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Image Encryption Technology Based on Fractional Two-Dimensional Triangle Function Combination Discrete Chaotic Map Coupled with Menezes-Vanstone Elliptic Curve Cryptosystem.

1. Introduction

Nowadays, image encryption plays a significant role with the development of security technology in the areas of network, communication, and cloud service. Multifarious chaos-based image encryption algorithms have been developed up to now, such as in [1-6]; however a few of them have referred to the image encryption algorithm based on fractional discrete chaotic map accompanied with Elliptic Curve Cryptography (ECC).

The theory of the fractional difference has been developed for decades [7-13]. Recently, Wu et al. [14-16] made a contribution to the application of the discrete fractional calculus (DFC) on an arbitrary time scale, and the theories of delta difference equations were utilized to reveal the discrete chaos behavior.

ECC is a widely used technology in data security and communication security; it can achieve the same level of security with smaller key sizes and higher computational efficiency [17]. Many famous public-key algorithms, such as Diffie-Hellman, EIGamal, and Schnorr, can be implemented by means of elliptic curves over finite fields. MVECC is one of the popular elliptic curve public-key cryptosystems [18] and we adopt it in our cryptosystem.

Many encryption methods based on fractional derivatives have been proposed in recent time, like fractional logistic maps [19], fractional-order chaos systems [20], and fractional form of hyperchaotic system [21].

In [22], a new image encryption algorithm based on one-dimensional fractional chaotic time series within fractional-order difference has been proposed; however, the two-dimensional discrete chaotic map has seldom been used in image encryption except [23, 24].

Our main purpose is to introduce a new two-dimensional discrete chaotic map based on fractional-order difference and apply it in image encryption. The rest of this paper is organized as follows. In Section 2, the definitions and the properties of the DFC are introduced. After that, the definitions and operation of ECC are given. Then, the working principle of MVECC is described in the next section. In Section 5, we give the fractional 2D-TFCDM on time scales from the discrete integral expression. The bifurcation diagrams and the phase portraits of the map are presented while the difference orders and the coefficients are changing; the largest Lyapunov exponent plots are also displayed. Afterwards, we apply the proposed map into image encryption and show several examples. In Section 7, the performance of the proposed image encryption method is analysed systematically. Finally, we have come to some conclusions.

2. Preliminaries

The definitions of the fractional sum and difference are given as follows. Let [N.sub.a] denote the isolated time scale and [N.sub.a] = {a, a + 1, a + 2, ...} (a [member of] R fixed). Within the DFC, the function f(t) is changed as a sequence f(n). The difference operator [DELTA] is defined as [DELTA]f(n) = f(n + 1) - f(n).

Definition 1 (see [25]). Let u: [N.sub.a] [right arrow] R and 0 < v be given. The vth fractional sum is defined by

[[DELTA].sup.-v.sub.a]u(t) = [1/[GAMMA](v)] [t-v.summation over (s=a)] [(t - s - i).sup.v-1]u(s), t [member of] [N.sub.a+v]. (1)

Note that a is the starting point; [t.sup.(v)] is the falling function defined as

[t.sup.(v)] = [[[GAMMA](t + 1)]/[[GAMMA](t + 1 - v)]]. (2)

Definition 2 (see [26]). For 0 < v, v [not member of] N, and u(t) defined on [N.sub.a], the v-order Caputo fractional difference is defined by

[mathematical expression not reproducible]. (3)

Theorem 3 (see [27]). For the delta fractional difference equation

[sup.C][[DELTA].sup.v.sub.a]u(t) = f(t + v - 1, u(t + v - 1)), [[DELTA].sup.k]u(a) = [u.sub.k], m = [v] + 1, k = 0, ..., m - 1 (4)

the equivalent discrete integral equation is

x(n) = [u.sub.0](t) + [1/[GAMMA](v)] [t-v.summation over (s=a+m-v)] [(t - s - 1).sup.(v-1)] x f(s + v - 1, u(s + v - 1)), t [member of] [N.sub.a+m], (5)

where

[u.sub.0] (t) = [m-1.summation over (k=0)][[(t - a).sup.(k)]/k!][[DELTA].sup.k]u(a). (6)

The complex difference equation with long-term memory is obtained here. It can reduce to the integer order one with the difference order v = 1 but the integer one does not hold the discrete memory. From (3) to (5), the domain is shifted from [N.sub.a+m-v] to [N.sub.a+m] and the function u(t) is preserved to be defined on the isolated time scale [N.sub.a] in the fractional sums.

3. Introduction to Elliptic Curve

Definition 4. An elliptic curve (EC) E over a prime field [F.sub.p] denoted by E([F.sub.p]) refers to the set of all points (x, y) that satisfy the equation

E : [y.sup.2] [equivalent to] [x.sup.3] + ax + b (mod p), (7)

together with a special point O at infinity, where a, b [member of] [F.sub.p], p [not equal to] 2, 3 and 4[a.sup.3] + 27[b.sup.2] [not equal to] 0 [28, 29].

3.1. Elliptic Curve Operations. If P = ([x.sub.1], [y.sub.1]), Q = ([x.sub.2], [y.sub.2]) [member of] E([F.sub.p]); then if [x.sub.1] = [x.sub.2] but [y.sub.1] [not equal to] [y.sub.2], P + Q = O; that is, Q = - P = ([x.sub.1], - [y.sub.1]) [29].

[mathematical expression not reproducible], (8)

where

[mathematical expression not reproducible]. (9)

The scalar multiplication over E([F.sub.p]) is defined by

[mathematical expression not reproducible], (10)

where k is an integer.

Definition 5. The order of an EC is defined by the number of points that lie on the EC denoted by #E [29].

Definition 6. Set P [member of] E([F.sub.p]), and then P is called a generator point if ord(P) = #E (ord(P) is the smallest positive integer n that makes nP = O) [29].

4. Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC)

MVECC is one of most significant extensions of ECC; the working principle of MVECC is as follows.

If Andy wants to encrypt and send the message M to Bob, they should do the step as mentioned hereunder:

(1) Andy and Bob make an agreement on an elliptic curve E([F.sub.p]) and the base point [alpha].

(2) Bob firstly selects a private key k to compute the public Key [gamma] = k x [alpha](0 [??] k < or d([alpha])).

(3) If Andy wants to send a message M = ([x.sub.1], [x.sub.2]) [member of] [Z.sup.*.sub.p] x [Z.sup.*.sub.p] to Bob, he firstly chooses a random private key d (0 [??] d < ord([alpha])) and then computes his public key [beta] = d x [alpha]. On the other hand, Andy calculates the secret key ([c.sub.1], [c.sub.2]) by

([c.sub.1], [c.sub.2]) = d x [gamma] = d x k x [alpha] = k x [beta]. (11)

He should compute the ciphered message afterwards by

[y.sub.1] = [x.sub.1] x [c.sub.1] mod p, [y.sub.2] = [x.sub.2] x [c.sub.2] mod p. (12)

(4) Then the ciphertext {[gamma], ([y.sub.1], [y.sub.2])} is sent to Bob. When Bob wants to get the plaintext ([x.sub.1], [x.sub.2]), firstly, he computes the secret key ([c.sub.1], [c.sub.2]) = k x [beta] = k x d x [alpha], and then he computes M = ([x.sub.1], [x.sub.2]) by

[x.sub.1] = [y.sub.1] x [c.sup.-1.sub.1] mod p, [x.sub.2] = [y.sub.2] x [c.sup.-1.sub.2] mod p, (13)

to get the plaintext [18].

Any adversary that only has [beta] and [gamma] without the private keys d and k very difficultly breaks the MVECC to get the plaintext M. What is more, if #E have only one big prime divisor, the EC is called a safe EC [29]; then, the MVECC can become an more efficient and secure cryptosystem.

5. Fractional 2D-TFCDM

From [14-16], we notice the application of the DFC in fractional generalizations of the discrete chaotic maps. Recently [30], the following 2D-TFCDM was proposed:

[x.sub.n+1] = [k.sub.1] cos ([x.sub.n] + [y.sub.n]), [k.sub.1] = 8 [y.sub.n+1] = [k.sub.2] sin ([x.sub.n] - [y.sub.n])> [k.sub.2] = 0.5. (14)

Now, consider the fractional generalization of x(n); the map was also studied in [31]:

[mathematical expression not reproducible]. (15)

From Theorem 3, we have the following equivalent discrete numerical formula for the variable [k.sub.1]: ([k.sub.2] = 0.5) with 0 < v < 1:

[mathematical expression not reproducible]. (16)

Let v = 1, x(0) = 0.19, y(0) = 0.06, n = 200, and [k.sub.1] be fixed. In what follows, Figure 1 is the bifurcation diagram where the step size of [k.sub.1] is 0.002. Figure 2 is the same bifurcation diagram except for v = 0.8.

In Figures 3 and 4, the largest Lyapunov exponent plots are drawn by use of the Jacobian matrix algorithm proposed in [32]. The largest Lyapunov exponent LE is positive somewhere; it is corresponding to the chaotic intervals in Figures 1 and 2.

By choosing 101 different initial values we can plot y(n) versus x(n) in one figure. The phase portraits of the integer map are derived from Figure 5. The cases of v = 0.8 and v = 0.6 are plotted in Figures 6 and 7, respectively.

6. Applications

The fractionalized chaotic map can be applied in image encryption. Exploit (16) into an algorithm, and set the initial values [x.sub.0], [y.sub.0], the order v, and the coefficients [k.sub.1], [k.sub.2] of chaotic system as keys. In this paper, we propose the encryption algorithm and divide it into 3 parts.

6.1. Generation of New Keys Based on Elliptic Curve in a Finite Field. Setting a = 1, b = 6, and p = 9996887 in (7), we can get E([F.sub.9996887]). Since #E = 10000721 is a prime number, according to [29], E([F.sub.9999887]) is a safe EC. Let [alpha] = (2,4), randomly select d = 9134417, k = 1269960 [member of] [1, #E]; then [beta] = d[alpha] = (6020909, 7282175), [gamma] = k[alpha] = (7495358, 7052635), and ([c.sub.1], [c.sub.2]) = k[beta] = (3049362, 3915118) = d[gamma]. The initial key v = 0.6026331, [x.sub.0] = 4.107532, [v.sub.01] = v x [10.sup.7] = 6026331, and [x.sub.01] = [x.sub.0] x [10.sup.6] = 4107532.

Calculate

[v'.sub.01] = [c.sub.1] x [v.sub.01] mod p = 3049362 x 6026331 mod 9996887 = 7123456 mod 9996887, [x'.sub.01] = [c.sub.2] x [x.sub.01] mod p = 3915118 x 4107532 mod 9996887 = 190000 mod 9996887. (17)

Then, the ciphertext is ((7495358, 7052635), 7123456, 190000), the enciphered key is v' = [v'.sub.01]/[10.sup.7] = 0.7123456, [x'.sub.0] = [x'.sub.01]/[10.sup.6] = 0.19.

Make [y.sub.0] = 3.650991, [k.sub.1] = 0.897029, and [k.sub.2] = 0.434264, and compute [y.sub.01] = [y.sub.0] x [10.sup.6], [k.sub.01] = [k.sub.1] x [10.sup.6], and [k.sub.02] = [k.sub.2] x [10.sup.6]; then

[mathematical expression not reproducible]. (18)

Set [y'.sub.0] = [y'.sub.01]/[10.sup.6] = 0.06, [k'.sub.1] = [k'.sub.01]/[10.sup.6] = 8, [k'.sub.2] = [k'.sub.02]/[10.sup.6] = 0.5, and then [x'.sub.0], [y'.sub.0], v', [k'.sub.1], [k'.sub.2] are taken as the keys of Section 6.2.

6.2. Permutation Procedure Based on Fractional 2D-TFCDM. Taking advantage of (16) with the initial values [x'.sub.0], [y'.sub.0], v', [k'.sub.1], and [k'.sub.2] generated in the last section, we can encrypt the image. The next step of encryption is permutation; it is subdivided into 4 steps:

(1) Set [x'.sub.0] as x(1); iterate (16) for MN - 1 times to generate the one-dimensional real number chaotic sequence x(i), i = 1, 2, ..., MN; here M and N denote the length and width of the original image V, respectively.

(2) Reorder x(k) by the bubble sort and get x'(k), and record the change of the subscript of x(k) as z(k).

(3) Change M x N original image V into 1 x MN sequence v(k), and rearrange v(k) according to z(k) to get the new sequence v'(k).

(4) Reshape v'(k) into M x N image as V'; V' is the permutated image we needed.

Reversing the above 4 steps, we can remove the effect of permutation to get the original image.

6.3. Encryption Method Based on Fractional 2D-TFCDM. (1) In Section 6.2 we get the chaotic sequence x(i) and image V'. Reshape M x N image V' into 1 x MN sequence u(i); that is i = N(m - 1) + n,(m = 1, 2, ..., M, n = 1, 2, ..., N). Another M x N image is used as a key image (K-image). Change the K-image also into 1 x MN sequence w(i).

(2) Set i = 0.

(3) Round x(i) x [10.sup.8] as [x.sub.1] (i), do modulus operation to [x.sub.1](i) in (19), and get [x.sub.2](i):

[x.sub.2](i) = mod ([x.sub.1](i), 256). (19)

(4) Do the following operation and get u'(i):

U'(i) = u(i) [direct sum] mod (w(i) + [x.sub.2](i), 256), (20)

where [direct sum] refers to the Xor operation, and u'(i) is the encrypted pixel value.

The inverse form of (20) is

u(i) = u'(i) [direct sum] mod (w(i) + [x.sub.2](i), 256). (21)

(5) Compute the iteration times k(i) according to

k(i) = 1 + mod (u'(i), 256). (22)

Then, iterate (16) for k(i) times to get x(i + 1), circle from step (3) to step (5), until getting x(MN).

(6) Change u'(i) into M x N image as V", which is the finally encrypted figure we need.

The decryption procedure is including 2 parts:

(1) Do all steps in encryption process except (20) which is replaced by (21).

(2) Reverse the procedure in Section 6.2. Then the decryption procedure is done.

Figure 8 shows the encryption process described in Sections 6.2 and 6.3 in a flow chart, and Figure 9 illustrates the iteration procedure of S box.

The original, encrypted, and decrypted images are shown in Figures 10-18. The proposed algorithm can encrypt any rectangular image.

The adopted cryptosystem in Section 6.1 is asymmetric; however, the ones in Sections 6.2 and 6.3 are symmetric.

7. Analysis of Results in Applications

7.1. Key Space. In the proposed algorithm, the initial values [x.sub.0], [y.sub.0], the order v, and the coefficients [k.sub.1], [k.sub.2] are taken as the secret keys; consequently there are 5 keys. Assume the precision of [x.sub.0], [y.sub.0], v, [k.sub.1], and [k.sub.2] are [10.sup.-16], 3 x [10.sup.-17], [10.sup.-16], [10.sup.-15], and [10.sup.-16], respectively; then the key's space is 1/3 x [10.sup.80] [approximately equal to] 1.12 x [2.sup.264]. If the size of the plaintext is 512 x 512, then the key space of K-image is also 512 x 512 x [2.sup.8] = [2.sup.26]. The total key space of the proposed algorithm is 1.12 x [2.sup.290].

7.2. Statistics Analysis. The quality against any statistical attack is important for a well-designed encryption method; it include 3 aspects as follows.

7.2.1. Correlation of the Plain- and Cipher-Images. In an ordinary image, the adjacent pixels are related; therefore the correlation coefficient of adjacent pixels is usually high. A good encryption algorithm should make the correlation coefficients of encrypted image nearly equal to zero. The closer to zero the correlation coefficients is, the better the encryption algorithm is. Formulas (23) calculate the correlation coefficient. The correlations along the x direction of both original and encrypted images are displayed in Figures 19-27 from Cameraman to Aerial. The correlation coefficients are displayed in Table 1.

[mathematical expression not reproducible]. (23)

With the sharp contrast of data between original image and encrypted image, Table 1 indicates that the encryption process make pixels of the encrypted image almost independent with each other. Consequently, the encryption algorithm is good at pixel value randomization.

Compared with other algorithm, we can observe that most correlation coefficients of encrypted image are nearer to 0 in Table 2. As a consequence of this, the proposed encryption algorithm is superior to others.

7.2.2. Histogram. Histogram reflects the distribution of colors inside the image. The adversary can get some effective information from the regularity of histogram. Therefore, a well-designed image encryption method should make the pixel value of encrypted image distribute uniformly. Figure 28 shows the histogram of Cameraman. Similarly, the histograms of the other 8 cases are drawn in Figures 29-36. It is illustrated that the proposed encryption method has a good effect on pixel value distribution uniformization.

7.2.3. Information Entropy. Information entropy defines the randomness and the unpredictability of information in an image. It is defined by

H(m) = [[2.sup.n]-1.summation over (i=0)]p([m.sub.i]) [log.sub.2] [1/p([m.sub.i])]. (24)

Here p([m.sub.i]) is the probability of [m.sub.i]; n is the number of bits that is required to represent the symbol [m.sub.i]. For the pixels values of the image are 0~255, according to (24) the information entropy is 8 bits for an ideally random image. Therefore, the closer to 8 bits the information entropy is, the better the encryption algorithm is. The information entropy of the 9 cases is gotten in Table 3; it indicates that the encrypted images are very close to the random images.

From Table 4, we can observe that the information entropy of proposed algorithm is nearer to 8 bits than other algorithms.

7.3. Sensitivity Analysis. The different range between two images is measured by two criteria: number of pixels change rate (NPCR) and unified average changing intensity (UACI). They are defined as follows:

[mathematical expression not reproducible]. (25)

Here W and H are the width and the height of [T.sub.1] and [T.sub.2].

7.3.1. Key Sensitivity. We encrypt the image by the keys [x.sub.0] = 0.19, [y.sub.0] = 0.06, v = 0.7123456, [k.sub.1] = 8, and [k.sub.2] = 0.5. Figure 37(a) is the decrypted image with the correct keys. Figure 37(b) represents the decrypted image under [10.sup.-16] adding to [x.sub.0] with other keys unchanged. Similarly, the secret keys [y.sub.0], v, [k.sub.1], [k.sub.2] are added as 3 x [10.sup.-17], [10.sup.- 16], [10.sup.-15] and [10.sup.-16] to decrypt the images separately with other keys unchanged. The results are shown in Figures 37(c)-37(f). The comparison of key space is shown in Table 5 and the NPCR and UACI between Figures 37(a) and 37(b)-37(f) are calculated in Table 6.

In contrast with other algorithm, the key space of proposed algorithm is larger than others.

Most NPCR are near to 99.61% and most of UACI are higher than 30% in Table 6. We cannot recognize the man inside from Figures 37(b)-37(f); therefore the encryption method is sensitive to the keys.

7.3.2. Plaintext Sensitivity. By encrypting two same images with only one pixel difference, the attackers can obtain effective information by comparing the two encrypted images. Therefore an encryption method designed against differential attack should ensure that the two encrypted images are completely different even if there is only a pixel difference in the original image.

In Table 7, Figure 10(a)(x, y) is the same as Figure 10(a) except for a pixel locating (x, y). After that, the 2 images are encrypted with the same keys and the NPCR and UACI between the 2 ciphertext images are calculated. Similarly, the data of other 8 cases are obtained in Tables 8-15.

From Table 16, the NPCR and UACI of proposed algorithm after 2-round encryption are nearer to the ideal values 99.61% and 33.46% [33] than others. Therefore the proposed method is better.

7.4. Resistance to Known-Plaintext and Chosen-Plaintext Attacks. In Section 6.3, the iteration times of the next round are decided by the encrypted pixel value of present round. In (20), [x.sub.2](i), generated from the fractional 2D-TFCDM, is dependent on k(i - 1) and determines k(i). Therefore, the corresponding keystream is different when different plaintext is encrypted. For the resultant information is related to the chosen-images, the attacker cannot get useful information after encrypting some special images. As a result, the attacks proposed in [34-41] become ineffective for our scheme. In a word, the proposed scheme can primely resist the known-plaintext and the chosen-plaintext attacks.

8. Conclusions

Fractional 2D-TFCDM is obtained from the 2D-TFCDM. After that, we found new chaotic dynamics behaviors from the fractionalized map. Moreover, the map can be converted into image encryption algorithm as an application. Finally, the encryption effect is analysed in 4 main aspects; we find the proposed scheme is superior to others almost anywhere in comparison. As far as we know, the proposed image encryption algorithm has never been reported before.

https://doi.org/10.1155/2018/4585083

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The project was supported by the National Natural Science Foundation of China (Grant nos. 61072147, 11271008).

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Zeyu Liu (iD), (1) Tiecheng Xia (iD), (1,2) and Jinbo Wang (2)

(1) Department of Mathematics, Shanghai University, Shanghai 200444, China

(2) Science and Technology on Communication Security Laboratory, Chengdu, Sichuan 610041, China

Correspondence should be addressed to Tiecheng Xia; xiatc@shu.edu.cn

Received 7 December 2017; Revised 10 February 2018; Accepted 22 February 2018; Published 23 April 2018

Academic Editor: Youssef N. Raffoul

Caption: Figure 1: The bifurcation diagram of the 2D-TFCDM of variable [k.sub.1] for v = 1.

Caption: Figure 2: The bifurcation diagram of the fractional 2D-TFCDM of variable [k.sub.1] for v = 0.8.

Caption: Figure 3: The largest Lyapunov exponent of the 2D-TFCDM of the variable [k.sub.1].

Caption: Figure 4: The largest Lyapunov exponent of the fractional 2D-TFCDM of the variable [k.sub.1].

Caption: Figure 5: The phase portraits of the 2D-TFCDM for [k.sub.1] = 8, [k.sub.2] = 0.5, and v = 1.

Caption: Figure 6: The phase portraits of the fractional 2D-TFCDM for [k.sub.1] = 8, [k.sub.2] = 0.5, and v = 0.8.

Caption: Figure 7: The phase portraits of the fractional 2D-TFCDM for [k.sub.1] = 8, [k.sub.2] = 0.5, and v = 0.6.

Caption: Figure 8: The proposed encryption method.

Caption: Figure 9: The S box.

Caption: Figure 10: Cameraman.

Caption: Figure 11: Lena.

Caption: Figure 12: Peppers.

Caption: Figure 13: Lake.

Caption: Figure 14: Dollar.

Caption: Figure 15: Columbia.

Caption: Figure 16: Lax.

Caption: Figure 17: Boat.

Caption: Figure 18: Aerial.

Caption: Figure 19: Cameraman.

Caption: Figure 20: Lena.

Caption: Figure 21: Peppers.

Caption: Figure 22: Lake.

Caption: Figure 23: Dollar.

Caption: Figure 24: Columbia.

Caption: Figure 25: Lax.

Caption: Figure 26: Boat.

Caption: Figure 27: Aerial.

Caption: Figure 28: Cameraman.

Caption: Figure 29: Lena.

Caption: Figure 30: Peppers.

Caption: Figure 31: Lake.

Caption: Figure 32: Dollar.

Caption: Figure 33: Columbia.

Caption: Figure 34: Lax.

Caption: Figure 35: Boat.

Caption: Figure 36: Aerial.

Caption: Figure 37: The test of key sensitivity.
Table 1: Correlation coefficients of image.

                      Original image

Image       Horizontal   Diagonal   Vertical

Cameraman     0.9276      0.9120     0.9597
Lena          0.9722      0.9527     0.9860
Peppers       0.9667      0.9382     0.9694
Lake          0.9768      0.9544     0.9748
Dollar        0.8035      0.6952     0.6938
Columbia      0.9727      0.9403     0.9705
Lax           0.7889      0.7151     0.8483
Boat          0.9407      0.9158     0.9545
Aerial        0.9135      0.7952     0.8677

                      Encrypted image

Image       Horizontal   Diagonal   Vertical

Cameraman     0.0119     -0.0021    -0.0025
Lena         -0.0140     -0.0086    -0.0034
Peppers      -0.0088      0.0080    -0.0054
Lake         -0.0155      0.0101    -0.0088
Dollar        0.0131     -0.0183     0.0263
Columbia      0.0060     -0.0104    -0.0093
Lax          -0.0107      0.0147     0.0107
Boat          0.0169     -0.0074    -0.0077
Aerial        0.0084     -0.0123    -0.0133

Table 2: Comparison of correlation coefficients of image.

                               Original image
Algorithm   Image
                    Horizontal   Vertical   Diagonal

Proposed    Lena      0.9722      0.9527     0.9860
[1]         Lena      0.9503      0.9755     0.9275
[2]         Lena     0.927970    0.926331   0.839072
[5]         Lena      0.946       0.973      0.921
[6]         Lena      0.9569      0.9236     0.9019

                      Encrypted image
Algorithm
            Horizontal   Vertical    Diagonal

Proposed     -0.0140      -0.0086     -0.0034
[1]          -0.0226      0.0041      0.0368
[2]         -0.010889    -0.018110   -0.006104
[5]          -0.0055      -0.0075     0.0026
[6]           0.0042      -0.0043     0.0163

Table 3: Information entropy.

Image       Original image   Encrypted image

Cameraman       7.0097           7.9974
Peppers         7.5739           7.9976
Dollar          6.9785           7.9992
Lax             6.8272           7.9993
Aerial          6.9940           7.9992
Lena            7.2185           7.9993
Lake            7.4845           7.9993
Columbia        7.2736           7.9992
Boat            6.9391           7.9972

Table 4: Comparison of information entropy.

Algorithm   Image   Original image   Encrypted image

Proposed    Lena        7.2185           7.9993
[1]         Lena        7.2072           7.9973
[4]         Lena      Undefined          7.9972
[19]        Lena      Undefined         7.987918
[20]        Lena       7.447144         7.988847

Table 5: Comparison of key spaces.

Algorithm          Proposed             [2]               [4]

Key spaces    2.23 x [10.sup.87]    [2.sup.128]   [approximately equal
             (1.12 x [2.sup.290])                   to] [2.sup.273]

Algorithm        [6]

Key spaces   [2.sup.276]

Table 6: NPCR and UACI between Figures 37(a) and 37(b)-37(f).

                  NPCR and UACI
Image
               NPCR(%)   UACI (%)

Figure 37(b)    99.61     31.26
Figure 37(c)    97.02     30.23
Figure 37(d)    99.60     31.03
Figure 37(e)    99.61     31.01
Figure 37(f)    99.62     31.27

Table 7: NPCR and UACI between cipher-images with slightly different
plain-images.

                            NPCR and UACI of Cameraman
Image
                        NPCR (1-round %)   UACI (1-round %)

Figure 10(a)(30, 30)          4.84               1.64
Figure 10(a)(50, 50)         81.43              27.39
Figure 10(a)(80, 80)         80.87              27.19
Figure 10(a)(100,100)         6.82               2.28

                            NPCR and UACI of Cameraman
Image
                        NPCR (2-round %)   UACI (2-round %)

Figure 10(a)(30, 30)         99.57              33.61
Figure 10(a)(50, 50)         99.62              33.56
Figure 10(a)(80, 80)         99.59              33.51
Figure 10(a)(100,100)        99.59              33.46

Table 8: NPCR and UACI between cipher-images with slightly different
plain-images.

                              NPCR and UACI of Lena
Image
                        NPCR (1-round %)   UACI (1-round %)

Figure 11(a)(30, 30)          1.21               0.41
Figure 11(a)(50, 50)         95.06              31.93
Figure 11(a)(80, 80)         94.90              31.93
Figure 11(a)(100,100)         1.71               0.58

                               NPCR and UACI of Lena
Image
                        NPCR (2-round %)   UACI (2-round %)

Figure 11(a)(30, 30)         99.59              33.39
Figure 11(a)(50, 50)         99.59              33.53
Figure 11(a)(80, 80)         99.60              33.48
Figure 11(a)(100,100)        99.63              33.40

Table 9: NPCR and UACI between cipher-images with slightly different
plain-images.

                             NPCR and UACI of Peppers
Image
                        NPCR (1-round %)   UACI (1-round %)

Figure 12(a)(30, 30)          4.83               1.64
Figure 12(a)(50, 50)         81.44              27.35
Figure 12(a)(80, 80)          6.12               2.03
Figure 12(a)(100,100)         6.83               2.32

                             NPCR and UACI of Peppers
Image
                        NPCR (2-round %)   UACI (2-round %)

Figure 12(a)(30, 30)         99.56              33.44
Figure 12(a)(50, 50)         99.62              33.50
Figure 12(a)(80, 80)         99.57              33.42
Figure 12(a)(100,100)        99.60              33.50

Table 10: NPCR and UACI between cipher-images with slightly different
plain-images.

                              NPCR and UACI of Lake
Image
                        NPCR (1-round %)   UACI (1-round %)

Figure 13(a)(30, 30)          1.21               0.41
Figure 13(a)(50, 50)         95.09              31.88
Figure 13(a)(80, 80)         94.92              31.89
Figure 13(a)(100,100)         1.71               0.58

                              NPCR and UACI of Lake
Image
                        NPCR (2-round %)   UACI (2-round %)

Figure 13(a)(30, 30)         99.61              33.46
Figure 13(a)(50, 50)         99.61              33.42
Figure 13(a)(80, 80)         99.60              33.46
Figure 13(a)(100,100)        99.59              33.47

Table 11: NPCR and UACI between cipher-images with slightly different
plain-images.

                             NPCR and UACI of Dollar
Image
                        NPCR (1-round %)   UACI (1-round %)

Figure 14(a)(30, 30)          1.21               0.41
Figure 14(a)(50, 50)         95.08              32.00
Figure 14(a)(80, 80)         94.90              31.93
Figure 14(a)(100,100)         1.71               0.57

                             NPCR and UACI of Dollar
Image
                        NPCR (2-round %)   UACI (2-round %)

Figure 14(a)(30, 30)         99.64              33.42
Figure 14(a)(50, 50)         99.60              33.49
Figure 14(a)(80, 80)         99.61              33.48
Figure 14(a)(100,100)        99.61              33.41

Table 12: NPCR and UACI between cipher-images with slightly different
plain-images.

                             NPCR and UACI of Columbia
Image
                        NPCR (1-round %)   UACI (1-round %)

Figure 15(a)(30, 30)         94.83              31.96
Figure 15(a)(50, 50)         93.48              31.40
Figure 15(a)(80, 80)          0.96               0.32
Figure 15(a)(100,100)         1.11               0.38

                             NPCR and UACI of Columbia
Image
                        NPCR (2-round %)   UACI (2-round %)

Figure 15(a)(30, 30)         99.61              33.47
Figure 15(a)(50, 50)         99.48              33.36
Figure 15(a)(80, 80)         99.60              33.45
Figure 15(a)(100,100)        99.61              33.51

Table 13: NPCR and UACI between cipher-images with slightly different
plain-images.

                               NPCR and UACI of Lax
Image
                        NPCR (1-round %)   UACI (1-round %)

Figure 16(a)(30, 30)          1.21               0.40
Figure 16(a)(50, 50)         95.06              31.99
Figure 16(a)(80, 80)         94.92              31.87
Figure 16(a)(100,100)         1.70               0.58

                               NPCR and UACI of Lax
Image
                        NPCR (2-round %)   UACI (2-round %)

Figure 16(a)(30, 30)         99.62              33.39
Figure 16(a)(50, 50)         99.61              33.49
Figure 16(a)(80, 80)         99.58              33.41
Figure 16(a)(100,100)        99.62              33.48

Table 14: NPCR and UACI between cipher-images with slightly different
plain-images.

                               NPCR and UACI of Boat
Image
                        NPCR (1-round %)   UACI (1-round %)

Figure 17(a)(30, 30)          4.83               1.60
Figure 17(a)(50, 50)         81.46              27.45
Figure 17(a)(80, 80)         80.82              27.24
Figure 17(a)(100,100)         6.82               2.32

                              NPCR and UACI of Boat
Image
                        NPCR (2-round %)   UACI (2-round %)

Figure 17(a)(30, 30)         99.59              33.47
Figure 17(a)(50, 50)         99.62              33.59
Figure 17(a)(80, 80)         99.58              33.48
Figure 17(a)(100,100)        99.62              33.61

Table 15: NPCR and UACI between cipher-images with slightly different
plain-images.

                              NPCR and UACI of Aerial
Image
                        NPCR (1-round %)   UACI (1-round %)

Figure 18(a)(30, 30)          1.21               0.41
Figure 18(a)(50, 50)         95.06              31.93
Figure 18(a)(80, 80)         94.91              31.88
Figure 18(a)(100,100)         1.71               0.57

                             NPCR and UACI of Aerial
Image
                        NPCR (2-round %)   UACI (2-round %)

Figure 18(a)(30, 30)         99.61              33.49
Figure 18(a)(50, 50)         99.62              33.43
Figure 18(a)(80, 80)         99.61              33.53
Figure 18(a)(100,100)        99.61              33.52

Table 16: Comparison of NPCR and UACI of image.

Algorithm    Image   NPCR (%)   UACI (%)

Proposed     Lena     99.60      33.48
[1]          Lena     99.61      33.53
[2]          Lena    99.6429    33.3935
[3]          Lena    99.6304    33.5989
[5]          Lena     99.932     39.520
[19]         Lena    75.62561   34.84288
[20]         Lena    99.6091    33.5038
[21]         Lena    99.6330    34.1319
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Title Annotation:Research Article
Author:Liu, Zeyu; Xia, Tiecheng; Wang, Jinbo
Publication:Discrete Dynamics in Nature and Society
Geographic Code:9CHIN
Date:Jan 1, 2018
Words:6627
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