Enhancing chaos in multistability regions of Duffing map for an asymmetric image encryption algorithm
Hayder Natiq, Animesh Roy, Santo Banerjee, A. P. Misra, N. A. A. Fataf
EEnhancing chaos in multistability regions of Duffing map for anasymmetric image encryption algorithm
Hayder Natiq a , Animesh Roy b , Santo Banerjee c , A. P. Misra b and N. A. A. Fataf d a Department of Computer Technology, Information Technology Collage, Imam JaâĂŹafar Al-Sadiq University, Iraq. b Department of Mathematics, Siksha Bhavana, Visva-Bharati University, Santiniketan, 731 235, India c Department of Mathematical Sciences, Giuseppe Luigi Lagrange, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy d Centre for Defence Foundation Studies, Universiti Pertahanan Nasional Malaysia, Sungai Besi, Malaysia
A R T I C L E I N F O
Keywords :2D Duffing mapcoexisting attractorshyperchaotic behaviorImage encryption
Abstract
We investigate the dynamics of a two-dimensional chaotic Duffing map which exhibits the occurrenceof coexisting chaotic attractors as well as periodic orbits with a typical set of system parameters. Suchunusual behaviors in low-dimensional maps is inadmissible especially in the applications of chaosbased cryptography. To this end, the Sine-Cosine chaotification technique is used to propose a mod-ified Duffing map in enhancing its chaos complexity in the multistable regions. Based on the en-hanced Duffing map, a new asymmetric image encryption algorithm is developed with the principlesof confusion and diffusion. While in the former, hyperchaotic sequences are generated for scramblingof plain-image pixels, the latter is accomplished by the elliptic curves, S-box and hyperchaotic se-quences. Simulation results and security analysis reveal that the proposed encryption algorithm caneffectively encrypt and decrypt various kinds of digital images with a high-level security.
1. Introduction
For the time being, various networks are used to transmita large amount of digital data [16]. Among these data, thereare several types of digital images that contain private infor-mation. It is, therefore, important to protect them from unse-cured channels and unauthorized users. In this regards, dif-ferent types of technologies have been improved to enhancethe protection of the transmitted digital images including wa-termarking [6], data hiding [14], and encryption [4, 12].Chaos-based image encryption is considered as one ofthe most interesting technology, and this is due to the manyunique properties of the chaotic maps, such as unpredictabil-ity, randomness, and complicated nonlinear behaviors [20].Therefore, numerous studies have developed different im-age encryption schemes using chaotic systems [2, 3, 13, 25,26, 27]. The corresponding researches have demonstratedthat the security level of a chaos-based encryption algorithmis highly dependent on the characteristics of the employedchaotic maps [1]. However, further investigations in recentyears have revealed that numerous employed maps could ex-hibit drawbacks, such as chaos degradation with finite preci-sion platforms, low complex performance, narrow and dis-continuous chaotic ranges [8]. Consequently, several stud-ies have devoted their efforts to improve the characteristicof chaotic maps by proposing different chaoticfication tech-niques. Some example are as follows. Hua et al. [9, 11]enhanced the chaotic behavior of Logistic map by modula-tion its output using nonlinear transformation. Natiq et al.[18] enhanced chaos complexity of the 2D Henon map us-ing Sine map for image encryption. Hua et al. [10] proposeda Cosine chaoticfication technique to generate robust chaotic ∗ Corresponding author [email protected] (S. Banerjee)
ORCID (s): maps for encrypting images.The endeavor of investigation the dynamics of chaoticsystems has revealed an interesting and appealing nonlin-ear phenomenon, namely multistability behaviors or coex-isting attractors [21]. For a specific set of system param-eters, multi-stable chaotic systems can generate more thanone chaotic or periodic attractor in case of choosing appro-priate different sets of initial conditions. Several multi-stablecontinuous-time systems have been presented for the last fewyear [22]. However, there are only few studies about thepresence of multistability in discrete-time chaotic systems(maps) [19]. To the best of our knowledge, discrete or con-tinuous multi-stable chaotic systems did not employ in imageencryption schemes, although these systems are more sensi-tive to the initial conditions than the single-stable chaoticsystems. In fact, this is due to the difficulty of choosing theareas where only coexisting chaotic attractors are occurred.However, we believe that this issue can be addressed by asuitable chaoticfication technique.In this paper, we further investigate the dynamics of a 2Ddiscrete chaotic system, namely the 2D-Duffing map. Nu-merical simulation results show that this map can generate acomplicated nonlinear phenomenon, where the occurrenceof multistability behavior can be observed. Two types ofmultistability behavior including the coexistence of chaoticand non-chaotic attractors, and the coexistence of two chaoticattractors can be observed with a specific set of the systemparameters. It is crucial to state here that this complicatedbehavior is rare in low-dimensional chaotic maps. How-ever, when an employed chaotic map has the coexistence ofchaotic with non-chaotic attractors, hence the chaos-basedimage encryption algorithm would be insecure. To handlethis problem, we introduce a chaotification technique basedon two trigonometric functions, which are employed to boostmore randomness to the outputs of Duffing map. Dynamical
Hayder Natiq et al.:
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Page 1 of 15 a r X i v : . [ n li n . PS ] S e p everaging social media news analysis shows that the proposed technique can successfullyconvert the chaotic regions to become hyperchaotic regions.Moreover, it can covert the non-chaotic regions as well tobecome chaotic or hyperchaotic regions. Performance eval-uations and comparisons demonstrate that the enhanced maphas more unpredictable behaviors, high randomness, and alarger hyperchaotic range than several 2D chaotic and hyper-chaotic maps. Furthermore, we employ the enhanced mapto design an asymmetric image encryption algorithm. Thesimulation results demonstrate that the proposed encryptionalgorithm can effectively encrypt various kinds of digital im-ages including Grey-scale, RGB, medical, and hand writingimages.This paper is arranged as follows: Section 2 investigatesthe dynamics of 2D-Duffing map. In Section 3, we intro-duce a chaotification approach to increase chaos complexityof the 2D-Duffing map, and then evaluate the performanceof the enhance map. Section 4 introduces an asymmetricimage encryption algorithm based on the enhanced Duffingmap. Section 5 simulates the proposed image encryption al-gorithm using different types of digital images. Section 6analyzes the security of the proposed encryption algorithm.Finally, Section 7 presents the conclusions.
2. The Duffing model
The Duffing system, which is also called Holmes map[5], is a 2D discrete-time chaotic system. The system can bewritten as { 𝑥 ( 𝑛 + 1) = 𝑥 ( 𝑛 ) ,𝑥 ( 𝑛 + 1) = − 𝛽𝑥 ( 𝑛 ) + 𝛼𝑥 ( 𝑛 ) − 𝑥 ( 𝑛 ) , (1)where 𝛽 , 𝛼 are positive parameters. From graphical point of view, 𝐸 is an equilibrium pointof the function 𝐺 ( 𝑥 ) only when 𝐺 𝑛 ( 𝐸 ) = 𝐸 . Thus, one canobtains the equilibrium points of system (1) by reducing itsdimension as follows 𝑥 ( 𝑣 )1 = − 𝛽𝑥 ( 𝑣 )1 + 𝛼𝑥 ( 𝑣 )1 − ( 𝑥 ( 𝑣 )1 ) , (2)where ( 𝑣 = 1 , , … ) . For the parameters 𝛽 = 0 . and 𝛼 =2 . , the equilibrium points are obtained as follows ⎧⎪⎨⎪⎩ 𝐸 = (0 , ,𝐸 = (1 . , . ,𝐸 = (−1 . , −1 . . The stability of the obtained equilibria is determined bythe Jacobian matrix of the Duffing map (1), which is definedas 𝐽 = ⎛⎜⎜⎜⎝ 𝜕𝑓 𝜕𝑥 𝜕𝑓 𝜕𝑥 𝜕𝑓 𝜕𝑥 𝜕𝑓 𝜕𝑥 ⎞⎟⎟⎟⎠ . L ya puno v ex pon e n t s Figure 1:
Lyapunov Exponents of the Duffing map (1) with theparameter 𝛽 = 0 . , and for the initial conditions (0 . , . . Using the above matrix, the Duffing map (1) can be Lin-earized at any arbitrary equilibrium point 𝐸 𝑖 = ( 𝑥 ∗1 , 𝑥 ∗2 ) by 𝐽 𝐸 𝑖 = ( 𝛽 𝛼 − 3( 𝑥 ∗2 ) ) . The corresponding eigenvalues at the equilibria 𝐸 𝑖 can beobtained by solving Eq. (3) det( 𝜆𝐼 − 𝐽 𝐸 𝑖 ) = 0 (3)that yields 𝜆 + ( 𝑥 ∗2 ) − 𝛼 ) 𝜆 + 𝛽 = 0 . Consequently, the eigenvalues are given by ⎧⎪⎪⎨⎪⎪⎩ 𝜆 = 3( 𝑥 ∗2 ) − 𝛼 − √( 𝑥 ∗2 ) − 𝛼 ) − 4 𝛽 𝜆 = −3( 𝑥 ∗2 ) + 𝛼 + √( 𝑥 ∗2 ) − 𝛼 ) − 4 𝛽 In the discrete dynamical systems, the stability of equilib-rium points is dependent on the eigenvalues. If an eigenvalue
Figure 2:
Chaotic attractor of the Duffing map (1) with theparameter 𝛽 = 0 . , 𝛼 = 2 . and for the initial conditions (0 . , . .Hayder Natiq et al.: Preprint submitted to Elsevier
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Table 1
The equilibria of Duffing map (1) and their stability for theparameters 𝛽 = 0 . and 𝛼 = 2 . .Equilibria 𝜆 𝜆 Stability analysis 𝐸 −2 . . unstable equilibrium 𝐸 . . stable equilibrium 𝐸 . . stable equilibrium is within the interval [−1 , , then the equilibrium point ex-hibits stable state. Otherwise, it shows unstable state. Forthe parameters 𝛽 = 0 . and 𝛼 = 2 . , the stability of theobtained equilibria of Duffing map (1) is illustrated in Ta-ble 1. Obviously, the Duffing map has one unstable equilib-rium point and two stable equilibria. To investigate the dynamical behavior of map (1), theLyapunov exponents (LE) are depicted in Figure 1 for theinitial conditions (0 . , . . As can be seen in this fig-ure, the map (1) exhibits two different behaviors in which itshows chaotic behavior when the largest Lyapunov exponent(LLE) is greater zero. Meanwhile, the periodic behavior isappeared when LLE is less than or equal to zero. Figure 2demonstrates the chaotic behaviors of the Duffing map whenthe parameters are set as 𝛽 = 0 . and 𝛼 = 2 . . Further-more, since the map (1) has one unstable equilibrium pointfor the parameters 𝛽 = 0 . and 𝛼 = 2 . , hence the gener-ated chaotic attractor in Figure 2 is self-excited [21]. LL E (b) Figure 3:
Coexisting two attractors with the parameter 𝛽 =0 . , and under two sets of initial states in which the blueand red orbits are initiated from (0 . , . and (0 . , . re-spectively: (a) Bifurcation diagram with respect to the variable 𝑥 ; (b) Largest Lyapunov exponents. x x (a) -1.5 -1 -0.5 0 0.5 1 1.5 x -1.5-1-0.500.511.5 x (b) Figure 4:
Multistability behaviors of the Duffing map (1) underthe initial conditions (0 . , . (red) and (0 . , . (blue):(a) the coexistence of chaotic and periodic orbit with the pa-rameters 𝛼 = 2 . , and 𝛽 = 0 . ; (b) the coexistence of twochaotic attractors with the parameters 𝛼 = 2 . , and 𝛽 = 0 . . Multistability behaviors or coexisting attractors indicatethat nonlinear dynamical systems can produce two or moreattractors by changing the initial conditions. In this subsec-tion, we demonstrate the existing of multistability behaviorin the Duffing map (1) by selecting an appropriate set of ini-tial conditions. To the best of our knowledge, this compli-cated behavior of the Duffing map has not been identified inthe previous studies.Denote 𝛽 = 0 . , when the parameter 𝛼 is varied from . to . , the coexisting bifurcation model and LE of Duff-ing map (1) for two sets of the initial conditions (0 . , (blue), (0 . , . (red) are plotted using MATLAB withthe step size . , as shown in Figure 3, respectively. Ascan be seen, the Duffing map (1) shows the coexistence ofself-excited chaotic attractor and periodic orbit mainly in theregion of . ≤ 𝛼 ≤ . . Furthermore, the coexis-tence of two self-excited chaotic attractors can be observedapproximately in the region of . ≤ 𝛼 ≤ . . To furthervisualize these interesting nonlinear phenomena, Figure 4(a) and (b) depict the phase spaces of the Duffing map (1)when the parameters 𝛼 is equal to . and . , respec-tively.It is important to state here that the keys in the chaos-based cryptosystems are typically formed by the initial con-ditions and the parameters of chaotic maps. It is, therefore,when the chaotic maps exhibit multistability behaviors suchas the coexistence of chaotic attractors with periodic orbits,the corresponding cryptosystems will be insecure. That means,it is quite interesting to introduce an efficient chaotificationtechnique on 2D Duffing map (1) for enhancing chaos in thenon-chaotic regions.
3. Sine-Cosine chaotification technique
This section proposes a new chaotification technique thatuses two trigonometric functions as nonlinear transform tothe outputs of Duffing map (1). Sine and Cosine functionsare applied to enhance the chaos and complexity of Duffingchaotic map in the chaotic region. Moreover, those functionscan also produce chaos in the non-chaotic regions.
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Equilibrium points E i g e n va l u e Figure 5:
Equilibrium points of the enhanced Duffing map (4) and its stability when the parameters 𝛼 = 2 . and 𝛽 = 0 . :1) for 𝐴 = 1 and 𝐵 = 1 , there is only one stable equilibriumpoint (green color); 2) for 𝐴 = 2 and 𝐵 = 3 , there are threeequilibria (red color) in which two stable and one unstable; 3)for 𝐴 = 15 and 𝐵 = 3 . , there are 25 equilibria (blue color) inwhich 15 stable and 10 unstable. The structure of the proposed chaotification technique isillustrated in Figure 6, where 𝑓 ( 𝑦 ( 𝑛 )) and 𝑔 ( 𝑥 ( 𝑛 ) , 𝑦 ( 𝑛 )) aretwo seed maps taken from Eq. (1) and Eq. (2) of the sys-tem (1), respectively. Meanwhile, Sine and Cosine functionsperform nonlinear transformation to 𝑓 ( 𝑦 ( 𝑛 )) and 𝑔 ( 𝑥 ( 𝑛 ) , 𝑦 ( 𝑛 )) ,respectively. Mathematically, the proposed technique can bedefined as follows { 𝑥 ( 𝑛 + 1) = 𝐴 sin( 𝑦 ( 𝑛 )) ,𝑦 ( 𝑛 + 1) = 𝐵 cos(− 𝛽𝑥 ( 𝑛 ) + 𝛼𝑦 ( 𝑛 ) − 𝑦 ( 𝑛 )) , (4)where 𝐴 , and 𝐵 are positive parameters. As can be seen in the previous section, the Duffing map(1) has three equilibria in which two of them are stable andone unstable, as illustrated in Table 1. Thus, it is interestingto know whether the enhanced Duffing map (4) generates thesame numbers or different numbers of equilibria when 𝛼 =2 . and 𝛽 = 0 . . Besides that, it is crucial to investigate thestability of these equilibria.Three cases are selected to investigate the number of equi-libria of the enhanced Duffing map (4) and its stability, asillustrated in Figure 5. Since the eigenvalues of enhancedDuffing map satisfy the relation 𝜆 = − 𝜆 , Figure 5 depictsthe equilibria of the enhanced Duffing map and the corre-sponding eigenvalue 𝜆 . The three cases are as follows: 1)when 𝐴 = 1 and 𝐵 = 1 , the enhanced Duffing map (4)has only one stable equilibrium point, as shown in the greencolor of Figure 5; 2) when 𝐴 = 2 and 𝐵 = 3 , the enhancedDuffing map (4) has three different equilibria in which two ofthem are stable and one unstable, as shown in the red color ofFigure 5. 3) when 𝐴 = 15 and 𝐵 = 3 . , the enhanced Duff-ing map (4) has 25 different equilibria in which 15 equilibriaare stable and 10 equilibria are unstable, as shown in the bluecolor of Figure 5.It can be concluded that dependent on the values of pa-rameters 𝐴 and 𝐵 , the enhanced Duffing map (4) can gener- Figure 6:
Structure of the enhanced Duffing map. ate less, or equal, or greater than the number of equilibria ofDuffing map (1).
It can be observed that the number of equilibria of theenhanced map (4) can be larger than the number of equilib-ria of the original map when the parameters 𝐴 and 𝐵 arelarge enough, as can be seen in the third case of Figure 5.Therefore, it is quite reasonable to assume that increasingthe equilibria of a dynamical system in a limited range canenhance chaos as well as overlapping coexisting attractors.To visually demonstrate the above concept, we set theparameters of the enhanced map (4) as 𝛽 = 0 . , 𝐴 = 𝐵 =4 . , hence the map has 37 equilibria distributed within therange [−3 . , . . Figure 7 (a) and (b) depict the coexist-ing bifurcation model and LLE of Duffing map (1) for twosets of the initial conditions (0 . , (blue), (0 . , . (red), respectively. Obviously, the chaotic regions of 2DDuffing map have enhanced, meanwhile, the non-chaotic re-gions have become chaotic. Furthermore, the two coexistingchaotic attractors of the enhanced map (4) have overlapped,and occupied a much larger region in the 2D phase space, ascan be seen in Figure 7 (c). A nonlinear dynamical system exhibits hyperchaotic be-havior only when it has at least two positive values of LE.Obviously, the 2D Duffing map (1) shows no hyperchaoticbehavior, because it exhibits only one positive value of LEfor some parameter values, as can be seen in Figure 1. Typi-cally, the trajectory of a dynamical system with hyperchaoticbehavior is more difficult to be predicted, thus it is more de-sired for cryptography applications.To examine the existence of hyperchaotic behavior in theenhanced Duffing map (4), we have depicted the LE whenthe two parameters 𝐴 and 𝐵 vary at the same time, as shownin Figure 8. The LLE of the enhanced map (4) is illustratedin Figure 8 (a), meanwhile, the lowest LE is illustrated inFigure 8 (b). It can be seen from this figure that the en-hanced map shows chaotic behavior with very high valuesof the LLE when 𝐵 ∈ (7 . , and for all values of param-eter 𝐴 . However, the hyperchaotic behavior of the enhancedmap is approximately observed in two regions as follows: 1) Hayder Natiq et al.:
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Dynamics of the enhanced Duffing map (4) under the initial conditions (0 . , . (red) and (0 . , . (blue) withthe parameters 𝛽 = 0 . , 𝐴 = 𝐵 = 4 . : (a) the coexisting bifurcation model; (b) Largest Lyapunov exponents; (c) phase spacewith 𝛼 = 1 . 𝐴 ∈ [12 , and 𝐵 ∈ [4 , . ; 2) 𝐴 ∈ [3 , and 𝐵 ∈ [2 , .To demonstrate that the hyperchaotic attractor of the en-hanced Duffing map (4) has high level of complexity, andspreads in a wide region of the 2D phase space, Figure 9 plotsthe hyperchaotic attractor of the enhanced map as well asthe attractors of other chaotic and hyperchaotic maps. Thesemaps are described as follows: the 2D-SLMM [11], 2D-SIMM [15], 2D-LASM [9], and 2D-LICM [2] are consid-ered as hyperchaotic maps. Meanwhile, the 2D Ushiki [7] isconsidered as chaotic map.Figure 9 can clearly illustrate that the attractor of the en-hanced map with the parameters 𝛼 = 1 , 𝛽 = 5 , 𝐴 = 15 , 𝐵 = 3 . occupies the whole 2D phase space ranging 𝑥 ∈[−15 , and 𝑦 ∈ [−3 . , . . This means that the enhancedmap generates extreme unpredictable sequences, and its er-godicity property is much better than other maps. Figure 8:
Hyperchaotic diagram of the enhanced Duffingmap (4) based on Lyapunov exponents with 𝛽 = 5 , 𝛼 = 1 :(a) the largest Lyapunov exponent; (b) the lowest Lyapunovexponent. Richman et al. [23] introduced an approach to developApproximate Entropy, which is widely used measure for es-timating the time series complexity. Several analysis havedemonstrated that the developed measure, namely, SampleEntropy (SamEn), is more accurate than Approximate En-tropy.To illustrate the complexity of the enhanced map (4),Figure 10 depicts the SamEn results of the enhanced mapand different chaotic and hyperchaotic maps. The parame-ters of the enhanced map (4) in this figure are set as 𝛽 = 5 , 𝐴 = 15 , and 𝐵 = 3 . . Obviously, the enhanced map (4) hasthe largest SamEn values, which means that one needs moreinformation to predict the generated sequences by this map. The randomness of the two generated hyperchaotic se-quences 𝑥 and 𝑦 by the enhanced Duffing map (4) can beevaluated by several randomness evaluation methods. Here,we use the software package of FIPS 140-2, which mainlyconsists of three different tests. For each test, the p- value isderived to reflect the randomness level. A chaotic sequencecan pass the test when the derived p-value is within a rangeof [10 −4 , −4 ] . The experimental results are shown inTables 2. As can be seen, the generated sequences 𝑥 and 𝑦 by the enhanced map (4) pass all the statistical tests, whichmeans that these two sequences are reliable PRNG, and haveexcellent randomness property.In summary, Lyapunov exponents, trajectory, SamEn,and FIPS 140-2 have demonstrated that the enhanced map (4)exhibits decent ergodicity property, wide hyperchaotic be-havior, high level of complexity and randomness. As a re-sult, the enhanced map would be very promising for cryp-tography applications.
4. Chaos based image encryption algorithm
This section introduces a new asymmetric image encryp-tion algorithm. Figure 11 illustrates the structure of the pro-posed algorithm, which achieves the confusion and diffu-sion processes by the hyperchaotic sequences and ellipticcurve over the Galois field 𝐺𝐹 . Specifically, the enhanced Hayder Natiq et al.:
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Chaotic and hyperchaotic attractors of different 2D maps: (a) 2D-SLMM [11]; (b) 2D-SIMM [15]; (c) 2D Ushiki map[7]; (d) 2D-LASM [9]; (e) 2D-LICM [2]; (f) the enhanced Duffing map (4) . Table 2
Randomness analysis test with FIPS-140-2 results of sequences 𝑥 and 𝑦 generated by themap 4. Tests Sub-tests 𝐱 𝐲 DecisionRuns test P-value 0.3489 0.1546 Pass0 runs, length 1 2499 2413 Pass0 runs, length 2 1235 1186 Pass0 runs, length 3 612 604 Pass0 runs, length 4 305 312 Pass0 runs, length 5 152 157 Pass0 runs, length 6 +
153 154 PassLongest run of 0 14 14 Pass1 runs, length 1 2592 2489 Pass1 runs, length 2 1287 1257 Pass1 runs, length 3 646 645 Pass1 runs, length 4 320 334 Pass1 runs, length 5 166 170 Pass1 runs, length 6 +
166 172 PassLongest run of 1 14 13 PassMonobit test P-value 0.3104 0.5489 PassNo. of 1s 20000-bitstream 10113 10039 PassPoker test p-value 0.1440 0.1926 PassY- value 13.1008 17.5232 Pass
Duffing map (4) is used to generate hyperchaotic sequencesfor scrambling the pixels of a plain-image through imagescrambling algorithm. Subsequently, the diffusion processis accomplished by field matrix, S-box, and hyperchaotic se-quence. Simulations results demonstrate that the proposedencryption algorithm gives the users a flexibility to encryptseveral kinds of images such as Grey scale, Medical, andRGB images with a higher level of security.
An efficient encryption algorithm should disassemble thehigh correlations between adjacent pixels. These high cor-relations can be de-correlated by scrambling adjacent pixelsto different positions. To ensure an efficient scrambling pro-cess, we divided the plain-images into -bit-plane. Then thepositions of all adjacent pixels are randomly scrambled bythe image scrambling algorithm. However, Algorithm 1 il- Hayder Natiq et al.:
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The elliptic curve is a set of points that satisfy the fol-lowing Weierstrass equation: 𝐸 ∶ 𝑦 + 𝑎 𝑥𝑦 + 𝑎 𝑦 = 𝑥 + 𝑎 𝑥 + 𝑎 𝑥 + 𝑎 , (5)where 𝑎 , 𝑎 , 𝑎 , 𝑎 , 𝑎 , and 𝑎 represent the parameters andthe initial conditions of the map (4) 𝐴, 𝛼, 𝐵, 𝛽, 𝐼𝐶 , and 𝐼𝐶 ,respectively.The elliptic curve equation over the field 𝐺𝐹 is definedas 𝐸 ( 𝑎, 𝑏 ) ∶ 𝑦 + 𝑥𝑦 = 𝑥 + 𝑎𝑥 + 𝑏, (6)which has obtained by the following transformation ( 𝑥, 𝑦 ) → ( 𝑎 𝑥 + 𝑎 𝑎 , 𝑎 𝑦 + 𝑎 𝑎 + 𝑎 𝑎 ) , where 𝑎 ≠ .To construct the field matrix 𝐹 , we should firstly extractsome points, which satisfy the equation (6), by the followingprimitive polynomial 𝑓 ( 𝑥 ) = 1 + 𝑥 + 𝑥 + 𝑥 + 𝑥 . Here, if the generator 𝑔 satisfies 𝑓 ( 𝑔 ) = 0 , one can obtainthat 𝑔 = 𝑔 + 𝑔 + 𝑔 + 1 , where 𝑔 equal to (00000010) .Consequently, the properties of elliptic curve 𝐸 ( 𝑎, 𝑏 ) isgiven by• Let us consider pair of points 𝑃 ( 𝑥, 𝑦 ) ∈ 𝐸 𝑛 ( 𝑎, 𝑏 ) sat-isfy Eq. (6) together with 𝑂 , which is the additive iden-tity, i.e. 𝑃 + 𝑂 = 𝑃 . Thus, if 𝑃 = ( 𝑥 𝑝 , 𝑦 𝑝 ) then − 𝑃 = ( 𝑥 𝑝 , 𝑥 𝑝 + 𝑦 𝑝 ) and 𝑃 − 𝑃 = 𝑂 .• If 𝑃 = ( 𝑥 𝑝 , 𝑦 𝑝 ) , 𝑄 = ( 𝑥 𝑞 , 𝑦 𝑞 ) and 𝑃 ≠ ± 𝑄 , then 𝑅 = 𝑃 + 𝑄 = ( 𝑥 𝑟 , 𝑦 𝑟 ) , where ( 𝑥 𝑟 , 𝑦 𝑟 ) = ( 𝛼 + 𝛼 + 𝑥 𝑝 + S a m E n The enhanced map2D-LICM2D-LASM2D-SIMM2D-SLMM
Figure 10:
SamEn results of different chaotic and hyperchaoticmaps, where parameter 𝜎 represents 𝛼, 𝑎, 𝑎 , 𝑎 , 𝛼 , for the en-hanced map (4) , 2D-LICM [2], 2D-LASM [9], 2D-SIMM [15],and 2D-SLMM [11], respectively. Algorithm 1:
Image scrambling at bit level
Input:
Plain-image of the size 𝑚 × 𝑛 . Output:
The scrambled and de-scrambled images Generate hyperchaotic sequences { 𝑥 } and { 𝑦 } withthe long of 𝑘 , where 𝑘 ≥ 𝑚 × 𝑛 ; Calculate 𝑆𝑥 = 𝑐𝑒𝑖𝑙 ( 𝑚𝑜𝑑 ( 𝑥 × 10 , and to formmatrix is of 𝑆𝑥 𝑚 × 𝑛 ← 𝑅𝑒𝑠ℎ𝑎𝑝𝑒 ( 𝑆𝑥, 𝑚, 𝑛 ) Calculate 𝑆𝑦 = 𝑐𝑒𝑖𝑙 ( 𝑚𝑜𝑑 ( 𝑦 × 10 , 𝑚 × 𝑛 )) and toform a sequence where each element don’t repeatand non zero % Image scrambling Reshape 𝐴 𝑚 × 𝑛 ⟵ { 𝑎 ( 𝑖 )} for 𝑖 = 1 , , ..., 𝑚 × 𝑛 anditerate 𝑎 ( 𝑆𝑦 ( 𝑖 )) Reshape 𝑎 ( 𝑆𝑦 ( 𝑖 )) ⟶ 𝐴 𝑚 × 𝑛 ) % Image de-scrambling Calculate 𝑖𝑛𝑣. ( 𝑆𝑦 ) from 𝑆𝑦 Reshape 𝐶𝐼 𝑚 × 𝑛 ⟵ { 𝑎 ( 𝑖 )} for 𝑖 = 1 , , ..., 𝑚 × 𝑛 anditerate 𝑎 ( 𝑖𝑛𝑣 ( 𝑆𝑦 ( 𝑖 ))) Reshape 𝑎 ( 𝑖𝑛𝑣 ( 𝑆𝑦 ( 𝑖 ))) ⟶ 𝐷𝑒𝑐𝑟𝑦𝑝𝑡𝑒𝑑𝑖𝑚𝑎𝑔𝑒 𝑚 × 𝑛 Divide the plain-image into -bit plane using i.e. 𝐴 𝑚 × 𝑛 (plain-image)= 𝐴 𝑚 × 𝑛 ×8 Divide 𝑆𝑥 into 𝑆𝑥 𝑚 × 𝑛 ×8 Initialize A matrix 𝐵 is of order 𝑚 × 𝑛 × 8 i.e. 𝐵 𝑚 × 𝑛 ×8 = 𝑧𝑒𝑟𝑜𝑠 ( 𝑚, 𝑛, for k=1 to do for j=1 to 𝑛 do for i=1 to 𝑚 do 𝐵 ( 𝑖, 𝑗, 𝑘 ) = 𝑏𝑖𝑡𝑥𝑜𝑟 ( 𝐴 ( 𝑖, 𝑗, 𝑘 ) , 𝑆𝑥 ( 𝑖, 𝑗, 𝑘 )) end end end Return 𝐵 Set the bit position at each plain using the previousplane Initialize a scramble matrix 𝑆𝑐 = 𝑧𝑒𝑟𝑜𝑠 ( 𝑚, 𝑛 ) for k=1 to do 𝑆𝑐 𝑘 = 𝑏𝑖𝑡𝑠𝑒𝑡 ( 𝐵 (∶ , ∶ , 𝑘 ) , 𝑆𝑐 𝑘 −1 ) end Return 𝑆𝑐 𝑚 × 𝑛 𝑥 𝑞 + 𝐴, 𝛼 ( 𝑥 𝑝 + 𝑥 𝑟 ) + 𝑥 𝑟 + 𝑦 𝑝 ) and 𝛼 = 𝑦 𝑞 + 𝑦 𝑝 𝑥 𝑝 + 𝑥 𝑞 . Again, if 𝑃 = 𝑄 , then 𝑃 = ( 𝑥 𝑟 , 𝑦 𝑟 ) = ( 𝜆 + 𝜆 + 𝐴, 𝑥 𝑝 +( 𝜆 +1) 𝑥 𝑟 ,where 𝜆 = 𝑥 𝑝 + 𝑦 𝑝 𝑥 𝑝 . To construct the S-box, we use the control parametersand initial conditions for generating two hyperchaotic se-quences { 𝑥 } , and { 𝑦 } by the enhanced Duffing map (4). Thesesequences are then transformed over the Galois field 𝐺𝐹 into new sequences as follows { 𝑥 = 𝑓 𝑙𝑜𝑜𝑟 ( 𝑚𝑜𝑑 ( 𝑥 × 10 , ,𝑦 = 𝑓 𝑙𝑜𝑜𝑟 ( 𝑚𝑜𝑑 ( 𝑦 × 10 , . Hayder Natiq et al.:
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Figure 11:
Schematic diagram of the proposed image encryption and decryption algorithm.
Algorithm 2:
The field matrix and its inverse.
Input:
The initial conditions 𝐼𝐶 , and 𝐼𝐶 , as wellas the primitive polynomial 𝑓 ( 𝑥 ) . Output:
The field matrix and inverse field matrix. Calculate 𝐹 (1) = 𝑚𝑜𝑑 ( 𝑑𝑒𝑐 𝑏𝑖𝑛 ( 𝐼𝐶 × 𝐼𝐶 × 10 ) , 𝑓 ( 𝑥 )) ; while 𝑖 = 1 , 𝑖 + + do 𝐹 ( 𝑖 + 1) = 𝑚𝑜𝑑 ( 𝑑𝑒𝑐 𝑏𝑖𝑛 ( 𝐹 ( 𝑖 ) × 𝐼𝐶 × 𝐼𝐶 × 10 ) , 𝑓 ( 𝑥 )) Calculate 𝐹 = [ 𝐹 (1) , 𝐹 (2) , ..., 𝐹 (8)] if 𝑑𝑒𝑡 ( 𝐹 ) ≠ over 𝑍 then return 𝐹 else Calculate 𝐹 = [ 𝐹 ( 𝑖 + 1) , 𝐹 ( 𝑖 + 2) , ..., 𝐹 ( 𝑖 + 8)] end end Get 𝐹 𝑏𝑖𝑡 ×8− 𝑏𝑖𝑡 To calculate inverse field matrix Calculate 𝐹 (1) = 𝑚𝑜𝑑 ( 𝑑𝑒𝑐 𝑏𝑖𝑛 ( 𝐼𝐶 × 𝐼𝐶 × 10 ) , 𝑓 ( 𝑥 )) while 𝑖 = 1 , 𝑖 + + do 𝑖𝑛𝑣.𝐹 ( 𝑖 + 1) = 𝑚𝑜𝑑 ( 𝑑𝑒𝑐 𝑏𝑖𝑛 ( 𝑖𝑛𝑣.𝐹 ( 𝑖 ) × 𝐼𝐶 × 𝐼𝐶 × 10 ) , 𝑓 ( 𝑥 )) Calculate 𝑖𝑛𝑣.𝐹 = [ 𝑖𝑛𝑣.𝐹 (1) , ..., 𝑖𝑛𝑣.𝐹 (8)] if 𝑑𝑒𝑡 ( 𝐹 × 𝑖𝑛𝑣.𝐹 ) = 1 over 𝑍 then return 𝑖𝑛𝑣.𝐹 else Calculate 𝑖𝑛𝑣.𝐹 = [ 𝑖𝑛𝑣.𝐹 ( 𝑖 + 1) , ..., 𝑖𝑛𝑣.𝐹 ( 𝑖 + 8)] end end Get 𝑖𝑛𝑣.𝐹 𝑏𝑖𝑡 ×8− 𝑏𝑖𝑡 Algorithm 3:
The S-Box and its inverse.
Input:
The sequences { 𝑥 } and { 𝑦 } , as well as theextract points 𝑃 , field matrix 𝐹 and 𝑓 ( 𝑥 ) . Output:
The S-Box and inverse S-Box. Calculate
𝐼𝑛𝑑𝑒𝑥 ← 𝑟𝑒𝑠ℎ𝑎𝑝𝑒 ( 𝑥 , , ; Calculate
𝐼𝑛𝑑𝑒𝑥 ← 𝑟𝑒𝑠ℎ𝑎𝑝𝑒 ( 𝑦 , , ; for j=1 to do for i=1 to do 𝑆𝑏 ( 𝑖 ) ← 𝐵𝑖𝑛 𝐷𝑒𝑐 ( 𝑚𝑜𝑑 ( 𝐷𝑒𝑐 𝐵𝑖𝑛 ( 𝐼𝑛𝑑𝑒𝑥 ( 𝑃 ( 𝑖, ⊗𝐷𝑒𝑐 𝐵𝑖𝑛 ( 𝑃 (1 , 𝑗 )) , 𝑓 ( 𝑥 )) ; end end Calculate 𝑆𝐵 ← 𝑟𝑒𝑠ℎ𝑎𝑝𝑒 ( 𝑆𝐵, , ; for j=1 to do for i=1 to do 𝑆 − 𝐵𝑜𝑥 ( 𝑖, 𝑗 ) = 𝐵𝑖𝑛 𝐷𝑒𝑐 ( 𝑚𝑜𝑑 ([ 𝐹 ] ×[ 𝐷𝑒𝑐 𝐵𝑖𝑛 ( 𝑆𝐵 ( 𝑖, 𝑗 )] ) , ⊕𝐷𝑒𝑐 𝐵𝑖𝑛 ( 𝐼𝑛𝑑𝑒𝑥 (16 .𝑖, .𝑗 ))) ; end end Return
S-Box; To calculate inverse S-Box Calculate 𝑖𝑛𝑣 [ 𝐹 ] ← 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 [ 𝐹 ] 𝑜𝑣𝑒𝑟 𝑍 ; for j=1 to do for i=1 to do 𝑖𝑛𝑣. ( 𝑆 − 𝐵𝑜𝑥 )( 𝑖, 𝑗 ) = 𝐵𝑖𝑛 𝐷𝑒𝑐 ( 𝑚𝑜𝑑 ( 𝑖𝑛𝑣. [ 𝐹 ] ×[ 𝐷𝑒𝑐 𝐵𝑖𝑛 ( 𝐼𝑛𝑑𝑒𝑥 (16 .𝑖, .𝑗 )] ) , ⊕𝐷𝑒𝑐 𝐵𝑖𝑛 ( 𝑆𝐵 ( 𝑖, 𝑗 ))) ; end end Return
Inv.(S-Box);
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The elliptic curve has solutions over the Galois field 𝐺𝐹 .Using the obtained solutions, we extract some points 𝑃 ( 𝑥, 𝑦 ) ,which are employed to construct S-Box along with the prim-itive polynomial 𝑓 ( 𝑥 ) and the field matrix 𝐹 , as illustrated inAlgorithm 3. Meanwhile, the field matrix is constructed bythe initial conditions 𝐼𝐶 , and 𝐼𝐶 of the enhanced map (4)and the primitive polynomial 𝑓 ( 𝑥 ) , as shown in Algorithm 2.It is crucial to state here that the field matrix is invertible over 𝑍 , which gives the possibility to generate the inverse S-Boxfor decryption process. An image encryption algorithm has the ability to defeatchosen-plaintext attack when it has an efficient diffusion pro-cess. Therefore, this section introduces a new asymmetricalgorithm based on the field matrix 𝐹 , S-Box, and the hyper-chaotic sequence 𝑦 . The detail diffusion process is describedas followsStep 1 : The pixel values of 𝑆𝑐 𝑚 × 𝑛 are divided into 𝑆𝑐 𝑖𝑘 × 𝑘 , where 𝑘 = 1 , , ..., , and 𝑖 = 1 , , ..., 𝑚 × 𝑛𝐾 .Step 2 : Using the generated hyperchaotic sequence { 𝑦 } by theenhanced map, 𝑆𝑦 ⟵ 𝑐𝑒𝑖𝑙 ( 𝑚𝑜𝑑 ( 𝑦 ×10 , can becalculated.Step 3 : The new scramble image matrix 𝑆𝐶 𝑖𝑘 × 𝑘 is generatedby the field matrix, which can be defined as 𝑆𝐶 𝑖𝑘 × 𝑘 ⟵ 𝐵𝑖𝑛 𝐷𝑒𝑐 ( 𝑚𝑜𝑑 ([ 𝐹 ] × 𝐷𝑒𝑐 𝐵𝑖𝑛 ( 𝑆𝑐 𝑖𝑘 × 𝑘 ( 𝑖 , 𝑗 , 𝑙 )) , ,where 𝑖 , 𝑗 = 1 , , ...𝑘 and 𝑙 = 1 , , ..., 𝑖 .Step 4 : The block cipher 𝐶𝐼 𝑖𝑘 × 𝑘 is obtained by the bitwise XORoperation between 𝑆𝐶 𝑖𝑘 × 𝑘 , S-Box and the sequence 𝑆 𝑦 .Step 5 : The cipher image is obtained by reshaping 𝐶𝐼 𝑖𝑘 × 𝑘 into 𝐶𝐼 𝑚 × 𝑛 as follows 𝐶𝐼 𝑚 × 𝑛 ⟵ 𝑟𝑒𝑠ℎ𝑎𝑝𝑒 ( 𝐶𝐼 𝑖𝑘 × 𝑘 , 𝑚, 𝑛 ) . The receiver section gets the cipher image along withthe initial condition and control parameters of the enhancedmap (4). After that, the receiver constructs the field matrix 𝐹 and S-box, which are considered as public key. Using ellipticcurve, hyperchaotic sequences, and public key, the receivercan construct the inverse field matrix and the inverse S-Box,which represent the private key. Now, the scrambled imagecan be obtained by the inverse of diffusion process. Finally,the plain-image is recovered by the inverse of confusion pro-cess.
5. Simulation results and key analysis
In this section, the proposed image encryption algorithmis simulated to demonstrate its efficiency. Moreover, the ro-bustness of employed key is investigated by calculating itsspace and sensitivity.
To illustrate the ability of the proposed image encryptionalgorithm for ciphering different types of images, Figure 12depicts the encryption results with uniformly distributed his-tograms of various kinds of plain-images including Grey scale,RGB, Sketch, and Hand writing images with the size of ( ). Moreover, this figure shows the encryption of a medi-cal image with the size of (
630 × 630 ) with its histogram. Itcan be seen from these experiment results that the proposedencryption algorithm can effectively encrypt various kindsof images.
Typically, the security key of chaos-based cryptographycontains two main components, which are the initial condi-tions and control parameters of the employed chaotic map.In the proposed encryption algorithm, the parameters andinitial conditions of the enhanced Duffing map (4) employas the main root of public and private keys. Each parameterand initial value takes to decimal places, which meansthat the complexity of each parameter and initial value is . Besides that, the hyperchaotic sequences of the en-hanced map (4) are generated by the parameters and ini-tial conditions for constructing the field matrix and S-box.Therefore, the key space in producing of 𝐹 and S-Box is ( 𝑓 𝑜𝑟 𝑏𝑖𝑡 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 )×2 ( 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 ) = 2 ,and the total key combinations is i.e. the size of key inthis proposed algorithm is bits. Consequently, the se-curity key of the proposed algorithm achieves the standardrequirement [1].Furthermore, the total time to break an encrypted imageis calculated as follows [17] 𝑌 = 𝑇 × 1000 𝐹 𝐿𝑂𝑃 𝑆 × 3153600 , where 𝑌 is the total years to break an encrypted image, and 𝑇 is the total security key space. A super computer has 𝐹 𝐿𝑂𝑃 𝑆 (floating-point operation per second). Therefore,the total time to break the encrypted image by the proposedalgorithm is approximately . years. The employed key of an image encryption scheme is con-sidered as highly sensitive when the encrypted image cannotbe recovered due to a slight difference in one of the key com-ponents. To visually show the key sensitivity of the proposedencryption algorithm, we set the main root of the public andprivate keys, which represent the initial conditions and pa-rameters of the map (4), as 𝐴 = 15 , 𝐵 = 3 . , 𝛽 = 5 , 𝛼 = 10 , 𝐼𝐶 = 0 . , 𝐼𝐶 = 0 . . Subsequently, we change 𝑡ℎ decimal places in the parameters, or initial conditions, orboth to obtain three other keys. Figure 13 demonstrates thekey sensitivity in decryption process with the original keyand the modified keys.
6. Security analysis
The most important indicator to evaluate an image en-cryption algorithm is the security performance of its encrypted
Hayder Natiq et al.:
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50 100 150 200 250 (c) (d)
50 100 150 200 25005001000150020002500 (e) (f)
50 100 150 200 250 (g)
50 100 150 200 25005001000150020002500 (h) (i) (j) (k)
50 100 150 200 25005001000150020002500 (l) (m) (n) (o) (p)
50 100 150 200 250 (q) (r) (s) (t)
50 100 150 200 250 (u) (v)0100020003000 50 100 150 200 250
100 200 300 400 500 (w) (x)
50 100 150 200 250
Figure 12:
Encryption results of different images: the first column depicts the plain-images including animal sketch image,hand writing image, Lena grey-scale image, medical image, animal color image, Barbara color image; the second column depictshistograms of the plain-images; the third column depicts the encryption of the plain-images; the fourth column depicts histogramsof the encrypted images.Hayder Natiq et al.:
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Figure 13:
Key sensitivity analysis: (a) plain-image; (b) the encrypted image; (c) the decrypted with the right key; (d)-(f) thedecrypted images with the wrong keys in which the change are slightly occurred in the parameters and initial conditions, only theparameters, only the initial condition, respectively. image. Therefore, this section introduces an analysis frame-work to investigate the security of the encrypted images bythe proposed algorithm.
Different kinds of noise and data lose can corrupt theencrypted images. Therefore, image encryption algorithmsshould be able to resist these kinds of noise and data lose.The first and second columns of Figure 14 demonstrate thequality results of the recovered image when the correspond-ing encrypted image undergoes Gaussian noise with den-sity, as well as salt and pepper noise with density. It canbe observed that although the encrypted images have noise,the corresponding recovered images contain the most visualinformation of the original images. Besides that, the pro-posed algorithm has successfully recovered the images withPSNR (Peak Signal-to-Noise Ratio) equal to . and MSE(Mean Squre Error) equal to . for adding Gaussiannoise. Meanwhile, the obtained PSNR and MSE for adding of salt and pepper noise are . and , respec-tively. Moreover, the third column of Figure 14 shows thatthe recovered images by the proposed algorithm can be stillrecognizable when the encrypted image has data loss.As a result, the proposed encryption algorithm can resist dif-ferent kids of noise and data loss. A vulnerable encryption scheme can be attacked by ob-serving the change in the encrypted images when small changeor modification are happened in the corresponding plain im-ages. This type of attack is called chosen-plaintext attack ordifferential attack. It is, therefore, the NPCR and UACI testsare usually employed to estimate the ability of encryptionalgorithms for resisting the differential attack. The NPCRindicates to the number of pixel change rate. Meanwhile,the UACI indicates to the unified averaged changed inten-sity. However, these two measures are given as follows
𝑁𝑃 𝐶𝑅 = ∑ 𝑚,𝑛𝑖,𝑗 𝐷 ( 𝑖, 𝑗 ) 𝑚 × 𝑛 × 100 , (7)where 𝐷 ( 𝑖, 𝑗 ) is the change of the pixel values from the plain-image to the encrypted image due to the encryption process,in which 𝐷 ( 𝑖, 𝑗 ) = { 𝑤ℎ𝑒𝑛 𝑃 ( 𝑖, 𝑗 ) = 𝐶𝐼 ( 𝑖, 𝑗 )1 𝑤ℎ𝑒𝑛 𝑃 ( 𝑖, 𝑗 ) ≠ 𝐶𝐼 ( 𝑖, 𝑗 ) . (8)As the pixel values are changed, the UACI can be used todetermine the average intensity of the difference between theoriginal and the encrypted image, where 𝑈 𝐴𝐶𝐼 = 1 𝑚 × 𝑛 𝑚,𝑛 ∑ 𝑖,𝑗 | 𝑃 ( 𝑖, 𝑗 ) − 𝐶𝐼 ( 𝑖, 𝑗 ) |
255 × 100 . (9) Hayder Natiq et al.:
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Figure 14:
Robustness analysis of noise and data loss: (a) and (d) depict the encrypted image with Gaussian noise, and thedecrypted image, respectively; (b) and (e) depict the encrypted image with salt and pepper noise, and the correspondingdecrypted image, respectively; (c) and (f) depict the encrypted image with data loss, and the decrypted image, respectively.
Table 3
The comparison results of NPCR and UACI of different image encryption schemes with ( 𝛼 = 0 . .Index Image size Our scheme LICM[2] ICMIE[3] Zhou[27] Wu[25] Liao[13]NPCR
256 × 256 ≥ .
512 × 512 ≥ . ≥ .
256 × 256 (33 . .
512 × 512 (33 . . . . However, Wu [24] presented a new standard of NPCR andUACI measures for better estimating the ability of an encryp-tion algorithm for resisting differential attack. In this regard,an image encryption algorithm can pass the NPCR test whenits NPCR value is bigger than a level 𝛼 , which is described in the following equation 𝑁 ∗ 𝛼 = 𝜇 𝑁 − Φ −1 ( 𝛼 ) √ 𝐹 ,𝑁 ∗ 𝛼 = 𝐹 − Φ −1 ( 𝛼 ) √ 𝐹 𝐹 + 1 , (10) Hayder Natiq et al.:
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Page 12 of 15everaging social media news (a) X Y (b) X Y (c) X Y (d) (e) X Y (f) X Y (g) X Y (h) Figure 15:
The correlation of the neighboring pixel pairs: the first column depicts the plain-image and the corresponding encryptedimage; the second column depicts the horizontal direction of plain-image and the encrypted image; the third column depicts thevertical direction of the plain-image and the encrypted image; the fourth column depicts the diagonal direction of the plain-imageand the encrypted image. where Φ −1 is the inverse CDF of the standard Normal dis-tribution 𝑁 (0 , , and 𝐹 is the largest supported pixel valuecompatible with the cipher text image format. Meanwhile,an encryption algorithm is successful exceed the UACI testwhen the simulation value is in the range of 𝑈 𝐴𝐶𝐼 ∈ ( 𝑈 − 𝛼 , 𝑈 + 𝛼 ) .Here, 𝑈 − 𝛼 and 𝑈 + 𝛼 are given by 𝑈 − 𝛼 = 𝜇 𝑈 − Φ −1 ( 𝛼 ∕2) 𝜎 𝑈 ,𝑈 + 𝛼 = 𝜇 𝑈 + Φ −1 ( 𝛼 ∕2) 𝜎 𝑈 , (11)where 𝜎 𝑈 = √ ( 𝐹 + 2)( 𝐹 + 2 𝐹 + 3)18( 𝐹 + 1) × 255 𝐹 ,𝜇 𝑈 = 𝐹 + 23 𝐹 + 3 . (12)In this test, for each plaint image, namely, 𝑃 𝐼 , we gen-erate another image, namely, 𝑃 𝐼 by selecting a pixel from 𝑃 𝐼 and changing its value by 1-bit. Subsequently, the UPCRand UACI values can be calculated by generating the en-crypted images of both 𝑃 𝐼 and 𝑃 𝐼 . The UPCR and UACIresults of different types of images, which have been en-crypted by the proposed algorithm and several other schemes,are illustrated in Table 3. It can be seen from these resultsthat the proposed algorithm has superior or competitive per-formance in defending the differential attack. The adjacent pixels of the original image are highly cor-related along the vertical, diagonal, and horizontal direc-tions. Thus, an efficient encryption algorithm can resist astatistical attack when the adjacent pixels of its encryptedimage is nearly zero. To calculate the pixels correlation, letus first define the covariance between a pair of pixel values 𝑥 and 𝑦 , which is given by 𝐶𝑜𝑣 ( 𝑥, 𝑦 ) = 𝐸 [( 𝑥 − 𝐸 ( 𝑥 ))( 𝑦 − 𝐸 ( 𝑦 ))] , (13)where 𝐸 ( 𝑥 ) and 𝐸 ( 𝑦 ) are the means. Now, the correlationcoefficients can be calculated by 𝜌 𝑥𝑦 = 𝐶𝑜𝑣 ( 𝑥, 𝑦 ) 𝜎 ( 𝑥 ) 𝜎 ( 𝑦 ) , 𝜎 ( 𝑥 ) , 𝜎 ( 𝑦 ) ≠ , (14)where 𝜎 ( 𝑥 ) and 𝜎 ( 𝑦 ) are the standard deviations of the dis-tribution of the pixel.In our experiment, adjacent pixels in horizontal, verti-cal, and diagonal directions are randomly chosen from bothplaint and encrypted images, as shown in 15. As can be seen,most of the pixels are close to the diagonal line of axis forthe plain image. Meanwhile, the pixels of the encrypted im-age distribute randomly on the whole space. Furthermore,quantitative and comparison results of adjacent pixels corre-lations of Lena image, which has encrypted by the proposedencryption algorithm and other existing schemes, are illus-trated in Table 4. Clearly, the 𝜌 𝑥𝑦 values of our scheme are Hayder Natiq et al.:
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Table 4
The comparison results of the correlation coefficients of the plain-image and the encryptedimage by various methods.Index Lena image Our scheme LICM[2] ICMIE[3] Zhou[27] Xu[26] Liao[13]Horizontal 0.971921627 -0.0009 0.0019 -0.0008 0.0102 0.0230 0.0127Vertical 0.9865777 0.0015 0.0012 -0.0013 -0.0053 0.0019 -0.0190Diagonal 0.96064343 -0.0010 0.0009 0.0018 -0.0161 0.0034 -0.0012
Table 5
The local Shannon entropy analysis with 𝛼 = 0 . 𝑘 = 30 , 𝑇 𝐵 = 1936 of images which arecollected from USC-SIPI Miscellaneous data-set. Image name Our scheme Wu [25] Zhou[27] Liao [13] ICMIE[3] LICM[2]5.1.09 7.902212 7.901985 7.903354 7.904191 7.902710 7.902385.1.10 7.901902 7.902731 7.902443 7.902371 7.902473 7.902645.1.11 7.902425 7.902446 7.902756 7.900799 7.902217 7.902335.1.12 7.902481 7.902556 7.901526 7.903374 7.903208 7.903255.1.13 7.902075 7.902688 7.945630 7.904566 7.902951 7.901925.1.14 7.902918 7.903474 7.902945 7.903111 7.901577 7.901765.2.08 7.903094 7.903953 7.902356 7.901762 7.902681 7.903015.2.09 7.902541 7.902233 7.899853 7.905854 7.902571 7.902595.2.10 7.902029 7.900714 7.902654 7.902768 7.902411 7.902995.3.01 7.902361 7.902727 7.902647 7.901040 7.903408 7.903985.3.02 7.903230 7.903182 7.900474 7.900981 7.903093 7.904217.1.01 7.901931 7.902173 7.902634 7.902145 7.901900 7.902227.1.02 7.902419 7.900879 7.901634 7.902157 7.903003 7.902247.1.03 7.902170 7.902543 7.905423 7.900645 7.902116 7.902797.1.04 7.903219 7.901126 7.902125 7.904141 7.902998 7.902537.1.05 7.902091 7.903579 7.883653 7.900027 7.903154 7.902607.1.06 7.902850 7.901930 7.902356 7.901736 7.902009 7.902317.1.07 7.902258 7.903000 7.902364 7.900802 7.903176 7.902567.1.08 7.902022 7.903197 7.904456 7.900944 7.902837 7.902517.1.09 7.902255 7.902308 7.903012 7.905658 7.902068 7.901957.1.10 7.902032 7.899542 7.901598 7.893848 7.903141 7.903227.2.01 7.902038 7.902772 7.901989 7.904525 7.902316 7.90325Boat.512 7.901863 7.901908 7.901879 7.900712 7.901920 7.02156Gray21.512 7.902807 7.900170 7.905107 7.902149 7.903359 7.90275House 7.90228701 7.903580 7.904581 7.902156 7.902568 7.90569Ruler.512 7.901977 7.903265 7.903001 7.901428 7.901889 7.90256Numbers.512 7.7292 7.903615 7.892351 7.903579 7.903379 7.90389Mean 7.902391 7.902381 7.903141 7.902128 7.902635 7.870211Pass Rate 27/27 17/27 18/27 10/27 27/27 22/27h 𝑙𝑒𝑓𝑡 = 𝑟𝑖𝑔ℎ𝑡 = more closer to and superior and competitive than those ofother schemes. The Local Shannon entropy, which quantitatively mea-sures the distribution of information, is used to estimated therandomness of an encrypted image. Mathematically, it is de-fined as 𝐻 𝑘,𝑇 𝐵 = 𝑘 ∑ 𝑖 =1 𝐻 ( 𝑠 𝑖 ) 𝑘 (15) where 𝑠 , 𝑠 , ..., 𝑠 𝑘 are 𝑘 selected blocks with 𝑇 𝐵 pixels of achosen image. If 𝐻 𝑘,𝑇 𝐵 is in the interval of ( ℎ 𝑙𝑒𝑓𝑡 , ℎ 𝑟𝑖𝑔ℎ𝑡 ) ,then the cipher text image will be considered as passing thetest. With respect to the 𝛼 level of significance in a 𝑍 − 𝑡𝑒𝑠𝑡 ,we calculate critical values ℎ 𝑙𝑒𝑓𝑡 and ℎ 𝑟𝑖𝑔ℎ𝑡 as follows ℎ 𝑙𝑒𝑓𝑡 = 𝜇 𝐻 𝑘,𝑇𝐵 − Φ −1 𝛼 ∕2 𝜎 𝐻 𝑘,𝑇𝐵 ℎ 𝑟𝑖𝑔ℎ𝑡 = 𝜇 𝐻 𝑘,𝑇𝐵 + Φ −1 𝛼 ∕2 𝜎 𝐻 𝑘,𝑇𝐵 (16)where Φ −1 is the inverse cumulative density function of thestandard normal distribution 𝑁 (0 , and 𝜇 𝐻 𝑘,𝑇𝐵 , and 𝜎 𝐻 𝑘,𝑇𝐵 Hayder Natiq et al.:
Preprint submitted to Elsevier
Page 14 of 15everaging social media news be the mean and variance of Local Shannon entropy calcu-lated in 𝑘 non overlapping blocks.In this test, we select 27 different images from USC-SIPIMiscellaneous dataset, and then encrypt these images by var-ious image encryption algorithms. Table5 lists Local Shan-non entropy results of the encrypted image. As can be seen,the results demonstrate that the proposed algorithm success-ful exceed the test.
7. Conclusion
In summary, this paper has demonstrated that low di-mensional discrete chaotic systems can provide complicatedmultistability behavior. As an example, we have shown theoccurrence of coexisting chaotic attractor with periodic or-bits as well as coexisting two chaotic attractors in the 2DDuffing map. Our finding can rise suspicion of employingexisting chaotic maps in cryptography applications. There-fore, we have introduced the Sine-Cosine chaotification tech-nique to enhance chaos complexity in the multistability re-gions of the 2D Duffing map. The proposed chaotificationtechnique can be easily generalized to other low-dimensionalchaotic maps. To illustrate its efficiency, several performanceevaluations including trajectory, Lyapunov exponents, bifur-cations, FIPS 140-2 test, and Sample entropy have demon-strated that the enhanced Duffing map exhibits a wide hyper-chaotic range, high randomness and extreme unpredictablebehavior. Besides that, its hyperchaotic sequences distributein a large area in the 2D phase space without exhibiting pe-riodic behaviors. Consequently, the enhanced Duffing mapcould be a better choice than other existing chaotic mapsfor cryptography applications. As an application, we furtherproposed a asymmetric image encryption algorithm, whichachieves the confusion and diffusion processes by hyperchaoticsequences, elliptic curve, and S-box. Simulations resultshave demonstrated that the proposed encryption algorithmgives the users a flexibility to encrypt several kinds of im-ages such as Grey scale, Medical, and RGB images with ahigher level of security.
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