Image Encryption Using Chebyshev Map and Rotation Equation

We propose a novel image encryption algorithm based on two pseudorandom bit generators: Chebyshev map based and rotation equation based. The first is used for permutation, and the second one for substitution operations. Detailed security analysis has been provided on the novel image encryption algorithm using visual testing, key space evaluation, histogram analysis, information entropy calculation, correlation coefficient analysis, differential analysis, key sensitivity test, and computational and complexity analysis. Based on the theoretical and empirical results the novel image encryption scheme demonstrates an excellent level of security.


Introduction
Over the past twenty years, the nonlinear chaotic systems have been ordinarily used in digital data encryption and transmission.In the imaginative work [1] of J. Fridrich is shown the good potential of the dynamical chaotic maps in symmetric image encryption.The paper highlights how to adapt nonlinear Baker map, Cat map and Standard map on a torus or on a rectangle in order of block encryption schemes.
An improved stochastic middle multi-bits quantification scheme based on Chebyshev map is proposed in [2].Novel image encryption scheme with Chebyshev map based diffusion operations is presented in [3].A novel design method of key stream by chaotic Chebyshev function is proposed in [4].In [5], a secure Diffie-Hellman key agreement protocol based on Chebyshev chaotic map is presented.
In [6], a chaotic cipher is proposed to encrypt color images through position permutation part and Logistic map based on substitution.By using Chebyshev map and Arnold map, a bit-level permutation image encryption algorithm is proposed [7].
In [8], based on the Lorenz attractor and perceptron algorithm, a chaotic image encryption system is proposed.an image encryption scheme using dynamic sequences generated by multiple chaotic systems is presented in [9].In [10], a bit-level permutation and Chen chaotic system are proposed to encrypt color images.
A chaos based image encryption scheme is proposed in this article.The novel algorithm is based on a simple multiple round substitution-permutation model.It is Chebyshev map based on permutation and rotation function based on substitution with motivation to maintain the high quality of the encrypted images.The novelty of our approach lies in the combination of two cryptographically strong pseudorandom generators.
In Section 2, we propose novel pseudo-random bit generator (PRBG) based on rotation function.In Section 2.2 in order to measure randomness of the bit sequence generated by the pseudo-random scheme, we use NIST, DIEHARD and ENT statistical packages.Section 4 presents the novel image encryption algorithm, and extended security cryptanalysis is given.Finally, the last section concludes the paper.

Proposed Pseudorandom Scheme
In this section, one real number of rotation formula is preprocessed to a binary pseudo-random sequence.
We are using rotation equations of the form [11,12] x t+1 = −a − (x t − a) cos θ + y t sin θ/r t y t+1 = −x t r t sin θ − y t cos θ r t = 0.5(x 2 t + x 4 t + 4y 2 t ), where the parameters are θ = 2 and a = 2.8.The rotation Equation (1) with initial conditions x 0 = 0.5, y 0 = 1.0 is graphed in Figure 1.This figure visually shows random-like positions of the points in the set.
The novel pseudorandom bit generation scheme consists of the following steps: Step 1: The initial values x 0 and y 0 from Equation (1) are determined.
Step 2: The rotation equation, Equation (1), is iterated for L 0 times, where L 0 is a constant.
Step 3: The iteration of the Equation (1) continues, and as a result, two decimal fractions x i and y i are generated.
Step 4: The number y i is post-processed as follows: where mod(x, y) returns the reminder after division, abs(x) returns the absolute value of x, and integer(x) returns the integer part of x, truncating the value at the decimal point.
The output bit s i is obtained.
Step 5: Return to Step 3 until pseudo-random bit stream limit is reached.
The rotation equations based pseudo-random bit scheme is implemented softwarely in C++ programming language, using the following initial values: x 0 = 0.2343214592, y 0 = −0.742190593,and L 0 = 140.

Statistical Test Analysis of the Pseudorandom Bit Generator Based on Rotation Equations
In order to measure randomness of the rotation equation based pseudo-random bit generator, we used NIST [13], DIEHARD [14], and ENT [15] statistical test suites.
Using the novel pseudo-random bit generator were produced 1000 sequences of 1,000,000 bits.The results from all tests are given in Table 1.The entire NIST test is passed successfully: all the p − values are distributed uniformly in the 10 subintervals and the pass rate is also in acceptable range.
The minimum pass rate for each statistical test with the exception of the random-excursion (variant) test is approximately 980 for a sample size of 1000 zero-one sequences.The minimum pass rate for the random excursion (variant) test is approximately 605 for a sample size of 619 binary sequences for Rotation equations based pseudorandom bit generator.The proposed scheme possesses random-like properties.
For the DIEHARD tests, we generated a file with 80 million bits from the proposed pseudorandom bit generator.The results are placed in Table 2.All P-values are in acceptable range of [0, 1).We tested the output of a string of 125,000,000 bytes of the proposed Rotation equations based pseudorandom bit generation scheme.The results are summarized in Table 3.The proposed pseudorandom bit generator passed all the tests of ENT.

Pseudo-Random Bit Generator Based on the Chebyshev Map
In this section we will describe the Chebyshev map [16,17] based pseudorandom bit generator proposed in [18].
In [19], a pseudorandom bit stream is generated by comparing the outputs of two Chebyshev maps.In [20], the real numbers of two Chebyshev polynomials are post-processed and combined with a simple threshold function to a binary pseudorandom stream.The described scheme modify the generators in [19] and [20] by simple avoiding the threshold functions and speed up the bit extracting process with increasing the number of chaotic maps.The scheme is based on the following four Chebyshev maps: where T 1 (x 1 ), T 1 (x 2 ), T 1 (x 3 ), and T 1 (x 4 ) are the initial values.The modified algorithm consists of the following steps: Step 1: The initial values T 1 (x 1 ), T 1 (x 2 ), T 1 (x 3 ), and T 1 (x 4 ) of the four Chebyshev maps from Equation (3) are determined.
Step 2: The four Chebyshev maps from Equation (3) are iterated for K, L, M and N times, respectively, to avoid the harmful effects of transitional procedures.
Step 3: The iteration of the Equation (3) continues, and as a result, four real fractions T i (x 1 ), T j (x 2 ), T k (x 3 ), and T l (x 4 ), are generated and post-processed as follows: s m 1 = mod(integer(abs(T i (x 1 ) × 10 15 )), 2) s m 2 = mod(integer(abs(T j (x 2 ) × 10 15 )), 2) where abs(x) returns the absolute value of x, integer(x) returns the integer part of x, truncating the value at the decimal point, and mod(x, y) returns the reminder after division.The four output bits s m 1 , s m 2 , s m 3 , and s m 4 are obtained.
Step 4: Return to Step 3 until the bit stream limit is reached.
The algorithm has good statistical properties measured by NIST, DIEHARD and ENT test packages.

Image Encryption Algorithm Based on Chebyshev Map and Rotation Equation
Here we describe an image encryption algorithm based on the proposed rotation equation based pseudo-random bit generator, Section 2, and Chebyshev map based pseudorandom bit generator [18].We also present security analysis of the encrypted images.

Encryption Algorithm
The novel image encryption algorithm is a simple modification of the substitution-permutation scheme [1].Here every single pixel encryption is based on pixel shuffling and pixel substitution, and on multiple overall rounds.
We consider plain images of n × n size.The binary length of n is n 0 .The encryption process is divided in two parts.In the first part we generate buffer image B of n × n size by relocating the pixels of the plain image P by Chebyshev map based PRBG.In the second part we generate ciphered image C of n × n size by transforming the buffer pixel values by rotation equation based PRBG.The image encryption begins, with an empty buffer image.The entire encryption process is given below: Step 1: The Chebyshev map based PRBG is iterated six times to produce 24 bits.These bits constitute a binary number b j .
Step 2: The j th column-vector is circularly shifted for mod (b j ) times.
Step 3: Repeating Steps 1-2 until all of the columns, j = 1 . . .n, are processed, and the buffered image B is produced.
Step 4: The rotation equation based PRBG is iterated to produce n × n × 8 bits for a monochrome image or n × n × 24 bits for a color image.
Step 5: Do XOR operation between the pseudo-random bit sequence and all of the buffer pixels in the buffered image to produce the encrypted image C ′ .
Step 6: Repeating Steps 1-5 for T ≥ 1 times, until encrypted image C is produced.

Security Analysis
The proposed image encryption algorithm is implemented in C++ programming language.All statistical results presented have been taken by T = 1.
Twenty eight 8-bit monochrome images and sixteen 24-bit color images have been encrypted for the statistical analysis.

Key Space Analysis
The set of all initial numbers compose the key space.The key space of the novel image encryption scheme has six secret key values x 0 = 0.2343214591, y 0 = −0.742190593, T 1 (x 1 ) = 0.702938119500914, T 1 (x 2 ) = −0.3001928364928377,T 1 (x 3 ) = 0.1385946382912478, and T 1 (x 4 ) = −0.2871955600387584,fixed as key K1.As stated in IEEE floating-point standard [21], the computational precision of the 64-bit double-precision number is about 10 −15 .The key space of the novel scheme is (10 15 ) 6 = 10 90 ≈ 2 298 .Furthermore, the initial iteration numbers K, L, M and N can also be used as a part of the key size.
Compared with similar image encryption algorithms [22][23][24][25][26][27], and [28] the proposed one has enough key space size, Table 4.The larger parameter space is based on the proposed combination of two different pseudorandom bit generators.

Visual Testing
The novel image encryption scheme is tested by using visual review.The inspection does not detect analogy between plain images and their corresponding encrypted images.As an example, Figure 2 shows the image 4.2.06Sailboat on lake, Figure 2a, and its encrypted version, Figure 2b.The encrypted image doesn't keep any segmented color clusters and source figures.

Histogram Analysis
Histogram analysis of three channels (red, green, and blue) of the plain and encrypted images is given.Figure 3  For mathematical quantity analysis of red, green and blue channels of the encrypted image 4.1.08Jelly beans, we employed Kolmogorov-Smirnov Goodness-of-Fit Test to evaluate an uniformity.The normality was detected with Normal Quantile Plots (NormQuant.xls,Dr Scott Guth at Mt San Antonio College).The Fugure 4 shows the the Normal quantile plots.The obtained correlation coefficients of 0.999198758 (red histogram), 0.998107709 (green histogram), and 0.997868951 (blue histogram) are larger than critical values 0.989519603 (α = 0.01), 0.995880763 (α = 0.05), and 0.99710484 (α = 0.1).Therefore, the data follow the normal distribution.Similar results of uniformity are obtained in [29].In addition of histogram analysis, the results of the average pixel intensity are given in Tables 5 and 6.They validate the uniformity in distribution of red, green and blue channels of the encrypted images.The information entropy H(X) is a statistical measure of uncertainty in communication theory [30].It is defined as follows: where X is a discrete random variable, p(x i ) is the probability density function of the occurrence of the symbol x i .Let us consider that there are 256 values of the information source in red, green, blue, and grey colors of the image with the same probability.We can get the perfect entropy H(X) = 8, corresponding to a truly random sample.The information entropy of red, green, blue and grey colors of the plain and their corresponding encrypted images are computed and displayed in Tables 7 and 8. From the obtained results it is clear that the entropies of red, green, blue, and grey colors of the encrypted images are very close to the ideal value, which is an indication that the new chaos-based image encryption scheme is secure and credible upon information entropy attempt.In addition, Table 9 summarizes the information entropy values for Lena and Peppers encrypted images compared with values in [7,[31][32][33].As we can see that although they are all close to the ideal entropy value, the results of the novel algorithm are larger than those of corresponding methods.

Correlation Coefficient Analysis
Because of the existing correlation either in horizontal, vertical, or diagonal direction of the plain image pixels, the correlation coefficient r between two adjacent pixels (a i , b i ) is computed [35]. where M is the total number of couples (a i , b i ), obtained from the image, and ā, b are the mean values of a i and b i , respectively.Correlation coefficient r can range in the interval of [−1.00, +1.00].
Table 10 shows the results of horizontal, vertical, and diagonal adjacent pixels correlation coefficients computations of the plain images and the corresponding encrypted images.It is clear that the novel chaos based image encryption scheme does not keep any linear dependencies between observed pixels in all three directions: the inspected horizontal, vertical and diagonal correlation coefficients of the encrypted images are close to 0. Overall, the correlation coefficients of the proposed algorithm are analogous with results of four other image encryption schemes [10,25,[27][28][29], Table 11.Proposed scheme -0.003761 0.001775 0.000686 Reference [25] 0.019732 0.002467 0.004438 Reference [10] -0.0574 -0.0035 0.0578 Reference [29] 0.0004 0.0021 -0.0038 Reference [27] 0.001354 -0.000254 -0.000327 Reference [28] 0.002016 -0.000916 0.001651 In addition, correlation coefficients between the corresponding pixel of the plain and their encrypted images are given in Table 13 (Columns 1, 2, 3, and 4).The computed correlation values are very close to 0.00.

Differential Attack
In general, a common characteristic of an image encryption scheme is to be sensitive to minor modifications in the plain images.Differential analysis allows that an adversary is able to create small changes in the plain image and revise the encrypted image.The alternation level can be computed by means of two formulae, namely, the number of pixels change rate (NPCR) and the unified average changing intensity (UACI) [35,36].
Let us assume encrypted images before and after one pixel modification in a plain image are C 1 and C 2 .The NPCR and UACI are defined as follows: where D is a two-dimensional set, having the same size as image C 1 or C 2 , and W and H are respectively the width and height of the image.The set D(i, j) is defined by C 1 (i, j) and C 2 (i, j), if C 1 (i, j) = C 2 (i, j) then D(i, j) = 1; otherwise, D(i, j) = 0.The NPCR and UACI test results from the proposed chaos based algorithm are shown in Tables 12 and 15 (Columns 2 and 3).The obtained NPCR and UACI values for all of the images are larger than the critical values proposed in [36] and similar to the values presented in [10].The values point out that the novel image encryption algorithm is vastly sensitive regarding to small changes in the plain images and has a vigorous strength of contrary differential cryptanalysis.

Key Sensitivity Test
The strong key sensitivity is another characteristic of the correlation analysis.a good image encryption scheme should be sensitive regarding the secret key i.e. a negligible change of the secret key.We encrypted the 48 images with two similar secret keys: K1 and K2 (x 0 = 0.2343214592, y 0 = −0.742190593,T 1 (x 1 ) = 0.7029381194009314, T 1 (x 2 ) = −0.3001928364928377,T 1 (x 3 ) = 0.1385946382912478, and T 1 (x 4 ) = −0.2871955600387584).The results are shown in Table 13 (Columns 5, 6, 7, and 8).It is clear that the novel image encryption method is strong key sensitive: the correlation coefficients are relatively close to zero.Moreover, another round of the NPCR and UACI tests are established.In this case, C 1 and C 2 are two encrypted images, obtained from one plain image by the novel encryption scheme using the keys K1 and K2.The results are displayed in Tables 14 and 15 (Columns 4 and 5).In addition, in Figure 5, the results of two tests are shown to decrypt the encrypted with key K1 4.1.07and 7.1.03images, Figure 5e and Figure 5f, with the secret key K2.
We observed that the two decrypted images Figure 5e and Figure 5f have no relation with the plain images 4.2.07 and 7.1.03,Figure 5a and Figure 5b, respectively.

Computational and Complexity Analysis
We have measured the average encryption time for 512 × 512 sized grayscale and color images by using the proposed image encryption algorithm.Computational analysis has been done on a 2.40 GHz Intel R Core TM i7-3630QM Dell Inspiron laptop.The results are provided in Table 16.One can see from Table 16 that the novel image encryption algorithm runs slower only than algorithm in Reference [28].The novel algorithm needs Θ(n 2 ) of pixel shuffling iterations.For analysis of theoretical complexity in substitutions, the time-consuming parts in computations are Θ(n 2 ) iterations of calculations of a sine and a cosine functions.Therefore, the proposed encryption scheme needs more theoretical time than the algorithms in [27,28].

Conclusions
A novel image encryption scheme based on the theory of chaos is proposed in this communication.The suggested technique combines Chebyshev polynomial based permutation, and rotation equation based substitution.a strict security analysis on the novel scheme is given.
Detailed security analysis has been provided on the novel image encryption algorithm using visual testing, key space evaluation, histogram analysis, information entropy calculation, correlation coefficient analysis, differential analysis, key sensitivity test, and computational and complexity analysis.
Based on the obtained results, we can conclude that the proposed chaos based image encryption algorithm is reasonable for the secure image communication.

Figure 2 .
Figure 2. Comparison of the plain image and the encrypted image: (a) original picture 4.2.06Sailboat on lake; (b) encrypted image of 4.2.06Sailboat on lake.

Figure 3 .
Figure 3. Histogram analysis of plain image and encrypted image: (a), (c), and (e) show the histograms of red, green, and blue channels of the plain picture 4.1.08Jelly beans; (b), (d), and (f) show the histograms of red, green, and blue channels of encrypted picture 4.1.08Jelly beans.

Figure 4 .
Figure 4. Kolmogorov-Smirnov test of the histograms of the encrypted image 4.1.08Jelly beans: (a), (b), and (c) show the Normal quantile plots of the red, green, and blue histograms.

Figure 5 .
Figure 5. Key sensitive analysis of the plain images 4.2.07 and 7.1.03,(a) and (b), encrypted with the key K1, (c) and (d), and decrypted with the key K2, (e) and (f).

Table 1 .
NIST test suite results.

Table 3 .
ENT statistical test results.

Table 5 .
Average pixel intensity of 24-bit color plain images and encrypted images.

Table 6 .
Average pixel intensity of 8-bit grayscale plain images and encrypted images.

Table 7 .
Entropy results of 24-bit plain images and encrypted images.

Table 8 .
Entropy results of 8-bit plain images and encrypted images.

Table 10 .
Horizontal, vertical and diagonal correlation coefficients of adjacent pixels in the plain and encrypted images.

Table 11 .
Horizontal, vertical and diagonal correlation coefficients comparisons.

Table 12 .
NPCR and UACI results of encrypted 24-bit plain images and encrypted with one pixel difference 24-bit plain images.

Table 13 .
Correlation coefficients between the corresponding pixels of the plain and encrypted images-Columns 1, 2, 3, and 4. Correlation coefficients between the corresponding pixels of the encrypted images with keys K1 and K2-Columns 5, 6, 7, and 8.

Table 14 .
NPCR and UACI results of encrypted 24-bit plain images with keys K1 and K2.

Table 15 .
NPCR and UACI results of encrypted 8-bit plain images and encrypted with one pixel difference 8-bit plane images-Column 2 and Column 3; NPCR and UACI results of encrypted 8-bit plain images with keys K1 and K2-Column 4 and Column 5.