# A Secure and Efficient Image Transmission Scheme Based on Two Chaotic Maps

Authors: Wei Feng; Jing Zhang; Zhentao Qin
Journal: Complexity
 (2021)

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## Abstract

The application of multimedia sensors is widespread, and people need to transmit images more securely and efficiently. In this paper, an image transmission scheme based on two chaotic maps is proposed. The proposed scheme consists of two parts, secure image transmission between sensor nodes and sink nodes (SIT-SS) and secure image transmission between sensor nodes and receivers (SIT-SR). For resource-constrained environments, SIT-SS utilizes Tent-Logistic Map (TLM) to generate chaotic sequences and adopts TLM-Driven permutation and transformation to confuse image pixels. Then the cipher image is obtained through TLM-Driven two-dimensional compressed sensing. Compared with existing schemes, the secret key design of SIT-SS is more reasonable and requires fewer hardware resources. When sampling ratio is greater than 0.6, its image reconstruction quality has obvious advantages. For environments with huge security threats, SIT-SR adopts dynamic permutation and confusion based on discrete logarithms to confuse the image and exploits dynamic diffusion based on discrete logarithms to generate final cipher image. Similarly, compared with some existing schemes, the design of SIT-SR is more practical, and the statistical characteristics of the cipher image are better. Finally, extensive simulation tests confirm the superiority of the proposed scheme.


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## Body

## 1. Introduction

Nowadays, the application of multimedia sensors is increasingly widespread in many fields, such as medicine, transportation, industry, education, and military. In these application scenarios, flexibly deployed sensors need to transmit massive images, such as medical and military images [1, 2]. Since it involves privacy protection, commercial and military security, etc., efficient and secure protection needs to be provided for these images. However, image data has several significant characteristics that are different from text data, such as large volume and strong pixel correlation [3]. And the hardware resources of sensors are limited. Therefore, traditional encryption schemes such as Advanced Encryption Standard (AES) are generally not suitable for heterogeneous application environments [4–7]. In order to continuously improve the efficiency and security of image transmission, researchers have been committed to designing new schemes based on emerging techniques and methods [3–28]. Among these new schemes, the ones based on compressed sensing (CS) and chaotic systems are favored by more and more researchers [11–13, 16–28].CS [29, 30] is a breakthrough signal acquiring paradigm, which can effectively capture and recover a signal with fewer nonadaptive samples. Once introduced, CS is quickly applied to image related information security applications [4–7, 10, 11, 31–37]. In the past decade, researchers have gradually introduced CS into information security applications in resource-constrained environments. In [4], a scheme called Diffie-Hellman-Hash-Compression was proposed. This scheme uses Semitensor Product (STP) CS to encrypt images of different dimensions and adopts hash algorithm and permutation operations to ensure secure image transmission. Taking into account the high privacy sensitivity and redundancy of medical images, Wang et al. [5] constructed a CS based medical image encryption scheme. This scheme carries out image encryption between sensor nodes by using a measurement matrix as the secret key and can realize image compression, privacy protection, and data aggregation simultaneously. In order to overcome the resource constraints of sensor nodes and ensure the security of data transmission, an image encryption system was exploited [6]. While enhancing the security of transmission process by integrating the quantization and diffusion operations, the system uses a new CS model and parallel reconstruction algorithm to shorten the encryption/decryption time. In [7], a flexible and secure data encryption system based on CS was proposed. The plain image is first sparsely represented through discrete wavelet transform and then permutated by Arnold scrambling. Finally, after CS and logistic chaotic permutation, the cipher image is obtained. Utilizing structurally random matrices, Unde et al. [10] presented an efficient scheme based on CS. In their scheme, artificial noise is injected into quantized CS measurements, thereby enhancing the ability to resist Chosen-Plaintext Attacks (CPAs).Chaotic systems have several characteristics that are very suitable for designing cryptosystems [1, 2]. Consequently, more and more researchers leverage chaotic systems to design various image encryption schemes. In [16], an image encryption scheme using memristive chaotic system was provided. This scheme uses Secure Hash Algorithm (SHA) to generate the secret key and calculate the initial value of the chaotic system. And it also introduces a dynamic Deoxyribonucleic Acid (DNA) encoding method to generate two regular DNA matrices for encoding images. In order to protect medical images, Moafimadani et al. [23] presented an image encryption scheme based on a chaotic system, which uses a fast permutation operation to scramble the plain image and utilizes an adaptive diffusion operation to generate the cipher image. In [24], a chaotic image encryption scheme using a new symmetric key generation system was proposed. This scheme exploits block-level permutation and improved zigzag transformation to achieve the confusion effect and adopts pixel shuffling to complete the pixel diffusion operation. With the goal of improving the security and efficiency of image encryption, Zhu et al. [25] proposed an efficient and simple S-box generation method using a new compound chaotic system and then introduced a new image encryption scheme based on double S-boxes. Based on dynamic DNA encoding and two chaotic systems, Zhou et al. [26] proposed an image encryption scheme with a two-round permutation-diffusion structure. This scheme exploits a two-dimensional (2D) rectangular transformation to complete the permutation operation, and before the diffusion operation, the hamming distances of DNA matrices are used to update the initial values of the chaotic systems.As can be seen from abovementioned works, in terms of designing image encryption schemes, reducing resource consumption and achieving higher security are key motivations. Although these schemes have advantages in some aspects, they all have room for further improvements. For example, the scheme proposed in [4] adopts SHA to resist CPAs. However, the implementation of SHA demands considerable hardware resources and would hinder the applicability of this scheme in resource-constrained environments. In addition, some encryption schemes adopt one-time pad secret key. When a large number of images need to be encrypted, such design is not practical. Therefore, while further improving the efficiency and security of image encryption, to overcome the shortcomings of these schemes, an image transmission scheme based on two chaotic maps, 2D-CS, dynamic perturbation, and discrete logarithms (ITS-CDD) is proposed. The proposed scheme consists of two parts, secure image transmission between sensor nodes and sink nodes (SIT-SS) and secure image transmission between sensor nodes and receivers (SIT-SR). Compared with some existing schemes, ITS-CDD has contributions summarized as follows:(1)
SIT-SS is designed for resource-constrained environment, whereas SIT-SR is designed for environments with huge security threats. Therefore, the applicability and practicability of ITS-CDD are higher.(2)
Dynamic perturbation parameters (DPPs) derived from system times and last encryption times are designed. So, ITS-CDD not only guarantees the diversity of equivalent key streams, but also does not rely on external algorithms.(3)
The secret key design of SIT-SS is more practical and requires fewer hardware resources.(4)
2D-CS based on lightweight chaotic map can reduce resource overhead.(5)
Discrete logarithms under finite multiplicative groupZ257∗ are introduced to ensure higher security.The remainder of this paper is organized as follows. 2D-CS, discrete logarithms, and two chaotic systems are introduced in Section2. ITS-CDD is described in Section 3. Simulation tests and theoretical analyses are carried out in Section 4. Finally, conclusions are drawn in Section 5.


## 2. Fundamental Knowledge

In SIT-SS, 2D-CS is introduced to realize image data compression and encryption. Discrete logarithms are used to enhance the security of SIT-SR. Two chaotic systems called Tent-Logistic Map (TLM) [38] and 2D Logistic-Sine-Coupling Map (2D-LSCM) [13] are adopted to generate the chaotic sequences.


### 2.1. 2D-CS

In terms of computational complexity and storage space, 2D-CS has obvious advantages over traditional CS [39, 40]. Assuming that A and B are random matrices, they both have the size of M×NM≪N. Then, one can obtain the 2D measurements Y∈RM×M of an image X∈RN×N. Specifically,(1)Y=AXBT,where A and B operate on the rows and columns of X, respectively.When decoding, one can regularize the image signal recovery by using signal prior information in the form of penalty:(2)X∧=argminfXX=12Y−AXBTF2+λJX,where λ is the regularization parameter, JX is a cost function which is used to handle the ill-posed problem, and 1/2Y−AXBTF2 is the l2 data-fidelity term. Moreover, researchers have proposed many 2D-CS reconstruction algorithms to solve the optimization problem mentioned above. In this paper, 2D projected gradient with embedding decryption (2DPG-ED) [12] algorithm is adopted.


### 2.2. Discrete Logarithms

Discrete logarithm calculation is a complex nonlinear calculation. In the encryption process, the use of discrete logarithms can improve its nonlinearity [14]. For the prime 257 and its corresponding finite multiplicative group Z257∗, one can define the discrete logarithms as follows: if d∈Z257∗ satisfies n≡gdmodp, then d is said to be the discrete logarithm of n∈Z257∗. Since Z257∗ has 128 generators, we can use them to enhance the diversity of equivalent key streams. To avoid complex discrete logarithm calculation, we calculate the discrete logarithm values under different generators in advance and save them to the 2D matrix DVM with the size of 128×256. Consequently, in ITS-CDD, discrete logarithm values can be obtained by directly accessing DVM. If one wants to calculate the discrete logarithm value of 107 under the generator 3, namely, calculating log3107mod257, one can access DVM1,107 to obtain the discrete logarithm value 31. Table 1 shows the discrete logarithm values of 101 to 107 under the first eight generators.Table 1
Discrete logarithm values of 101 to 107 under the first eight generators.
Row index ofDVMCorresponding generatorgn (column index of DVM)101 (101)102 (102)103 (103)104 (104)105 (105)106 (106)107 (107)1375169201250141137312514131255214916389365924925262921779473151651889101163510125111792462351431376124373105581734112771414321311750105538381910315761281253203


### 2.3. TLM and 2D-LSCM

To save hardware resources, TLM is adopted in SIT-SS, which is easy to implement and has good chaotic performance. TLM can be defined as(3)xi=r1r2xi−11−r2xi−1,xi−1<0.5,r1r21−xi−11−r21−xi−1,xi−1≥0.5,where xi is generated by the i-th iteration, xi−1 is the input of the i-th iteration, x0∈0,1 is the initial state, and r1∈3.57,4,r2∈1,2 are the control parameters. Figure 1 shows the 2D bifurcation diagram and Lyapunov Exponents (LE) diagrams of TLM.Figure 1
2D bifurcation diagram and LE diagrams of TLM: (a) 2D bifurcation diagram; (b) LE diagram versus parameterr1; (c) LE diagram versus parameter r2.
(a)(b)(c)Compared with TLM, 2D-LSCM has better chaotic performance, but its structure is more complex, so it is more suitable for environments with more hardware resources. 2D-LSCM can be defined as(4)xi=sinπ4γxi−11−xi−1+1−γsinπyi−1,yi=sinπ4γyi−11−yi−1+1−γsinπxi−1,where xi,yi is the system state generated by the i-th iteration, xi−1,yi−1 is the input of the i-th iteration, x0,y0 is the initial state, and γ is the control parameter. The value ranges of all these parameters are 0,1. Figure 2 shows the 2D bifurcation diagram and LE diagram of 2D-LSCM.Figure 2
2D bifurcation diagram and LE diagram of 2D-LSCM: (a) 2D bifurcation diagram; (b) LE diagram versus parameterγ.
(a)(b)


## 2.1. 2D-CS

In terms of computational complexity and storage space, 2D-CS has obvious advantages over traditional CS [39, 40]. Assuming that A and B are random matrices, they both have the size of M×NM≪N. Then, one can obtain the 2D measurements Y∈RM×M of an image X∈RN×N. Specifically,(1)Y=AXBT,where A and B operate on the rows and columns of X, respectively.When decoding, one can regularize the image signal recovery by using signal prior information in the form of penalty:(2)X∧=argminfXX=12Y−AXBTF2+λJX,where λ is the regularization parameter, JX is a cost function which is used to handle the ill-posed problem, and 1/2Y−AXBTF2 is the l2 data-fidelity term. Moreover, researchers have proposed many 2D-CS reconstruction algorithms to solve the optimization problem mentioned above. In this paper, 2D projected gradient with embedding decryption (2DPG-ED) [12] algorithm is adopted.


## 2.2. Discrete Logarithms

Discrete logarithm calculation is a complex nonlinear calculation. In the encryption process, the use of discrete logarithms can improve its nonlinearity [14]. For the prime 257 and its corresponding finite multiplicative group Z257∗, one can define the discrete logarithms as follows: if d∈Z257∗ satisfies n≡gdmodp, then d is said to be the discrete logarithm of n∈Z257∗. Since Z257∗ has 128 generators, we can use them to enhance the diversity of equivalent key streams. To avoid complex discrete logarithm calculation, we calculate the discrete logarithm values under different generators in advance and save them to the 2D matrix DVM with the size of 128×256. Consequently, in ITS-CDD, discrete logarithm values can be obtained by directly accessing DVM. If one wants to calculate the discrete logarithm value of 107 under the generator 3, namely, calculating log3107mod257, one can access DVM1,107 to obtain the discrete logarithm value 31. Table 1 shows the discrete logarithm values of 101 to 107 under the first eight generators.Table 1
Discrete logarithm values of 101 to 107 under the first eight generators.
Row index ofDVMCorresponding generatorgn (column index of DVM)101 (101)102 (102)103 (103)104 (104)105 (105)106 (106)107 (107)1375169201250141137312514131255214916389365924925262921779473151651889101163510125111792462351431376124373105581734112771414321311750105538381910315761281253203


## 2.3. TLM and 2D-LSCM

To save hardware resources, TLM is adopted in SIT-SS, which is easy to implement and has good chaotic performance. TLM can be defined as(3)xi=r1r2xi−11−r2xi−1,xi−1<0.5,r1r21−xi−11−r21−xi−1,xi−1≥0.5,where xi is generated by the i-th iteration, xi−1 is the input of the i-th iteration, x0∈0,1 is the initial state, and r1∈3.57,4,r2∈1,2 are the control parameters. Figure 1 shows the 2D bifurcation diagram and Lyapunov Exponents (LE) diagrams of TLM.Figure 1
2D bifurcation diagram and LE diagrams of TLM: (a) 2D bifurcation diagram; (b) LE diagram versus parameterr1; (c) LE diagram versus parameter r2.
(a)(b)(c)Compared with TLM, 2D-LSCM has better chaotic performance, but its structure is more complex, so it is more suitable for environments with more hardware resources. 2D-LSCM can be defined as(4)xi=sinπ4γxi−11−xi−1+1−γsinπyi−1,yi=sinπ4γyi−11−yi−1+1−γsinπxi−1,where xi,yi is the system state generated by the i-th iteration, xi−1,yi−1 is the input of the i-th iteration, x0,y0 is the initial state, and γ is the control parameter. The value ranges of all these parameters are 0,1. Figure 2 shows the 2D bifurcation diagram and LE diagram of 2D-LSCM.Figure 2
2D bifurcation diagram and LE diagram of 2D-LSCM: (a) 2D bifurcation diagram; (b) LE diagram versus parameterγ.
(a)(b)


## 3. Proposed Image Transmission Scheme

Different from some existing schemes, ITS-CDD consists of two parts, secure image transmission between sensor nodes and sink nodes (SIT-SS) and secure image transmission between sensor nodes and receivers (SIT-SR). Figure3 shows the secure image transmission between sensor nodes and sink nodes.Figure 3
Secure image transmission between sensor nodes and sink nodes.Compared with the existing schemes, SIT-SS has two main innovations. One is introducing TLM to save the hardware resources of sensors, and the other is introducing DPPs to enhance the ability to resist CPAs. Figure4 shows the secure image transmission between sink nodes and receivers.Figure 4
Secure image transmission between sink nodes and receivers.Considering that sink nodes have more resources, there are huge security threats in the process of transmitting images to receivers through the media cloud. We have adopted some measures to improve the security of image transmission, such as the adoption of 2D-LSCM with better chaotic performance and the introduction of discrete logarithms.


### 3.1. Transmission between Sensor Nodes and Sink Nodes

To save space, in this subsection, we mainly introduce the improvements to 2DCS-ETC [12].


#### 3.1.1. DPP Generation

According to previous cryptanalysis works, the main reason why some schemes cannot resist CPAs is that equivalent key streams only depend on the secret key [41–47]. Therefore, some researchers use the hash value of the plain image to ensure the diversity of equivalent key streams. However, the implementation of hash algorithm is not suitable for sensor nodes with limited resources. Considering that system times and last encryption times are constantly changing and would be affected by many factors, they are used to generate DPPs. The specific generation process of DPPs is as follows:(i)
Step 1: obtain the system timeTs in milliseconds.(ii)
Step 2: get the timeTe spent in the last encryption process in milliseconds. If it is the first time to encrypt, set Te to an initial value Ti.(iii)
Step 3: one DPP is obtained byβ=Ts+Temod256.(iv)
Step 4: repeat Step 1 through Step 3 until 32 DPPs are obtained, namely,β1,β2,…,β32.In this way, we can obtain a set of DPPs. Like the hash value, DPPs can ensure that the equivalent key streams used when encrypting different images are different, thereby effectively resisting CPAs. More importantly, no complicated calculations are required to obtain DPPs, and even if the same plain image is encrypted, different equivalent key streams would be generated.


#### 3.1.2. TLM-Driven Global Permutation

Obviously, confusion is the requirement that must be considered when designing modern cryptosystems. Confusion means that each bit of the secret key should affect as many cipher image bits as possible [48]. Permutation operations are commonly used to achieve confusion, but permutation-only image encryption schemes have been proven to be insecure [49]. Therefore, SIT-SS introduces DPPs in the permutation process. This makes the permutation process not only dependent on the secret key, but also dependent on the DPPs that will inevitably change every time the plain image is encrypted. Compared with 2DCS-ETC using the random permutation matrix to complete the permutation and treat it as secret key, we use TLM and DPPs to complete the permutation. This can not only reduce the resource overhead of sensor nodes, but also improve the ability to resist CPAs. The specific process of TLM-Driven Global Permutation (GP) is as follows:(i)
Step 1: use the parametersr11,r21,x1 to iterate TLM N×N+r31 times. In order to avoid negative effects, discard the first r31 chaotic state values.(ii)
Step 2: convert the obtained chaotic sequenceS of length N×N into the integer sequence(5)Si=floorSi×1015modN×N+1,wherei=1,2,…,N×N, floor⋅ returns the integer part of an operand.(iii)
Step 3: stretch the plain imageP of size N×N into the 1D sequence P˜.(iv)
Step 4: calculate the index(6)I=Simod32+1,of 32 DPPs and the permutation position(7)α=Si+βImodN×N+1,wherei=1,2,…,N×N. Swap two pixels of P˜ according to α.


#### 3.1.3. TLM-Driven Negative-Positive Transformation

A nonlinear operation called Negative-Positive Transformation (NPT) is introduced by 2DCS-ETC to improve security. Similarly, we use TLM and DPPs to complete NPT instead of using a random matrix in the form of secret key. This can further reduce the resource overhead of sensor nodes and improve the ability to resist CPAs.(i)
Step 1: use parameterr12,r22,x2 to iterate TLM N×N+r32 times. In order to avoid negative effects, discard the first r32 chaotic state values.(ii)
Step 2: convert the obtained chaotic sequenceS of length N×N into the bit sequence(8)Si=floorSi×1015mod2,wherei=1,2,…,N×N.(iii)
Step 3: according toS, perform the following NPT operation on P˜.(9)Ci=P˜i,Si=1,255−P˜i,Si=0,wherei=1,2,…,N×N.(iv)
Step 4: reshapeC into the 2D cipher image.


#### 3.1.4. TLM-Driven 2D-CS

If the chaotic sequence generated by the chaotic system is assembled into a complete measurement matrix, its performance is usually almost the same as other commonly used random matrices [11]. Moreover, compared with directly using a random matrix and treating it as secret key, the chaotic measurement matrix can significantly save the resource overhead of sensor nodes. In SIT-SS, TLM is used to generate the measurement matrices required for 2D-CS. Suppose the size of the measurement matrices A and B to be created is M×NM≪N; the specific process of TLM-Driven 2D-CS is as follows:(i)
Step 1: use the parametersr13,r23,x3 to iterate TLM N×N+r33 times. In order to avoid negative effects, discard the first r33 chaotic state values.(ii)
Step 2: arrange the obtained chaotic sequence into the square matrixS∧ of size N×N.(iii)
Step 3: takeM rows from the orthogonal basis of S∧ as the measurement matrix A.(iv)
Step 4: repeat Step 1 through Step 3; create the measurement matrixB in a similar manner.(v)
Step 5: useA and B to obtain the 2D measurements of the cipher image C.In addition to the improvements made above, the other steps of SIT-SS are basically the same as those of 2DCS-ETC, which are not repeated here. Since we have introduced TLM and DPPs in SIT-SS, the security of image transmission between sensor nodes and sink nodes has become higher, and the resource requirements for sensors are also lower. Significantly, SIT-SS still maintains the advantages of 2DCS-ETC, which is demonstrated and discussed in Section4.1. To save hardware resources, we directly use r11,r21,x1,r12,r22,x2,r13,r23,x3 as the secret key of SIT-SS.


### 3.2. Transmission between Sink Nodes and Receivers

In SIT-SR, we use 2D-LSCM [13] which has better chaotic performance to generate chaotic sequences. Moreover, discrete logarithms and DPPs are introduced to achieve secure image transmission between sink nodes and receivers. It should be noted that through the use of discrete logarithms and our targeted design, DPPs can be directly sent out in plaintext form by sink nodes. When decrypting, receivers can directly use DPPs that arrived in plaintext form. In other words, DPPs are not one-time pad secret keys, nor are they secret parameters. Next, we introduce the specific process of SIT-SR, as shown in Figure 5.Figure 5
Flowchart of SIT-SR.


#### 3.2.1. Secret Key and Chaotic System Parameters

In order to avoid the secret key issues pointed out in some cryptanalysis works and simplify the generation process of chaotic system parameters [14, 41, 42], we set the secret key K in this stage as a binary sequence with the length of 270 bits. Namely, K=a1a2⋯a270. In specific implementation, we directly use nine 32-bit unsigned integers b1, b2, b3, b4, b5, b6, b7, b8, b9 to generate three sets of parameters x∧01,y∧01,r∧1, x∧02,y∧02,r∧2, x∧03,y∧03,r∧3 for 2D-LSCM. As shown in equation (11), this means that the 30×9 bits of K correspond to the 30 bits of each unsigned integer, respectively.(10)x∧01=b1×2+1×2−32,y∧01=b2×2+1×2−32,r∧01=b3×2+1×2−32,x∧02=b4×2+1×2−32,y∧02=b5×2+1×2−32,r∧02=b6×2+1×2−32,x∧03=b7×2+1×2−32,y∧03=b8×2+1×2−32,r∧03=b9×2+1×2−32,where(11)b1=a1a2⋯a30,b2=a31a32⋯a60,b3=a61a62⋯a90,b4=a91a92⋯a120,b5=a121a122⋯a150,b6=a151a152⋯a180,b7=a181a182⋯a210,b8=a211a212⋯a240,b9=a241a242⋯a270.Besides, these three sets of chaotic system parametersx∧01,y∧01,r∧1, x∧02,y∧02,r∧2, x∧03,y∧03,r∧3 are used to generate chaotic matrices for the encryption process.


#### 3.2.2. DPP Generation

The generation process of DPPs in SIT-SR is exactly the same as SIT-SS. And we mark 32 DPPs used in SIT-SR asH.


#### 3.2.3. Dynamic Permutation and Confusion Based on Discrete Logarithms

As mentioned above, discrete logarithms and DPPs are introduced in dynamic permutation and confusion based on discrete logarithms (DPC-D), so as to enhance the security of image transmission. Specifically, compared with some existing permutation operations, DPC-D has the following advantages:(1)
UseH to further perturb the permutation results and adopt different perturbation strategies for the row index and column index. Therefore, the permutation results depend not only on the secret key, but also on H.(2)
Based on discrete logarithms,H and the sorting results of the chaotic matrix S are used to nonlinearly transform the pixel value of each plain image pixel, thereby further improving the security of image transmission.In order to better describe the specific steps of DPC-D, an algorithm is provided in Algorithm1.Algorithm 1: DPC-D algorithm.
Require: the plain image P with the size of N×N, the chaotic matrix S with the size of N×N, the dynamic perturbation parameters H with the size of 1×32 and the discrete logarithm value matrixDVM with the size of 128×256.(1)
SetT∈NN×N;(2)
Set the sumhs of the dynamic perturbation parameters to 0;(3)
Set the row index valueg used to represent the adopted generator to 0;(4)
Set the index valueidx used to represent the adopted dynamic perturbation parameters to 0;(5)
Sort each column ofS in ascending order, thus get the column index matrix O and sorted result B;(6)
fori=1 to 32 do(7)
hs=hs+Hi;(8)
end for(9)
Calculate the row index value of the generator to be used, namely letg=hsmod128+1;(10)
fori=1 to Ndo(11)
SortBi in ascending order and obtain the row index vector v;(12)
forj=1 to Ndo(13)
idx=i−1×N+jmod32+1;(14)
TOi,j+HidxmodN+1,j+hsmodN+1=DVMg,POi,vj,vj+Hidx+vjmod256+1−1;(15)
end for(16)
end forEnsure: the permuted and transformed image T.


### 3.3. Dynamic Diffusion Based on Discrete Logarithms

To further improve security, dynamic diffusion based on discrete logarithms (DD-D) also adopts discrete logarithms andH. Specifically, compared with some existing diffusion operations, DD-D has the following advantages:(1)
Considering that multipixel diffusion is of little significance, single-pixel diffusion is adopted, thereby reducing the amount of computation(2)
The nonlinearity of the diffusion process is improved by introducing discrete logarithms; thus the security of image transmission is further improvedIn order to better describe the specific steps of DD-D, an algorithm is provided in Algorithm2.Algorithm 2: DD-D algorithm.
Require: the permuted and transformed image T with the size of N×N, the chaotic matrix R with the size of N×N, the dynamic perturbation parameters H with the size of 1×32 and the discrete logarithm value matrix DVM with the size of 128×256.(1)
SetC∈NN×N;(2)
ConvertR into the integer matrix IR with the same format as the pixels of T, namely IR=⌊R×232⌋mod256;(3)
Set the bitwise XOR resulthx of the dynamic perturbation parameters to 0;(4)
Set the row index valuesg1, g2 used to represent the adopted generators to 0 s;(5)
fori=1 to 32 do(6)
hx=bitxorhx,Hi;(7)
end for(8)
Calculate the row index values of the generators to be used, letg1=bitxorhx,IR1,1mod128+1, g2=bitxorg1,IRN,Nmod128+1;(9)
tmp:,1=T:,1+DVMg1,T:,N+1+DVMg2,IR:,1+1mod256;(10)
fori=2 to Ndo(11)
tmp:,i=T:,i+DVMg1,tmp:,i−1+1+DVMg2,IR:,i+1mod256;(12)
end for(13)
C1,:=tmp1,:+DVMg1,tmpN,:+1+DVMg2,IR1,:+1mod256;(14)
fori=2 to Ndo(15)
Ci,:=tmpi,:+DVMg1,Ci−1,:+1+DVMg2,IRi,:+1mod256;(16)
end forEnsure: the diffused image C.Since a symmetric encryption structure is adopted in SIT-SR, the decryption process is actually constituted by the corresponding inverse operations of the encryption operations. With the received DPPs and the agreed secret keyK, receivers can decrypt the plain image from the cipher image. To save space, these inverse operations are not repeated here.


## 3.1. Transmission between Sensor Nodes and Sink Nodes

To save space, in this subsection, we mainly introduce the improvements to 2DCS-ETC [12].


### 3.1.1. DPP Generation

According to previous cryptanalysis works, the main reason why some schemes cannot resist CPAs is that equivalent key streams only depend on the secret key [41–47]. Therefore, some researchers use the hash value of the plain image to ensure the diversity of equivalent key streams. However, the implementation of hash algorithm is not suitable for sensor nodes with limited resources. Considering that system times and last encryption times are constantly changing and would be affected by many factors, they are used to generate DPPs. The specific generation process of DPPs is as follows:(i)
Step 1: obtain the system timeTs in milliseconds.(ii)
Step 2: get the timeTe spent in the last encryption process in milliseconds. If it is the first time to encrypt, set Te to an initial value Ti.(iii)
Step 3: one DPP is obtained byβ=Ts+Temod256.(iv)
Step 4: repeat Step 1 through Step 3 until 32 DPPs are obtained, namely,β1,β2,…,β32.In this way, we can obtain a set of DPPs. Like the hash value, DPPs can ensure that the equivalent key streams used when encrypting different images are different, thereby effectively resisting CPAs. More importantly, no complicated calculations are required to obtain DPPs, and even if the same plain image is encrypted, different equivalent key streams would be generated.


### 3.1.2. TLM-Driven Global Permutation

Obviously, confusion is the requirement that must be considered when designing modern cryptosystems. Confusion means that each bit of the secret key should affect as many cipher image bits as possible [48]. Permutation operations are commonly used to achieve confusion, but permutation-only image encryption schemes have been proven to be insecure [49]. Therefore, SIT-SS introduces DPPs in the permutation process. This makes the permutation process not only dependent on the secret key, but also dependent on the DPPs that will inevitably change every time the plain image is encrypted. Compared with 2DCS-ETC using the random permutation matrix to complete the permutation and treat it as secret key, we use TLM and DPPs to complete the permutation. This can not only reduce the resource overhead of sensor nodes, but also improve the ability to resist CPAs. The specific process of TLM-Driven Global Permutation (GP) is as follows:(i)
Step 1: use the parametersr11,r21,x1 to iterate TLM N×N+r31 times. In order to avoid negative effects, discard the first r31 chaotic state values.(ii)
Step 2: convert the obtained chaotic sequenceS of length N×N into the integer sequence(5)Si=floorSi×1015modN×N+1,wherei=1,2,…,N×N, floor⋅ returns the integer part of an operand.(iii)
Step 3: stretch the plain imageP of size N×N into the 1D sequence P˜.(iv)
Step 4: calculate the index(6)I=Simod32+1,of 32 DPPs and the permutation position(7)α=Si+βImodN×N+1,wherei=1,2,…,N×N. Swap two pixels of P˜ according to α.


### 3.1.3. TLM-Driven Negative-Positive Transformation

A nonlinear operation called Negative-Positive Transformation (NPT) is introduced by 2DCS-ETC to improve security. Similarly, we use TLM and DPPs to complete NPT instead of using a random matrix in the form of secret key. This can further reduce the resource overhead of sensor nodes and improve the ability to resist CPAs.(i)
Step 1: use parameterr12,r22,x2 to iterate TLM N×N+r32 times. In order to avoid negative effects, discard the first r32 chaotic state values.(ii)
Step 2: convert the obtained chaotic sequenceS of length N×N into the bit sequence(8)Si=floorSi×1015mod2,wherei=1,2,…,N×N.(iii)
Step 3: according toS, perform the following NPT operation on P˜.(9)Ci=P˜i,Si=1,255−P˜i,Si=0,wherei=1,2,…,N×N.(iv)
Step 4: reshapeC into the 2D cipher image.


### 3.1.4. TLM-Driven 2D-CS

If the chaotic sequence generated by the chaotic system is assembled into a complete measurement matrix, its performance is usually almost the same as other commonly used random matrices [11]. Moreover, compared with directly using a random matrix and treating it as secret key, the chaotic measurement matrix can significantly save the resource overhead of sensor nodes. In SIT-SS, TLM is used to generate the measurement matrices required for 2D-CS. Suppose the size of the measurement matrices A and B to be created is M×NM≪N; the specific process of TLM-Driven 2D-CS is as follows:(i)
Step 1: use the parametersr13,r23,x3 to iterate TLM N×N+r33 times. In order to avoid negative effects, discard the first r33 chaotic state values.(ii)
Step 2: arrange the obtained chaotic sequence into the square matrixS∧ of size N×N.(iii)
Step 3: takeM rows from the orthogonal basis of S∧ as the measurement matrix A.(iv)
Step 4: repeat Step 1 through Step 3; create the measurement matrixB in a similar manner.(v)
Step 5: useA and B to obtain the 2D measurements of the cipher image C.In addition to the improvements made above, the other steps of SIT-SS are basically the same as those of 2DCS-ETC, which are not repeated here. Since we have introduced TLM and DPPs in SIT-SS, the security of image transmission between sensor nodes and sink nodes has become higher, and the resource requirements for sensors are also lower. Significantly, SIT-SS still maintains the advantages of 2DCS-ETC, which is demonstrated and discussed in Section4.1. To save hardware resources, we directly use r11,r21,x1,r12,r22,x2,r13,r23,x3 as the secret key of SIT-SS.


## 3.1.1. DPP Generation

According to previous cryptanalysis works, the main reason why some schemes cannot resist CPAs is that equivalent key streams only depend on the secret key [41–47]. Therefore, some researchers use the hash value of the plain image to ensure the diversity of equivalent key streams. However, the implementation of hash algorithm is not suitable for sensor nodes with limited resources. Considering that system times and last encryption times are constantly changing and would be affected by many factors, they are used to generate DPPs. The specific generation process of DPPs is as follows:(i)
Step 1: obtain the system timeTs in milliseconds.(ii)
Step 2: get the timeTe spent in the last encryption process in milliseconds. If it is the first time to encrypt, set Te to an initial value Ti.(iii)
Step 3: one DPP is obtained byβ=Ts+Temod256.(iv)
Step 4: repeat Step 1 through Step 3 until 32 DPPs are obtained, namely,β1,β2,…,β32.In this way, we can obtain a set of DPPs. Like the hash value, DPPs can ensure that the equivalent key streams used when encrypting different images are different, thereby effectively resisting CPAs. More importantly, no complicated calculations are required to obtain DPPs, and even if the same plain image is encrypted, different equivalent key streams would be generated.


## 3.1.2. TLM-Driven Global Permutation

Obviously, confusion is the requirement that must be considered when designing modern cryptosystems. Confusion means that each bit of the secret key should affect as many cipher image bits as possible [48]. Permutation operations are commonly used to achieve confusion, but permutation-only image encryption schemes have been proven to be insecure [49]. Therefore, SIT-SS introduces DPPs in the permutation process. This makes the permutation process not only dependent on the secret key, but also dependent on the DPPs that will inevitably change every time the plain image is encrypted. Compared with 2DCS-ETC using the random permutation matrix to complete the permutation and treat it as secret key, we use TLM and DPPs to complete the permutation. This can not only reduce the resource overhead of sensor nodes, but also improve the ability to resist CPAs. The specific process of TLM-Driven Global Permutation (GP) is as follows:(i)
Step 1: use the parametersr11,r21,x1 to iterate TLM N×N+r31 times. In order to avoid negative effects, discard the first r31 chaotic state values.(ii)
Step 2: convert the obtained chaotic sequenceS of length N×N into the integer sequence(5)Si=floorSi×1015modN×N+1,wherei=1,2,…,N×N, floor⋅ returns the integer part of an operand.(iii)
Step 3: stretch the plain imageP of size N×N into the 1D sequence P˜.(iv)
Step 4: calculate the index(6)I=Simod32+1,of 32 DPPs and the permutation position(7)α=Si+βImodN×N+1,wherei=1,2,…,N×N. Swap two pixels of P˜ according to α.


## 3.1.3. TLM-Driven Negative-Positive Transformation

A nonlinear operation called Negative-Positive Transformation (NPT) is introduced by 2DCS-ETC to improve security. Similarly, we use TLM and DPPs to complete NPT instead of using a random matrix in the form of secret key. This can further reduce the resource overhead of sensor nodes and improve the ability to resist CPAs.(i)
Step 1: use parameterr12,r22,x2 to iterate TLM N×N+r32 times. In order to avoid negative effects, discard the first r32 chaotic state values.(ii)
Step 2: convert the obtained chaotic sequenceS of length N×N into the bit sequence(8)Si=floorSi×1015mod2,wherei=1,2,…,N×N.(iii)
Step 3: according toS, perform the following NPT operation on P˜.(9)Ci=P˜i,Si=1,255−P˜i,Si=0,wherei=1,2,…,N×N.(iv)
Step 4: reshapeC into the 2D cipher image.


## 3.1.4. TLM-Driven 2D-CS

If the chaotic sequence generated by the chaotic system is assembled into a complete measurement matrix, its performance is usually almost the same as other commonly used random matrices [11]. Moreover, compared with directly using a random matrix and treating it as secret key, the chaotic measurement matrix can significantly save the resource overhead of sensor nodes. In SIT-SS, TLM is used to generate the measurement matrices required for 2D-CS. Suppose the size of the measurement matrices A and B to be created is M×NM≪N; the specific process of TLM-Driven 2D-CS is as follows:(i)
Step 1: use the parametersr13,r23,x3 to iterate TLM N×N+r33 times. In order to avoid negative effects, discard the first r33 chaotic state values.(ii)
Step 2: arrange the obtained chaotic sequence into the square matrixS∧ of size N×N.(iii)
Step 3: takeM rows from the orthogonal basis of S∧ as the measurement matrix A.(iv)
Step 4: repeat Step 1 through Step 3; create the measurement matrixB in a similar manner.(v)
Step 5: useA and B to obtain the 2D measurements of the cipher image C.In addition to the improvements made above, the other steps of SIT-SS are basically the same as those of 2DCS-ETC, which are not repeated here. Since we have introduced TLM and DPPs in SIT-SS, the security of image transmission between sensor nodes and sink nodes has become higher, and the resource requirements for sensors are also lower. Significantly, SIT-SS still maintains the advantages of 2DCS-ETC, which is demonstrated and discussed in Section4.1. To save hardware resources, we directly use r11,r21,x1,r12,r22,x2,r13,r23,x3 as the secret key of SIT-SS.


## 3.2. Transmission between Sink Nodes and Receivers

In SIT-SR, we use 2D-LSCM [13] which has better chaotic performance to generate chaotic sequences. Moreover, discrete logarithms and DPPs are introduced to achieve secure image transmission between sink nodes and receivers. It should be noted that through the use of discrete logarithms and our targeted design, DPPs can be directly sent out in plaintext form by sink nodes. When decrypting, receivers can directly use DPPs that arrived in plaintext form. In other words, DPPs are not one-time pad secret keys, nor are they secret parameters. Next, we introduce the specific process of SIT-SR, as shown in Figure 5.Figure 5
Flowchart of SIT-SR.


### 3.2.1. Secret Key and Chaotic System Parameters

In order to avoid the secret key issues pointed out in some cryptanalysis works and simplify the generation process of chaotic system parameters [14, 41, 42], we set the secret key K in this stage as a binary sequence with the length of 270 bits. Namely, K=a1a2⋯a270. In specific implementation, we directly use nine 32-bit unsigned integers b1, b2, b3, b4, b5, b6, b7, b8, b9 to generate three sets of parameters x∧01,y∧01,r∧1, x∧02,y∧02,r∧2, x∧03,y∧03,r∧3 for 2D-LSCM. As shown in equation (11), this means that the 30×9 bits of K correspond to the 30 bits of each unsigned integer, respectively.(10)x∧01=b1×2+1×2−32,y∧01=b2×2+1×2−32,r∧01=b3×2+1×2−32,x∧02=b4×2+1×2−32,y∧02=b5×2+1×2−32,r∧02=b6×2+1×2−32,x∧03=b7×2+1×2−32,y∧03=b8×2+1×2−32,r∧03=b9×2+1×2−32,where(11)b1=a1a2⋯a30,b2=a31a32⋯a60,b3=a61a62⋯a90,b4=a91a92⋯a120,b5=a121a122⋯a150,b6=a151a152⋯a180,b7=a181a182⋯a210,b8=a211a212⋯a240,b9=a241a242⋯a270.Besides, these three sets of chaotic system parametersx∧01,y∧01,r∧1, x∧02,y∧02,r∧2, x∧03,y∧03,r∧3 are used to generate chaotic matrices for the encryption process.


### 3.2.2. DPP Generation

The generation process of DPPs in SIT-SR is exactly the same as SIT-SS. And we mark 32 DPPs used in SIT-SR asH.


### 3.2.3. Dynamic Permutation and Confusion Based on Discrete Logarithms

As mentioned above, discrete logarithms and DPPs are introduced in dynamic permutation and confusion based on discrete logarithms (DPC-D), so as to enhance the security of image transmission. Specifically, compared with some existing permutation operations, DPC-D has the following advantages:(1)
UseH to further perturb the permutation results and adopt different perturbation strategies for the row index and column index. Therefore, the permutation results depend not only on the secret key, but also on H.(2)
Based on discrete logarithms,H and the sorting results of the chaotic matrix S are used to nonlinearly transform the pixel value of each plain image pixel, thereby further improving the security of image transmission.In order to better describe the specific steps of DPC-D, an algorithm is provided in Algorithm1.Algorithm 1: DPC-D algorithm.
Require: the plain image P with the size of N×N, the chaotic matrix S with the size of N×N, the dynamic perturbation parameters H with the size of 1×32 and the discrete logarithm value matrixDVM with the size of 128×256.(1)
SetT∈NN×N;(2)
Set the sumhs of the dynamic perturbation parameters to 0;(3)
Set the row index valueg used to represent the adopted generator to 0;(4)
Set the index valueidx used to represent the adopted dynamic perturbation parameters to 0;(5)
Sort each column ofS in ascending order, thus get the column index matrix O and sorted result B;(6)
fori=1 to 32 do(7)
hs=hs+Hi;(8)
end for(9)
Calculate the row index value of the generator to be used, namely letg=hsmod128+1;(10)
fori=1 to Ndo(11)
SortBi in ascending order and obtain the row index vector v;(12)
forj=1 to Ndo(13)
idx=i−1×N+jmod32+1;(14)
TOi,j+HidxmodN+1,j+hsmodN+1=DVMg,POi,vj,vj+Hidx+vjmod256+1−1;(15)
end for(16)
end forEnsure: the permuted and transformed image T.


## 3.2.1. Secret Key and Chaotic System Parameters

In order to avoid the secret key issues pointed out in some cryptanalysis works and simplify the generation process of chaotic system parameters [14, 41, 42], we set the secret key K in this stage as a binary sequence with the length of 270 bits. Namely, K=a1a2⋯a270. In specific implementation, we directly use nine 32-bit unsigned integers b1, b2, b3, b4, b5, b6, b7, b8, b9 to generate three sets of parameters x∧01,y∧01,r∧1, x∧02,y∧02,r∧2, x∧03,y∧03,r∧3 for 2D-LSCM. As shown in equation (11), this means that the 30×9 bits of K correspond to the 30 bits of each unsigned integer, respectively.(10)x∧01=b1×2+1×2−32,y∧01=b2×2+1×2−32,r∧01=b3×2+1×2−32,x∧02=b4×2+1×2−32,y∧02=b5×2+1×2−32,r∧02=b6×2+1×2−32,x∧03=b7×2+1×2−32,y∧03=b8×2+1×2−32,r∧03=b9×2+1×2−32,where(11)b1=a1a2⋯a30,b2=a31a32⋯a60,b3=a61a62⋯a90,b4=a91a92⋯a120,b5=a121a122⋯a150,b6=a151a152⋯a180,b7=a181a182⋯a210,b8=a211a212⋯a240,b9=a241a242⋯a270.Besides, these three sets of chaotic system parametersx∧01,y∧01,r∧1, x∧02,y∧02,r∧2, x∧03,y∧03,r∧3 are used to generate chaotic matrices for the encryption process.


## 3.2.2. DPP Generation

The generation process of DPPs in SIT-SR is exactly the same as SIT-SS. And we mark 32 DPPs used in SIT-SR asH.


## 3.2.3. Dynamic Permutation and Confusion Based on Discrete Logarithms

As mentioned above, discrete logarithms and DPPs are introduced in dynamic permutation and confusion based on discrete logarithms (DPC-D), so as to enhance the security of image transmission. Specifically, compared with some existing permutation operations, DPC-D has the following advantages:(1)
UseH to further perturb the permutation results and adopt different perturbation strategies for the row index and column index. Therefore, the permutation results depend not only on the secret key, but also on H.(2)
Based on discrete logarithms,H and the sorting results of the chaotic matrix S are used to nonlinearly transform the pixel value of each plain image pixel, thereby further improving the security of image transmission.In order to better describe the specific steps of DPC-D, an algorithm is provided in Algorithm1.Algorithm 1: DPC-D algorithm.
Require: the plain image P with the size of N×N, the chaotic matrix S with the size of N×N, the dynamic perturbation parameters H with the size of 1×32 and the discrete logarithm value matrixDVM with the size of 128×256.(1)
SetT∈NN×N;(2)
Set the sumhs of the dynamic perturbation parameters to 0;(3)
Set the row index valueg used to represent the adopted generator to 0;(4)
Set the index valueidx used to represent the adopted dynamic perturbation parameters to 0;(5)
Sort each column ofS in ascending order, thus get the column index matrix O and sorted result B;(6)
fori=1 to 32 do(7)
hs=hs+Hi;(8)
end for(9)
Calculate the row index value of the generator to be used, namely letg=hsmod128+1;(10)
fori=1 to Ndo(11)
SortBi in ascending order and obtain the row index vector v;(12)
forj=1 to Ndo(13)
idx=i−1×N+jmod32+1;(14)
TOi,j+HidxmodN+1,j+hsmodN+1=DVMg,POi,vj,vj+Hidx+vjmod256+1−1;(15)
end for(16)
end forEnsure: the permuted and transformed image T.


## 3.3. Dynamic Diffusion Based on Discrete Logarithms

To further improve security, dynamic diffusion based on discrete logarithms (DD-D) also adopts discrete logarithms andH. Specifically, compared with some existing diffusion operations, DD-D has the following advantages:(1)
Considering that multipixel diffusion is of little significance, single-pixel diffusion is adopted, thereby reducing the amount of computation(2)
The nonlinearity of the diffusion process is improved by introducing discrete logarithms; thus the security of image transmission is further improvedIn order to better describe the specific steps of DD-D, an algorithm is provided in Algorithm2.Algorithm 2: DD-D algorithm.
Require: the permuted and transformed image T with the size of N×N, the chaotic matrix R with the size of N×N, the dynamic perturbation parameters H with the size of 1×32 and the discrete logarithm value matrix DVM with the size of 128×256.(1)
SetC∈NN×N;(2)
ConvertR into the integer matrix IR with the same format as the pixels of T, namely IR=⌊R×232⌋mod256;(3)
Set the bitwise XOR resulthx of the dynamic perturbation parameters to 0;(4)
Set the row index valuesg1, g2 used to represent the adopted generators to 0 s;(5)
fori=1 to 32 do(6)
hx=bitxorhx,Hi;(7)
end for(8)
Calculate the row index values of the generators to be used, letg1=bitxorhx,IR1,1mod128+1, g2=bitxorg1,IRN,Nmod128+1;(9)
tmp:,1=T:,1+DVMg1,T:,N+1+DVMg2,IR:,1+1mod256;(10)
fori=2 to Ndo(11)
tmp:,i=T:,i+DVMg1,tmp:,i−1+1+DVMg2,IR:,i+1mod256;(12)
end for(13)
C1,:=tmp1,:+DVMg1,tmpN,:+1+DVMg2,IR1,:+1mod256;(14)
fori=2 to Ndo(15)
Ci,:=tmpi,:+DVMg1,Ci−1,:+1+DVMg2,IRi,:+1mod256;(16)
end forEnsure: the diffused image C.Since a symmetric encryption structure is adopted in SIT-SR, the decryption process is actually constituted by the corresponding inverse operations of the encryption operations. With the received DPPs and the agreed secret keyK, receivers can decrypt the plain image from the cipher image. To save space, these inverse operations are not repeated here.


## 4. Simulation Tests and Analyses

In this section, extensive simulation tests are performed to demonstrate the superiority of ITS-CDD. ITS-CDD is an image transmission scheme composed of two parts, and the resource conditions and design goals of each part are different. Therefore, SIT-SS is compared with 2DCS-ETC for resource-constrained environments, whereas SIT-SR is compared with more versatile schemes for general application environments. Without loss of generality, randomly generated secret keys are used to complete the tests. Table2 lists the hardware and software configurations used in the tests.Table 2
Software and hardware configurations used in simulation tests.
Configuration itemDescriptionCPUIntel Xeon CPU E3-1231 v3 3.40 GHzMemory capacity8 GBOperating systemWindows 7 Ultimate (64 bit)Test platformMATLAB R2017a (9.2.0538062)


### 4.1. Simulation Tests for SIT-SS

Since reducing the resource consumption of sensors and improving the security of image transmission is our goal in designing SIT-SS, the analysis and verification of SIT-SS are mainly focused on these two aspects. The test images used are eight images used in [12].


#### 4.1.1. Encryption and Decryption

Four plain images Lena, Boats, House, and Parrots are shown in Figure6. Their corresponding cipher images and decrypted images generated in SIT-SS are also provided. As can be seen from these images, the cipher images are similar to noise, attackers cannot obtain useful information from them, and there are no significant visual differences between the decrypted images and corresponding plain images.Figure 6
Encryption and reconstruction of four plain images: (a) plain image Lena; (b) cipher image of (a); (c) compressed cipher image of (a); (d) reconstructed image of (c); (e) plain image of Boats; (f) cipher image of (e); (g) compressed cipher image of (e); (h) reconstructed image of (g); (i) plain image of House; (j) cipher image of (i); (k) compressed cipher image of (i); (l) reconstructed image of (k); (m) plain image of Parrots; (n) cipher image of (m); (o) compressed cipher image of (m); (p) reconstructed image of (o).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)


#### 4.1.2. Reconstruction Quality

Researchers often use Peak Signal-to-Noise Ratio (PSNR) to evaluate image reconstruction quality. Generally, a higher PSNR value indicates a better reconstruction quality. The definition of PSNR is as follows:(12)PSNR=10×log1025521/M×N∑i=1M∑i=1NRi,j−Pi,j2,where M×N is the size of the reconstructed image R and original image P. PSNR versus sampling ratio for 2DCS-ETC and SIT-SS is listed in Table 3. As can be seen from Table 3, SIT-SS can achieve the same or slightly different PSNR values as 2DCS-ETC. And when sampling ratio is greater than 0.6, its image reconstruction quality has obvious advantages.Table 3
PSNR (dB) of 2DCS-ETC and SIT-SS under different sampling ratios.
ImageSchemeSampling ratio0.10.20.30.40.50.60.70.80.9Lena2DCS-ETC25.160830.723833.789436.340738.516939.042940.308841.175941.7017SIT-SS25.523230.667533.768036.392638.643940.493542.616544.956247.9543Barbara2DCS-ETC22.013525.858928.896731.716634.214735.631537.339939.091640.2334SIT-SS22.199025.826028.817531.659334.184136.485338.879741.525645.0071Boats2DCS-ETC24.629029.791332.811235.301137.404738.077839.180940.343841.5691SIT-SS24.109529.690332.865735.340537.421339.337141.356143.587646.6325Cameraman2DCS-ETC22.312528.766531.838934.263036.495336.975138.223139.023739.7931SIT-SS21.832428.704631.820134.257536.444938.653440.950043.328746.3889Foreman2DCS-ETC29.593035.853738.155838.541339.411139.857040.264140.361640.6835SIT-SS29.757235.786638.102539.852441.497842.947744.539546.317548.4910House2DCS-ETC28.989934.120936.091137.588338.942839.140840.048941.120242.1401SIT-SS28.376434.118836.138637.565438.971740.354642.049244.175747.0669Monarch2DCS-ETC22.362429.005432.572935.617537.989038.135939.316440.051540.9473SIT-SS22.007128.812332.632335.386437.800539.961142.137744.565747.4934Parrots2DCS-ETC25.986033.096835.794137.834438.087339.113939.787040.358940.6380SIT-SS25.788132.961835.644337.795439.594341.295643.065545.029547.6101Each bold value means that one of the two compared schemes has a higher PSNR value than the other.


#### 4.1.3. Secret Key

In 2DCS-ETC, the random permutation matrix and random binary integer matrix are used as the secret key, thereby obtaining a huge key space. However, in resource-constrained environments, it is not suitable to use such secret key that requires a large amount of storage space. For example, if the size of the plain image is1024×1024, the secret key used would be at least 2 228 224 bytes in length. In addition, in the encryption and decryption process, the generation and storage of two measurement matrices also bring significant resource requirements. However, in SIT-SS, we only need to store six floating-point numbers used as the secret key. Meanwhile, SIT-SS also enjoys a large enough key space, which is about 10133. Apparently, such a large key space is sufficient to resist brute force attacks.


#### 4.1.4. Chosen-Plaintext Attack

As we know, CPAs are the reasons why some encryption schemes are cracked. It is generally believed that a secure encryption scheme should be able to resist CPAs. Derived from system times and last encryption times, DPPs always change dynamically and will be affected by many factors, thereby ensuring the diversity and unpredictability of equivalent key streams. Without relying on external algorithms such as hash algorithms, the diversity of equivalent key streams brings excellent resistance to CPAs.


### 4.2. Simulation Tests for SIT-SR

The simulation tests presented in this subsection are to demonstrate the superiority of SIT-SR in terms of security and encryption efficiency. The test images used for SIT-SR are from The USC-SIPI Image Database (https://sipi.usc.edu/database/).


#### 4.2.1. Encryption and Decryption of Different Types of Images

For different types of images, we uniformly treat them as 8-bit grayscale images in ITS-CDD. Specifically, for images with a pixel depth of less than 8 bits, we directly process them as 8-bit grayscale images, whereas for images with pixel depth greater than 8 bits, we encrypt them in groups of 8 bits. For example, if we need to encrypt an image with a pixel depth of 16 bits, we can encrypt the lower 8 bits and higher 8 bits of each plain image pixel separately. Figure7 shows the encryption and decryption effects of SIT-SR for different types of images. One can see that SIT-SR has excellent encryption effects for different types of images. The generated cipher images are very similar to noise images, and attackers cannot directly obtain any valuable information from these cipher images.Figure 7
Encryption and decryption effects for different types of images: (a) f15.png; (b) cipher image of (a); (c) decrypted image of (b); (d) 5.2.09.tiff; (e) cipher image of (d); (f) decrypted image of (e); (g) gray21.512.tiff; (h) cipher image of (g); (i) decrypted image of (h); (j) 4.2.04.tiff; (k) cipher image of (j); (l) decrypted image of (k); (m) f15.png; (n) cipher image of (m); (o) decrypted image of (n); (p) f15.png; (q) cipher image of (p); (r) decrypted image of (q); (s) f15.png; (t) cipher image of (s); (u) decrypted image of (t); (v) f15.png; (w) cipher image of (v); (x) decrypted image of (w).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


#### 4.2.2. Key Space and Key Sensitivity

Since the key space would affect the ability to resist brute force attacks, a secure encryption scheme should have a sufficiently large key space [50]. We carefully design the secret key, which not only solves the issues about equivalent secret keys, but also expands the key space to 2270. Therefore, SIT-SR is excellent in terms of the ability to resist brute force attacks.It is well known that a secure encryption scheme should be able to achieve superior confusion effect [50, 51]. That is, one smallest change in the secret key should make the cipher image change significantly. To evaluate the performance of SIT-SR in this regard, we randomly generated the secret key(13)KR=03D7D6F3E884829564EDBA77D8683B811AE4FC8655CC7EE7BC305537BFEA2EE2238E.UsingKR, we encrypted elaine.512.tiff to obtain the corresponding cipher image ČR. Next, we changed one bit of KR to get two secret keys KR1, KR2 with single smallest changes. These two changed secret keys also were used to encrypt elaine.512.tiff, thus obtaining the corresponding cipher images ČR1, ČR2. Finally, the difference images between ČR1, ČR2, and ČR were calculated. As can be seen from Figure 8, the difference images between ČR1, ČR2, and ČR look similar to an ordinary cipher image. This means that the key sensitivity of SIT-SR in the encryption process is extraordinary.Figure 8
Key sensitivity test for encryption process: (a) elaine.512.tiff; (b)ČR obtained by KR; (c) ČR1 obtained by KR1; (d) ČR2 obtained by KR2; (e) difference image between ČR and ČR1; (f) difference image between ČR and ČR2.
(a)(b)(c)(d)(e)(f)Similarly, in order to verify the key sensitivity of SIT-SR in the decryption process,KR1 and KR2 were adopted to decrypt ČR. The test results are shown in Figure 9. Once again, judging from the difference image between the wrongly decrypted images, the key sensitivity of SIT-SR in the decryption process is excellent.Figure 9
Key sensitivity test for decryption process: (a)ČR obtained by KR; (b) decrypted image ČR obtained by KR; (c) decrypted image ČR1 obtained by KR1; (d) decrypted image ČR2 obtained by KR2; (e) difference image between ČR1 and ČR2.
(a)(b)(c)(d)(e)For measuring the degree of changes between images, NPCR (Number of Pixel Change Ratio) and UACI (Unified Average Change in Intensity) are commonly used indicators [13]. Among them,(14)NPCR=1M×N∑i,jDi,j×100%,refers to the ratio of the pixels that change, whereas(15)UACI=1M×N∑i,jCi,j1−Ci,j2255×100%,refers to the average intensity of the pixel value changes. In equations (14) and (15), M×N is the size of the images, i=1,2,…,M, j=1,2,…,N, and D is the difference matrix between the image C1 before the change and the image C2 after the change. If Ci,j1≠Ci,j2, then Di,j=1. Otherwise, Di,j=0. In order to quantitatively analyze the key sensitivity of SIT-SR, we calculated the NPCR and UACI values between the cipher images before and after the secret key changes. As one can see from Table 4, all the test results are very close to the ideal values, which means that SIT-SR does have extremely high key sensitivity.Table 4
UPCR and UACI values between cipher images when secret key changes.
ChangeNPCR (%)UACI (%)Lowest bit ofb1 is inverted99.612133.4601Lowest bit ofb2 is inverted99.602833.4625Lowest bit ofb3 is inverted99.609733.4837Lowest bit ofb4 is inverted99.601333.4633Lowest bit ofb5 is inverted99.600933.4764Lowest bit ofb6 is inverted99.617433.4512Lowest bit ofb7 is inverted99.607533.4810Lowest bit ofb8 is inverted99.612233.4706Lowest bit ofb9 is inverted99.613033.4562Random image99.609433.4635The bold values are the ideal values.


#### 4.2.3. Pixel Value Distribution

Unlike plain images, cipher images must have extremely uniform pixel value distributions; otherwise attackers will have the opportunity to conduct statistics based attacks [50, 51]. In order to visually demonstrate the pixel value distribution characteristics of the plain images and the cipher images generated by SIT-SR, the pixel value distribution histograms of these images are plotted. As can be seen from Figure 10, the pixel distributions of the plain images are relatively concentrated, whereas the pixels of the cipher images are extremely uniformly distributed throughout the entire value range.Figure 10
Pixel value distribution histograms of elaine.512.tiff, 7.1.03.tiff, gray21.512.tiff, and their corresponding cipher images: (a) elaine.512.tiff; (b) histogram of (a); (c) cipher image of (a); (d) histogram of (c); (e) 7.1.03.tiff; (f) histogram of (e); (g) cipher image of (e); (h) histogram of (g); (i) gray21.512.tiff; (j) histogram of (i); (k) cipher image of (i); (l) histogram of (k).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)In addition, the histogram variance analysis and chi-square test analysis are also performed on the cipher images to qualitatively analyze the ability of SIT-SR to resist statistical attacks. In general, if the histogram variance of a cipher image is smaller, then the uniformity of the cipher image is higher. The histogram variance of an 8-bit grayscale image can be defined as follows:(16)VH=12562∑i=1256∑j=125612pi−pj2,where H=p1,p2,…,p256; pi and pj are the numbers of the pixels whose grayscale values are equal to i−1 and j−1. Table 5 lists the histogram variance test results of some images. These images include one random image, the 8-bit grayscale image Lena, and its cipher images generated by different schemes.Table 5
Histogram variance test results of some images.
AlgorithmImageVarianceLena256.bmp33860.0546[13]Cipher image266.7578[16]Cipher image260.7188[17]Cipher image276.3906SIT-SRCipher image257.1094Random image253.8946The bold value means that SIT-SR has the best test result among four compared schemes.From Table5, one can see that the histogram variance of the plain image is very large, which means that its pixel value distribution is extremely uneven, whereas among the cipher images generated by the four image encryption schemes, the cipher image of SIT-SR has the smallest histogram variance, which indicates that this cipher image has the most uniform pixel value distribution and is closest to the random image.Another way to quantitatively analyze the uniformity of a cipher image is the chi-square test. The chi-square value of a cipher image can be calculated as follows:(17)χ2=∑i=1nsi−H×W×p2H×W×p,where H×W is the height and width of the cipher image, si is the number of pixels whose pixel value is i−1, n is the number of all possible pixel values (for 8-bit grayscale images, n=256), and p=1/n. Next, the critical value χ0.052255 of the chi-square test at the significant level α=0.05 can be determined, which is 293.2478. If a cipher image has a chi-square value less than 293.2478, then this image can be considered to have passed the chi-square test; that is, its pixel value distribution is statistically uniform. Consequently, the smaller the chi-square value of a cipher image is, the better its uniformity is. As can be seen from Table 6, for some commonly used test images, the cipher images generated by SIT-SR have all passed the chi-square test. And in most cases, SIT-SR performs better than another scheme.Table 6
Chi-square values of different cipher images.
FilenameSizeChi-square value[13]SIT-SRLena256.bmp256×256255.8555253.30355.1.10.tiff256×256261.3125254.19535.1.12.tiff256×256256.2578244.53285.1.13.tiff256×256274.8750245.37975.2.08.tiff512×512252.7471247.54345.2.09.tiff512×512274.3906257.34345.3.01.tiff1024×1024236.3027229.81257.1.02.tiff512×512252.9141226.21977.1.03.tiff512×512248.8984257.23247.1.04.tiff512×512281.2773258.40437.1.05.tiff512×512275.1055263.8584boat.512.tiff512×512230.2256232.7012elaine.512.tiff512×512266.6377230.0078gray21.512.tiff512×512244.8789245.3027ruler.512.tiff512×512290.8057223.2813testpat.1k.tiff1024×1024258.6455239.7627


#### 4.2.4. Plain Image Sensitivity

When the plain image changes, the corresponding change degree of the cipher image is plain image sensitivity. Generally speaking, to effectively resist differential attacks, an image encryption scheme must have excellent plain image sensitivity. To intuitively show the plain image sensitivity of SIT-SR, we first encrypted some commonly used test images. Next, after changing one pixel bit for each plain image, the plain images with the smallest changes were encrypted. At last, we calculated the difference images between the cipher images obtained before and after the smallest changes. The relevant test results are shown in Figure11. As one can see from Figure 11, each plain image has undergone only one smallest change, but almost all cipher pixels have changed. In addition to that, these significant differences are independent of where the plain images change and are very close to random images.Figure 11
Plain image sensitivity test results for SIT-SR: (a1) 5.1.10.tiff; (a2) cipher image of (a1); (a3) the least significant bit of the pixel at (1,1) in (a1) is reversed; (a4) cipher image of (a3); (a5) difference image between (a2) and (a4); (b1) boat.512.tiff; (b2) cipher image of (b1); (b3) the least significant bit of the pixel at (256,256) in (b1) is reversed; (b4) cipher image of (b3); (b5) difference image between (b2) and (b4).UPCR and UACI are also utilized to qualitatively analyze the plain image sensitivity of SIT-SR. The UPCR and UACI values between the cipher images obtained before and after the smallest changes of 15 common test plain images are shown in Table7. Judging from the test results, SIT-SR has excellent plain image sensitivity. The test results of SIT-SR are closer to the ideal values 99.6094% and 33.4635% and perform better in terms of stability.Table 7
NPCR and UACI test results on plain image sensitivity.
Filename[13]SIT-SRNPCR (%)UACI (%)NPCR (%)UACI (%)5.1.10.tiff99.601433.477499.616233.46455.1.12.tiff99.622233.466899.609333.46425.1.13.tiff99.609133.478299.611933.45975.2.08.tiff99.613833.459699.608033.45895.2.09.tiff99.607233.449699.611733.45215.3.01.tiff99.610333.455199.610733.45957.1.02.tiff99.608833.474999.614033.46357.1.03.tiff99.602633.493099.608133.48367.1.04.tiff99.611733.469999.602533.46457.1.05.tiff99.610133.476699.608333.4643boat.512.tiff99.604533.461899.609833.4825elaine.512.tiff99.613933.491899.609833.4521gray21.512.tiff99.606933.466099.612233.4713ruler.512.tiff99.611333.439499.611533.4453testpat.1k.tiff99.605433.457199.609133.4624Average99.608033.469599.611333.4598Std. Dev.0.00420.01610.00280.0045The bold values here emphasize that SIT-SR has better performance than the other scheme.


#### 4.2.5. Information Entropy

Information entropy is an indicator that can be used to measure the randomness or disorder of an information source. If the information entropy of the information source is higher, it can be considered that the information source has higher randomness or disorder [18–20]. When it comes to an 8-bit grayscale image, the information entropy of the grayscale image can be calculated as follows:(18)HS=−∑i=1npsi×log2psi,where n is the total number of symbols si; psi is the occurrence probability of symbol si. For the 8-bit grayscale cipher images, the ideal value of the information entropy is 8 [18–20]. From Table 8, one can see that the information entropy of each cipher image generated by SIT-SR is very close to the ideal value 8. As shown in Table 9, compared with several image encryption schemes, the information entropy of the Lena cipher image generated by SIT-SR is closest to the ideal value 8. Therefore, SIT-SR performs best in terms of the information entropy.Table 8
Information entropy test results of plain images and cipher images.
FilenamePlain imageCipher image5.2.08.tiff7.20107.99935.2.09.tiff6.99407.99945.3.01.tiff7.52377.99987.1.02.tiff4.00457.99937.1.03.tiff5.49577.99947.1.04.tiff6.10747.99937.1.05.tiff6.56327.9994boat.512.tiff7.19147.9994elaine.512.tiff7.50607.9994gray21.512.tiff4.39237.9993ruler.512.tiff0.50007.9994testpat.1k.tiff4.40777.9998Table 9
Information entropy test results of Lena cipher images.
Scheme[13][18][19][20]SIT-SRInformation entropy7.99927.99797.99097.99917.9994The bold value here emphasizes that SIT-SR has better performance than other schemes.In order to measure the randomness of cipher images more comprehensively, a measure named Local Shannon Entropy (LSE) is proposed [52]. This measure is increasingly adopted to verify the randomness of cipher images [13]. Mathematically, LSE can be defined as follows:(19)Lq,sr=∑i=1qHriq,where r1,r2,…,rq are q randomly selected nonoverlapping image blocks, s is the number of pixels in each block, and Hri is the information entropy of ri. According to the test method suggested in [52], we carried out the LSE test on the cipher images generated by SIT-SR, and the relevant test results are shown in Table 10. Compared with two recent image encryption schemes, SIT-SR has the best performance in terms of standard deviation and pass rate.Table 10
LSE test results of different schemes.
Filename[13][15]SIT-SR5.2.087.90237.90247.90225.2.097.90207.90217.90237.1.027.90207.90157.90217.1.037.90267.90197.90247.1.047.90197.90217.9023boat.5127.90187.90227.9024gray21.5127.90267.90267.9025ruler.5127.90417.90287.9026Std.Dev.0.00070.00040.0002Pass/All6/87/88/8The bold values indicate that compared with the other two schemes, SIT-SR has the best performance in terms of standard deviation and pass rate.


#### 4.2.6. Pixel Correlation

The extremely high correlation between adjacent pixels is one of the salient features of plain images and also one of the important reasons why traditional encryption schemes are not suitable for image encryption [50]. Therefore, a secure image encryption scheme should eliminate the correlation between adjacent pixels as much as possible. CC (correlation coefficient) is an effective indicator to measure the correlation between adjacent pixels, and its definition is as follows:(20)CCab=Ea−Ea×b−EbDa×Db,where a and b are the grayscale values of two adjacent pixels; Ea and Da are the expectation and variance of the grayscale value a. In order to verify the performance of SIT-SR in terms of the pixel correlation, for the horizontal, vertical, and diagonal directions, we have randomly selected 20,000 pairs of adjacent pixels from each plain image and its corresponding cipher image to calculate the CCs. The relevant test results are shown in Table 11.Table 11
Correlation test results for adjacent pixels of plain images and their cipher images.
FilenameHorizontalVerticalDiagonalPlain imageCipher imagePlain imageCipher imagePlain imageCipher image5.2.08.tiff0.89060.00080.9322−0.00380.84520.00285.2.09.tiff0.8591−0.00040.9000−0.00180.80070.00205.3.01.tiff0.98170.00150.97760.00440.8981−0.00337.1.02.tiff0.9446−0.00240.9431−0.00160.90030.00017.1.03.tiff0.9317−0.00210.94360.00020.9059−0.00177.1.04.tiff0.96720.00380.97710.00280.9552−0.00597.1.05.tiff0.91080.00380.94250.00580.89190.0024boat.512.tiff0.97110.00300.93940.00480.92450.0002elaine.512.tiff0.9720−0.00110.97610.00100.96960.0016gray21.512.tiff0.9998−0.00230.9968−0.00360.99660.0031ruler.512.tiff0.4702−0.00260.45240.0004−0.0312−0.0023testpat.1k.tiff0.79920.00350.75010.00510.69970.0005From Table11, one can see that there are very high correlations between adjacent pixels of the plain images; that is, the absolute values of CCs are extremely high, whereas in the cipher images generated by SIT-SR, there is almost no correlation between adjacent pixels; that is, the absolute values of CCs are extremely low <0.006.In addition, in order to more intuitively show the correlation changes between adjacent pixels caused by the encryption of SIT-SR, the correlation distribution charts of the plain image elaine.512.tiff and its corresponding cipher image are drawn. As can be seen from Figure12, after the encryption processing of SIT-SR, there is almost no correlation between adjacent pixels in each direction.Figure 12
Pixel correlation distribution charts of elaine.512.tiff and its cipher image: (a, d) distribution charts in horizontal direction; (b, e) distribution charts in vertical direction; (c, f) distribution charts in diagonal direction.
(a)(b)(c)(d)(e)(f)


#### 4.2.7. Chosen-Plaintext Attack

In fact, almost all simulation tests related to security analysis can only ensure the security of image encryption schemes under ciphertext-only attacks (COAs) [51, 53]. This is exactly why some image encryption schemes have been broken. Among the four types of attacks, which are COAs, Known-Plaintext Attacks (KPAs), CPAs, and Chosen-Ciphertext Attacks (CCAs), CCAs are the most threatening ones, but the attack conditions required by them are practically meaningless [3, 50]. If attackers can choose cipher images arbitrarily, then they do not need to crack at all, because for any cipher image, they can directly recover its plain image. Therefore, it is generally believed that CPAs are the most threatening ones among common practical attacks. Actually, in the cryptanalysis works about image encryption, the vast majority of them adopt CPAs [3, 51]. Next, from the perspective of attackers, the ability of SIT-SR to resist CPAs is analyzed.Apparently, attackers will encounter several problems when they try to break SIT-SR with CPAs. Firstly, we assume that they could obtain the equivalent key streams of the encryption process from chosen plain images and corresponding cipher images. However, because SIT-SR introduce DPPs in serval encryption steps, the equivalent key streams they obtained cannot be used to recover other ordinary plain images, which are encrypted under different DPPs. Secondly, SIT-SR also performs nonlinear transformations on the plain image pixels during the permutation process, so the common attack method that ignores the permutation process by the chosen plain images composed of single-value pixels cannot work. Thirdly, SIT-SR adopts a nonlinear diffusion structure; that is, it adopts the discrete logarithms based on two different generators, which makes the encryption process cannot be simplified by chosen plain images. To sum up, SIT-SR can effectively resist CPAs.


#### 4.2.8. Encryption Efficiency

Improving encryption efficiency is one of the most important motivations to design new image encryption schemes. SIT-SR introduce DPPs and discrete logarithms, but in fact, discrete logarithms can be calculated in advance, and the calculation process of DPP is very simple, so the impact on encryption efficiency is very small. In addition, SIT-SR uses single-pixel diffusion and only performs three iterations in the encryption process. These also help to reduce the total number of primitive operations that need to be executed. Table12 shows the average time required by SIT-SR and some other recent image encryption schemes to encrypt the 8-bit grayscale image Lena 256×256. As can be seen from Table 12, although the time complexity of each image encryption scheme is OM×N, the scheme proposed in [13] requires the least number of primitive operations, so it has the highest encryption efficiency, whereas for SIT-SR, it adds a certain number of primitive operations to ensure the security, but it still maintains the significant advantage of high encryption efficiency. That is, in terms of encryption efficiency, SIT-SR is significantly better than the remaining four image encryption schemes.Table 12
Comparison of test results obtained by different encryption schemes to encrypt Lena image.
SchemeSIT-SR[13][18][19][21][22]Time (s)0.0890.0720.4170.6830.2750.264Throughput (Mbps)5.61806.94441.19900.73211.81821.8939The bold values here emphasize that SIT-SR has better performance than other schemes.


## 4.1. Simulation Tests for SIT-SS

Since reducing the resource consumption of sensors and improving the security of image transmission is our goal in designing SIT-SS, the analysis and verification of SIT-SS are mainly focused on these two aspects. The test images used are eight images used in [12].


### 4.1.1. Encryption and Decryption

Four plain images Lena, Boats, House, and Parrots are shown in Figure6. Their corresponding cipher images and decrypted images generated in SIT-SS are also provided. As can be seen from these images, the cipher images are similar to noise, attackers cannot obtain useful information from them, and there are no significant visual differences between the decrypted images and corresponding plain images.Figure 6
Encryption and reconstruction of four plain images: (a) plain image Lena; (b) cipher image of (a); (c) compressed cipher image of (a); (d) reconstructed image of (c); (e) plain image of Boats; (f) cipher image of (e); (g) compressed cipher image of (e); (h) reconstructed image of (g); (i) plain image of House; (j) cipher image of (i); (k) compressed cipher image of (i); (l) reconstructed image of (k); (m) plain image of Parrots; (n) cipher image of (m); (o) compressed cipher image of (m); (p) reconstructed image of (o).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)


### 4.1.2. Reconstruction Quality

Researchers often use Peak Signal-to-Noise Ratio (PSNR) to evaluate image reconstruction quality. Generally, a higher PSNR value indicates a better reconstruction quality. The definition of PSNR is as follows:(12)PSNR=10×log1025521/M×N∑i=1M∑i=1NRi,j−Pi,j2,where M×N is the size of the reconstructed image R and original image P. PSNR versus sampling ratio for 2DCS-ETC and SIT-SS is listed in Table 3. As can be seen from Table 3, SIT-SS can achieve the same or slightly different PSNR values as 2DCS-ETC. And when sampling ratio is greater than 0.6, its image reconstruction quality has obvious advantages.Table 3
PSNR (dB) of 2DCS-ETC and SIT-SS under different sampling ratios.
ImageSchemeSampling ratio0.10.20.30.40.50.60.70.80.9Lena2DCS-ETC25.160830.723833.789436.340738.516939.042940.308841.175941.7017SIT-SS25.523230.667533.768036.392638.643940.493542.616544.956247.9543Barbara2DCS-ETC22.013525.858928.896731.716634.214735.631537.339939.091640.2334SIT-SS22.199025.826028.817531.659334.184136.485338.879741.525645.0071Boats2DCS-ETC24.629029.791332.811235.301137.404738.077839.180940.343841.5691SIT-SS24.109529.690332.865735.340537.421339.337141.356143.587646.6325Cameraman2DCS-ETC22.312528.766531.838934.263036.495336.975138.223139.023739.7931SIT-SS21.832428.704631.820134.257536.444938.653440.950043.328746.3889Foreman2DCS-ETC29.593035.853738.155838.541339.411139.857040.264140.361640.6835SIT-SS29.757235.786638.102539.852441.497842.947744.539546.317548.4910House2DCS-ETC28.989934.120936.091137.588338.942839.140840.048941.120242.1401SIT-SS28.376434.118836.138637.565438.971740.354642.049244.175747.0669Monarch2DCS-ETC22.362429.005432.572935.617537.989038.135939.316440.051540.9473SIT-SS22.007128.812332.632335.386437.800539.961142.137744.565747.4934Parrots2DCS-ETC25.986033.096835.794137.834438.087339.113939.787040.358940.6380SIT-SS25.788132.961835.644337.795439.594341.295643.065545.029547.6101Each bold value means that one of the two compared schemes has a higher PSNR value than the other.


### 4.1.3. Secret Key

In 2DCS-ETC, the random permutation matrix and random binary integer matrix are used as the secret key, thereby obtaining a huge key space. However, in resource-constrained environments, it is not suitable to use such secret key that requires a large amount of storage space. For example, if the size of the plain image is1024×1024, the secret key used would be at least 2 228 224 bytes in length. In addition, in the encryption and decryption process, the generation and storage of two measurement matrices also bring significant resource requirements. However, in SIT-SS, we only need to store six floating-point numbers used as the secret key. Meanwhile, SIT-SS also enjoys a large enough key space, which is about 10133. Apparently, such a large key space is sufficient to resist brute force attacks.


### 4.1.4. Chosen-Plaintext Attack

As we know, CPAs are the reasons why some encryption schemes are cracked. It is generally believed that a secure encryption scheme should be able to resist CPAs. Derived from system times and last encryption times, DPPs always change dynamically and will be affected by many factors, thereby ensuring the diversity and unpredictability of equivalent key streams. Without relying on external algorithms such as hash algorithms, the diversity of equivalent key streams brings excellent resistance to CPAs.


## 4.1.1. Encryption and Decryption

Four plain images Lena, Boats, House, and Parrots are shown in Figure6. Their corresponding cipher images and decrypted images generated in SIT-SS are also provided. As can be seen from these images, the cipher images are similar to noise, attackers cannot obtain useful information from them, and there are no significant visual differences between the decrypted images and corresponding plain images.Figure 6
Encryption and reconstruction of four plain images: (a) plain image Lena; (b) cipher image of (a); (c) compressed cipher image of (a); (d) reconstructed image of (c); (e) plain image of Boats; (f) cipher image of (e); (g) compressed cipher image of (e); (h) reconstructed image of (g); (i) plain image of House; (j) cipher image of (i); (k) compressed cipher image of (i); (l) reconstructed image of (k); (m) plain image of Parrots; (n) cipher image of (m); (o) compressed cipher image of (m); (p) reconstructed image of (o).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)


## 4.1.2. Reconstruction Quality

Researchers often use Peak Signal-to-Noise Ratio (PSNR) to evaluate image reconstruction quality. Generally, a higher PSNR value indicates a better reconstruction quality. The definition of PSNR is as follows:(12)PSNR=10×log1025521/M×N∑i=1M∑i=1NRi,j−Pi,j2,where M×N is the size of the reconstructed image R and original image P. PSNR versus sampling ratio for 2DCS-ETC and SIT-SS is listed in Table 3. As can be seen from Table 3, SIT-SS can achieve the same or slightly different PSNR values as 2DCS-ETC. And when sampling ratio is greater than 0.6, its image reconstruction quality has obvious advantages.Table 3
PSNR (dB) of 2DCS-ETC and SIT-SS under different sampling ratios.
ImageSchemeSampling ratio0.10.20.30.40.50.60.70.80.9Lena2DCS-ETC25.160830.723833.789436.340738.516939.042940.308841.175941.7017SIT-SS25.523230.667533.768036.392638.643940.493542.616544.956247.9543Barbara2DCS-ETC22.013525.858928.896731.716634.214735.631537.339939.091640.2334SIT-SS22.199025.826028.817531.659334.184136.485338.879741.525645.0071Boats2DCS-ETC24.629029.791332.811235.301137.404738.077839.180940.343841.5691SIT-SS24.109529.690332.865735.340537.421339.337141.356143.587646.6325Cameraman2DCS-ETC22.312528.766531.838934.263036.495336.975138.223139.023739.7931SIT-SS21.832428.704631.820134.257536.444938.653440.950043.328746.3889Foreman2DCS-ETC29.593035.853738.155838.541339.411139.857040.264140.361640.6835SIT-SS29.757235.786638.102539.852441.497842.947744.539546.317548.4910House2DCS-ETC28.989934.120936.091137.588338.942839.140840.048941.120242.1401SIT-SS28.376434.118836.138637.565438.971740.354642.049244.175747.0669Monarch2DCS-ETC22.362429.005432.572935.617537.989038.135939.316440.051540.9473SIT-SS22.007128.812332.632335.386437.800539.961142.137744.565747.4934Parrots2DCS-ETC25.986033.096835.794137.834438.087339.113939.787040.358940.6380SIT-SS25.788132.961835.644337.795439.594341.295643.065545.029547.6101Each bold value means that one of the two compared schemes has a higher PSNR value than the other.


## 4.1.3. Secret Key

In 2DCS-ETC, the random permutation matrix and random binary integer matrix are used as the secret key, thereby obtaining a huge key space. However, in resource-constrained environments, it is not suitable to use such secret key that requires a large amount of storage space. For example, if the size of the plain image is1024×1024, the secret key used would be at least 2 228 224 bytes in length. In addition, in the encryption and decryption process, the generation and storage of two measurement matrices also bring significant resource requirements. However, in SIT-SS, we only need to store six floating-point numbers used as the secret key. Meanwhile, SIT-SS also enjoys a large enough key space, which is about 10133. Apparently, such a large key space is sufficient to resist brute force attacks.


## 4.1.4. Chosen-Plaintext Attack

As we know, CPAs are the reasons why some encryption schemes are cracked. It is generally believed that a secure encryption scheme should be able to resist CPAs. Derived from system times and last encryption times, DPPs always change dynamically and will be affected by many factors, thereby ensuring the diversity and unpredictability of equivalent key streams. Without relying on external algorithms such as hash algorithms, the diversity of equivalent key streams brings excellent resistance to CPAs.


## 4.2. Simulation Tests for SIT-SR

The simulation tests presented in this subsection are to demonstrate the superiority of SIT-SR in terms of security and encryption efficiency. The test images used for SIT-SR are from The USC-SIPI Image Database (https://sipi.usc.edu/database/).


### 4.2.1. Encryption and Decryption of Different Types of Images

For different types of images, we uniformly treat them as 8-bit grayscale images in ITS-CDD. Specifically, for images with a pixel depth of less than 8 bits, we directly process them as 8-bit grayscale images, whereas for images with pixel depth greater than 8 bits, we encrypt them in groups of 8 bits. For example, if we need to encrypt an image with a pixel depth of 16 bits, we can encrypt the lower 8 bits and higher 8 bits of each plain image pixel separately. Figure7 shows the encryption and decryption effects of SIT-SR for different types of images. One can see that SIT-SR has excellent encryption effects for different types of images. The generated cipher images are very similar to noise images, and attackers cannot directly obtain any valuable information from these cipher images.Figure 7
Encryption and decryption effects for different types of images: (a) f15.png; (b) cipher image of (a); (c) decrypted image of (b); (d) 5.2.09.tiff; (e) cipher image of (d); (f) decrypted image of (e); (g) gray21.512.tiff; (h) cipher image of (g); (i) decrypted image of (h); (j) 4.2.04.tiff; (k) cipher image of (j); (l) decrypted image of (k); (m) f15.png; (n) cipher image of (m); (o) decrypted image of (n); (p) f15.png; (q) cipher image of (p); (r) decrypted image of (q); (s) f15.png; (t) cipher image of (s); (u) decrypted image of (t); (v) f15.png; (w) cipher image of (v); (x) decrypted image of (w).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


### 4.2.2. Key Space and Key Sensitivity

Since the key space would affect the ability to resist brute force attacks, a secure encryption scheme should have a sufficiently large key space [50]. We carefully design the secret key, which not only solves the issues about equivalent secret keys, but also expands the key space to 2270. Therefore, SIT-SR is excellent in terms of the ability to resist brute force attacks.It is well known that a secure encryption scheme should be able to achieve superior confusion effect [50, 51]. That is, one smallest change in the secret key should make the cipher image change significantly. To evaluate the performance of SIT-SR in this regard, we randomly generated the secret key(13)KR=03D7D6F3E884829564EDBA77D8683B811AE4FC8655CC7EE7BC305537BFEA2EE2238E.UsingKR, we encrypted elaine.512.tiff to obtain the corresponding cipher image ČR. Next, we changed one bit of KR to get two secret keys KR1, KR2 with single smallest changes. These two changed secret keys also were used to encrypt elaine.512.tiff, thus obtaining the corresponding cipher images ČR1, ČR2. Finally, the difference images between ČR1, ČR2, and ČR were calculated. As can be seen from Figure 8, the difference images between ČR1, ČR2, and ČR look similar to an ordinary cipher image. This means that the key sensitivity of SIT-SR in the encryption process is extraordinary.Figure 8
Key sensitivity test for encryption process: (a) elaine.512.tiff; (b)ČR obtained by KR; (c) ČR1 obtained by KR1; (d) ČR2 obtained by KR2; (e) difference image between ČR and ČR1; (f) difference image between ČR and ČR2.
(a)(b)(c)(d)(e)(f)Similarly, in order to verify the key sensitivity of SIT-SR in the decryption process,KR1 and KR2 were adopted to decrypt ČR. The test results are shown in Figure 9. Once again, judging from the difference image between the wrongly decrypted images, the key sensitivity of SIT-SR in the decryption process is excellent.Figure 9
Key sensitivity test for decryption process: (a)ČR obtained by KR; (b) decrypted image ČR obtained by KR; (c) decrypted image ČR1 obtained by KR1; (d) decrypted image ČR2 obtained by KR2; (e) difference image between ČR1 and ČR2.
(a)(b)(c)(d)(e)For measuring the degree of changes between images, NPCR (Number of Pixel Change Ratio) and UACI (Unified Average Change in Intensity) are commonly used indicators [13]. Among them,(14)NPCR=1M×N∑i,jDi,j×100%,refers to the ratio of the pixels that change, whereas(15)UACI=1M×N∑i,jCi,j1−Ci,j2255×100%,refers to the average intensity of the pixel value changes. In equations (14) and (15), M×N is the size of the images, i=1,2,…,M, j=1,2,…,N, and D is the difference matrix between the image C1 before the change and the image C2 after the change. If Ci,j1≠Ci,j2, then Di,j=1. Otherwise, Di,j=0. In order to quantitatively analyze the key sensitivity of SIT-SR, we calculated the NPCR and UACI values between the cipher images before and after the secret key changes. As one can see from Table 4, all the test results are very close to the ideal values, which means that SIT-SR does have extremely high key sensitivity.Table 4
UPCR and UACI values between cipher images when secret key changes.
ChangeNPCR (%)UACI (%)Lowest bit ofb1 is inverted99.612133.4601Lowest bit ofb2 is inverted99.602833.4625Lowest bit ofb3 is inverted99.609733.4837Lowest bit ofb4 is inverted99.601333.4633Lowest bit ofb5 is inverted99.600933.4764Lowest bit ofb6 is inverted99.617433.4512Lowest bit ofb7 is inverted99.607533.4810Lowest bit ofb8 is inverted99.612233.4706Lowest bit ofb9 is inverted99.613033.4562Random image99.609433.4635The bold values are the ideal values.


### 4.2.3. Pixel Value Distribution

Unlike plain images, cipher images must have extremely uniform pixel value distributions; otherwise attackers will have the opportunity to conduct statistics based attacks [50, 51]. In order to visually demonstrate the pixel value distribution characteristics of the plain images and the cipher images generated by SIT-SR, the pixel value distribution histograms of these images are plotted. As can be seen from Figure 10, the pixel distributions of the plain images are relatively concentrated, whereas the pixels of the cipher images are extremely uniformly distributed throughout the entire value range.Figure 10
Pixel value distribution histograms of elaine.512.tiff, 7.1.03.tiff, gray21.512.tiff, and their corresponding cipher images: (a) elaine.512.tiff; (b) histogram of (a); (c) cipher image of (a); (d) histogram of (c); (e) 7.1.03.tiff; (f) histogram of (e); (g) cipher image of (e); (h) histogram of (g); (i) gray21.512.tiff; (j) histogram of (i); (k) cipher image of (i); (l) histogram of (k).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)In addition, the histogram variance analysis and chi-square test analysis are also performed on the cipher images to qualitatively analyze the ability of SIT-SR to resist statistical attacks. In general, if the histogram variance of a cipher image is smaller, then the uniformity of the cipher image is higher. The histogram variance of an 8-bit grayscale image can be defined as follows:(16)VH=12562∑i=1256∑j=125612pi−pj2,where H=p1,p2,…,p256; pi and pj are the numbers of the pixels whose grayscale values are equal to i−1 and j−1. Table 5 lists the histogram variance test results of some images. These images include one random image, the 8-bit grayscale image Lena, and its cipher images generated by different schemes.Table 5
Histogram variance test results of some images.
AlgorithmImageVarianceLena256.bmp33860.0546[13]Cipher image266.7578[16]Cipher image260.7188[17]Cipher image276.3906SIT-SRCipher image257.1094Random image253.8946The bold value means that SIT-SR has the best test result among four compared schemes.From Table5, one can see that the histogram variance of the plain image is very large, which means that its pixel value distribution is extremely uneven, whereas among the cipher images generated by the four image encryption schemes, the cipher image of SIT-SR has the smallest histogram variance, which indicates that this cipher image has the most uniform pixel value distribution and is closest to the random image.Another way to quantitatively analyze the uniformity of a cipher image is the chi-square test. The chi-square value of a cipher image can be calculated as follows:(17)χ2=∑i=1nsi−H×W×p2H×W×p,where H×W is the height and width of the cipher image, si is the number of pixels whose pixel value is i−1, n is the number of all possible pixel values (for 8-bit grayscale images, n=256), and p=1/n. Next, the critical value χ0.052255 of the chi-square test at the significant level α=0.05 can be determined, which is 293.2478. If a cipher image has a chi-square value less than 293.2478, then this image can be considered to have passed the chi-square test; that is, its pixel value distribution is statistically uniform. Consequently, the smaller the chi-square value of a cipher image is, the better its uniformity is. As can be seen from Table 6, for some commonly used test images, the cipher images generated by SIT-SR have all passed the chi-square test. And in most cases, SIT-SR performs better than another scheme.Table 6
Chi-square values of different cipher images.
FilenameSizeChi-square value[13]SIT-SRLena256.bmp256×256255.8555253.30355.1.10.tiff256×256261.3125254.19535.1.12.tiff256×256256.2578244.53285.1.13.tiff256×256274.8750245.37975.2.08.tiff512×512252.7471247.54345.2.09.tiff512×512274.3906257.34345.3.01.tiff1024×1024236.3027229.81257.1.02.tiff512×512252.9141226.21977.1.03.tiff512×512248.8984257.23247.1.04.tiff512×512281.2773258.40437.1.05.tiff512×512275.1055263.8584boat.512.tiff512×512230.2256232.7012elaine.512.tiff512×512266.6377230.0078gray21.512.tiff512×512244.8789245.3027ruler.512.tiff512×512290.8057223.2813testpat.1k.tiff1024×1024258.6455239.7627


### 4.2.4. Plain Image Sensitivity

When the plain image changes, the corresponding change degree of the cipher image is plain image sensitivity. Generally speaking, to effectively resist differential attacks, an image encryption scheme must have excellent plain image sensitivity. To intuitively show the plain image sensitivity of SIT-SR, we first encrypted some commonly used test images. Next, after changing one pixel bit for each plain image, the plain images with the smallest changes were encrypted. At last, we calculated the difference images between the cipher images obtained before and after the smallest changes. The relevant test results are shown in Figure11. As one can see from Figure 11, each plain image has undergone only one smallest change, but almost all cipher pixels have changed. In addition to that, these significant differences are independent of where the plain images change and are very close to random images.Figure 11
Plain image sensitivity test results for SIT-SR: (a1) 5.1.10.tiff; (a2) cipher image of (a1); (a3) the least significant bit of the pixel at (1,1) in (a1) is reversed; (a4) cipher image of (a3); (a5) difference image between (a2) and (a4); (b1) boat.512.tiff; (b2) cipher image of (b1); (b3) the least significant bit of the pixel at (256,256) in (b1) is reversed; (b4) cipher image of (b3); (b5) difference image between (b2) and (b4).UPCR and UACI are also utilized to qualitatively analyze the plain image sensitivity of SIT-SR. The UPCR and UACI values between the cipher images obtained before and after the smallest changes of 15 common test plain images are shown in Table7. Judging from the test results, SIT-SR has excellent plain image sensitivity. The test results of SIT-SR are closer to the ideal values 99.6094% and 33.4635% and perform better in terms of stability.Table 7
NPCR and UACI test results on plain image sensitivity.
Filename[13]SIT-SRNPCR (%)UACI (%)NPCR (%)UACI (%)5.1.10.tiff99.601433.477499.616233.46455.1.12.tiff99.622233.466899.609333.46425.1.13.tiff99.609133.478299.611933.45975.2.08.tiff99.613833.459699.608033.45895.2.09.tiff99.607233.449699.611733.45215.3.01.tiff99.610333.455199.610733.45957.1.02.tiff99.608833.474999.614033.46357.1.03.tiff99.602633.493099.608133.48367.1.04.tiff99.611733.469999.602533.46457.1.05.tiff99.610133.476699.608333.4643boat.512.tiff99.604533.461899.609833.4825elaine.512.tiff99.613933.491899.609833.4521gray21.512.tiff99.606933.466099.612233.4713ruler.512.tiff99.611333.439499.611533.4453testpat.1k.tiff99.605433.457199.609133.4624Average99.608033.469599.611333.4598Std. Dev.0.00420.01610.00280.0045The bold values here emphasize that SIT-SR has better performance than the other scheme.


### 4.2.5. Information Entropy

Information entropy is an indicator that can be used to measure the randomness or disorder of an information source. If the information entropy of the information source is higher, it can be considered that the information source has higher randomness or disorder [18–20]. When it comes to an 8-bit grayscale image, the information entropy of the grayscale image can be calculated as follows:(18)HS=−∑i=1npsi×log2psi,where n is the total number of symbols si; psi is the occurrence probability of symbol si. For the 8-bit grayscale cipher images, the ideal value of the information entropy is 8 [18–20]. From Table 8, one can see that the information entropy of each cipher image generated by SIT-SR is very close to the ideal value 8. As shown in Table 9, compared with several image encryption schemes, the information entropy of the Lena cipher image generated by SIT-SR is closest to the ideal value 8. Therefore, SIT-SR performs best in terms of the information entropy.Table 8
Information entropy test results of plain images and cipher images.
FilenamePlain imageCipher image5.2.08.tiff7.20107.99935.2.09.tiff6.99407.99945.3.01.tiff7.52377.99987.1.02.tiff4.00457.99937.1.03.tiff5.49577.99947.1.04.tiff6.10747.99937.1.05.tiff6.56327.9994boat.512.tiff7.19147.9994elaine.512.tiff7.50607.9994gray21.512.tiff4.39237.9993ruler.512.tiff0.50007.9994testpat.1k.tiff4.40777.9998Table 9
Information entropy test results of Lena cipher images.
Scheme[13][18][19][20]SIT-SRInformation entropy7.99927.99797.99097.99917.9994The bold value here emphasizes that SIT-SR has better performance than other schemes.In order to measure the randomness of cipher images more comprehensively, a measure named Local Shannon Entropy (LSE) is proposed [52]. This measure is increasingly adopted to verify the randomness of cipher images [13]. Mathematically, LSE can be defined as follows:(19)Lq,sr=∑i=1qHriq,where r1,r2,…,rq are q randomly selected nonoverlapping image blocks, s is the number of pixels in each block, and Hri is the information entropy of ri. According to the test method suggested in [52], we carried out the LSE test on the cipher images generated by SIT-SR, and the relevant test results are shown in Table 10. Compared with two recent image encryption schemes, SIT-SR has the best performance in terms of standard deviation and pass rate.Table 10
LSE test results of different schemes.
Filename[13][15]SIT-SR5.2.087.90237.90247.90225.2.097.90207.90217.90237.1.027.90207.90157.90217.1.037.90267.90197.90247.1.047.90197.90217.9023boat.5127.90187.90227.9024gray21.5127.90267.90267.9025ruler.5127.90417.90287.9026Std.Dev.0.00070.00040.0002Pass/All6/87/88/8The bold values indicate that compared with the other two schemes, SIT-SR has the best performance in terms of standard deviation and pass rate.


### 4.2.6. Pixel Correlation

The extremely high correlation between adjacent pixels is one of the salient features of plain images and also one of the important reasons why traditional encryption schemes are not suitable for image encryption [50]. Therefore, a secure image encryption scheme should eliminate the correlation between adjacent pixels as much as possible. CC (correlation coefficient) is an effective indicator to measure the correlation between adjacent pixels, and its definition is as follows:(20)CCab=Ea−Ea×b−EbDa×Db,where a and b are the grayscale values of two adjacent pixels; Ea and Da are the expectation and variance of the grayscale value a. In order to verify the performance of SIT-SR in terms of the pixel correlation, for the horizontal, vertical, and diagonal directions, we have randomly selected 20,000 pairs of adjacent pixels from each plain image and its corresponding cipher image to calculate the CCs. The relevant test results are shown in Table 11.Table 11
Correlation test results for adjacent pixels of plain images and their cipher images.
FilenameHorizontalVerticalDiagonalPlain imageCipher imagePlain imageCipher imagePlain imageCipher image5.2.08.tiff0.89060.00080.9322−0.00380.84520.00285.2.09.tiff0.8591−0.00040.9000−0.00180.80070.00205.3.01.tiff0.98170.00150.97760.00440.8981−0.00337.1.02.tiff0.9446−0.00240.9431−0.00160.90030.00017.1.03.tiff0.9317−0.00210.94360.00020.9059−0.00177.1.04.tiff0.96720.00380.97710.00280.9552−0.00597.1.05.tiff0.91080.00380.94250.00580.89190.0024boat.512.tiff0.97110.00300.93940.00480.92450.0002elaine.512.tiff0.9720−0.00110.97610.00100.96960.0016gray21.512.tiff0.9998−0.00230.9968−0.00360.99660.0031ruler.512.tiff0.4702−0.00260.45240.0004−0.0312−0.0023testpat.1k.tiff0.79920.00350.75010.00510.69970.0005From Table11, one can see that there are very high correlations between adjacent pixels of the plain images; that is, the absolute values of CCs are extremely high, whereas in the cipher images generated by SIT-SR, there is almost no correlation between adjacent pixels; that is, the absolute values of CCs are extremely low <0.006.In addition, in order to more intuitively show the correlation changes between adjacent pixels caused by the encryption of SIT-SR, the correlation distribution charts of the plain image elaine.512.tiff and its corresponding cipher image are drawn. As can be seen from Figure12, after the encryption processing of SIT-SR, there is almost no correlation between adjacent pixels in each direction.Figure 12
Pixel correlation distribution charts of elaine.512.tiff and its cipher image: (a, d) distribution charts in horizontal direction; (b, e) distribution charts in vertical direction; (c, f) distribution charts in diagonal direction.
(a)(b)(c)(d)(e)(f)


### 4.2.7. Chosen-Plaintext Attack

In fact, almost all simulation tests related to security analysis can only ensure the security of image encryption schemes under ciphertext-only attacks (COAs) [51, 53]. This is exactly why some image encryption schemes have been broken. Among the four types of attacks, which are COAs, Known-Plaintext Attacks (KPAs), CPAs, and Chosen-Ciphertext Attacks (CCAs), CCAs are the most threatening ones, but the attack conditions required by them are practically meaningless [3, 50]. If attackers can choose cipher images arbitrarily, then they do not need to crack at all, because for any cipher image, they can directly recover its plain image. Therefore, it is generally believed that CPAs are the most threatening ones among common practical attacks. Actually, in the cryptanalysis works about image encryption, the vast majority of them adopt CPAs [3, 51]. Next, from the perspective of attackers, the ability of SIT-SR to resist CPAs is analyzed.Apparently, attackers will encounter several problems when they try to break SIT-SR with CPAs. Firstly, we assume that they could obtain the equivalent key streams of the encryption process from chosen plain images and corresponding cipher images. However, because SIT-SR introduce DPPs in serval encryption steps, the equivalent key streams they obtained cannot be used to recover other ordinary plain images, which are encrypted under different DPPs. Secondly, SIT-SR also performs nonlinear transformations on the plain image pixels during the permutation process, so the common attack method that ignores the permutation process by the chosen plain images composed of single-value pixels cannot work. Thirdly, SIT-SR adopts a nonlinear diffusion structure; that is, it adopts the discrete logarithms based on two different generators, which makes the encryption process cannot be simplified by chosen plain images. To sum up, SIT-SR can effectively resist CPAs.


### 4.2.8. Encryption Efficiency

Improving encryption efficiency is one of the most important motivations to design new image encryption schemes. SIT-SR introduce DPPs and discrete logarithms, but in fact, discrete logarithms can be calculated in advance, and the calculation process of DPP is very simple, so the impact on encryption efficiency is very small. In addition, SIT-SR uses single-pixel diffusion and only performs three iterations in the encryption process. These also help to reduce the total number of primitive operations that need to be executed. Table12 shows the average time required by SIT-SR and some other recent image encryption schemes to encrypt the 8-bit grayscale image Lena 256×256. As can be seen from Table 12, although the time complexity of each image encryption scheme is OM×N, the scheme proposed in [13] requires the least number of primitive operations, so it has the highest encryption efficiency, whereas for SIT-SR, it adds a certain number of primitive operations to ensure the security, but it still maintains the significant advantage of high encryption efficiency. That is, in terms of encryption efficiency, SIT-SR is significantly better than the remaining four image encryption schemes.Table 12
Comparison of test results obtained by different encryption schemes to encrypt Lena image.
SchemeSIT-SR[13][18][19][21][22]Time (s)0.0890.0720.4170.6830.2750.264Throughput (Mbps)5.61806.94441.19900.73211.81821.8939The bold values here emphasize that SIT-SR has better performance than other schemes.


## 4.2.1. Encryption and Decryption of Different Types of Images

For different types of images, we uniformly treat them as 8-bit grayscale images in ITS-CDD. Specifically, for images with a pixel depth of less than 8 bits, we directly process them as 8-bit grayscale images, whereas for images with pixel depth greater than 8 bits, we encrypt them in groups of 8 bits. For example, if we need to encrypt an image with a pixel depth of 16 bits, we can encrypt the lower 8 bits and higher 8 bits of each plain image pixel separately. Figure7 shows the encryption and decryption effects of SIT-SR for different types of images. One can see that SIT-SR has excellent encryption effects for different types of images. The generated cipher images are very similar to noise images, and attackers cannot directly obtain any valuable information from these cipher images.Figure 7
Encryption and decryption effects for different types of images: (a) f15.png; (b) cipher image of (a); (c) decrypted image of (b); (d) 5.2.09.tiff; (e) cipher image of (d); (f) decrypted image of (e); (g) gray21.512.tiff; (h) cipher image of (g); (i) decrypted image of (h); (j) 4.2.04.tiff; (k) cipher image of (j); (l) decrypted image of (k); (m) f15.png; (n) cipher image of (m); (o) decrypted image of (n); (p) f15.png; (q) cipher image of (p); (r) decrypted image of (q); (s) f15.png; (t) cipher image of (s); (u) decrypted image of (t); (v) f15.png; (w) cipher image of (v); (x) decrypted image of (w).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


## 4.2.2. Key Space and Key Sensitivity

Since the key space would affect the ability to resist brute force attacks, a secure encryption scheme should have a sufficiently large key space [50]. We carefully design the secret key, which not only solves the issues about equivalent secret keys, but also expands the key space to 2270. Therefore, SIT-SR is excellent in terms of the ability to resist brute force attacks.It is well known that a secure encryption scheme should be able to achieve superior confusion effect [50, 51]. That is, one smallest change in the secret key should make the cipher image change significantly. To evaluate the performance of SIT-SR in this regard, we randomly generated the secret key(13)KR=03D7D6F3E884829564EDBA77D8683B811AE4FC8655CC7EE7BC305537BFEA2EE2238E.UsingKR, we encrypted elaine.512.tiff to obtain the corresponding cipher image ČR. Next, we changed one bit of KR to get two secret keys KR1, KR2 with single smallest changes. These two changed secret keys also were used to encrypt elaine.512.tiff, thus obtaining the corresponding cipher images ČR1, ČR2. Finally, the difference images between ČR1, ČR2, and ČR were calculated. As can be seen from Figure 8, the difference images between ČR1, ČR2, and ČR look similar to an ordinary cipher image. This means that the key sensitivity of SIT-SR in the encryption process is extraordinary.Figure 8
Key sensitivity test for encryption process: (a) elaine.512.tiff; (b)ČR obtained by KR; (c) ČR1 obtained by KR1; (d) ČR2 obtained by KR2; (e) difference image between ČR and ČR1; (f) difference image between ČR and ČR2.
(a)(b)(c)(d)(e)(f)Similarly, in order to verify the key sensitivity of SIT-SR in the decryption process,KR1 and KR2 were adopted to decrypt ČR. The test results are shown in Figure 9. Once again, judging from the difference image between the wrongly decrypted images, the key sensitivity of SIT-SR in the decryption process is excellent.Figure 9
Key sensitivity test for decryption process: (a)ČR obtained by KR; (b) decrypted image ČR obtained by KR; (c) decrypted image ČR1 obtained by KR1; (d) decrypted image ČR2 obtained by KR2; (e) difference image between ČR1 and ČR2.
(a)(b)(c)(d)(e)For measuring the degree of changes between images, NPCR (Number of Pixel Change Ratio) and UACI (Unified Average Change in Intensity) are commonly used indicators [13]. Among them,(14)NPCR=1M×N∑i,jDi,j×100%,refers to the ratio of the pixels that change, whereas(15)UACI=1M×N∑i,jCi,j1−Ci,j2255×100%,refers to the average intensity of the pixel value changes. In equations (14) and (15), M×N is the size of the images, i=1,2,…,M, j=1,2,…,N, and D is the difference matrix between the image C1 before the change and the image C2 after the change. If Ci,j1≠Ci,j2, then Di,j=1. Otherwise, Di,j=0. In order to quantitatively analyze the key sensitivity of SIT-SR, we calculated the NPCR and UACI values between the cipher images before and after the secret key changes. As one can see from Table 4, all the test results are very close to the ideal values, which means that SIT-SR does have extremely high key sensitivity.Table 4
UPCR and UACI values between cipher images when secret key changes.
ChangeNPCR (%)UACI (%)Lowest bit ofb1 is inverted99.612133.4601Lowest bit ofb2 is inverted99.602833.4625Lowest bit ofb3 is inverted99.609733.4837Lowest bit ofb4 is inverted99.601333.4633Lowest bit ofb5 is inverted99.600933.4764Lowest bit ofb6 is inverted99.617433.4512Lowest bit ofb7 is inverted99.607533.4810Lowest bit ofb8 is inverted99.612233.4706Lowest bit ofb9 is inverted99.613033.4562Random image99.609433.4635The bold values are the ideal values.


## 4.2.3. Pixel Value Distribution

Unlike plain images, cipher images must have extremely uniform pixel value distributions; otherwise attackers will have the opportunity to conduct statistics based attacks [50, 51]. In order to visually demonstrate the pixel value distribution characteristics of the plain images and the cipher images generated by SIT-SR, the pixel value distribution histograms of these images are plotted. As can be seen from Figure 10, the pixel distributions of the plain images are relatively concentrated, whereas the pixels of the cipher images are extremely uniformly distributed throughout the entire value range.Figure 10
Pixel value distribution histograms of elaine.512.tiff, 7.1.03.tiff, gray21.512.tiff, and their corresponding cipher images: (a) elaine.512.tiff; (b) histogram of (a); (c) cipher image of (a); (d) histogram of (c); (e) 7.1.03.tiff; (f) histogram of (e); (g) cipher image of (e); (h) histogram of (g); (i) gray21.512.tiff; (j) histogram of (i); (k) cipher image of (i); (l) histogram of (k).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)In addition, the histogram variance analysis and chi-square test analysis are also performed on the cipher images to qualitatively analyze the ability of SIT-SR to resist statistical attacks. In general, if the histogram variance of a cipher image is smaller, then the uniformity of the cipher image is higher. The histogram variance of an 8-bit grayscale image can be defined as follows:(16)VH=12562∑i=1256∑j=125612pi−pj2,where H=p1,p2,…,p256; pi and pj are the numbers of the pixels whose grayscale values are equal to i−1 and j−1. Table 5 lists the histogram variance test results of some images. These images include one random image, the 8-bit grayscale image Lena, and its cipher images generated by different schemes.Table 5
Histogram variance test results of some images.
AlgorithmImageVarianceLena256.bmp33860.0546[13]Cipher image266.7578[16]Cipher image260.7188[17]Cipher image276.3906SIT-SRCipher image257.1094Random image253.8946The bold value means that SIT-SR has the best test result among four compared schemes.From Table5, one can see that the histogram variance of the plain image is very large, which means that its pixel value distribution is extremely uneven, whereas among the cipher images generated by the four image encryption schemes, the cipher image of SIT-SR has the smallest histogram variance, which indicates that this cipher image has the most uniform pixel value distribution and is closest to the random image.Another way to quantitatively analyze the uniformity of a cipher image is the chi-square test. The chi-square value of a cipher image can be calculated as follows:(17)χ2=∑i=1nsi−H×W×p2H×W×p,where H×W is the height and width of the cipher image, si is the number of pixels whose pixel value is i−1, n is the number of all possible pixel values (for 8-bit grayscale images, n=256), and p=1/n. Next, the critical value χ0.052255 of the chi-square test at the significant level α=0.05 can be determined, which is 293.2478. If a cipher image has a chi-square value less than 293.2478, then this image can be considered to have passed the chi-square test; that is, its pixel value distribution is statistically uniform. Consequently, the smaller the chi-square value of a cipher image is, the better its uniformity is. As can be seen from Table 6, for some commonly used test images, the cipher images generated by SIT-SR have all passed the chi-square test. And in most cases, SIT-SR performs better than another scheme.Table 6
Chi-square values of different cipher images.
FilenameSizeChi-square value[13]SIT-SRLena256.bmp256×256255.8555253.30355.1.10.tiff256×256261.3125254.19535.1.12.tiff256×256256.2578244.53285.1.13.tiff256×256274.8750245.37975.2.08.tiff512×512252.7471247.54345.2.09.tiff512×512274.3906257.34345.3.01.tiff1024×1024236.3027229.81257.1.02.tiff512×512252.9141226.21977.1.03.tiff512×512248.8984257.23247.1.04.tiff512×512281.2773258.40437.1.05.tiff512×512275.1055263.8584boat.512.tiff512×512230.2256232.7012elaine.512.tiff512×512266.6377230.0078gray21.512.tiff512×512244.8789245.3027ruler.512.tiff512×512290.8057223.2813testpat.1k.tiff1024×1024258.6455239.7627


## 4.2.4. Plain Image Sensitivity

When the plain image changes, the corresponding change degree of the cipher image is plain image sensitivity. Generally speaking, to effectively resist differential attacks, an image encryption scheme must have excellent plain image sensitivity. To intuitively show the plain image sensitivity of SIT-SR, we first encrypted some commonly used test images. Next, after changing one pixel bit for each plain image, the plain images with the smallest changes were encrypted. At last, we calculated the difference images between the cipher images obtained before and after the smallest changes. The relevant test results are shown in Figure11. As one can see from Figure 11, each plain image has undergone only one smallest change, but almost all cipher pixels have changed. In addition to that, these significant differences are independent of where the plain images change and are very close to random images.Figure 11
Plain image sensitivity test results for SIT-SR: (a1) 5.1.10.tiff; (a2) cipher image of (a1); (a3) the least significant bit of the pixel at (1,1) in (a1) is reversed; (a4) cipher image of (a3); (a5) difference image between (a2) and (a4); (b1) boat.512.tiff; (b2) cipher image of (b1); (b3) the least significant bit of the pixel at (256,256) in (b1) is reversed; (b4) cipher image of (b3); (b5) difference image between (b2) and (b4).UPCR and UACI are also utilized to qualitatively analyze the plain image sensitivity of SIT-SR. The UPCR and UACI values between the cipher images obtained before and after the smallest changes of 15 common test plain images are shown in Table7. Judging from the test results, SIT-SR has excellent plain image sensitivity. The test results of SIT-SR are closer to the ideal values 99.6094% and 33.4635% and perform better in terms of stability.Table 7
NPCR and UACI test results on plain image sensitivity.
Filename[13]SIT-SRNPCR (%)UACI (%)NPCR (%)UACI (%)5.1.10.tiff99.601433.477499.616233.46455.1.12.tiff99.622233.466899.609333.46425.1.13.tiff99.609133.478299.611933.45975.2.08.tiff99.613833.459699.608033.45895.2.09.tiff99.607233.449699.611733.45215.3.01.tiff99.610333.455199.610733.45957.1.02.tiff99.608833.474999.614033.46357.1.03.tiff99.602633.493099.608133.48367.1.04.tiff99.611733.469999.602533.46457.1.05.tiff99.610133.476699.608333.4643boat.512.tiff99.604533.461899.609833.4825elaine.512.tiff99.613933.491899.609833.4521gray21.512.tiff99.606933.466099.612233.4713ruler.512.tiff99.611333.439499.611533.4453testpat.1k.tiff99.605433.457199.609133.4624Average99.608033.469599.611333.4598Std. Dev.0.00420.01610.00280.0045The bold values here emphasize that SIT-SR has better performance than the other scheme.


## 4.2.5. Information Entropy

Information entropy is an indicator that can be used to measure the randomness or disorder of an information source. If the information entropy of the information source is higher, it can be considered that the information source has higher randomness or disorder [18–20]. When it comes to an 8-bit grayscale image, the information entropy of the grayscale image can be calculated as follows:(18)HS=−∑i=1npsi×log2psi,where n is the total number of symbols si; psi is the occurrence probability of symbol si. For the 8-bit grayscale cipher images, the ideal value of the information entropy is 8 [18–20]. From Table 8, one can see that the information entropy of each cipher image generated by SIT-SR is very close to the ideal value 8. As shown in Table 9, compared with several image encryption schemes, the information entropy of the Lena cipher image generated by SIT-SR is closest to the ideal value 8. Therefore, SIT-SR performs best in terms of the information entropy.Table 8
Information entropy test results of plain images and cipher images.
FilenamePlain imageCipher image5.2.08.tiff7.20107.99935.2.09.tiff6.99407.99945.3.01.tiff7.52377.99987.1.02.tiff4.00457.99937.1.03.tiff5.49577.99947.1.04.tiff6.10747.99937.1.05.tiff6.56327.9994boat.512.tiff7.19147.9994elaine.512.tiff7.50607.9994gray21.512.tiff4.39237.9993ruler.512.tiff0.50007.9994testpat.1k.tiff4.40777.9998Table 9
Information entropy test results of Lena cipher images.
Scheme[13][18][19][20]SIT-SRInformation entropy7.99927.99797.99097.99917.9994The bold value here emphasizes that SIT-SR has better performance than other schemes.In order to measure the randomness of cipher images more comprehensively, a measure named Local Shannon Entropy (LSE) is proposed [52]. This measure is increasingly adopted to verify the randomness of cipher images [13]. Mathematically, LSE can be defined as follows:(19)Lq,sr=∑i=1qHriq,where r1,r2,…,rq are q randomly selected nonoverlapping image blocks, s is the number of pixels in each block, and Hri is the information entropy of ri. According to the test method suggested in [52], we carried out the LSE test on the cipher images generated by SIT-SR, and the relevant test results are shown in Table 10. Compared with two recent image encryption schemes, SIT-SR has the best performance in terms of standard deviation and pass rate.Table 10
LSE test results of different schemes.
Filename[13][15]SIT-SR5.2.087.90237.90247.90225.2.097.90207.90217.90237.1.027.90207.90157.90217.1.037.90267.90197.90247.1.047.90197.90217.9023boat.5127.90187.90227.9024gray21.5127.90267.90267.9025ruler.5127.90417.90287.9026Std.Dev.0.00070.00040.0002Pass/All6/87/88/8The bold values indicate that compared with the other two schemes, SIT-SR has the best performance in terms of standard deviation and pass rate.


## 4.2.6. Pixel Correlation

The extremely high correlation between adjacent pixels is one of the salient features of plain images and also one of the important reasons why traditional encryption schemes are not suitable for image encryption [50]. Therefore, a secure image encryption scheme should eliminate the correlation between adjacent pixels as much as possible. CC (correlation coefficient) is an effective indicator to measure the correlation between adjacent pixels, and its definition is as follows:(20)CCab=Ea−Ea×b−EbDa×Db,where a and b are the grayscale values of two adjacent pixels; Ea and Da are the expectation and variance of the grayscale value a. In order to verify the performance of SIT-SR in terms of the pixel correlation, for the horizontal, vertical, and diagonal directions, we have randomly selected 20,000 pairs of adjacent pixels from each plain image and its corresponding cipher image to calculate the CCs. The relevant test results are shown in Table 11.Table 11
Correlation test results for adjacent pixels of plain images and their cipher images.
FilenameHorizontalVerticalDiagonalPlain imageCipher imagePlain imageCipher imagePlain imageCipher image5.2.08.tiff0.89060.00080.9322−0.00380.84520.00285.2.09.tiff0.8591−0.00040.9000−0.00180.80070.00205.3.01.tiff0.98170.00150.97760.00440.8981−0.00337.1.02.tiff0.9446−0.00240.9431−0.00160.90030.00017.1.03.tiff0.9317−0.00210.94360.00020.9059−0.00177.1.04.tiff0.96720.00380.97710.00280.9552−0.00597.1.05.tiff0.91080.00380.94250.00580.89190.0024boat.512.tiff0.97110.00300.93940.00480.92450.0002elaine.512.tiff0.9720−0.00110.97610.00100.96960.0016gray21.512.tiff0.9998−0.00230.9968−0.00360.99660.0031ruler.512.tiff0.4702−0.00260.45240.0004−0.0312−0.0023testpat.1k.tiff0.79920.00350.75010.00510.69970.0005From Table11, one can see that there are very high correlations between adjacent pixels of the plain images; that is, the absolute values of CCs are extremely high, whereas in the cipher images generated by SIT-SR, there is almost no correlation between adjacent pixels; that is, the absolute values of CCs are extremely low <0.006.In addition, in order to more intuitively show the correlation changes between adjacent pixels caused by the encryption of SIT-SR, the correlation distribution charts of the plain image elaine.512.tiff and its corresponding cipher image are drawn. As can be seen from Figure12, after the encryption processing of SIT-SR, there is almost no correlation between adjacent pixels in each direction.Figure 12
Pixel correlation distribution charts of elaine.512.tiff and its cipher image: (a, d) distribution charts in horizontal direction; (b, e) distribution charts in vertical direction; (c, f) distribution charts in diagonal direction.
(a)(b)(c)(d)(e)(f)


## 4.2.7. Chosen-Plaintext Attack

In fact, almost all simulation tests related to security analysis can only ensure the security of image encryption schemes under ciphertext-only attacks (COAs) [51, 53]. This is exactly why some image encryption schemes have been broken. Among the four types of attacks, which are COAs, Known-Plaintext Attacks (KPAs), CPAs, and Chosen-Ciphertext Attacks (CCAs), CCAs are the most threatening ones, but the attack conditions required by them are practically meaningless [3, 50]. If attackers can choose cipher images arbitrarily, then they do not need to crack at all, because for any cipher image, they can directly recover its plain image. Therefore, it is generally believed that CPAs are the most threatening ones among common practical attacks. Actually, in the cryptanalysis works about image encryption, the vast majority of them adopt CPAs [3, 51]. Next, from the perspective of attackers, the ability of SIT-SR to resist CPAs is analyzed.Apparently, attackers will encounter several problems when they try to break SIT-SR with CPAs. Firstly, we assume that they could obtain the equivalent key streams of the encryption process from chosen plain images and corresponding cipher images. However, because SIT-SR introduce DPPs in serval encryption steps, the equivalent key streams they obtained cannot be used to recover other ordinary plain images, which are encrypted under different DPPs. Secondly, SIT-SR also performs nonlinear transformations on the plain image pixels during the permutation process, so the common attack method that ignores the permutation process by the chosen plain images composed of single-value pixels cannot work. Thirdly, SIT-SR adopts a nonlinear diffusion structure; that is, it adopts the discrete logarithms based on two different generators, which makes the encryption process cannot be simplified by chosen plain images. To sum up, SIT-SR can effectively resist CPAs.


## 4.2.8. Encryption Efficiency

Improving encryption efficiency is one of the most important motivations to design new image encryption schemes. SIT-SR introduce DPPs and discrete logarithms, but in fact, discrete logarithms can be calculated in advance, and the calculation process of DPP is very simple, so the impact on encryption efficiency is very small. In addition, SIT-SR uses single-pixel diffusion and only performs three iterations in the encryption process. These also help to reduce the total number of primitive operations that need to be executed. Table12 shows the average time required by SIT-SR and some other recent image encryption schemes to encrypt the 8-bit grayscale image Lena 256×256. As can be seen from Table 12, although the time complexity of each image encryption scheme is OM×N, the scheme proposed in [13] requires the least number of primitive operations, so it has the highest encryption efficiency, whereas for SIT-SR, it adds a certain number of primitive operations to ensure the security, but it still maintains the significant advantage of high encryption efficiency. That is, in terms of encryption efficiency, SIT-SR is significantly better than the remaining four image encryption schemes.Table 12
Comparison of test results obtained by different encryption schemes to encrypt Lena image.
SchemeSIT-SR[13][18][19][21][22]Time (s)0.0890.0720.4170.6830.2750.264Throughput (Mbps)5.61806.94441.19900.73211.81821.8939The bold values here emphasize that SIT-SR has better performance than other schemes.


## 5. Conclusions

In order to improve the efficiency and security of image transmission, an image transmission scheme based on two chaotic maps is proposed in this paper. The proposed scheme divides the image transmission from sensor nodes to receivers into two stages and carries out a targeted design, which can better adapt to heterogeneous application environments. For image transmission between sensor nodes and sink nodes, the proposed scheme reduces the requirements for hardware resources and improves the image reconstruction quality by introducing a lightweight chaotic map. Besides, the design of dynamic perturbation improves the security of image transmission at this stage, whereas for image transmission between sink nodes and receivers, the proposed scheme improves the security and efficiency of image transmission by introducing another chaotic map with better chaotic performance and discrete logarithms. In order to verify and demonstrate the excellent performance of the proposed scheme, extensive simulation tests and theoretical analyses are carried out. These tests and analyses show that, compared with some recent schemes, the proposed scheme has higher feasibility, security, and practicability. In the future, we will extend the proposed scheme to video transmission.



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*Source: 1898998-2021-11-25.xml*