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Novel Image Encryption Scheme Based on Chebyshev Polynomial and Duffing Map Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 283639, 11 pages http://dx.doi.org/10.1155/2014/283639 Research Article Novel Image Encryption Scheme Based on Chebyshev Polynomial and Duffing Map Borislav Stoyanov and Krasimir Kordov Faculty of Mathematics and Informatics, Konstantin Preslavski University of Shumen, 9712 Shumen, Bulgaria Correspondence should be addressed to Borislav Stoyanov; [email protected] Received 10 December 2013; Accepted 4 March 2014; Published 26 March 2014 Academic Editors: T. Cao, M. K. Khan, and F. Yu Copyright © 2014 B. Stoyanov and K. 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. We present a novel image encryption algorithm using Chebyshev polynomial based on permutation and substitution and Duffing map based on substitution. Comprehensive security analysis has been performed on the designed scheme using key space analysis, visual testing, histogram analysis, information entropy calculation, correlation coefficient analysis, differential analysis, key sensitivity test, and speed test. The study demonstrates that the proposed image encryption algorithm shows advantages of 113 more than 10 key space and desirable level of security based on the good statistical results and theoretical arguments. 1. Introduction chaotic cryptographic algorithm with good complex struc- ture is designed [5]. In [6], an improved stochastic middle In recent years, the dynamical chaotic systems have been multibits quantification algorithm based on Chebyshev poly- commonly used for the design of cryptographic primitives nomial is proposed. Three-party key agreement protocols featuring chaotic behaviour and random-like properties. In using the enhanced Chebyshev polynomial are proposed in his seminal work [1], Shannon pointed out the excellent [7, 8]. possibilities of the dynamical chaotic maps in the communi- Fridrich [9] describes how to adapt Baker map, Cat map, cations. He identified two basic properties that the good data and Standard map on a torus or on a rectangle for the pur- encryption systems should have to prevent (resist) statistical pose of substitution-permutation image encryption. In [10], attacks: diffusion and confusion. Diffusion can propagate a new permutation-substitution image encryption scheme the change over the whole encrypted data, and confusion using logistic, tent maps, and Tompkins-Paige algorithm is can hide the relationship between the original data and proposed. In [11], chaotic cipher is proposed to encrypt color the encrypted data. Permutation, which rearranges objects, images through position permutation part and Logistic map is the simplest method of diffusion, and substitution, that based on substitution. Yau et al. [12]proposedanimage replaces an object with another one, is the simplest type of encryption scheme based on Sprott chaotic circuit. In [13], Fu confusion. The consistent use of dynamical chaotic system et al. proposed a digital image encryption method by using based permutation and substitution methods is in the deep Chirikov standard map based permutation and Chebyshev cryptographic fundamental. polynomial based diffusion operations. The authors of [2]usedChebyshevpolynomialtocon- In [14], a bit-level permutation scheme using chaotic struct secure El Gamal-like and RSA-like algorithms. A new sequence sorting has been proposed for image encryption. more practical and secure Diffie-Hellman key agreement The operations are completed by Chebyshev polynomial protocol based on Chebyshev polynomial is presented in and Arnold Cat map. An image encryption algorithm in [3]. In [4], a stream cipher constructed by Duffing map whichthekeystreamisgeneratedbyChebyshevfunctionis based message-embedded scheme is proposed. By mixing the presented in [15]. Simulation results are given to confirm the Lorenz attractor and Duffing map, a new six-dimensional necessary level of security. In [16], a new image encryption 2 The Scientific World Journal scheme, based on Chebyshev polynomial, Sin map, Cubic where abs() returns the absolute value of ,floor() returns map, and 2D coupled map lattice, is proposed. The experi- the value of to the nearest integers less than or equal to , mentalresultsshowthesecurityofthealgorithm. and mod(, ) returns the reminder after division. In [17], a color image encryption scheme based on skew tent map and hyper chaotic system of 6th-order CNN is Step 5. The following threshold function from (3)isapplied: presented. An image encryption scheme based on rotation 1, if () >() , matrix bit-level permutation and block diffusion is proposed (() ,())={ 0, () ≤() , (3) in [18]. if A new chaos based image encryption scheme is suggested and a pseudorandom bit is produced. in this paper. The algorithm is a simple improvement of one round substitution-permutation model. The encryption Step 6. Return to Step 3 until pseudorandom bit stream limit process is divided in two major parts: Chebyshev polynomial is reached. based on permutation and substitution and Duffing map based on substitution. In Section 2, we propose two pseu- 2.2. Pseudorandom Bit Generator Based on the Duffing Map. dorandom bit generators (PRBGs): one based on Chebyshev In this section, the real numbers of two Duffing maps are polynomial and the other based on Duffing map. In Section 3, preprocessedandcombinedwithasimplethresholdfunction in order to measure randomness of the bit sequence generated to a binary pseudorandom sequence. by the two pseudorandom schemes, we use NIST, DIEHARD, The Duffing map is a 2D discrete dynamical system which and ENT statistical packages. Section 4 presents the proposed takes a point (, V) intheplaneandmapsittoanewpoint. image encryption algorithm, and some security cryptanalysis The proposed pseudorandom bit generator is based on two is given. Finally, the last section concludes the paper. Duffing maps, given by the following equations: 1,+1 = V1,, 2. Proposed Pseudorandom Bit Generators 3 V1,+1 =−1, +V1, − V1,, 2.1. Pseudorandom Bit Generator Based on the Chebyshev (4) Polynomial. In this section, the real numbers of two Cheby- 2,+1 = V2,, shev polynomials [2, 19] are preprocessed and combined V =− +V − V3 . with a simple threshold function to a binary pseudorandom 2,+1 2, 2, 2, sequence. The maps depend on the two constants and .Theseare The proposed pseudorandom bit generator is based on usually set to = 2.75 and =0.2to produce chaotic nature. two Chebyshev polynomials, as described by The initial values 1,0, V1,0, 2,0,andV2,0 areusedasthekey. Step 1.Theinitialvalues1,0, V1,0, 2,0,andV2,0 of the two (+1) = ( )= ( × ( )) , cos arc cos Duffing maps from4 ( ) are determined. (1) (+1) =( )= ( × ( )) , cos arc cos Step 2.ThefirstandthesecondDuffingmapsfrom(4)are iterated for 0 and 0 times, respectively, to avoid the harm- ful effects of transitional procedures, where 0 and 0 are where (,) ∈ [−1, 1] and (, ) ∈ [2, ∞) are control parameters. The initial values (0) and (0) and parameters different constants. (, ) areusedasthekey. Step 3. The iteration of (4)continues,and,asaresult,tworeal () () Step 1.Theinitialvalues(0), (0), ,and of the two fractions and are generated. Chebyshev polynomials from (1) are determined. Step 4. The following threshold function ℎ from (5)isapplied: Step 2.ThefirstandthesecondChebyshevpolynomialsfrom 1, V > V , ℎ(V , V )={ if 1, 2, (1)areiteratedfor0 and 0 times to avoid the harmful effects 1, 2, (5) 0, if V1, ≤ V2,, of transitional procedures, respectively, where 0 and 0 are different constants. and a pseudorandom bit is produced. Step 3. The iteration of (1)continues,and,asaresult,two Step 5. Return to Step 3 until pseudorandom bit stream limit decimal fractions () and () are generated. is reached. Step 4. These decimal fractions are preprocessed as follows: 3. Statistical Test Analysis of the Proposed Pseudorandom Bit Generators () = ( ( ( () ×1014)),2) mod floor abs In order to measure randomness of the zero-one sequence (2) () = ( ( ( () ×1014)),2), generated by the new pseudorandom generators, we used mod floor abs NIST, DIEHARD, and ENT statistical packages. The Scientific World Journal 3 2 The Chebyshev polynomial and the Duffing map based -value =(9/2,/2),where is the pseudorandom bit schemes are implemented by software complemented incomplete gamma statistical function. If simulation in C++ language, using the following initial seeds: -value ≥0.0001 , then the sequences can be deemed to be (0) = 0.9798292345345, (0) = −0.4032920230495034, uniformly distributed. = 2.995, = 3.07, 1,0 = −0.04, V1,0 = 0.2, 2,0 = 0.23,and Using the proposed pseudorandom Using the proposed V2,0 = −0.13,statedasakeyK1. pseudorandom bit generators were generated 1000 sequences of 1000000 bits. The results from all statistical tests are given 3.1. NIST Statistical Test Analysis. The NIST statistical test in Table 1. suite (version 2.1.1) is proposed by the National Institute The entire NIST test is passed successfully: all the - × of Standards and Technology [20]. The suite includes 15 values from all 2 1000 sequences are distributed uniformly tests, which focus on a variety of different types of non- inthe10subintervalsandthepassrateisalsoinacceptable randomness that could exist in a sequence. These tests range. Theminimumpass rate for each statisticaltest with the are frequency (monobit), block-frequency, cumulative sums, exception
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