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

A new fractional two-dimensional triangle function combination discrete chaotic map (2D-TFCDM) with the discrete fractional difference is proposed. We observe the bifurcation behaviors and draw the bifurcation diagrams, the largest Lyapunov exponent plot, and the phase portraits of the proposed map, respectively. On the application side, we apply the proposed discrete fractional map into image encryption with the secret keys ciphered by Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC). Finally, the image encryption algorithm is analysed in four main aspects that indicate the proposed algorithm is better than others.


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][2][3][4][5][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][8][9][10][11][12][13]. Recently, Wu et al. [14][15][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 onedimensional fractional chaotic time series within fractionalorder difference has been proposed; however, the twodimensional 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.
Definition 1 (see [25]).Let : N  → R and 0 < ] be given.The ]th fractional sum is defined by Note that  is the starting point;  (]) is the falling function defined as Definition 2 (see [26]).For 0 < ], ] ∉ N, and () defined on N  , the ]-order Caputo fractional difference is defined by (3) Theorem 3 (see [27]).For the delta fractional difference equa- the equivalent discrete integral equation is where

Introduction to Elliptic Curve
Definition 4.An elliptic curve (EC)  over a prime field   denoted by (  ) refers to the set of all points (, ) that satisfy the equation together with a special point  at infinity, where ,  ∈   ,  ̸ = 2, 3 and 4 3 + 27 2 ̸ = 0 [28,29]. where The scalar multiplication over (  ) is defined by where  is an integer.
Definition 5.The order of an EC is defined by the number of points that lie on the EC denoted by # [29].

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  to Bob, they should do the step as mentioned hereunder: (1) Andy and Bob make an agreement on an elliptic curve (  ) and the base point .
Any adversary that only has  and  without the private keys  and  very difficultly breaks the MVECC to get the plaintext .What is more, if # 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.
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 () versus () in one figure.The phase portraits of the integer map are derived from Figure 5.The cases of ] = 0.8 and ] = 0.6 are plotted in Figures 6 and 7, respectively.

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

Permutation Procedure Based on Fractional 2D-TFCDM.
Taking advantage of (16) with the initial values   0 ,   0 , ]  ,   1 , and   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   0 as (1); iterate (16) for −1 times to generate the one-dimensional real number chaotic sequence (),  = 1, 2, . . ., ; here  and  denote the length and width of the original image , respectively.
(2) Reorder () by the bubble sort and get   (), and record the change of the subscript of () as ().
(3) Change × original image  into 1× sequence V(), and rearrange V() according to () to get the new sequence V  ().
(4) Reshape V  () into  ×  image as   ;   is the permutated image we needed.
Reversing the above 4 steps, we can remove the effect of permutation to get the original image.
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.

Analysis of Results in Applications
7.1.Key Space.In the proposed algorithm, the initial values  0 ,  0 , the order ], and the coefficients  1 ,  2 are taken as the   , respectively; then the key's space is 1/3 × 10 80 ≈ 1.12 × 2 264 .If the size of the plaintext is 512 × 512, then the key space of K-image is also 512 × 512 × 2 8 = 2 26 .The total key space of the proposed algorithm is 1.12 ×2 290 .

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

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     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.

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.

Information Entropy. Information entropy defines the randomness and the unpredictability of information in an image. It is defined by
Here (  ) is the probability of   ;  is the number of bits that is required to represent the symbol   .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.

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: Here  and  are the width and the height of  1 and  2 .
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.In Table 7, Figure 10(a)(, ) is the same as Figure 10(a) except for a pixel locating (, ).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.

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),  2 (), generated from the fractional 2D-TFCDM, is dependent on ( − 1) and determines ().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][35][36][37][38][39][40][41] become ineffective for our scheme.In a word, the proposed scheme can primely resist the knownplaintext and the chosen-plaintext attacks.

Figure 3 :
Figure 3: The largest Lyapunov exponent of the 2D-TFCDM of the variable  1 .

Figure 37 (
a) is the decrypted image with the correct keys.

Table 1 :
Correlation coefficients of image.

Table 4 :
Comparison of information entropy.

Table 5 :
Comparison of key spaces.

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

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

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

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

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

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

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

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

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

Table 16 :
Comparison of NPCR and UACI of image.