A Multi-Image Encryption with Super-Lager-Capacity Based on Spherical Diffraction and Filtering Diffusion
A Multi-Image Encryption with Super-Lager-Capacity Based on Spherical Diffraction and Filtering...
Wu, Hanmeng;Wang, Jun;Zhang, Ziyi;Chen, Xudong;Zhu, Zheng
2020-08-17 00:00:00
applied sciences Article A Multi-Image Encryption with Super-Lager-Capacity Based on Spherical Diraction and Filtering Diusion Hanmeng Wu, Jun Wang * , Ziyi Zhang, Xudong Chen and Zheng Zhu School of Electronics and Information Engineering, Sichuan University, Chengdu 610065, China; hanmeng_wu@163.com (H.W.); ziyi1040363061@163.com (Z.Z.); helloxudong@outlook.com (X.C.); zzhu0420@163.com (Z.Z.) * Correspondence: jwang@scu.edu.cn Received: 18 July 2020; Accepted: 14 August 2020; Published: 17 August 2020 Abstract: A multi-image encryption with super-large-capacity is proposed by using spherical diraction and filtering diusion. In the proposed method, initial images are processed sequentially by filtering diusion and chaos scrambling. The images are combined into one image using XOR operation. The combined image is encrypted by improved equal modulus decomposition after spherical diraction. There are three main contributions of the proposed method—(1) resisting phase-retrieval attack due to the asymmetry of spherical diraction; (2) high flexibility of decrypting images individually; and (3) super-large encryption capacity of the product of image resolution and grayscale level, which is the most significant advantage. The feasibility and eectiveness of the proposed encryption are verified by numerical simulation results. Keywords: multi-image encryption; spherical diraction; filtering diusion 1. Introduction With the development of information technology, how to ensure the security of image information becomes an essential point. In image encryption research, optical information processing technology and digital technology have been widely used in image encryption [1–7]. Refregier and Javidi proposed double random phase encoding (DRPE) in 1995, which pioneered the field of optical encryption [8]. However, because of the linear symmetry of DRPE technology [9], it is proved to be vulnerable to the known-plaintext attack [10] and chosen-plaintext attack [11]. To improve security, several nonlinear cryptosystems are proposed to improve the weaknesses of traditional symmetric cryptosystems, such as Fresnel domain [12], cylindrical diraction transform [13–15], gyrator transform domain [16], and fractional Mellin transform [17]. However, these image encryption methods are usually applied to encrypt one image, which has made it challenging to meet the growing information needs. Recently, more and more scholars have begun to study multi-image encryption [18–23]. Multi-image encryption can be divided into two categories according to the encryption process. One of the multi-image encryption technologies is based on cascading [24–26]. Sui et al. [27] used cascaded fractional Fourier transform to encrypt multiple images. As the number of images increases, the quality of decryption decreases rapidly. Li et al. [28] proposed a multiple-image encryption method using the cascaded fractional Fourier transform, which cannot get a decrypted image individually from the ciphertext. Hence, the decryption flexibility is poor, and the number of encryption images is severely limited. The other multi-image encryption technology is based on multiplexing, such as space multiplexing [29], region multiplexing [30], wavelength multiplexing [31,32], and position multiplexing [33]. For instance, Deepan et al. [34] proposed a multi-image cryptosystem based on compressive sensing and the DPRE technique. By this method, the space multiplexing method is also used to integrate multiple-image data. Zhao et al. [35] came up with a multi-image encryption Appl. Sci. 2020, 10, 5691; doi:10.3390/app10165691 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 14 [31,32], and position multiplexing [33]. For instance, Deepan et al. [34] proposed a multi-image Appl. Sci. 2020, 10, 5691 2 of 14 cryptosystem based on compressive sensing and the DPRE technique. By this method, the space multiplexing method is also used to integrate multiple-image data. Zhao et al. [35] came up with a multi-image encryption by using the position multiplexing. Besides, Hu et al. [36] proposed an by using the position multiplexing. Besides, Hu et al. [36] proposed an asymmetric multi-image asymmetric multi-image encryption using the convolution multiplexing. Due to crosstalk issues, this encryption using the convolution multiplexing. Due to crosstalk issues, this method can only encrypt method can only encrypt up to 12 images. Moreover, Liu et al. [37] present a double image encryption up to 12 images. Moreover, Liu et al. [37] present a double image encryption using fractional order using fractional order multiplexing to combine two initial images. Because of the characteristics of multiplexing to combine two initial images. Because of the characteristics of multiplexing technology, multiplexing technology, it can decrypt images individually, which has high decryption flexibility. it can decrypt images individually, which has high decryption flexibility. In the process of information In the process of information multiplexing, the information of different images occupies the same multiplexing, the information of dierent images occupies the same channel. However, the traditional channel. However, the traditional multiplexing method will lead to data crosstalk and the limited multiplexing method will lead to data crosstalk and the limited encryption capacity with the number encryption capacity with the number of images increasing. of images increasing. In order to solve the problem, a multi-image encryption using spherical diffraction and filtering In order to solve the problem, a multi-image encryption using spherical diraction and filtering diffusion is proposed. Owing to the asymmetric characteristics of spherical diffraction, the proposed diusion is proposed. Owing to the asymmetric characteristics of spherical diraction, the proposed encryption can effectively resist the phase retrieval attack. Moreover, each image can be decrypted encryption can eectively resist the phase retrieval attack. Moreover, each image can be decrypted individually with its private key, which has high decryption flexibility. The most significant individually with its private key, which has high decryption flexibility. The most significant advantage advantage of this algorithm is the super-large encryption capacity. The encryption capacity is the of this algorithm is the super-large encryption capacity. The encryption capacity is the product of 8 8 product of image resolution and grayscale level, which is super-large of 2 × m × n in case of m × n image resolution and grayscale level, which is super-large of 2 m n in case of m n points of an 8 8 points of an image with 2 grayscale level. Numerical simulation results verify the feasibility and image with 2 grayscale level. Numerical simulation results verify the feasibility and eectiveness of effectiveness of the proposed encryption. the proposed encryption. The rest of this paper is organized as follows. Section 2 introduces the spherical diffraction and The rest of this paper is organized as follows. Section 2 introduces the spherical diraction and filtering diffusion technology. Section 3 presents principle of encryption and decryption. Section 4 filtering diusion technology. Section 3 presents principle of encryption and decryption. Section 4 shows simulation results. Section 5 draws a conclusion. shows simulation results. Section 5 draws a conclusion. 2. Principles of the Method 2. Principles of the Method 2.1. Encryption in Spherical Diraction Domain 2.1. Encryption in Spherical Diffraction Domain Since spherical diraction (SpD) is an asymmetry diraction propagation model [38], the SpD-based Since spherical diffraction (SpD) is an asymmetry diffraction propagation model [38], the SpD- encryption should overcome the shortage of symmetry cryptosystem, which is based on traditional based encryption should overcome the shortage of symmetry cryptosystem, which is based on diraction. According to the spherical diraction theory [38–42], the object and the observation traditional diffraction. According to the spherical diffraction theory [38–42], the object and the surfaces are concentric spheres. r and R represent the radii of the inner and outer surfaces of the observation surfaces are concentric spheres. r and R represent the radii of the inner and outer surfaces concentric sphere, respectively. In this diraction model, there are two models of propagation, namely of the concentric sphere, respectively. In this diffraction model, there are two models of propagation, the inside-out propagation (IOP) model and the outside-in propagation (OIP) model, as shown in namely the inside-out propagation (IOP) model and the outside-in propagation (OIP) model, as Figure 1a,b, respectively. shown in Figure 1a and Figure 1b, respectively. (a) (b) Figure 1. Spherical diffraction model: (a) inside-out propagation (IOP) model, (b) outside-in Figure 1. Spherical diraction model: (a) inside-out propagation (IOP) model, (b) outside-in propagation propagation (OIP) model. (OIP) model. We use Q ( , ' ) and P ( , ' ) in the spherical coordinates to denote the object and the We use QR (θR, φR) and Pr (θr, φr) in the spherical coordinates to denote the object and the R R R r r r observation points for the OIP model, respectively. The spherical coordinate of Q ( , ' ) and P observation points for the OIP model, respectively. The spherical coordinate of Qr (θr r, φrr) an r d PR (θR, ( , ' ) denote the object and the observation points for the IOP model, respectively. and are φR) denote the object and the observation points for the IOP model, respectively. θr and θR are in the R R r R in the range of to , ' and ' are in the range of /2 to /2. If we represent the diraction R Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 14 Appl. Sci. 2020, 10, 5691 3 of 14 range of −π to π, φr and φR are in the range of −π/2 to π/2. If we represent the diffraction distributions on the inner distributions and outer on the inner surfaces by and outer ur (θ surfaces r, φr) and by uR u (θR ( , φ ,R' ), the di ) and ffuracti ( o,n ' integral ), the f di o rmu raction las based on integral r r r R R R the R formulas ayleibased gh–Sommerf on the el Rayleigh–Sommerfeld d integral equation ofintegral the two equation models ca ofn b the e wri twotten models as can be written as exp(ikd ) OP Rr exp(ikd ) O P uC θϕ,, = u θ ϕcosα R rdθdϕ=SpDu θ,ϕ ……OIP , () ( ) () () (1) rr r R R R R R r r r r u ( ,' ) = C u ( ,' ) cos d d' = SpD (u ( ,' )) :::::: OIP, (1) r r r s r r r r R R R R R OP d Rr O P R r rR−− cos(θθ ) cos(ϕϕ− ) R rR r cos α = , r R cos( ) cos(' ' ) R r R r (2) cos = d , (2) OP Rr O P R r dd==d = r+R−2c Rr os(θθ− )cos(ϕϕ− ) , (3) OP O P 2 2 R r R r Rr r R d = d = d = r + R 2Rr cos( ) cos(' ' ), (3) r r O P O P R R R r r R x exp(ikd ) OP exp(ikd ) rR O P r R uC (, θϕ )== u (θ ,ϕ ) dθdϕ SpDu (θ ,ϕ ) ……IOP , () (4) u ( ,RR ' ) =R C u (r ,r' )r dr d'r = SpD R R(uR(R ,' )) :::::: IOP, (4) R R R sr r r r r R R R R d OP O rR P r R where C denotes a constant and k denotes the wavenumber of the incident light. d denotes the where C denotes a constant and k denotes the wavenumber of the incident light. d denotes the distance di between stance between two poi two points of P and nts of Q P on and the Q object on the object and observation and observ surfaces. ation surfaces. s represents s represents the the object sur object face. surface. 2.2. Filtering Diusion Technology 2.2. Filtering Diffusion Technology Image filtering is a digital image processing technology, which is mainly used to eliminate image noise or extract image features. The method performs convolution operation on a 2D image block and Image filtering is a digital image processing technology, which is mainly used to eliminate image a 2D matrix, named kernel. To be specific, the additions of all the multiplications of the kernel pixels noise or extract image features. The method performs convolution operation on a 2D image block and image pixels are used as the value of the current pixel. The current pixel value corresponds to the and a 2D matrix, named kernel. To be specific, the additions of all the multiplications of the kernel center element of kernel, and the other elements correspond to adjacent pixels. R is the selected image pixels and image pixels are used as the value of the current pixel. The current pixel value corresponds block, the central pixel R is the pixel to be processed, and M is the kernel. Assuming the kernel size to the center element of ker x ,ynel, and the other elements correspond to adjacent pixels. R is the selected is (2n + 1) (2n + 1), the calculation of image filtering can be defined as image block, the central pixel Rx,y is the pixel to be processed, and M is the kernel. Assuming the kernel size is (2n + 1) × (2n + 1), the calculation of image filtering can be defined as n n X X nn R = M R . (5) i+n+1,j+n+1 x+i,y+j x,y RM = R . (5) x,1 yi ++n,j+n+1x+i,y+j i = n j = n in =− j =−n Selecting a suitable kernel to filter can emphasize some features or remove unwanted parts in the Selecting a suitable kernel to filter can emphasize some features or remove unwanted parts in image. However, we can also use a randomly generated kernel to diuse the image for encryption. the image. However, we can also use a randomly generated kernel to diffuse the image for Since the slight change of the current pixel would aect the value of each subsequent pixel, it has a encryption. Since the slight change of the current pixel would affect the value of each subsequent good diusion eect. pixel, it has a good diffusion effect. To make the filtering invertible, we set the center pixel of the kernel as one and set other pixels as To make the filtering invertible, we set the center pixel of the kernel as one and set other pixels integers. To get a better diusion eect, the lower-right corner of the kernel is set to correspond to the as integers. To get a better diffusion effect, the lower-right corner of the kernel is set to correspond to current pixel, namely M = 1. The current pixel R is processed by the upper and left adjacent 2n+1,2n+1 x ,y the current pixel, namely M2n+1,2n+1 = 1. The current pixel Rx,y is processed by the upper and left adjacent pixels. Figure 2 shows the filtering diusion process. pixels. Figure 2 shows the filtering diffusion process. Figure 2. The filtering diusion process. Figure 2. The filtering diffusion process. According to filtering diffusion theory [43,44], the calculation of filtering diffusion can be expressed as Appl. Sci. 2020, 10, 5691 4 of 14 Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 14 According to filtering diusion theory [43,44], the calculation of filtering diusion can be expressed as nn 0 1 n n R= M R mod F , B X X C (6) x, y i+n+11 , j+n+ x+i, y+ j B C B C i=-n j=-n B C R = B M R CmodF, (6) i+n+1,j+n+1 x+i,y+j x,y B C @ A i = n j = n 0 1 RR=− M R modF . (7) xy,, xy i++ n1, j++ n1 x+i,y+ j B C X B C ij , ∈−n,...,n ∩(i , j )≠ ( 0,0) B {} C B 0 C B C R = R M R modF. (7) x,y B i+n+1,j+n+1 x+i,y+jC x,y B C @ A i,j2f n,:::,ng\(i,j),(0,0) Before filtering the current pixel, the upper and left pixels should be processed. When doing inverse filtering, it is also required that the upper and left adjacent pixels have not been restored. Before filtering the current pixel, the upper and left pixels should be processed. When doing inverse Therefore, the processing order of inverse filtering should be reversed from that of the forward filtering, it is also required that the upper and left adjacent pixels have not been restored. Therefore, operation. the processing order of inverse filtering should be reversed from that of the forward operation. 3. Princ 3. Principle iple o of f Encryp Encryption tion and D and Decryption ecryption In the process of encryptio In the process of encryption, n, fi filtering ltering di diff usion usion and c and chaotic haotic scr scrambling ambling are are fir first st used t used to o di diff u use se and and scramble scramble encryp encrypted ted i images, mages, r respectively espectively. . S Second, econd, multiple images are superimposed multiple images are superimposed together together usin using g the the XO XOR R operation. operation. Then, the Then, the sup superimposed erimposed im image age iis s ttransformed ransformed int into o the spheri the spherical cal di diffra raction ction domain domain. . Fin Finally ally, , t the he improv improved ed e equal qual mod modulus ulus decomp decomposition osition (E (EMD) MD) is is emplo employed yed t to o obt obtain ain ciph ciphertext ertext and p and private rivate k key ey. The . The p pr rocess ocess o of f decr decryption yption is the inve is the inverse rse oper operation ation of of encr encryption. yption. 3.1. Process of Encryption 3.1. Process of Encryption The multi-image encryption based on filtering diusion and spherical diraction is shown in The multi-image encryption based on filtering diffusion and spherical diffraction is shown in Figure 3. Suppose the original gray images are represented by I (x, y), the proposed method can be Figure 3. Suppose the original gray images are represented by IN N (x, y), the proposed method can be described as follows: described as follows: Figure 3. Process of encryption. Figure 3. Process of encryption. Step 1: First, the images are processed by filtering diusion using the 2D kernel of size 2 2 Step 1: First, the images are processed by filtering diffusion using the 2D kernel of size 2 × 2 indiv individually idually.. We set We set M M2,2 as one, and the as one, and the other three numbers other three numbers are randomly gene are randomly generated rated intege integers rs from from 2,2 0 to 255. In the filtering diusion process, the leftmost column and the topmost row of the image 0 to 255. In the filtering diffusion process, the leftmost column and the topmost row of the image do not do not have enough pixe have enough pixels ls for calc for calculation. ulation. Hence, t Hence, he rightmost a the rightmost nd and bottommost bottommost pixel pixels s can be canused be used for for expansion [43]. Starting from the upper left corner of the image, each pixel is processed in turn. expansion [43]. Starting from the upper left corner of the image, each pixel is processed in turn. The processed The processed pixepixel l value c value an b can e expressed be expressed as as 0 1 B X C B C Ix (,y) = B M(i,j)I (i,j) CmodF , 0 (8) NN B C I (x, y) = B M(i, j)I (i, j)CmodF, (8) B C N ij,{ ∈1,2} @ A i,j2f1,2g where F denotes the grayscale level. where F denotes the grayscale level. Step 2: To get a better encryption effect, the images are scrambled by the chaotic sequence individually. The chaotic sequence is obtained by x =− μxx (1 ) . (9) nn +1 n The range of xn is (0, 1) and the range of μ is (0, 4). Appl. Sci. 2020, 10, 5691 5 of 14 Step 2: To get a better encryption eect, the images are scrambled by the chaotic sequence individually. The chaotic sequence is obtained by x = x (1 x ). (9) n n n+1 The range of x is (0, 1) and the range of is (0, 4). Step 3: Combining the images into one image using the XOR operation. The superimposed result can be written as 0 0 0 f(x, y) = I (x, y) I (x, y) ::: I (x, y). (10) 2 N At the same time, P can be obtained by P (x, y) = f(x, y) I (x, y), (11) where P is reserved as the private key of each image for decryption. Step 4: The combined image f (x, y) is mapped on the spherical surface with a radius of R after it is modulated by the random phase mask RPM (x, y) = exp [iR (x, y)]. R is a random array uniformly 1 1 distributed between 0 and 2. According to the OIP model, the modulated image is transformed into a spherical coordinate. If we use U (x, y) to represent the complex amplitude after spherical diraction, the expression can be written as U(x, y) = SpD [ f(x, y)RPM(x, y)]. (12) Step 5: To improve the anti-attack ability, we use the improved EMD [45] to decompose complex amplitude. The distribution of complex amplitude can be written as U = A exp(i ) = C [exp(iR ) + exp(iPK)], (13) R = 2 rand(), (14) where A is the amplitude and represents the phase of U, rand() denotes a random function. The ciphertext C and the private key PK can be expressed as follows: C = 0.5A [cos( R )] , (15) PK = 2 R , (16) where the ciphertext C and the private key PK are both real-valued, so that they are convenient for recording and transmission. 3.2. Process of Decryption Figure 4 shows process of decryption. The detailed steps are as follows: Step 1: According to the Equation (13), we can get the complex amplitude U. The U is modulated by RPM* after spherical diraction. The combined image f (x, y) can be obtained by f(x, y) = ISpD[U(x, y)]RPM , (17) where ISpD [] resents the inverse process of the spherical diraction calculation. Then f (x, y) is rounded to unify the data types. Step 2: Based on the Equation (11), image I (x, y) can be obtained by I (x, y) = f(x, y) P (x, y). (18) N Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 14 Step 3: Combining the images into one image using the XOR operation. The superimposed result can be written as '' ' f (, xy)=⊕ I (x, y) I (xy , )⊕I (x, y) . (10) 12 N At the same time, PN can be obtained by P (,xy)=⊕ f (x, y) I (,xy) , (11) NN where PN is reserved as the private key of each image for decryption. Step 4: The combined image f (x, y) is mapped on the spherical surface with a radius of R after it is modulated by the random phase mask RPM (x, y) = exp [iR1 (x, y)]. R1 is a random array uniformly distributed between 0 and 2π. According to the OIP model, the modulated image is transformed into a spherical coordinate. If we use U (x, y) to represent the complex amplitude after spherical diffraction, the expression can be written as U(, x y) = SpD f ( x, y)RPM(, x y) . [ ] (12) Step 5: To improve the anti-attack ability, we use the improved EMD [45] to decompose complex amplitude. The distribution of complex amplitude can be written as U =× A exp(iβ )= C× exp(iR )+ exp(iPK ) , [ ] (13) R =× 2r π and() , (14) where A is the amplitude and β represents the phase of U, rand() denotes a random function. The ciphertext C and the private key PK can be expressed as follows: −1 CA =× 0.5 [] cos(β−R ) , (15) PKR =− 2β , (16) where the ciphertext C and the private key PK are both real-valued, so that they are convenient for Appl. Sci. 2020, 10, 5691 6 of 14 recording and transmission. 3.2. Process of Decryption Step 3: According to the Equation (8), the decrypted images can be obtained by the inverse chaotic scrambling and the inverse filtering diusion in this scheme. Figure 4 shows process of decryption. The detailed steps are as follows: Figure 4. Process of decryption. Figure 4. Process of decryption. 4. Simulation Results Step 1: According to the Equation (13), we can get the complex amplitude U. The U is modulated by RPM* after spherical diffraction. The combined image f (x, y) can be obtained by 4.1. Encryption and Decryption Results f (, x y) = ISpD[U(, x y)]RPM , (17) The numerical simulation for the proposed encryption has been carried out on a PC with Intel (R) Core(TM) i5-7200U CPU 2.50 GHz and 8 GB memory capacity. Python 3.7 is used for this numerical simulation. The outer spherical R, inner spherical r and the wavelength are set as 500 10 m, 3 6 50 10 m and 250 10 m, respectively, as shown in Table 1. According to the sampling theorem in the spherical diraction calculation [41], the pixel numbers in the azimuthal and vertical directions are at least 2514 and 1257, respectively. If we take 1/5 of the surface for spherical diraction, the minimum number of pixels required in the azimuthal and vertical directions are 503 and 252, respectively. Therefore, the size of the original image is selected as 512 512 pixels. Figure 5a–d show the original images with 512 512 pixels for the proposed method. Figure 5e,f shows the private key of the first image P and the combined image f, respectively. Figure 5g,h shows the ciphertext C and the private key PK, respectively. Figure 5a1–d1 shows decrypted images with correct keys. Table 1. The algorithm’s parameters. R r x 3 3 6 500 10 m 50 10 m 250 10 m 0.50 3.70 To evaluate the decryption quality, the peak signal-to-noise (PSNR) and the correlation coecient (CC) between the decrypted and original images are calculated. The formula of PSNR and CC can be given by 2 3 6 7 6 M N 255 7 6 7 6 7 PSNR(I , I ) = 10 log , (19) d 6 P 7 4 5 [ ] I (x, y) I (x, y) 8x,y d E [I E[I ]][I E[I ]] o o d d CC = q q , (20) n o n o 2 2 E [I E[I ]] E [I E[I ]] o o d d where M N is the size of image and E{} denotes the expected value of the function, I and I represent the original and decrypted images, respectively. The PSNR and the CC values between the original images and corresponding decrypted images are infinity and 1, respectively, which shows the decrypted images are lossless. Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 14 where ISpD [•] resents the inverse process of the spherical diffraction calculation. Then f (x, y) is rounded to unify the data types. Step 2: Based on the Equation (11), image IN (x, y) can be obtained by I (, xy)=⊕ f (x, y) P (, x y) . (18) NN Step 3: According to the Equation (8), the decrypted images can be obtained by the inverse chaotic scrambling and the inverse filtering diffusion in this scheme. 4. Simulation Results 4.1. Encryption and Decryption Results The numerical simulation for the proposed encryption has been carried out on a PC with Intel (R) Core(TM) i5-7200U CPU 2.50GHz and 8GB memory capacity. Python 3.7 is used for this numerical −3 simulation. The outer spherical R, inner spherical r and the wavelength λ are set as 500 × 10 m, 50 × −3 −6 10 m and 250 × 10 m, respectively, as shown in Table 1. According to the sampling theorem in the spherical diffraction calculation [41], the pixel numbers in the azimuthal and vertical directions are at least 2514 and 1257, respectively. If we take 1/5 of the surface for spherical diffraction, the minimum number of pixels required in the azimuthal and vertical directions are 503 and 252, respectively. Therefore, the size of the original image is selected as 512 × 512 pixels. Figure 5a–d show the original images with 512 × 512 pixels for the proposed method. Figure 5e,f shows the private key of the first image P1 and the combined image f, respectively. Figure 5g,h shows the ciphertext C and the private key PK, respectively. Figure 5a1–d1 shows decrypted images with correct keys. Table 1. The algorithm’s parameters. R r λ xn μ −3 −3 −6 500 × 10 m 50 × 10 m 250 × 10 m 0.50 3.70 To evaluate the decryption quality, the peak signal-to-noise (PSNR) and the correlation coefficient (CC) between the decrypted and original images are calculated. The formula of PSNR and CC can be given by 2 MN ×× 255 PSNR(, I I ) =10log , (19) od 10 Ix (,y) −I (x,y) [] od ∀xy , EI−− E I I E I [] [ ] { } oo d d CC = , (20) E I−− EI EI EI [] [] {} oo {} d d where M × N is the size of image and E{•} denotes the expected value of the function, Io and Id represent the original and decrypted images, respectively. The PSNR and the CC values between the original images and corresponding decrypted images are infinity and 1, respectively, which shows Appl. Sci. 2020, 10, 5691 7 of 14 the decrypted images are lossless. Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 14 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 14 (a) (b) (c) (d) (e) (f) (g) (h) (e) (f) (g) (h) (a1) (b1) (c1) (d1) (a1) (b1) (c1) (d1) Figure 5. Encryption and decryption results: (a–d) original images, (e) private key for the first image Figure Figure 5. 5. Encry Encryption ption and decr and decryption yption resu results: lts: ( (a a– –d d) original i ) original images, mages, ( (e e) priv ) private ate key for the key for the first first image image P1, (f) combined image f, (g) ciphertext C, (h) private key PK, (a1–d1) decrypted images. P P1, ( , (f f) combine ) combined d image image ff, ( , (g g) c ) ciphertext iphertext C C, ( , (h h) private ) private key key PK PK,, ( (a1 a1– –d1 d1) ) decry decrypted pted images. images. 4.2. Key Sensitivity and Key Space Analysis 4.2. Key Sensitivity and Key Space Analysis 4.2. Key Sensitivity and Key Space Analysis The key sensitivity is usually used to test whether the proposed method is stable. For brevity, only The key sensitivity is usually used to test whether the proposed method is stable. For brevity, The key sensitivity is usually used to test whether the proposed method is stable. For brevity, the decryption image “cameraman” is shown. The dependences of CC on the change of r, R, and are only the decryption image “cameraman” is shown. The dependences of CC on the change of r, R, and only the decryption image “cameraman” is shown. The dependences of CC on the change of r, R, and 6 6 9 shown in Figure 6. When r, R, and value changes 2 10 , 4 10 , 4 10 respectively, CC value −6 −6 −9 −6 −6 −9 λ are shown in Figure 6. When r, R, and λ value changes 2 × 10 , 4 × 10 , 4 × 10 respectively, CC λ are shown in Figure 6. When r, R, and λ value changes 2 × 10 , 4 × 10 , 4 × 10 respectively, CC would change dramatically. It shows that this method is extremely sensitive to the keys. From the value would change dramatically. It shows that this method is extremely sensitive to the keys. From value would change dramatically. It shows that this method is extremely sensitive to the keys. From 6 3 18 above analysis, the key space of this method is (10 ) = 6 10 3 . 18 6 3 18 the above analysis, the key space of this method is (10 ) = 10 . the above analysis, the key space of this method is (10 ) = 10 . (a) (b) (a) (b) Figure 6. Correlation coefficient (CC) values of decrypted images with incorrect keys: (a) Δr/ΔR Figure Figure 6. 6. Correlation coefficient (CC) values of Correlation coecient (CC) values of decrypted decrypted im images ages with i with incorr ncorrect k ect keys: eys: ( (a a) ) DΔr r/ /DΔR R (mm/100), (b) Δλ (μm/100). (mm (mm/100), ( /100), (b b)) DΔλ ( (μm m/100). /100). 4. 4.3. Hi 3. Histo stog gram ram The histogram shows the effectiveness and security of the encryption. The ideal histogram The histogram shows the effectiveness and security of the encryption. The ideal histogram shoul should d be be a al lmost most di differ ffere en nt t from t from th he o e or ri ig gina inal l ima imag ge. e. The hist The histog ogra rams of ms of ori orig gin ina al l im image ages s are are s sh hown in own in Figure 7a–d, respectively. Figure 7e shows the histogram of the encrypted image, which is very Figure 7a–d, respectively. Figure 7e shows the histogram of the encrypted image, which is very different fr different fro om m the origin the original im al image. age. Fig Figu ur re e 7f 7f shows th shows the other encry e other encryp pted image ted image using d using di ifferent fferent origin original al images, which is similar to Figure 7e. images, which is similar to Figure 7e. Appl. Sci. 2020, 10, 5691 8 of 14 4.3. Histogram The histogram shows the eectiveness and security of the encryption. The ideal histogram should be almost dierent from the original image. The histograms of original images are shown in Figure 7a–d, respectively. Figure 7e shows the histogram of the encrypted image, which is very dierent from the original image. Figure 7f shows the other encrypted image using dierent original images, which is similar to Figure 7e. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 14 (a) (b) (c) (d) (e) (f) Figure Figure 7. 7. Histogram: Histogram: ( (aa –– d d ))original original images, images, ( ( ee )) encrypted encrypted image, image, ( (ff )) en encrypted crypted image image using different using dierent original originalimages. images. 4.4. Adjacent Pixel Correlation 4.4. Adjacent Pixel Correlation Table 2 shows the adjacent pixels correlation of the four original images, the encrypted image Table 2 shows the adjacent pixels correlation of the four original images, the encrypted image and the encrypted image in other multi-image encryption method. An ecient encryption should and the encrypted image in other multi-image encryption method. An efficient encryption should significantly reduce the adjacent pixel correlation of the original images. It can be defined as significantly reduce the adjacent pixel correlation of the original images. It can be defined as E[(X )(Y )] EX [( −− μμ )(Y )] X Y XY CCCC (X,(, Y X) Y=) = , , (21) (21) σσ X Y XY where X and Y are two dierent pixel sequences, in which each pixel is the adjacent pixel of X. and where X and Y are two different pixel sequences, in which each pixel is the adjacent pixel of X. μ and denote the mean value and the standard derivation, respectively. E [] represents the mathematical σ denote the mean value and the standard derivation, respectively. E [•] represents the mathematical expectation. expectation. Table 2 shows that the correlation between adjacent pixels of the ciphertext is much lower than Table 2. Adjacent pixel correlation. that of the original images, and it’s also lower than the results of other methods. Correlation Coecient Horizontal Vertical Diagonal Table 2. Adjacent pixel correlation. cameraman 0.966649 0.978250 0.978232 peppers 0.973320 0.976367 0.980875 Correlation Coefficient Horizontal Vertical Diagonal baboon 0.879297 0.799324 0.799965 cameraman 0.966649 0.978250 0.978232 coco 0.989351 0.990687 0.990696 peppers 0.973320 0.976367 0.980875 ciphertext 0.000046 0.000203 0.002142 baboon 0.879297 0.799324 0.799965 Ref. [36] 0.002800 0.034100 0.054700 coco 0.989351 0.990687 0.990696 Ref. [22] 0.026800 0.013500 0.010300 Ref. [23] 0.025300 0.019700 0.026900 ciphertext 0.000046 0.000203 0.002142 Ref. [36] 0.002800 0.034100 −0.054700 Ref. [22] −0.026800 0.013500 0.010300 Table 2 shows that the correlation between adjacent pixels of the ciphertext is much lower than Ref. [23] 0.025300 0.019700 0.026900 that of the original images, and it’s also lower than the results of other methods. 4.5. Noise and Occlusion Attacks Images are susceptible to noise pollution during transmission. It is necessary to ensure that the encryption has a certain anti-noise attack capability. In this section, the multiplicative noise is added to the encrypted image to test the effects of noise. The multiplicative noise model is given by PP=+ (1 kG) , (22) Appl. Sci. 2020, 10, 5691 9 of 14 4.5. Noise and Occlusion Attacks Images are susceptible to noise pollution during transmission. It is necessary to ensure that the encryption has a certain anti-noise attack capability. In this section, the multiplicative noise is added to the encrypted image to test the eects of noise. The multiplicative noise model is given by P = P(1 + kG), (22) Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 14 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 14 where P and P denotes the encrypted image and the encrypted image with noise, respectively. G ′′ where P and P denotes the encrypted image and the encrypted image with noise, respectively. G where P and P denotes the encrypted image and the encrypted image with noise, respectively. G represents the white Gaussian noise with zero-mean and 0.01 standard deviation, k represents the represents th represents the white Gaussian noise w e white Gaussian noise wiith th zero-m zero-me ea an n and and 0. 0.0 01 1 st sta an ndard dard dev deviiat atiio on, n, k k r re epr pres esents ents the the noise strength. noise strength. noise strength. Figure 8 shows the decrypted results of the encrypted image with dierent strength noise. Fig Figu ure re 8 sho 8 show ws the s the decryp decrypted results of t ted results of th he encryp e encrypted ted i im ma ag ge wi e with dif th diff fe erent strength noi rent strength nois se. Fi e. Figure gure Figure 8a–c shows the results of the intensity k = 0.05, 0.07, 0.10, respectively, and the corresponding 8a–c shows the results of the intensity k = 0.05, 0.07, 0.10, respectively, and the corresponding PSNR 8a–c shows the results of the intensity k = 0.05, 0.07, 0.10, respectively, and the corresponding PSNR PSNR values are 30.96, 27.07, 24.36 dB, respectively. v va allu ues es are are 3 30 0..9 96 6,, 27 27.0 .07, 7, 2 24 4..3 36 dB 6 dB, re , resp spect ectiiv ve ely. ly. (a) (b) (c) (a) (b) (c) Figure Figure 8. 8. The The d decrypted ecrypted re results sults oof f the the en encrypted crypted im image age with with different dierent stren strength gth noise: ( noise: a) (k a)= 0.05 k = 0.05, , (b) Figure 8. The decrypted results of the encrypted image with different strength noise: (a) k = 0.05, (b) (k b= 0 ) k .= 07, (c) 0.07, k (c) = 0.10. k = 0.10. k = 0.07, (c) k = 0.10. Figure 9a–c shows the decryption results of the ciphertext center being occluded by 4 4, 6 6, Fig Figu ure re 9a– 9a–c c shows the shows the decrypti decryption resul on result ts of s of the ci the cipherte phertext xt cent center b er be eing ing occl occlude uded d b by y 4 4 × × 4, 4, 6 6 × × 6 6,, 8 8 pixels, respectively, and the corresponding PSNR values are 31.24, 25.51, 24.00 dB, respectively. 8 × 8 pixels, respectively, and the corresponding PSNR values are 31.24, 25.51, 24.00 dB, respectively. 8 × 8 pixels, respectively, and the corresponding PSNR values are 31.24, 25.51, 24.00 dB, respectively. ( (a a) ( ) (b b) ( ) (c c) ) Figure Figure 9. Figure 9. 9. The The The d ddecrypted e ecry cryp pted re ted rersu su esults lt lts o s of fof the the the en enencrypted cry cryp pted ted im image with image age with with different p different p dierent iixels xels pixels oc occl clocclusion: us usiio on: ( n: (a a) ) 4 × 4 × (a 4, ( 4, ( ) 4 b b) 6 ) 6 4, ( × 6, ( × 6, ( b) 6 c c ) ) 8 × 8 × 6, ( 8. 8. c) 8 8. 4.6. Potential Attack Analysis 4.6. Potential Attack Analysis 4.6. Potential Attack Analysis Usually, if the encryption can resist the chosen-plaintext attack, it can resist other common attacks. Usua Usuall lly, if y, if the encrypti the encryption on ca can resist the chos n resist the chosen-p en-pla lain intte ext xt at attta ack, ck, it it can r can re esi sist st ot other common her common Based on the proposed method, the illegal user uses the encryption to encrypt arbitrary images to attacks. Based on the proposed method, the illegal user uses the encryption to encrypt arbitrary attacks. Based on the proposed method, the illegal user uses the encryption to encrypt arbitrary obtain fake private keys, which would be used to decrypt the original image. At the same time, imag images t es to o obt obta ain in fa fake priv ke priva at te keys, wh e keys, whic ich would be u h would be us sed t ed to o decryp decrypt t t th he origina e originall imag image. At e. At t th he same e same the encryption key such as R, r, , and the 2D kernel is known by the illegal user. Figure 10a–d shows ti time, the encrypti me, the encryption key such a on key such as s R R,, r, r, λλ, , and the 2D ker and the 2D kern nel is known el is known by the illeg by the illegal al user user. Figur . Figure e 10a–d 10a–d the decrypted image, which doesn’t have any information about the original image. It shows that the shows the shows the decrypted i decrypted im ma ag ge, whi e, whic ch doesn’t ha h doesn’t have ve a an ny y inf info orma rmat tiio on about the origina n about the originall i im ma ag ge. It e. It shows shows proposed method can resist the chosen-plaintext attack. tha that t the proposed method the proposed method ca can resist the n resist the chosen- chosen-p plai lain ntext a text atta ttack. ck. ( (a a) ) ( (b b) ) ( (c c) ) ( (d d) ) Figure 10. Decryption results of chosen-plaintext attack without key (a) R, (b) r, (c) λ, (d) 2D kernel. Figure 10. Decryption results of chosen-plaintext attack without key (a) R, (b) r, (c) λ, (d) 2D kernel. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 14 where P and P denotes the encrypted image and the encrypted image with noise, respectively. G represents the white Gaussian noise with zero-mean and 0.01 standard deviation, k represents the noise strength. Figure 8 shows the decrypted results of the encrypted image with different strength noise. Figure 8a–c shows the results of the intensity k = 0.05, 0.07, 0.10, respectively, and the corresponding PSNR values are 30.96, 27.07, 24.36 dB, respectively. (a) (b) (c) Figure 8. The decrypted results of the encrypted image with different strength noise: (a) k = 0.05, (b) k = 0.07, (c) k = 0.10. Figure 9a–c shows the decryption results of the ciphertext center being occluded by 4 × 4, 6 × 6, 8 × 8 pixels, respectively, and the corresponding PSNR values are 31.24, 25.51, 24.00 dB, respectively. (a) (b) (c) Figure 9. The decrypted results of the encrypted image with different pixels occlusion: (a) 4 × 4, (b) 6 × 6, (c) 8 × 8. 4.6. Potential Attack Analysis Usually, if the encryption can resist the chosen-plaintext attack, it can resist other common attacks. Based on the proposed method, the illegal user uses the encryption to encrypt arbitrary images to obtain fake private keys, which would be used to decrypt the original image. At the same time, the encryption key such as R, r, λ, and the 2D kernel is known by the illegal user. Figure 10a–d shows the decrypted image, which doesn’t have any information about the original image. It shows Appl. Sci. 2020, 10, 5691 10 of 14 that the proposed method can resist the chosen-plaintext attack. (a) (b) (c) (d) Figure 10. Figure 10. Decr Decryption yption resu results lts of chosen-pla of chosen-plaintext intext attack attack wit without hout k key ey ( (a a) R, ) R, ( (b b) r, ( ) r, (c c) ) λ,, ( (d d) ) 2 2D D kernel. kernel. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 14 Phase retrieval attack is usually used to test the security of the encryption method. Figure 11 Phase retrieval attack is usually used to test the security of the encryption method. Figure 11 shows the relation between the CC value of the combined image and iteration numbers. It shows that shows the relation between the CC value of the combined image and iteration numbers. It shows that the CC value quickly converges to around 0 with the increase of iteration numbers, which indicates the CC value quickly converges to around 0 with the increase of iteration numbers, which indicates that the proposed method can resist phase retrieval attack eectively. that the proposed method can resist phase retrieval attack effectively. Figure 11. CC values change along with the iteration number. Figure 11. CC values change along with the iteration number. An encryption system with good encryption eect should be able to resist dierential attack. An encryption system with good encryption effect should be able to resist differential attack. Generally, the number of pixel change rate (NPCR) and unified average changing intensity (UACI) are Generally, the number of pixel change rate (NPCR) and unified average changing intensity (UACI) used to evaluate the ability to resist the dierential attack, which are defined as follows: are used to evaluate the ability to resist the differential attack, which are defined as follows: MN M N X X NPCR=× D(, i j) 100% , (23) NPCR = D(i, j) 100%, (23) MN × ij == 11 M N i = 1 j = 1 1, E (ij , ) ≠ E (i, j ) Di (, j) = (24) 1, E(i, j) , E (i, j) 0, E (ij , ) = E (i, j) D(i, j) = , (24) 0, E(i, j) = E (i, j) MN ' 1| Ei(, j) − E(i, j)| UACI=× M N 100% , 0 (25) X X E(i, j) E (i, j) MN × 255 ij == 11 UACI = 100%, (25) M N 255 i = 1 j = 1 where E and E are two ciphertexts, generated from two original images that have only one different pixel, and M × N is the size of image. To obtain NPCR and UACI, we choose one of the images to where E and E are two ciphertexts, generated from two original images that have only one dierent change a random pixel, and encrypt them with the same key. The results of NPCR and UACI are pixel, and M N is the size of image. To obtain NPCR and UACI, we choose one of the images to shown in Table 3. It shows that the proposed encryption can resist differential attack very well. change a random pixel, and encrypt them with the same key. The results of NPCR and UACI are shown in Table 3. It shows that the proposed encryption can resist dierential attack very well. Table 3. The values of the number of pixel change rate (NPCR) and unified average changing intensity (UACI). Image NPCR (%) UACI (%) cameraman 99.6046 33.1825 peppers 99.6046 33.9021 baboon 99.6046 33.3199 coco 99.6046 34.1323 4.7. Time and Computational Complexity Analysis In this analysis, we use the original images of size 265 × 256 and 512 × 512 to test the time to encrypt different numbers of images. The results are shown in Table 4. It shows that as the number of encryption increases, the encryption time also increases. At the same time, the computational complexity of the proposed encryption is O (n ). Appl. Sci. 2020, 10, 5691 11 of 14 Table 3. The values of the number of pixel change rate (NPCR) and unified average changing intensity (UACI). Image NPCR (%) UACI (%) cameraman 99.6046 33.1825 peppers 99.6046 33.9021 baboon 99.6046 33.3199 coco 99.6046 34.1323 4.7. Time and Computational Complexity Analysis In this analysis, we use the original images of size 265 256 and 512 512 to test the time to encrypt dierent numbers of images. The results are shown in Table 4. It shows that as the number of encryption increases, the encryption time also increases. At the same time, the computational complexity of the proposed encryption is O (n ). Table 4. The time of encrypting dierent numbers of images. Size One Image Two Images Three Images Four Images 256 256 0.6393 s 1.2354 s 1.8577 s 2.4489 s 512 512 2.5149 s 4.9587 s 7.4733 s 9.7806 s 4.8. Capacity of Image Encryption For a multi-image encryption, the number of encryption images is an essential indicator of measurement. In this proposed encryption, the bit-wise XOR operation for two images will maintain the pixel value range of original images. That is, the pixel value of the original images is in the range of 0–255, and the pixel value of the combined image is still in the range of 0–255 after using the XOR operation. Therefore, multi-image encryption through XOR multiplexing should have no decryption crosstalk theoretically. Since the private key of each dierent image generated in the encryption process must be dierent, the encryption capacity is limited by the maximum number of possible private keys, which is the product of the maximum grayscale level and the resolution of the private key image. Therefore, the encryption capacity EC can be expressed as EC = 2 m n, (26) where w represents the bit-depth of the encrypted image, m n represents the size of the encrypted image. If the size of the encrypted image is m n, and the grayscale level is 8, the encryption capacity would be 2 m n. We preprocess the images to ensure that each image is dierent. Figure 12 shows the decryption results of 200, 500, 1000, and 2000 dierent images with 64 64 pixels and their CC values, respectively. It can be seen that the decryption quality is still excellent when the number of encryption images reaches 2000. This indicates that the proposed encryption breaks through the number limitation of the conventional multi-image encryptions and greatly increases their encryption capacity. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 14 Table 4. The time of encrypting different numbers of images. Size One Image Two Images Three Images Four Images 256 × 256 0.6393 s 1.2354 s 1.8577 s 2.4489 s 512 × 512 2.5149 s 4.9587 s 7.4733 s 9.7806 s 4.8. Capacity of Image Encryption For a multi-image encryption, the number of encryption images is an essential indicator of measurement. In this proposed encryption, the bit-wise XOR operation for two images will maintain the pixel value range of original images. That is, the pixel value of the original images is in the range of 0–255, and the pixel value of the combined image is still in the range of 0–255 after using the XOR operation. Therefore, multi-image encryption through XOR multiplexing should have no decryption crosstalk theoretically. Since the private key of each different image generated in the encryption process must be different, the encryption capacity is limited by the maximum number of possible private keys, which is the product of the maximum grayscale level and the resolution of the private key image. Therefore, the encryption capacity EC can be expressed as (26) EC=2 ×× m n , where w represents the bit-depth of the encrypted image, m × n represents the size of the encrypted image. If the size of the encrypted image is m × n, and the grayscale level is 8, the encryption capacity would be 2 × m × n. We preprocess the images to ensure that each image is different. Figure 12 shows the decryption results of 200, 500, 1000, and 2000 different images with 64 × 64 pixels and their CC values, respectively. It can be seen that the decryption quality is still excellent when the number of encryption images reaches 2000. This indicates that the proposed encryption breaks through the number limitation of the conventional multi-image encryptions and greatly increases their encryption Appl. Sci. 2020, 10, 5691 12 of 14 capacity. Figure 12. Decryption results of 200, 500, 1000, and 2000 different images with 64 × 64 pixels and their Figure 12. Decryption results of 200, 500, 1000, and 2000 dierent images with 64 64 pixels and their CC value. CC value. 5. Conclusions 5. Conclusions In this article, we propose a large-capacity multi-image encryption using spherical diraction In this article, we propose a large-capacity multi-image encryption using spherical diffraction and filtering diusion. In this encryption, original images are combined into one image using the and filtering diffusion. In this encryption, original images are combined into one image using the XOR operation after filtering diusion. Then, the superimposed image is transformed into the XOR operation after filtering diffusion. Then, the superimposed image is transformed into the spherical diraction domain and encrypted by the improved EMD. Due to the remainder operation spherical diffraction domain and encrypted by the improved EMD. Due to the remainder operation in the XOR operation, the range of the combined image is always maintained in the original image in the XOR operation, the range of the combined image is always maintained in the original image range, which would not exceed the limitation of image range or cause data crosstalk. Therefore, the range, which would not exceed the limitation of image range or cause data crosstalk. Therefore, the encryption capacity is the product of image resolution and grayscale level, which is super-large of encryption capacity is the product of image resolution and grayscale level, which is super-large of 2 8 8 2 m n in case of m n points of an image with 2 grayscale level. Moreover, the proposed method × m × n in case of m × n points of an image with 2 grayscale level. Moreover, the proposed method is highly flexible due to individual decryption with the individual private key for each plaintext is highly flexible due to individual decryption with the individual private key for each plaintext image. Besides, it can resisting phase-retrieval attack due to the asymmetry of spherical diraction. The encryption system significantly increases the capacity of the multi-image encryption system, though it has weak anti-occlusion ability. In further work, the encryption method can be used in the color multi-image encryption system. The feasibility and eectiveness of the proposed encryption have been demonstrated by the simulation results. Author Contributions: Conceptualization, H.W. and J.W.; methodology, H.W.; software, H.W., X.C. and Z.Z. (Zheng Zhu); validation, H.W. and J.W.; formal analysis, Z.Z. 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