Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 6, 287-293 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.517 Signature Attractor Based Pseudorandom Generation Algorithm Krasimir Kordov Department of Computer Informatics Faculty of Mathematics and Informatics Konstantin Preslavski University of Shumen, 9712 Shumen, Bulgaria Copyright c 2015 Krasimir Kordov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We study a chaotic attractor based pseudorandom generation algorithm. The novel scheme use a signature attractor and the logical XOR function. The output binary digits are analysed by NIST, ENT and DIEHARD statistical applications. Subject Classification: 03.67.Dd, 05.90.+m, 43.60.Cg, 46.65.+g Keywords: Signature attractor, XOR function, pseudorandom generation algorithm 1 Introduction The pseudorandom generators are main part in symmetric key encryption and public key encryption schemes. Large class of symmetric key algorithms is based on feedback shift registers. The Shrinking generator and the selfshrinking generator, based on linear feedback shift registers, are proposed in [5] and [8], respectively. Different variants of shrinking pseudorandom schemes, based on feedback with carry shift registers, are presented in [2], [3], [4], [13], [16], [17], and [19]. Another large symmetric key encryption class is based on chaotic maps. In [6], secure communications via chaotic synchronization is experimentally
288 Krasimir Kordov demonstrated using Chua s circuit. A method and system for secure encryption based on the Bernoulli Shift and the Logistic Map is proposed in [9]. A chaotic cryptographic pseudorandom generator constructed from the solutions of the Lorenz attractor, filtered by 32-bit bent Boolean function is presented in [12]. Novel pseudorandom generation algorithm based on Chebyshev polynomial and Tinkerbell map is proposed in [14]. In [15], a modified Chebyshev polynomial based pseudorandom generation scheme is constructed. The use of Circle map chaotic iterations to build pseudorandom number generator is presented in [18]. The aim of the paper is to present a pseudorandom bit generator based Signature attractor. The novel algorithm has excellent statistical properties. 2 Signature Attractor based Pseudorandom Generation Algorithm 2.1 Proposed scheme The Signature attractor is presented in [10], Eq. (1): x t+1 = x t cos θ t y t sin θ t + 1 0.8x t z t y t+1 = x t sin θ t + y t cos θ t z t+2 = 1.4z t+1 + 0.3z t (1 z t ) 1 θ t = 5.5. x 2 t + yt 2 + zt 2 (1) The novel algorithm is based on the following steps: Step 1: The initial values x 0, y 0, z 0, and z 1 from Eq. (1) are determined. Step 2: The attractor from Eq. (1) is iterated for L 1 times. Step 3: The iteration of the Eq. (1) continues, and as a result, two real fractions y i and z i 1, are generated and post-processed as follows: s 1 = mod(abs(integer(y i 10 7 )), 2) s 2 = mod(abs(integer(z i 1 10 7 )), 2), where integer(x) returns the integer part of x, truncating the value at the decimal point, abs(x) returns the absolute value of x, and mod(x, y) returns the reminder after division. Step 4: Perform logical XOR between s 1 and s 2 to get a single output bit. Step 5: Return to Step 3 until the bit stream limit is reached.
Signature attractor based PRG algorithm 289 The proposed bit generator is implemented in C++, using the following initial values: x 0 = 0.5390212, y 0 = 2.1918441, z 0 = 0.157331, and z 1 = 1.380591, and L 1 = 150. 2.2 Key space evaluation The secret key space is composed by the four secret values x 0, y 0, z 0, and z 1. With computational precision of about 10 15 [21] the proposed key space is more than 2 199, which is sufficient enough against brute-force attacks [1]. 2.3 Statistical tests Three software test programs are used in order to measure the behaviour of the output binary streams. The DIEHARD package [7] includes 19 statistical tests, the NIST software application [11] is a set of 15 statistical tests, and the ENT package [20] includes 6 statistical tests. The test results are given in Table 1, Table 2, and Table 3, respectively. All of statistical tests are passed successfully. DIEHARD Proposed Generator statistical test P-value Birthday spacings 0.642348 Overlapping 5-permutation 0.349805 Binary rank (31 x 31) 0.839956 Binary rank (32 x 32) 0.904074 Binary rank (6 x 8) 0.405068 Bitstream 0.503717 OPSO 0.515470 OQSO 0.508207 DNA 0.489971 Stream count-the-ones 0.769666 Byte count-the-ones 0.433502 Parking lot 0.697197 Minimum distance 0.485176 3D spheres 0.527554 Squeeze 0.796239 Overlapping sums 0.566081 Runs up 0.733746 Runs down 0.654742 Craps 0.177186 Table 1: DIEHARD statistical test results for two 80 million bits sequences generated by the proposed generator
290 Krasimir Kordov NIST Proposed Generator statistical test P-value Pass rate Frequency (monobit) 0.026410 989/1000 Block-frequency 0.217857 994/1000 Cumulative sums (Forward) 0.116065 987/1000 Cumulative sums (Reverse) 0.051281 991/1000 Runs 0.044797 991/1000 Longest run of Ones 0.278461 988/1000 Rank 0.191687 992/1000 FFT 0.248014 986/1000 Non-overlapping templates 0.440342 989/1000 Overlapping templates 0.990138 986/1000 Universal 0.190654 982/1000 Approximate entropy 0.743915 986/1000 Random-excursions 0.478173 605/611 Random-excursions Variant 0.477893 606/611 Serial 1 0.007918 991/1000 Serial 2 0.729870 990/1000 Linear complexity 0.347257 990/1000 Table 2: NIST Statistical test suite results for 1000 sequences of size 10 6 -bit each generated by the proposed generator ENT Proposed Generator statistical test results Entropy 7.999998 bits per byte Optimum compression OC would reduce the size of this 125000000 byte file by 0 %. χ 2 distribution For 125000000 samples is 295.49, and randomly would exceed this value 4.14 % of the time. Arithmetic mean value 127.4987 (127.5 = random) Monte Carlo π estim. 3.141381554 (error 0.01 %) Serial correl. coeff. 0.000113 (totally uncorrelated = 0.0) Table 3: ENT statistical test results for two 80 million bits sequences generated by the proposed generator.
Signature attractor based PRG algorithm 291 3 Conclusion We have designed a pseudorandom number generation scheme based on the Signature attractor. Our cryptanalysis showed that the new algorithm design has enough key space and good statistical results. Acknowledgements. This paper is supported by the Project BG051PO001-3.3.06-0003 Building and steady development of PhD students, post-phd and young scientists in the areas of the natural, technical and mathematical sciences. The Project is realized by the financial support of the Operative Program Development of the human resources of the European social fund of the European Union. References [1] G. Alvarez, S. Li, Some Basic Cryptographic Requirements for Chaos- Based Cryptosystems, International Journal of Bifurcation and Chaos, 16 (2006), 2129-2151. http://dx.doi.org/10.1142/s0218127406015970 [2] Arnault, F., Berger, T.: F-FCSR: design of a new class of stream ciphers. In: Gilbert, H., Handshuh, H. (eds.) FSE 2005. LNCS, vol. 3557, pp. 83 97. Springer-Verlag Berlin Heidelberg (2005). http://dx.doi.org/10.1007/11502760 6 [3] Arnault, F., Berger,T., Design and properties of a new pseudorandom generator based on a filtered FCSR automaton, IEEE Transactions on Computers, 54 (2005), 1374-1383. http://dx.doi.org/10.1109/tc.2005.181 [4] Arnault, F., Berger, T., Minier, M., Lauradoux, C., X-FCSR: a New Software Oriented Stream Cipher Based Upon FCSRs, In: Srinathan, K., Pandu Rangan, C., Yung, M. (eds.) Progress in Cryptology - Indocrypt 2007, LNCS, vol. 4859, pp. 341 350, Springer-Verlag Berlin Heidelberg. http://dx.doi.org/10.1007/978-3-540-77026-8 26 [5] D. Coppersmith, H. Krawczyk, Y. Mansour, The shrinking generator, in Advances in Cryptology - CRYPTO 93, LNCS 773, pp. 22 39. http://dx.doi.org/10.1007/3-540-48329-2 3 [6] Lj. Kocarev, K.S. Halle, K. Eckert, L.O. Chua, U. Parlitz, Experimental Demonstration of Secure Communications via Chaotic Synchronization, International Journal of Bifurcation and Chaos, Vol. 2, No. 3 (1992), 709 713. http://dx.doi.org/10.1142/s0218127492000823
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