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Design of Clothing with Encrypted Information of Lost Children Information Based on Chaotic System and DNA Theory

Publicado en línea: 01 Oct 2022
Volumen & Edición: AHEAD OF PRINT
Páginas: -
Detalles de la revista
License
Formato
Revista
eISSN
2300-0929
Primera edición
19 Oct 2012
Calendario de la edición
4 veces al año
Idiomas
Inglés
Introduction

In recent years, the problem of lost children has become a hot topic around the world. The topic of losing of children has frequently made headlines, which has become a social issue of public concern. Every year, countries will formulate some corresponding measures for child safety issues, but the loss of children still cannot be avoided. With the improvement of science and technology, children's tracking devices, such as children's smart watches and anti-loss schoolbags, have emerged in the wearable market.

Through locating positions of children, warning of danger, and recording, tracking devices can help parents to acquire the position and confirm the safety of their children. However, there are also some disadvantages, such as the instability of electronic components, and the network environment may have deviations on positioning. The life of the battery is another issue. The volume of the existing intelligent positioning equipment is usually small, which limits the battery capacity. In addition, such a locator has a large amount of radiation. Moreover, the compliance of the locator also makes it inconvenient to clean clothes and shoes [1,2,3,4,5,6,7,8,9,10]. Therefore, compared with electronic tags and identification cards that rely on radio frequency identification (RFID) technology, we prefer a nonradiation, confidential identification technology that does not need to rely on network storage. The encryption algorithm proposed in this paper can be used in any environment to decrypt the child's identifying information.

Nowadays, chaotic system theory has been widely used in communication encryption. It is very suitable for image encryption because of its sensitivity to parameter values and initial values, as well as uncertain random motion in the sequence [1,2,3,4,5]. Therefore, we have designed an information encryption system for lost children based on chaotic systems. The children's photos and their relevant information are encrypted and stored in the washing labels. Multi-image encryption can encrypt more than one image, which ensures that the pixel of each encryption effect is more efficient, and the whole processing time is shorter. Multi-image encryption based on chaos theory has already existed; however, problems of low security performance and low encryption efficiency are apparent. Therefore, we propose a multi-image encryption algorithm based on both fractional and DNA theory.

On the basis of the integer-order chaotic system, we decompose its fractional order, because the fractional-order system has more complex dynamic characteristics, and the security performance of the encryption system designed on this basis is also higher [6,7,8,9]. For instance, Dong et al. designed a color image encryption algorithm based on a fractional hyperchaotic system and introduced the compression theory of DNA encoding [10]. Yang et al. analyzed the dynamic behavior of the fractional-order memory delay system and applied it to the image encryption design [11]. Among the existing decomposition methods, the ADM decomposition method has the advantages of fast convergence, low resource consumption, and fast calculation speed. It is widely used to solve chaotic systems [12,13,14,15,16]. Lei et al. used the Adomian decomposition method to analyze the dynamics of the fractional-order chaotic system and realized the circuit simulation [17]. He et al. used CADM to analyze the dynamic characteristics of a fractional-order hyperchaotic system [18]. At present, there is little research on using fractional-order chaotic systems for communication security [19,20,21]. Therefore, this paper uses the ADM decomposition method to analyze the fractional-order dynamics of the new Lü chaotic system and applies it to the image encryption scheme. The simulation is carried out on the DSP platform, and t he results show that the design proposed in this paper not only satisfies the requirements of concealment, flexibility, and security of confidential communication, but also has the advantages of low cost, portability, and low maintenance.

In addition, when performing multiple image encryption tasks, the speed of a single-image encryption design is relatively fast, but the overall efficiency is low [22,23,24,25,26]. The multi-image encryption design can perform the encryption operation of multiple images at the same time and ensure the phase diagram of each encryption effect; the efficiency is higher, and the overall process is shorter in time. At present, there are already some multi-image encryption designs based on chaos theory [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47]. Xiao et al. designed a low-cost and secure multi-image encryption scheme based on p-tensor product compressed sensing [27]. Sinha et al. designed a multi-image encryption scheme based on chaos theory, using multiple scrambling rules of canonical transformation [28]. However, some existing encryption algorithms have problems such as low security performance or low encryption efficiency [29,30,31,32,33]. For this reason, this paper proposes a multi-image encryption algorithm based on fractional order and DNA theory. Fractional chaotic systems can have more complex system dynamics and faster calculation efficiency. The DNA theory and Arnold transform scrambling images can increase the security performance of the algorithm and thus improve the efficiency of the encryption design as a whole.

The overall structure of the paper is arranged as follows: The second section introduces the research background of this design. In Section 3, the dynamic characteristics of the chaotic system are analyzed and DSP simulation is carried out. Section 4 introduces the content of multi-image encryption design in detail, including two parts: DNA theory and Arnold transform scrambling. In Section 5, the performance of the new encryption algorithm is comprehensively analyzed and compared with the existing algorithm. Section 6 gives conclusions.

Development Background and Practical Application

At present, the identification technology of information tends to be the RFID system. This system uses electronic tags to identify objects. The electronic tags exchange data with the reader through radio waves, and the data returned by the electronic tags can be transmitted to the host for identification. Therefore, RFID technology must have a host data exchange and management system to complete the storage, management and control of electronic tag data information. The system proposed in this paper does not require a network, and the encrypted text itself contains picture information and children's personal information, which greatly reduces the cost and increases the practicability and flexibility of the application.

These photos and related information on children are stored in two different pictures; then, those two pictures will be combined and encoded. The encrypted information is embedded in the washing labels of clothing of children. When a child is lost, the police can decrypt the relevant information through specific readers. Then information about the missing child will be obtained accurately, and the guardians of lost children will be informed promptly. This method of encryption greatly increases the difficulty in decoding for criminals, prohibiting them from accessing the information on the children, which may improve the probability of recovering of lost children.

The child's information is shown in Figure 1(a), and the child's photo is shown in Figure 1(b). Through encryption, the encrypted information is shown in Figure 1(c), and the effect image printed on the clothes is shown in Figure 1(d)–(e).

Figure 1

(a) Lost children information; (b) lost children photo; (c) encrypted image; (d) assembling effect picture (front); (e) assembling effect picture (back).

Fractional Order Lü Chaotic System
A New Improved Lü Chaotic System

According to the system description of the three-dimensional Lü system, the variable U and the constant k are additionally added to the system, and a new four-dimensional chaotic system is established as follows: {x˙=a(xy)+uy˙=byxz+kz˙=cz+xyu˙=dy \left\{ {\matrix{ {\dot x = a\left( {x - y} \right) + u} \hfill \cr {\dot y = by - xz + k} \hfill \cr {\dot z = - cz + xy} \hfill \cr {\dot u = dy} \hfill \cr } } \right. in which x, y, z, and u are four state variables, and positive parameters are a, b, c, d, and k.

For the above system, when a = 40, b = 30, c = 4, d = 20, and k = 90 and initial condition (1, 1, 30, 80), the maximum Lyapunov exponent L1 = 1.9450, L2 = 0, L3 = −1.5795, and L4 = −10.6985. At this time, the hidden attractors are chaotic. The hidden attractor graph is shown in Figure 2.

Figure 2

Attractor phase diagram of Lü system. (a) x–y; (b) x–z; (c) xy–z.

Scheduling Simplified Lü Chaotic System Solving

Using the ADM algorithm, the fractional differential equation of System 1 is {Dt0qx1=a(x1x2)+x4Dt0qx2=bx2x1x3+kDt0qx3=cx3+x1x2Dt0qx4=dx2 \left\{ {\matrix{ {D_{{t_0}}^q{x_1} = a\left( {{x_1} - {x_2}} \right) + {x_4}} \hfill \cr {D_{{t_0}}^q{x_2} = b{x_2} - {x_1}{x_3} + k} \hfill \cr {D_{{t_0}}^q{x_3} = - c{x_3} + {x_1}{x_2}} \hfill \cr {D_{{t_0}}^q{x_4} = d{x_2}} \hfill \cr } } \right.

Decompose the linear items and nonlinear items in the system, [Lx1Lx2Lx3Lx4]=[a(x1x2)+x4bx2+kcx3dx2][Nx1Nx2Nx3Nx4]=[0x1x3x1x20][g1g2g3g4]=[0000] \left[ {\matrix{ {L{x_1}} \hfill \cr {L{x_2}} \hfill \cr {L{x_3}} \hfill \cr {L{x_4}} \hfill \cr } } \right] = \left[ {\matrix{ {a\left( {{x_1} - {x_2}} \right) + {x_4}} \hfill \cr {b{x_2} + k} \hfill \cr { - c{x_3}} \hfill \cr {d{x_2}} \hfill \cr } } \right]\,\left[ {\matrix{ {N{x_1}} \hfill \cr {N{x_2}} \hfill \cr {N{x_3}} \hfill \cr {N{x_4}} \hfill \cr } } \right] = \left[ {\matrix{ 0 \hfill \cr { - {x_1}{x_3}} \hfill \cr {{x_1}{x_2}} \hfill \cr 0 \hfill \cr } } \right]\,\left[ {\matrix{ {{g_1}} \hfill \cr {{g_2}} \hfill \cr {{g_3}} \hfill \cr {{g_4}} \hfill \cr } } \right] = \left[ {\matrix{ 0 \hfill \cr 0 \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right]\,\,

Furthermore, decompose the nonlinear items available, {A20=x10x30A21=x10x31x10x31A22=x12x30x11x31x10x32A23=x13x30x12x31x11x32x10x33A24=x44x10x43x11x42x12x41x13x40x14A25=x15x30x14x31x13x32x12x33x11x34x10x35A26=x16x30x15x31x14x32x13x33x12x34x11x35x10x36 \left\{ {\matrix{ {A_2^0 = - x_1^0x_3^0} \hfill \cr {A_2^1 = - x_1^0x_3^1 - x_1^0x_3^1} \hfill \cr {A_2^2 = - x_1^2x_3^0 - x_1^1x_3^1 - x_1^0x_3^2} \hfill \cr {A_2^3 = - x_1^3x_3^0 - x_1^2x_3^1 - x_1^1x_3^2 - x_1^0x_3^3} \hfill \cr {A_2^4 = - x_4^4x_1^0 - x_4^3x_1^1 - x_4^2x_1^2 - x_4^1x_1^3 - x_4^0x_1^4} \hfill \cr {A_2^5 = - x_1^5x_3^0 - x_1^4x_3^1 - x_1^3x_3^2 - x_1^2x_3^3 - x_1^1x_3^4 - x_1^0x_3^5} \hfill \cr {A_2^6 = - x_1^6x_3^0 - x_1^5x_3^1 - x_1^4x_3^2 - x_1^3x_3^3 - x_1^2x_3^4 - x_1^1x_3^5 - x_1^0x_3^6} \hfill \cr } } \right. {A30=x20x10A31=x20x11+x20x11A32=x22x10+x21x11+x20x12A33=x23x10+x22x11+x21x12+x20x13A34=x24x10+x23x11+x22x12+x21x13+x20x14A35=x25x10+x24x11+x23x12+x22x13+x21x14+x20x15A36=x26x10+x25x11+x24x12+x23x13+x22x14+x21x15+x20x16 \left\{ {\matrix{ {A_3^0 = x_2^0x_1^0} \hfill \cr {A_3^1 = x_2^0x_1^1 + x_2^0x_1^1} \hfill \cr {A_3^2 = x_2^2x_1^0 + x_2^1x_1^1 + x_2^0x_1^2} \hfill \cr {A_3^3 = x_2^3x_1^0 + x_2^2x_1^1 + x_2^1x_1^2 + x_2^0x_1^3} \hfill \cr {A_3^4 = x_2^4x_1^0 + x_2^3x_1^1 + x_2^2x_1^2 + x_2^1x_1^3 + x_2^0x_1^4} \hfill \cr {A_3^5 = x_2^5x_1^0 + x_2^4x_1^1 + x_2^3x_1^2 + x_2^2x_1^3 + x_2^1x_1^4 + x_2^0x_1^5} \hfill \cr {A_3^6 = x_2^6x_1^0 + x_2^5x_1^1 + x_2^4x_1^2 + x_2^3x_1^3 + x_2^2x_1^4 + x_2^1x_1^5 + x_2^0x_1^6} \hfill \cr } } \right.

Given the initial state, {x0=x0(t0+)=c10y0=y0(t0+)=c20z0=z0(t0+)=c30u0=u0(t0+)=c40 \left\{ {\matrix{ {{x^0} = {x_0}\left( {t_0^ + } \right) = c_1^0} \hfill \cr {{{\rm{y}}^0} = {y_0}\left( {t_0^ + } \right) = c_2^0} \hfill \cr {{z^0} = {z_0}\left( {t_0^ + } \right) = c_3^0} \hfill \cr {{u^0} = {u_0}\left( {t_0^ + } \right) = c_4^0} \hfill \cr } } \right.

Assign the coefficient to the corresponding variable, and then the coefficients of the other five terms of x can be deduced as {C11=a(c20c10)+c40C21=bc20c10c30+kC31=cc30+c10c20C41=bc20 \left\{ {\matrix{ {C_1^1 = a\left( {c_2^0 - c_1^0} \right) + c_4^0} \hfill \cr {C_2^1 = bc_2^0 - c_1^0c_3^0 + k} \hfill \cr {C_3^1 = - c \cdot c_3^0 + c_1^0c_2^0} \hfill \cr {C_4^1 = bc_2^0} \hfill \cr } } \right. {C12=a(c21c11)+c41C22=bc21(c11c30+c10c31)+kC32=cc31+(c11c20+c10c21)C42=dc21 \left\{ {\matrix{ {C_1^2 = a\left( {c_2^1 - c_1^1} \right) + c_4^1} \hfill \cr {C_2^2 = bc_2^1 - \left( {c_1^1c_3^0 + c_1^0c_3^1} \right) + k} \hfill \cr {C_3^2 = - c \cdot c_3^1 + \left( {c_1^1c_2^0 + c_1^0c_2^1} \right)} \hfill \cr {C_4^2 = dc_2^1} \hfill \cr } } \right. {C13=a(c22c12)+c42C23=bc22(c12c30+c11c31Γ(2q+1)Γ2(q+1)+c10c32)+kC33=cc32+c12c20+c11c21Γ(2q+1)Γ2(q+1)+c10c22C43=dc22 \left\{ {\matrix{ {C_1^3 = a\left( {c_2^2 - c_1^2} \right) + c_4^2} \hfill \cr {C_2^3 = bc_2^2 - \left( {c_1^2c_3^0 + c_1^1c_3^1\,{{\Gamma \left( {2q + 1} \right)} \over {{\Gamma ^2}\left( {q + 1} \right)}} + c_1^0c_3^2} \right) + k} \hfill \cr {C_3^3 = - c \cdot c_3^2 + c_1^2c_2^0 + c_1^1c_2^1{{\Gamma \left( {2q + 1} \right)} \over {{\Gamma ^2}\left( {q + 1} \right)}} + c_1^0c_2^2} \hfill \cr {C_4^3 = dc_2^2} \hfill \cr } } \right. {C14=a(c23c13)+c43C24=bc23[c13c30+(c12c31+c11c32)Γ(3q+1)Γ(2q+1)(q+1)+c10c33]+kC34=cc33+c13c20+(c12c21+c11c22)Γ(3q+1)Γ(2q+1)(q+1)+c10c23C44=dc23 \left\{ {\matrix{ {C_1^4 = a\left( {c_2^3 - c_1^3} \right) + c_4^3} \hfill \cr {C_2^4 = bc_2^3 - \left[ {c_1^3c_3^0 + \left( {c_1^2c_3^1 + c_1^1c_3^2} \right)\,{{\Gamma \left( {3q + 1} \right)} \over {\Gamma \left( {2q + 1} \right)\left( {q + 1} \right)}} + c_1^0c_3^3} \right] + k} \hfill \cr {C_3^4 = - c \cdot c_3^3 + c_1^3c_2^0 + \left( {c_1^2c_2^1 + c_1^1c_2^2} \right){{\Gamma \left( {3q + 1} \right)} \over {\Gamma \left( {2q + 1} \right)\left( {q + 1} \right)}} + c_1^0c_2^3} \hfill \cr {C_4^4 = dc_2^3} \hfill \cr } } \right. {C15=a(c24c14)+c44C25=bc24[c14c30+c12c32Γ(4q+1)Γ2(2q+1)+(c13c31+c11c33)Γ(4q+1)Γ(3q+1)(q+1)+c10c35]+kC35=cc34+c14c20+c12c22Γ(4q+1)Γ2(2q+1)+(c13c21+c11c23)Γ(4q+1)Γ(3q+1)(q+1)+c10c25C45=dc24 \left\{ {\matrix{ {C_1^5 = a\left( {c_2^4 - c_1^4} \right) + c_4^4} \hfill \cr {C_2^5 = bc_2^4 - \left[ {c_1^4c_3^0 + c_1^2c_3^2\,{{\Gamma \left( {4q + 1} \right)} \over {{\Gamma ^2}\left( {2q + 1} \right)}} + \left( {c_1^3c_3^1 + c_1^1c_3^3} \right){{\Gamma \left( {4q + 1} \right)} \over {\Gamma \left( {3q + 1} \right)\left( {q + 1} \right)}}+c_1^0c_3^5} \right] + k} \hfill \cr {C_3^5 = - c \cdot c_3^4 + c_1^4c_2^0 + c_1^2c_2^2{{\Gamma \left( {4q + 1} \right)} \over {{\Gamma ^2}\left( {2q + 1} \right)}} + \left( {c_1^3c_2^1 + c_1^1c_2^3} \right){{\Gamma \left( {4q + 1} \right)} \over {\Gamma \left( {3q + 1} \right)\left( {q + 1} \right)}} + c_1^0c_2^5} \hfill \cr {C_4^5 = dc_2^4} \hfill \cr } } \right. {C16=a(c25c15)+c45C26=bc25[c15c30+(c14c31+c11c34)Γ(5q+1)Γ(4q+1)(q+1)+(c13c32+c12c33)Γ(5q+1)Γ(3q+1)(2q+1)+c10c35]+kC36=cc35+c15c20+(c14c21+c11c24)Γ(5q+1)Γ(4q+1)(q+1)+(c13c22+c12c23)Γ(5q+1)Γ(3q+1)(2q+1)+c10c25C46=dc25 \left\{ {\matrix{ {C_1^6 = a\left( {c_2^5 - c_1^5} \right) + c_4^5} \hfill \cr {C_2^6 = bc_2^5 - \left[ {c_1^5c_3^0 + \left( {c_1^4c_3^1 + c_1^1c_3^4} \right)\,{{\Gamma \left( {5q + 1} \right)} \over {\Gamma \left( {4q + 1} \right)\left( {q + 1} \right)}} + \left( {c_1^3c_3^2 + c_1^2c_3^3} \right){{\Gamma \left( {5q + 1} \right)} \over {\Gamma \left( {3q + 1} \right)\left( {2q + 1} \right)}} + c_1^0c_3^5} \right] + k} \hfill \cr {C_3^6 = - c \cdot c_3^5 + c_1^5c_2^0 + \left( {c_1^4c_2^1 + c_1^1c_2^4} \right){{\Gamma \left( {5q + 1} \right)} \over {\Gamma \left( {4q + 1} \right)\left( {q + 1} \right)}} + \left( {c_1^3c_2^2 + c_1^2c_2^3} \right){{\Gamma \left( {5q + 1} \right)} \over {\Gamma \left( {3q + 1} \right)\left( {2q + 1} \right)}} + c_1^0c_2^5} \hfill \cr {C_4^6 = dc_2^5} \hfill \cr } } \right.

The solution to the chaotic system is obtained, xj(t)=cj0+cj1hqq+cj2h2q2q2+cj3h3q6q3+cj4h2q24q4+cj5h5q120q5+cj66hq720q6 {x_j}\left( t \right) = c_j^0 + c_j^1{{{h^q}} \over q} + c_j^2{{{h^{2q}}} \over {2{q^2}}} + c_j^3{{{h^{3q}}} \over {6{q^3}}} + c_j^4{{{h^{2q}}} \over {24{q^4}}} + c_j^5{{{h^{5q}}} \over {120{q^5}}} + c_j^6{{6{h^q}} \over {720{q^6}}}

Analysis of Fractional Chaos System

To further analyze the influence of system parameters on the fractional-order chaotic system, the influence of the main parameters on the fractional-order critical chaotic system is studied.

Change the parameter a corresponding to the fractional chaotic system, take q=0.9, a ∈ [20, 40], and the result in the bifurcation diagram and Lyapunov diagram of the system is shown in Figure 3. It can be seen from Figure 3 that the system enters chaotic behavior as a period-doubling bifurcation (PDB) behavior. With the gradual increase of the parameter a, the period doubles into a chaotic state. When a ∈ [34.4 40], most of them belong to the chaotic state. It can be seen from the bifurcation diagram that chaos is basically present in this interval. The bifurcation diagram of the system is basically the same as the Lyapunov value of the system. To further verify the above conclusions, the phase diagram of the system under certain specific values of parameter a is shown in Figure 4.

Figure 3

The bifurcation diagram and Lyapunov index spectrum of System (1) in a change.

Figure 4

Phase diagram of System (1) in a change. (a) When a=36, the system is in a state of chaos; (b) when a=33, the system is in a periodic state; (c) when a=37.6, the system is in a periodic state.

Change the parameter c, set q=0.9, and take c ∈ [0, 40]. The resulting bifurcation diagram and Lyapunov index spectrum of the system are shown in Figure 4. It can be seen from Figure 5 that the system enters a chaotic state in a quasi-periodic process. c ∈ [0, 0.9] or [1.5, 2] belong to the periodic state; the remaining regional system belongs to the chaotic state. It can be seen that the bifurcation diagram of the system is basically consistent with the Lyapunov exponential spectrum.

Figure 5

The bifurcation diagram and Lyapunov index spectrum of System (1) in c change.

The fractional order of a simplified Lü system occurs when the order q changes. The bifurcation diagram is shown in Figure 6. It can be seen from Figure 6(a) that when q is greater than 0.5, the system is a chaotic system. The introduction of fractional order changes makes the dynamic performance of the chaotic system more abundant, and the application in actual encryption and secure communication is also more suitable.

Figure 6

The bifurcation diagram and Lyapunov index spectrum of System (1) in q change.

Figure 7

(a) Console and (b) simulation results.

Implementation of Chaos System Based on DSP

DSP is an important branch in the development of microcomputers. The digital chaotic secure communication system implemented with DSP not only meets the requirements of concealment, flexibility, and confidentiality of secure communication, but also has the advantages of low cost, easy portability, and low maintenance. In this paragraph, the newly proposed chaotic system can be implemented. The console and simulation results are shown in the figure. The results show that the mixed degree system is easy to implement, has a stable structure, and can meet various performance indicators such as the complexity and randomness of chaos.

Proposed Multi-Image Encryption Algorithm
DNA Operations

Following the same idea as the binary principle, the addition and subtraction operation of DNA base pairs can be obtained. Therefore, the eight encoding rules (given in Table 1) correspond to the eight DNA manipulation rules. Taking Rule 1 in Table 1 as an example, the addition and subtraction operations of the DNA code are shown in Table 2. Adding base pairing based on addition and subtraction adds to the randomness of the encryption.

The law of encoding

Rule 1 2 3 4 5 6 7 8
00 A A T T G G C C
01 C G C G T A T A
10 G C G C A T A T
11 T T A A C C G G

Addition and subtraction rules

+ A C G T - A C G T
A A C G T A A T G C
C C G T A C C A T G
G G T A C G G C A T
T T A C G T T G C A
Encryption Steps

To encrypt the two images of the children's information page and the children's photos at the same time, this paper proposes a new multi-image encryption algorithm based on DNA theory and fractional chaotic sequence. First, the two pictures are merged. Then, scrambling and DNA expansion operations are performed on the 256 × 512 image. The process of the proposed encryption scheme is shown in Figure 8.

Figure 8

Flowchart of image encryption.

The detailed process is described as follows:

Step 1: Combine the two original images into a large 256 × 512 image.

Step 2: Randomly choose a ∈ (35, 45), b ∈ (25, 35), c ∈ (0, 5), d= ∈ (17, 23), k= ∈ (87, 100), q ∈ (0.6, 1), and [x0 y0 w0 z0] as the initial values and control parameters of the chaotic system, respectively. The chaotic sequence C can be generated by the following formula: {A=abs(x×1016)B=floor(Ai)C=mod(Bi,512×256)) \left\{ {\matrix{ {A = {\rm{abs}}\left( {{\rm{x}} \times {{10}^{16}}} \right)} \hfill \cr {B = {\rm{floor}}\left( {Ai} \right)} \hfill \cr {\left. {C = {\rm{mod}} \left( {Bi,512 \times 256} \right)} \right)} \hfill \cr } } \right.

Step 3: Scrambling image elements

This design uses Arnold transform. The pseudo-random sequence is obtained by setting the parameters of the mixed system. Two sequences, a and b, are extracted from the quantized random sequence, and the index sequence q is obtained from this.

After the synthesized image is scrambled, a preliminary disturbed image can be obtained, which is then formed into an image matrix. The scrambling algorithm is as follows: {a=X(x+1:x+512×256)b=X(y+1:y+512×256)q=mod((b+a×(1:512×256)),512×256)+1 \left\{ {\matrix{ {a = X\left( {x + 1:x + 512 \times 256} \right)} \hfill \cr {b = X\left( {y + 1:\,y + 512 \times 256} \right)} \hfill \cr {q = {\rm{mod}} \left( {\left( {b + a \times \left( {1:512 \times 256} \right)} \right),\,512 \times 256} \right) + 1} \hfill \cr } } \right.

Step 4: DNA encryption sequence

Convert the pixel matrix into a binary matrix, and randomly select an encryption method in Table 1 to convert the binary matrix into a DNA sequence matrix.

Step 5: DNA addition, subtraction, and pairing

Reset the initial value and parameter value of the system. According to formula (14), a set of chaotic sequence k1 is obtained again. Re-encode the obtained sequence in Step 4 to obtain a new set of DNA sequence matrices k2; the size of k2 is the same as the size of k1.

Then randomly select addition or subtraction in Table 2 and perform addition and subtraction operations on the k2 matrix and k1 to obtain the matrix k3. Then choose a rule from Table 3 to perform complementary operations on k3.

Step 6: Convert DNA sequence to decimal sequence

Key space size comparison

Encryption Scheme Proposed Algorithm Ref. [2] Ref. [4] Ref. [29] Ref. [33]
Key space 2506 2399 2407 2412 2396

The final DNA sequence matrix is reverse-encoded and converted back into a decimal matrix.

Step 7: Output encrypted image

Decryption Steps

For the decryption process in the design, the preset initial values and control parameters are used to obtain the same chaotic sequence pair to encrypt the image and decrypt it. The decryption process is the reverse process of the encryption process.

Experimental Results and Analyses
Verify the Validity of the Encryption System

To verify the effectiveness of the proposed dual-image encryption algorithm, this section tests the design. The step length, parameters, initial value, and order were preset. A set of regular images and a set of lost children information images were tested, respectively. The encrypted image completely hides the original image information, the children information group completely conceals the information on the text and photos, and the decrypted image has no difference from the original image, which proves that the algorithm proposed in this paper is feasible. The simulation results are shown in Figure 9.

Figure 9

Verify the validity of the encryption system. (a) Conventional image encryption; (b) lost children information encryption.

Analysis of Key Space

The key of the encryption algorithm should have enough space to resist brute force cracking. The encryption key of this algorithm is mainly composed of nine control parameters and 228 choices of DNA encryption theory. Therefore, the total key space is 2478 + 228 = 2506. Table 3 compares the size of the key space of some existing dual-image encryption algorithms. Therefore, the encryption algorithm proposed in this paper can withstand brute force attacks.

Analysis of Key Sensitivity

Encryption design needs to be sensitive to the preset secret key parameters, that is, when there is a slight difference, it can be identified and decryption fails. Figure 10 takes the image “Fruits” as an example. When any parameter is slightly disturbed, the image will fail to decrypt. The wrongly decrypted image can no longer identify any valid information. It shows that the sensitivity of each parameter is good, and the design of this article has strong secret key sensitivity.

Figure 10

Analysis of key sensitivity. (a) a = 40 + 10−15; (b) b = 30 + 10−15; (c) c = 4 + 10−15; (d) d = 20 + 10−14; (e) k = 90 + 10−14; (f) q = 0.9 + 10−16.

Figure 11

Analysis of the histogram. (a), (b), and (c) are the histograms of the images “Fruits,” “Candy,” and the corresponding cipher image. (d), (e), and (f) are the histograms of the image “Lost children information,’ “Lost children photos,” and the corresponding cipher image.

Analysis of the Histogram

The histogram can be used to count the frequency of the grey value of the grey image. The grey value distribution of the image before encryption is uneven and relatively concentrated, and it is vulnerable to statistical attacks. After the encryption design proposed in this article, the pixel distribution of the encrypted image is very uniform. Subsequently, we quantified the anti-cracking ability of the encryption method through a chi-square test. Table 4 compares the chi-square values of some existing encryption algorithms. It can be seen from the data that the chi-square value of the encrypted cipher image is very low, indicating that the image has been randomly scrambled. The comparison with other algorithms proves that our algorithm can make the pixels more chaotic, which shows that the program has a stronger ability to resist statistical attacks.

Chi-square value analysis of existing encryption algorithms

Our Algorithm Ref. [22] Ref. [23] Ref. [24] Ref. [36]
Fruits 246.80 270.36 251.58 249.86 258.96
Baboon 244.74 264.47 248.36 250.49 268.49
Average 245.77 267.92 249.97 250.18 263.72
Analysis of the Correlation of Adjacent Pixels

Disrupting the correlation between pixels is also one of the criteria for evaluating encryption algorithms. A good encryption algorithm should make the correlation between the pixels of the encrypted image as low as possible. In this way, the encryption scheme can hide the original image information. “Fruits” and “Candy” were used to test the correlation change of the image before and after encryption. Figure 12 intuitively shows the horizontal, vertical, and diagonal distributions of the two original images and the combined and encrypted image. Before encryption, the correlation between image points and points is very high, forming a regular distribution. The encrypted image becomes evenly distributed. The distribution is quantified and compared with the existing algorithms. As shown in Table 5, the data prove that the design in this paper can disrupt the correlation between image pixels.

Figure 12

Analysis of the correlation of adjacent pixels. (a)–(c) are the correlation distributions of “Fruits” images in the horizontal, vertical, and diagonal directions. (d)–(f) are the correlation distributions of “Candy” images in the horizontal, vertical, and diagonal directions, respectively. (g)–(i) and (j)–(l) are the correlation distributions of the “encrypted” image in the horizontal, vertical, and diagonal directions, respectively.

Quantification and comparison of the correlation of adjacent pixels

Channel Direction Cipher Image Average Ref. [1] Ref. [3] Ref. [30] Ref. [31]
Fruits H 0.00264 0.01045 −0.00429 0.00946 −0.01441
V 0.00069 −0.00286 0.00059 0.05929 0.06589
D −0.00113 0.00858 −21.1398 −0.04235 0.00528
Candy H 0.00737 0.02013 −0.00572 −0.00869 0.01628
V 0.00064 0.00011 −0.00121 −0.02904 0.00205
D 0.00102 −0.00429 0.00429 0.06402 0.01628
Analysis of the Plaintext Sensitivity

Plaintext sensitivity refers to the difference comparison of different images after encryption under the same set of secret keys. The main indicators are pixel change rate (NPCR) and uniform average change intensity (UACI). The calculation formula is k(i,j)=p1(i,j)p2(i,j) k\left( {i,j} \right) = {p_1}\left( {i,j} \right) - {p_2}\left( {i,j} \right) NPCR(%)=Sign(k(i,j))H×W NPCR\left( \% \right) = {{{\rm{Sign}}\left( {k\left( {i,j} \right)} \right)} \over {H \times W}} UCAI(%)=|k(i,j)|255×H×W UCAI\left( \% \right) = {{\left| {k\left( {i,j} \right)} \right|} \over {255 \times H \times W}} where the encrypted pixel size at pixel point (i, j) is P1 and P2, respectively.

The test results are listed in Table 6. The NPCR and UACI values are close to the expected theoretical values (NPCR=99.6094% and UACI=33.4635%). In addition, we have listed the average values of NPCR and UACI of some existing encryption algorithms. The comparison result shows that the plaintext sensitivity of the proposed encryption algorithm is superior to that of existing algorithms. It further proves that the encryption design proposed in this paper is very sensitive to plaintext and can effectively resist differential attacks.

Analysis of the plaintext sensitivity

Image Candy Ref. [2] Ref. [4] Ref. [29] Ref. [33]
NPCR(%) 99.6048 99.5846 99.5978 99.5887 99.4978
UACI(%) 33.5343 33.5248 33.5259 33.5213 33.4968
Analysis of the Information Entropy

Information entropy is a reference value that reflects the degree of data confusion. The higher the information entropy of the picture information, the stronger the randomness. Therefore, we use information entropy to evaluate whether the randomness of the image encrypted by the algorithm is large enough. For 256 × 256 photos, the theoretical value of information entropy after encryption should be close to 8. The algorithm in this article is 7.997, and the encryption algorithm performs well. Table 7 analyzes some existing multi-image encryption algorithms and proves that the algorithm proposed in this paper can effectively disturb the pixels.

Analysis of the information entropy

Encryption Scheme Ciher Image
Our scheme average 7.9976
Ref. [1] average 7.9969
Ref. [3] average 7.9965
Ref. [5] average 7.9968
Ref. [30] average 7.9973
Ref. [32] average 7.9971

Analysis of the running time

Our Algorithm Ref. [22] Ref. [24] Ref. [25] Ref. [28]
Encryption 1.568 1.958 1.862 2.035 2.158
Decryption 1.642 2.003 - - 1.988
Analysis of the Noise Attack

In the process of image encryption and decryption, information pollution or data loss may occur. This section uses Gaussian noise in the encryption process to test the resistance of the encryption algorithm to noise pollution. The result is shown in Figure 13. Although the decrypted image has received some pollution, the content of the image can still be clearly distinguished. In general, our scheme has strong anti-interference ability against Gaussian noise attacks.

Figure 13

Gaussian noise attack analysis results. (a) With 0.05; (b) with 0.06.

Analysis of the Time Efficiency

Time efficiency is also one of the criteria for evaluating the performance of encryption algorithms. We take “Fruits” and “Candy” as examples to test the encryption and decryption time of the algorithm in this paper, and compare them with existing algorithms. Experiments have proved that the algorithm has a short encryption time and high encryption efficiency and can be used in actual encryption applications.

Conclusion

This paper proposes a new multi-image lost children information encryption algorithm that combines the theory based on fractional-order chaotic systems and DNA theory. First, we analyze the newly proposed chaotic system, and the result proves that the fractional-order Lü chaotic system has good dynamic characteristics and is suitable for image encryption. Afterwards, a simulation was carried out on the DSP to prove that the system is achievable and conducive to capture and is suitable for actual communication. On this basis, we introduced DNA theory into the multi-image encryption algorithm. The quality of the algorithm encryption and decryption process is satisfactory, and the security performance is greatly improved. Finally, the performance of the encryption algorithm is analyzed. The new encryption algorithm can perform encryption operations on multiple pictures at the same time. While ensuring the encryption efficiency, it also ensures the security performance. The new encryption algorithm has a larger key space and can effectively resist multiple cracking attacks. In summary, the multi-image encryption algorithm proposed in this paper has good encryption performance and provides a theoretical basis for practical applications.

Figure 1

(a) Lost children information; (b) lost children photo; (c) encrypted image; (d) assembling effect picture (front); (e) assembling effect picture (back).
(a) Lost children information; (b) lost children photo; (c) encrypted image; (d) assembling effect picture (front); (e) assembling effect picture (back).

Figure 2

Attractor phase diagram of Lü system. (a) x–y; (b) x–z; (c) x–y–z.
Attractor phase diagram of Lü system. (a) x–y; (b) x–z; (c) x–y–z.

Figure 3

The bifurcation diagram and Lyapunov index spectrum of System (1) in a change.
The bifurcation diagram and Lyapunov index spectrum of System (1) in a change.

Figure 4

Phase diagram of System (1) in a change. (a) When a=36, the system is in a state of chaos; (b) when a=33, the system is in a periodic state; (c) when a=37.6, the system is in a periodic state.
Phase diagram of System (1) in a change. (a) When a=36, the system is in a state of chaos; (b) when a=33, the system is in a periodic state; (c) when a=37.6, the system is in a periodic state.

Figure 5

The bifurcation diagram and Lyapunov index spectrum of System (1) in c change.
The bifurcation diagram and Lyapunov index spectrum of System (1) in c change.

Figure 6

The bifurcation diagram and Lyapunov index spectrum of System (1) in q change.
The bifurcation diagram and Lyapunov index spectrum of System (1) in q change.

Figure 7

(a) Console and (b) simulation results.
(a) Console and (b) simulation results.

Figure 8

Flowchart of image encryption.
Flowchart of image encryption.

Figure 9

Verify the validity of the encryption system. (a) Conventional image encryption; (b) lost children information encryption.
Verify the validity of the encryption system. (a) Conventional image encryption; (b) lost children information encryption.

Figure 10

Analysis of key sensitivity. (a) a = 40 + 10−15; (b) b = 30 + 10−15; (c) c = 4 + 10−15; (d) d = 20 + 10−14; (e) k = 90 + 10−14; (f) q = 0.9 + 10−16.
Analysis of key sensitivity. (a) a = 40 + 10−15; (b) b = 30 + 10−15; (c) c = 4 + 10−15; (d) d = 20 + 10−14; (e) k = 90 + 10−14; (f) q = 0.9 + 10−16.

Figure 11

Analysis of the histogram. (a), (b), and (c) are the histograms of the images “Fruits,” “Candy,” and the corresponding cipher image. (d), (e), and (f) are the histograms of the image “Lost children information,’ “Lost children photos,” and the corresponding cipher image.
Analysis of the histogram. (a), (b), and (c) are the histograms of the images “Fruits,” “Candy,” and the corresponding cipher image. (d), (e), and (f) are the histograms of the image “Lost children information,’ “Lost children photos,” and the corresponding cipher image.

Figure 12

Analysis of the correlation of adjacent pixels. (a)–(c) are the correlation distributions of “Fruits” images in the horizontal, vertical, and diagonal directions. (d)–(f) are the correlation distributions of “Candy” images in the horizontal, vertical, and diagonal directions, respectively. (g)–(i) and (j)–(l) are the correlation distributions of the “encrypted” image in the horizontal, vertical, and diagonal directions, respectively.
Analysis of the correlation of adjacent pixels. (a)–(c) are the correlation distributions of “Fruits” images in the horizontal, vertical, and diagonal directions. (d)–(f) are the correlation distributions of “Candy” images in the horizontal, vertical, and diagonal directions, respectively. (g)–(i) and (j)–(l) are the correlation distributions of the “encrypted” image in the horizontal, vertical, and diagonal directions, respectively.

Figure 13

Gaussian noise attack analysis results. (a) With 0.05; (b) with 0.06.
Gaussian noise attack analysis results. (a) With 0.05; (b) with 0.06.

Chi-square value analysis of existing encryption algorithms

Our Algorithm Ref. [22] Ref. [23] Ref. [24] Ref. [36]
Fruits 246.80 270.36 251.58 249.86 258.96
Baboon 244.74 264.47 248.36 250.49 268.49
Average 245.77 267.92 249.97 250.18 263.72

Addition and subtraction rules

+ A C G T - A C G T
A A C G T A A T G C
C C G T A C C A T G
G G T A C G G C A T
T T A C G T T G C A

The law of encoding

Rule 1 2 3 4 5 6 7 8
00 A A T T G G C C
01 C G C G T A T A
10 G C G C A T A T
11 T T A A C C G G

Analysis of the running time

Our Algorithm Ref. [22] Ref. [24] Ref. [25] Ref. [28]
Encryption 1.568 1.958 1.862 2.035 2.158
Decryption 1.642 2.003 - - 1.988

Key space size comparison

Encryption Scheme Proposed Algorithm Ref. [2] Ref. [4] Ref. [29] Ref. [33]
Key space 2506 2399 2407 2412 2396

Quantification and comparison of the correlation of adjacent pixels

Channel Direction Cipher Image Average Ref. [1] Ref. [3] Ref. [30] Ref. [31]
Fruits H 0.00264 0.01045 −0.00429 0.00946 −0.01441
V 0.00069 −0.00286 0.00059 0.05929 0.06589
D −0.00113 0.00858 −21.1398 −0.04235 0.00528
Candy H 0.00737 0.02013 −0.00572 −0.00869 0.01628
V 0.00064 0.00011 −0.00121 −0.02904 0.00205
D 0.00102 −0.00429 0.00429 0.06402 0.01628

Analysis of the plaintext sensitivity

Image Candy Ref. [2] Ref. [4] Ref. [29] Ref. [33]
NPCR(%) 99.6048 99.5846 99.5978 99.5887 99.4978
UACI(%) 33.5343 33.5248 33.5259 33.5213 33.4968

Analysis of the information entropy

Encryption Scheme Ciher Image
Our scheme average 7.9976
Ref. [1] average 7.9969
Ref. [3] average 7.9965
Ref. [5] average 7.9968
Ref. [30] average 7.9973
Ref. [32] average 7.9971

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