is examined to ensure uniformity. For this, the interval between 0 and 1 is divided into 10 Table 2. Examination of the proportion of sequences that pass the 14 first statistical tests and the distribution of p_values. S.NO. Statistical Test P_valueT Proportion of sequences pass- ing the test 1. FT 0.739918 0.9700 2. BFT (m = 128) 0.181557 1.0000 3. RT 0.897763 0.9900 4. LROT 0.275709 0.9900 5. MRT 0.554420 0.9900 6. SPT 0.759756 1.0000 7. NOTMT (m = 9) Template = 00000 0 0 0 1 0.153763 0.9800 Template = 00000 0 0 1 1 0.319084 0.9900 Template = 00000 1 0 1 1 0.678686 0.9700 Template = 00000 1 1 0 1 0.037566 1.0000 Template = 00010 0 0 1 1 0.275709 0.9900 Template = 00010 0 1 0 1 0.616305 0.9900 Template = 00100 0 0 1 1 0.090936 0.9900 Template = 00100 0 1 0 1 0.437274 0.9900 Template = 11110 0 0 0 0 0.924076 1.0000 Template = 11111 1 1 1 0 0.534146 0.9900 8. OTMT (Template = 111111111) 0.085587 1.0000 9. MUST (L = 7, Q = 1280) 0.191687 0.9700 10. LZT 0.171867 0.9800 11. LCT (M = 500) 0.554420 0.9900 12. ST (m = 16) 0.867692 0.9800 13. AET (m = 10) 0.304126 0.9800 14. CST Forward 0.719747 0.9700 Reverse 0.574903 0.9800 Table 3. Examination of the proportion of sequences that pass the 15th and 16th statistical tests and the distribution of p_values. S.NO. Statistical Test P_valueT Proportio n of sequences passing the test 15. RET x = -4 0.534146 1.0000 x = -3 0.455937 1.0000 x = -2 0.739918 0.9828 x = -1 0.122325 0.9800 x = 1 0.171867 1.0000 x = 2 0.911413 1.0000 x = 3 0.534146 1.0000 x = 4 0.262249 1.0000 16. REVT x = -9 0.075719 1.0000 x = -8 0.534146 1.0000 x = -7 0.045675 1.0000 x = -6 0.122325 1.0000 x = -5 0.383827 1.0000 x = -4 0.574903 1.0000 x = -3 0.045675 1.0000 x = -2 0.122325 1.0000 x = -1 0.419021 1.0000 x = 1 0.236810 0.9828 x = 2 0.455937 1.0000 x = 3 0.066882 1.0000 x = 4 0.262249 1.0000 x = 5 0.816537 1.0000 x = 6 0.004981 1.0000 x = 7 0.383827 1.0000 x = 8 0.534146 1.0000 x = 9 0.020548 1.0000 sub-interva ls, and the fo llowing 2 X value for each test is calculated: 2 10 2 1 10 10 i i m F Xm , An Efficient PRBG Based on Chaotic Map and Engel Continued Fractions Copyright © 2010 SciRes. JSEA 1146 Figure 1. Proportions of the sequences passing the tests for all tests of NIST suite. The region between two horizontal lines is the acceptable range of proportion. Figure 2. P_values for all tests of NIST suite. The horizontal line represe nts the threshold value0001.0 . Figure 3. Histograms of p_values. where i F is the number of p_value’s in the sub-interval i and m is the sample size. Next, a p_value of the p_value’s is calculated as: 29 _ ,, 22 TX p valueigamc where igamc is the incomplete gamma function, (for more detail you can refer [29,30]). If _ 0.0001 T pvalue then those p_value's can be considered uniformly distri- buted. Table 2 and 3 show the p_valueT of the 16 tests ob- tained by 100m binary sequences, each containing 000.000.1 random bits. The sequences were generated from the proposed PRBG by using different keys. Refer- ring to Table 2 and 3, it is clear that the p_values are uniformly distributed. In Figure 2, we have graphically presented the computed p_valueT for all tests with the threshold value0.0001. In addition, Figure 3 shows the histograms of p_values for runs and serial tests. 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