Pseudo-random Bit Generation Algorithm Based on Chebyshev Polynomial and Tinkerbell Map

Transcription

1 Applied Mathematical Sciences, Vol. 8, 2014, no. 125, HIKARI Ltd, Pseudo-random Bit Generation Algorithm Based on Chebyshev Polynomial and Tinkerbell Map Borislav Stoyanov Department of Computer Informatics Faculty of Mathematics and Informatics Konstantin Preslavski University of Shumen, 9712 Shumen, Bulgaria Copyright c 2014 Borislav Stoyanov. 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 This paper proposes a novel pseudo-random bit generation algorithm combining Chebyshev polynomial and Tinkerbell map. We calculated the key space of the proposed scheme. The output zero-one bits are statistically tested with three packages: NIST, DIEHARD and ENT. The experimental results show that the data are uniformly distributed with sufficient enough statistical properties to disturb brute-force attacks. Keywords: Chebyshev polynomial, Tinkerbell map, pseudo-random bit generation scheme 1 Introduction Pseudo-random number generators based on chaotic maps are widely studied in the last twenty-five years. The need of novel ones is constantly increasing in the digital multimedia broadcasting [8]. Chebyshev polynomial and Tinkerbell map are intensively used in security algorithms. In [9], a pseudo-random generator based on two Chebyshev polynomials is designed. Modified Chebyshev map based key encryption scheme is proposed in [4]. Image encryption based on Chebyshev polynomial is presented in [10]. A method for chaos based encryption of data items by an arithmetic operation with Tinkerbell map as an example, is provided in [5]. The aim of the paper is to propose a new pseudo-random bit generator based on a mix of polynomial and chaotic function: Chebyshev polynomial and Tinkerbell map. The proposed scheme possesses suitable properties for cryptographic applications.

2 6206 Borislav Stoyanov 2 Basic Primitives 2.1 Description of Chebyshev Polynomial The Chebyshev polynomial [3] T n (x) : R R of degree n is recursively defined as T n (x) = 2xT n 1 (x) T n 2 (x), (1) where n 2, x [ 1, 1], T 0 (x) = 1 and T 1 (x) = x. The polynomial T n (x) can be expressed using cosine: x(n + 1) = T k (x n ) = cos(k arccos(x n )), (2) where x n [ 1, 1] and k [2, ). Restricted in the interval [ 1, 1] the Chebyshev polynomial is a commonly used chaotic map T n (x) : [ 1, 1] [ 1, 1] for all n > Description of Tinkerbell Map The Tinkerbell map [1] is a dynamical system given by: y m+1 = y 2 m z 2 m + ay m + bz m z m+1 = 2y m z m + cy m + dz m, (3) where a = 0.9, b = , c = 2.0 and d = Pseudo-random Bit Generation Algorithm Based on Chebyshev Polynomial and Tinkerbell Map 3.1 Proposed Pseudo-Random Bit Generation Algorithm The new pseudo-random bit generation algorithm consists of the following steps: Step 1: The initial values x 0 and k of the the eq.(2), y 0 and z 0 of the eq.(3), and a bit stream limit L are determined. Step 2: The Chebyshev polynomial and Tinkerbell map are iterated for L 1 and L 2 times, respectively. Step 3: The iteration of the eq. (2) and eq. (3) continues, and as a result, two decimal fractions x n and y m, are generated and post-processed as follows: s i = mod(integer(x n 10 7 )), 2) s j = mod(integer(y m 10 7 )), 2), where integer(x) returns the integer part of x, truncating the value at the decimal point, and mod(x, y) returns the reminder after division.

3 PRBG algorithm based on Chebyshev polynomial and Tinkerbell map 6207 Step 4: Perform XOR operation between s i and s j to get a single output bit s k. Step 5: Return to Step 3 until the bit stream limit L is reached. The novel bit generator is implemented in C++, using the following initial values: x 0 = , k = 2.89, y 0 = , z 0 = , L 1 = 100, and L 2 = Key space The set of all posible initial values creates the key space. The novel algorithm has six secret keys x 0, k, y 0, z 0, L 1, and L 2. As stated in [12], the computational precision of the 64-bit double-precision number is about We work with a precision of 10 7 and the proposed key space is more than The key space is secure against exhaustive key search [2]. 3.3 Experimental statistical tests Three statistical packages to evaluate the pseudo-random properties of the output bits are used. The NIST package [7] includes 15 statistical tests. One thousand sequences of bits were generated using the proposed algorithm. The results are given in Table 1. The entire NIST package is passed successfully. The minimum pass rate for each statistical test with the exception of the Random excursion (variant) test is approximately of 980 for a sample size of 1000 zeroone sequences. The minimum pass rate for the random excursion (variant) test is approximately equal to 611 for a sample size 625 zero-one sequences. NIST Proposed Algorithm statistical test P-value Pass rate Frequency (monobit) /1000 Block-frequency /1000 Cumulative sums (Forward) /1000 Cumulative sums (Reverse) /1000 Runs /1000 Longest run of Ones /1000 Rank /1000 FFT /1000 Non-overlapping templates /1000 Overlapping templates /1000 Universal /1000 Approximate entropy /1000 Random-excursions /625 Random-excursions Variant /625 Serial /1000 Serial /1000 Linear complexity /1000 Table 1: NIST statistical package results. The DIEHARD statistical suite [6] consists of 19 tests. The result is given in Table 2. All calculated P-values are in a range of [0, 1). All tests are passed

4 6208 Borislav Stoyanov successfully. DIEHARD Proposed Algorithm statistical test P-value Birthday spacings Overlapping 5-permutation Binary rank (31 x 31) Binary rank (32 x 32) Binary rank (6 x 8) Bitstream OPSO OQSO DNA Stream count-the-ones Byte count-the-ones Parking lot Minimum distance D spheres Squeeze Overlapping sums Runs up Runs down Craps Table 2: DIEHARD statistical test results. The ENT package [11] includes 6 statistical tests. We tested output string of bytes of the proposed scheme. The result is in Table 3. The novel pseudo-random generation algorithm passed all the tests of ENT. ENT Proposed Algorithm statistical test results Entropy bits per byte Optimum compression OC would reduce the size of this byte file by 0 %. χ 2 distribution For samples is , and randomly would exceed this value % of the time. Arithmetic mean value (127.5 = random) Monte Carlo π estim (error 0.00 %) Serial correl. coeff (totally uncorrelated = 0.0) Table 3: ENT statistical test results. 4 Conclusion We have presented a novel pseudo-random generation algorithm based on Chebyshev polynomial and Tinkerbell map. Based on the provided analysis the scheme has suitable properties for cryptographic applications. Acknowledgements. This work is partially supported by the Scientific research fund of Konstantin Preslavski University of Shumen under the grant No. RD /

5 PRBG algorithm based on Chebyshev polynomial and Tinkerbell map 6209 References [1] K.T. Alligood, T.D. Sauer, J.A. Yorke, CHAOS: An Introduction to Dynamical Systems, Springer-Verlag, Berlin, [2] G. Alvarez, S. Li, Some Basic Cryptographic Requirements for Chaos- Based Cryptosystems, International Journal of Bifurcation and Chaos, 16 (2006), [3] L. Kocarev, J. Makraduli, P. Amato, Public-key encryption based on chebyshev polynomials, Circuits, Systems and Signal Processing, 24 (2005), [4] K.M. Kordov, Modified Chebyshev Map Based Pseudo-Random Bit Generator, in 6th AMiTaNS 14, AIP CP, accepted for publication. [5] H.S. Lambert, Method and Apparatus for Encryption of Data, US Patent B2, Nov. 7, [6] G. Marsaglia, DIEHARD: a Battery of Tests of Randomness, [7] A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Application, NIST Special Publication , Revision 1a (Revised: April 2010), Lawrence E. Bassham III, 2010, [8] Z. Shu, G. Zhang, and J. Jiang, Multimedia Security: A Survey of Chaos- Based Encryption Technology, Multimedia - A Multidisciplinary Approach to Complex Issues 5, [9] B.P. Stoyanov, Pseudo-random bit generator based on Chebyshev map, in 5th AMiTaNS 13, AIP CP, 1561 (2013), [10] B. Stoyanov, K. Kordov, Novel Image Encryption Scheme Based on Chebyshev Polynomial and Duffing Map, The Scientific World Journal 2014, Article ID , 2014, [11] J. Walker, ENT: A Pseudorandom Number Sequence Test Program, [12] IEEE Computer Society, IEEE standard for binary floating-point arithmetic, ANSI/IEEE Std. 754, 1985.

6 6210 Borislav Stoyanov Received: August 1, 2014