Synchronization control based on memristor neural network and method thereof in image encryption

Abstract

The application discloses a synchronization control method based on a memristor neural network and an image encryption method thereof. When a driving system and a response system in a fractional order memristor neural network reach synchronization, a chaotic sequence is generated by the driving system, a global scrambling operation and a diffusion operation with disturbance are performed on an image by using a scrambling and diffusion technology, and finally, the encrypted image is obtained through mutual diffusion between components of the image. Decryption is performed by using the chaotic sequence of the response system. The image encryption method based on the synchronization control of the chaotic system overcomes the limitations of traditional encryption methods, and the chaotic sequence generated by the chaotic system has good randomness. The security and efficiency of the encryption scheme are determined by the chaotic sequence itself and the combination of the chaotic sequence and the image pixels.

Description

Technical Field

[0001] The present invention relates to a synchronous control based on a memristor neural network in the field of nonlinear dynamics and digital image encryption and an application method thereof in image encryption, and in particular to a Mittag-Leffer synchronous control method based on a memristor neural network and an application method thereof in image encryption. Background Art

[0002] The memristor, a fourth circuit element distinct from the other three basic circuit elements, possesses the ability to remember dynamic history. First proposed by Chinese-American Professor Shaotang Cai, it wasn't until 2008 that HP Labs successfully implemented and constructed a memristor device. The addition of memristors to traditional artificial neural networks based on MP neuron models has resulted in a new memristor-based neural network model. Using memristors to simulate the human brain will be more realistic. Recent research has yielded numerous achievements in the field of memristor neural networks.

[0003] Considering the infinite memory capacity of fractional-order systems, by introducing fractional-order calculus into memristive neural networks (FMNNs), they can exhibit more complex dynamical behavior. Because fractional-order systems offer more precise descriptions in physical modeling and practical engineering, they have long been an effective tool for describing scientific and engineering applications. In recent years, researchers have generalized FMNNs to include complex-valued state variables, such as FCVMNNs, and quaternary state variables, such as FQVMNNs. Typically, theoretical analysis employs a separation approach, splitting a complex-valued system into two real-valued subsystems or a quaternary system into four real-valued subsystems. Classical real-valued neural network theory is then used to analyze the dynamical behavior of these multiple subsystems.

[0004] Quaternions are formed by expanding an imaginary part of a complex-valued domain into three imaginary parts. Neural networks, based on complex-valued domains, have constructed quaternion-valued neural network models that efficiently process multidimensional data. Quaternions have been slow to develop due to their non-commutative nature, but they offer unique advantages in multi-layer perception and image processing. Consequently, researchers have recently devoted significant attention to multi-dimensional algebras such as quaternions.

[0005] It is well known that synchronization plays a crucial role in dynamic behavior, given the potential applications of system control in image encryption. To improve system performance and reliability, researchers have been diligently researching various effective synchronization techniques. To date, research has involved numerous synchronization methods, including anti-synchronization, Mittag-Leffler synchronization, exponential synchronization, and finite-time synchronization. Furthermore, due to speed constraints imposed by amplifiers within circuits, varying communication delays within neurons are crucial. The varying speeds of signal transmission between neurons can lead to discontinuities in neural network functions. Summary of the Invention

[0006] To address the limitations of traditional encryption methods, the present invention provides a Mittag-Leffer synchronization control method based on a memristive neural network and its application in image encryption. This image encryption method, based on the synchronization control of a chaotic system, overcomes the limitations of traditional encryption methods by leveraging the randomness of the chaotic sequences generated by the chaotic system. The chaotic sequences themselves, as well as their combination with image pixels, determine the security and efficiency of the encryption scheme.

[0007] The present invention is implemented by the following technical solution: a Mittag-Leffle synchronization control method based on a quaternary memristor neural network, which is used to control the drive system and the response system in the fractional-order memristor neural network to achieve synchronization. The synchronization control method includes the following steps:

[0008] First, define the error function e of the fractional-order memristor neural network that changes with time t p (t), let:

[0009] e p (t) = y p (t)-x p (t)

[0010] Where x p (t) represents the state variable of the driving system; y p (t) represents the state variable of the response system;

[0011] Next, design the open-loop controller u p (t) is:

[0012]

[0013] Where η p ,ξ p ,λ p Both represent real numbers, τ β represents the βth delay from one neuron to another, β = 1, 2, ..., n;

[0014] The model design of the drive system is as follows:

[0015]

[0016] Where, The symbol for fractional derivatives, 0<α<1, p represents the number of rows of neurons, q represents the number of columns of neurons, p=1,2,…,n, q=1,2,…,n, x p (t) represents the neuron state of the p-th row of neurons in the driving system changing with time t, c p represents the self-feedback coefficient of the p-th row of neurons, a pq (x q (t)) represents the memristor weight of a neuron in the p-th row and q-th column, x q (t) represents the state of the qth column of neurons, f q (y q (t)) represents the activation function of the qth column neuron; β represents the type of delay from one neuron to another neuron, b pq (x q (t)) represents another memristor weight of the connection between the p-th row and q-th column neurons, g q (x q (t-τ β )) represents the activation function of the neuron in the qth column with the βth delay, I p (t) represents the external interference of the p-th row of neurons changing with time t;

[0017] The model design of the response system is:

[0018]

[0019] Where, The symbol for fractional derivatives, 0<α<1, p represents the number of rows of neurons, q represents the number of columns of neurons, p=1,2,…,n, q=1,2,…,n, y p (t) represents the neuron state of the p-th row of neurons in the response system changing with time t, c p represents the self-feedback coefficient of the p-th row of neurons, a pq (y q (t)) represents the memristor weight of a connection between the p-th row and q-th column neurons, y q (t) represents the state of the qth column of neurons, f q (y q (t)) represents the activation function of the qth column neuron; β represents the type of delay from one neuron to another neuron, b pq (y q(t)) represents another memristor weight of the connection between the p-th row and q-th column neurons, g q (y q (t-τ β )) represents the activation function of the neuron in the qth column with the βth delay, I p (t) represents the external interference of the p-th row of neurons changing with time t; u p (t) represents the controller applied to the neurons in the pth row;

[0020] The model of the error function is:

[0021]

[0022] They are the results after differential inclusion and extreme value mapping respectively.

[0023] The present invention also provides an image encryption method based on a fractional-order memristor neural network. When the driving system and the response system in the fractional-order memristor neural network are synchronized, the driving system is used to generate a chaotic sequence. The scrambling diffusion technique is used to perform a global scrambling operation and a diffusion operation with added disturbance on the image. Finally, the components of the image diffuse with each other to obtain an encrypted image. Decryption is performed by performing the inverse operation using the chaotic sequence of the response system.

[0024] Step 1: The grayscale image matrix P representing the plaintext image m×n , dimensionality reduction processing is converted into a one-dimensional vector sequence P σ ={P1,P2,...,P m×n}, where m and n represent the rows and columns of the matrix respectively;

[0025] Step 2: For the chaotic random sequence w σ ={w1,w2,...,w m×n} Arrange in ascending order to obtain index sequence B σ ={B1,B2,...,B m×n};

[0026] [~,A σ ]=sort(w σ )

[0027] Step 3: Use index sequence A as follows σ Descrambled sequence P σ , get the scrambled gray sequence B σ ={B1,B2,...,B m×n};

[0028] The diffusion is:

[0029] Step 1: The iterative chaotic random sequence z σ ={z1,z2,...,z m×n}Use m sequence s σ ={s1,s2,...,s m×n} is disturbed to obtain the chaotic sequence C σ ={C1,C2,...,C m×n};

[0030]

[0031] Step 2: Scramble the sequence B σ ={B1,B2,...,B m×n} and chaotic sequence C σ ={C1,C2,...,C m×n} Perform XOR operation to get D σ ={D1,D2,...,D m×n};

[0032] Step 3: XOR the sequence D σ ={D1,D2,...,D m×n}Restore it to a matrix of image size m×n, and obtain the encrypted image E.

[0033] The present invention also provides an image decryption method based on a fractional-order memristor neural network, which is used to decrypt an image encrypted by the above-mentioned image encryption method based on a fractional-order memristor neural network, and uses the chaotic sequence of the response system to perform inverse operations: inverse diffusion and inverse scrambling;

[0034] Reverse diffusion:

[0035] Step 1: Transform the ciphertext image matrix V m×n , dimensionality reduction processing is converted into a one-dimensional vector sequence V σ ={V1,V2,...,V m×n}.

[0036] Step 2: The chaotic random sequence r obtained by iterating the response system σ ={r1,r2,...,r m×n}Use m sequence s σ ={s1,s2,...,s m×n} to descramble and obtain the chaotic sequence G σ ={G1,G2,...,G m×n};

[0037]

[0038] Step 3: Sequence V obtained by ciphertext imageσ ={V1,V2,...,V m×n} and chaotic sequence G σ ={G1,G2,...,G m×n} Perform XOR operation to get H σ ={H1,H2,...,H m×n};

[0039] Reverse scrambling:

[0040] Step 1: For the chaotic random sequence o σ ={o1,o2,...,o m×n} Arrange in ascending order to obtain index sequence J σ ={J1,J2,...,J m×n};

[0041] Step 2: Use the index sequence J as follows σ Descrambled sequence H m×n , get the scrambled gray sequence L σ ={L1,L2,...,L m×n};

[0042] [~,J σ ]=sort(o σ )

[0043] Step 3: Reverse the scrambled sequence L σ ={L1,L2,...,L m×n}Restore it to a matrix of image size m×n, and get the decrypted image Q.

[0044] Q = reshape(L σ ,m,n).

[0045] The present invention also provides a fractional-order memristor neural network, which uses the above-mentioned image encryption method based on the fractional-order memristor neural network to encrypt the image, and uses the above-mentioned image decryption method based on the fractional-order memristor neural network to decrypt the image.

[0046] The present invention also provides a computer terminal, which includes a memory, a processor, and a computer program stored in the memory and runnable on the processor. When the processor executes the program, the steps of the above-mentioned image encryption method based on the fractional-order memristor neural network and the steps of the above-mentioned image decryption method based on the fractional-order memristor neural network are implemented.

[0047] The present invention also provides a readable storage medium, which stores computer program instructions. When the computer program instructions are read and executed by a processor, the steps of the above-mentioned image encryption method based on the fractional-order memristor neural network and the steps of the above-mentioned image decryption method based on the fractional-order memristor neural network are implemented.

[0048] The basic encryption principle of this invention is as follows: after the drive system and the response system achieve synchronization, the drive system generates a chaotic sequence. Using scrambling and diffusion techniques, a global scrambling operation and a diffusion operation with perturbations are performed on the image. Finally, the image components diffuse with each other to produce an encrypted image. Decryption is performed by performing the inverse operation using the chaotic sequence of the response system. Various performance analyses of the algorithm were subsequently conducted to address its security flaws. Testing and performance analysis have shown that this encryption algorithm exhibits good encryption effectiveness and a high level of security, resisting various attacks.

[0049] Compared with the current prior art, the present invention has the following advantages:

[0050] (1) Based on the real-valued neural network (RVNN), the fractional-order quaternion-valued memristor neural network (FOQVMNN) is decomposed into four real-valued neural networks using separation technology, and a reasonable Lyapunov function is constructed, which eliminates the need to solve the original system. In order to analyze the universality of the method, the number domain is expanded to the high-dimensional system of the quaternion domain. By designing a new state feedback controller, the Mittag-Leffler synchronization criterion of the FOQVMNN with multiple time delays is obtained. The relevant conclusions can be well extended to existing research.

[0051] (2) A novel state feedback control scheme is proposed to solve the image encryption and decryption problem. In the proposed image encryption framework, the chaotic sequence generated by the neural network is used to scramble and replace the original image pixel sequence. The simulation results show that the proposed image encryption algorithm is effective in terms of security, correlation, and robustness against noise attacks. Therefore, the research in this paper is of certain help to the integration.

[0052] (3) The application scope provided by the present invention is more general and can be extended to image encryption applications of other memristive neural networks. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 This is the overall structural diagram of encrypted communication.

[0054] Figure 2 This is the overall flow chart of the design of the Mittag-Leffler synchronization control method for the fractional-order quaternion-valued memristor neural network based on state feedback control and its encrypted communication.

[0055] Figure 3 1(t) and 1(t) are the trajectory diagrams of x1(t) and y1(t) in embodiment 1 of the present invention without adding a state feedback controller.

[0056] Figure 4 2(t) and y2(t) are the trajectory diagrams of x2(t) and y2(t) in embodiment 1 of the present invention without adding a state feedback controller.

[0057] Figure 5 This is a state trajectory diagram of the error system without a state feedback controller in embodiment 1 of the present invention.

[0058] Figure 6 This is a trajectory diagram of x1(t) and y1(t) in embodiment 1 of the present invention with a state feedback controller added.

[0059] Figure 7 This is a trajectory diagram of x2(t) and y2(t) in embodiment 1 of the present invention with a state feedback controller added.

[0060] Figure 8 This is a state trajectory diagram of the error system with a state feedback controller added in embodiment 1 of the present invention.

[0061] Figure 9 This is a flowchart of grayscale image encryption and decryption in embodiment 2 of the present invention.

[0062] Figure 10 This is a histogram of plaintext and ciphertext during the security analysis process in implementation 2 of the present invention.

[0063] Figure 11 This is a correlation distribution diagram of adjacent pixel points in the horizontal, vertical and diagonal directions of the sampled pixel points in the plaintext and ciphertext images during the security analysis process in embodiment 2 of the present invention.

[0064] Figure 12 This is the encryption result of the all-black and all-white image during the security analysis process in implementation 2 of the present invention.

[0065] Figure 13 These are the decryption results of sample images with pixel loss ratios of 1 / 16, 1 / 4, and 1 / 2 of the original ciphertext image during the security analysis process in implementation 2 of the present invention. DETAILED DESCRIPTION

[0066] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0067] Example 1

[0068] See also Figure 1 and Figure 2 ,in, Figure 1 The overall structural block diagram of encrypted communication is the overall structural block diagram of the fractional-order quaternion-valued memristor neural network based on state feedback control. Figure 2 The present invention provides a flowchart for designing a Mittag-Leffler synchronization control method for a fractional-order quaternion-valued memristor neural network based on state feedback control in embodiment 1 of the present invention. Figure 4 The trajectory diagram of x2(t) and y2(t) without state feedback controller in embodiment 1 of the present invention (the figure contains a real part Figure 3 imaginary part diagram).

[0069] This embodiment provides a Mittag-Leffler synchronization control method for a fractional-order quaternary memristor neural network based on an open-loop controller. The method is used to achieve Mittag-Leffler synchronization between a driving network and a response network of a fractional-order quaternary memristor neural network system. The specific steps of the synchronization control method are as follows:

[0070] First, define the error function e of the system p (t), let:

[0071] e p (t) = y p (t)-x p (t)

[0072] Among them, x p (t) represents the state variable of the driving system; y p (t) represents the state variable of the response system.

[0073] Next, design the open-loop controller u p (t) is:

[0074]

[0075] In this embodiment, the driving system model of the fractional-order quaternion-valued memristor neural network is:

[0076]

[0077] Where, The symbol for fractional derivatives, 0<α<1, p represents the number of rows of neurons, q represents the number of columns of neurons, p=1,2,…,n, q=1,2,…,n, x p (t) represents the neuron state of the p-th row of neurons in the driving system changing with time t, c p represents the self-feedback coefficient of the p-th row of neurons, a pq (xq (t)) represents the memristor weight of a neuron in the p-th row and q-th column, x q (t) represents the state of the qth column of neurons, f q (y q (t)) represents the activation function of the qth column neuron; β represents the type of delay from one neuron to another neuron, τ β represents the βth delay from one neuron to another, b pq (x q (t)) represents another memristor weight of the connection between the p-th row and q-th column neurons, g q (x q (t-τ β )) represents the activation function of the neuron in the qth column with the βth delay, I p (t) represents the external interference of the p-th row of neurons changing with time t.

[0078] In this embodiment, the response system model of the fractional-order quaternion-valued memristor neural network is:

[0079]

[0080] Where, The symbol for fractional derivatives, 0<α<1, p represents the number of rows of neurons, q represents the number of columns of neurons, p=1,2,…,n, q=1,2,…,n, y p (t) represents the neuron state of the p-th row of neurons in the response system changing with time t, c p represents the self-feedback coefficient of the p-th row of neurons, a pq (y q (t)) represents the memristor weight of a connection between the p-th row and q-th column neurons, y q (t) represents the state of the qth column of neurons, f q (y q (t)) represents the activation function of the qth column neuron; β represents the type of delay from one neuron to another neuron, τ β represents the βth delay from one neuron to another, b pq (y q (t)) represents another memristor weight of the connection between the p-th row and q-th column neurons, g q (y q (t-τ β )) represents the activation function of the neuron in the qth column with the βth delay, I p (t) represents the external interference of the p-th row of neurons changing with time t; u p (t) represents the controller applied to the neurons in the pth row.

[0081] 1. Theoretical Proof

[0082] 1. Quaternion Theory

[0083] The quaternion field is derived from the extension of the real and complex number fields, first discovered by mathematician Hamilton in 1843. It is different from both real and complex numbers. In four-dimensional space, quaternions are restricted by non-commutativity and distinguish between real and complex multiplication. It contains a real part, which can be generalized to three imaginary parts. The specific description is as follows:

[0084]

[0085] The state variable x p ∈Q, And the three imaginary units are i, j, k.

[0086] Satisfies Hamilton's law:

[0087] i 2 =j 2 =k 2 =ijk=-1,

[0088] ij=k=-ji, jk=i=-kj, ki=j=-ik.

[0089] State variable x p The conjugate expression of is described as Modulo it has

[0090] From the above Hamilton law, we can see that the commutative law of multiplication does not apply to quaternions. Suppose there is a four-dimensional vector The corresponding simple operation rules are:

[0091] (1) Addition operation:

[0092]

[0093] (2) Multiplication operation:

[0094]

[0095] 2. Conditional assumptions:

[0096] First, the study of Mittag-Leffle synchronization in quaternary memristor neural networks typically involves dividing the system into four subsystems, including one real system and three imaginary systems. However, due to the introduction of quaternary sign functions and related fractional-order inequalities in this paper, this method eliminates the need to divide the system into four subsystems and instead considers the system as a whole.

[0097] Secondly, considering the discontinuity of the memristor weight, in the sense of the Filioppov solution, according to the differential inclusion and extreme value mapping theory, we can obtain the drive-response system and the error system as follows:

[0098] Drive system:

[0099]

[0100] Response System:

[0101]

[0102] Error system:

[0103]

[0104] Where, They are the results after differential inclusion and extreme value mapping respectively.

[0105] in,

[0106]

[0107]

[0108]

[0109]

[0110] Among them, T q represents the switching jump time, and T q >0.

[0111] Statement: The upper right subscripts R, I, J, and K in the following text represent the components of R, I, J, and K respectively. Their specific meanings have been explained in the previous section. What is not specified are the directions of R, I, J, and K, which are vectors in four directions.

[0112] For the driving network and response network of any system, if Mittag-Leffle synchronization is to be achieved through the designed state feedback controller, the following conditions must be met:

[0113]

[0114] Among them, δ=min{δ1, δ2, δ3, δ4}, ρ=max{ρ1, ρ2, ρ3, ρ4}.,

[0115]

[0116]

[0117]

[0118]

[0119]

[0120]

[0121] 3. Derivation and Proof

[0122] Construct a Lyapunov function as:

[0123]

[0124] according to

[0125]

[0126] have

[0127]

[0128] According to the Lipschitz condition:

[0129]

[0130]

[0131] in is the Lipschitz constant. Then the following appears etc. also refer to the Lipschitz constant in the RR direction. The subscripts and subscripts do not change its meaning. They all represent variables in four directions, namely a real part R and three imaginary parts I, J, K, such as Figure 3 As shown, Figure 3 The trajectory diagram of x1(t) and y1(t) without state feedback controller in embodiment 1 of the present invention (the figure contains a real part Figure 3 imaginary part diagram).

[0132] Then, using the properties and operation rules of quaternions, the fractional derivative in the R direction is simplified:

[0133]

[0134] Similarly, the fractional derivatives in the I, J, and K directions can be obtained.

[0135] according to We can further obtain:

[0136]

[0137] We assume that:

[0138]

[0139]

[0140]

[0141]

[0142]

[0143]

[0144] The above parts are sorted out as follows:

[0145]

[0146] Among them, δ=min{δ1, δ2, δ3, δ4} and ρ=max{ρ1, ρ2, ρ3, ρ4}.

[0147] Based on the Razumikhin technique, any solution e(t) of the system has We define the constant Can be rewritten as:

[0148]

[0149] when Thus we can get V(e(t))≤(V(0)+ε)E α (-ψt α ), where V(0) is the initial value, and ψ and ε are arbitrary constants greater than zero. This satisfies the Mittag-Leffle definition of synchronization.

[0150] If the error system e(t) satisfies V(e(t))≤(V(0)+ε)E α (-ψt α ) condition, the drive-response system is said to have achieved Mittag-Leffler synchronization.

[0151] according to Then you can launch Synchronization is achieved and the proof is complete.

[0152] Please combine Figures 5 to 8 , Figure 5 This is a state trajectory diagram of the error system without a state feedback controller in embodiment 1 of the present invention. Figure 6 The trajectory diagram of x1(t) and y1(t) under the state feedback controller in embodiment 1 of the present invention (the figure contains a real part Figure 3 imaginary part diagram). Figure 7The trajectory diagram of x2(t) and y2(t) under the state feedback controller in embodiment 1 of the present invention (the figure contains a real part Figure 3 imaginary part diagram). Figure 8 This is a state trajectory diagram of the error system with a state feedback controller added in embodiment 1 of the present invention.

[0153] Note: To put it simply, the error curves of the response system and the drive system gradually approach 0, and the two reach consistency. See the error curve for a schematic diagram of the effect. Figure 5 and Figure 8 The response system variable curve will be tracked by the controller and the variable curve of the drive system will be overlapped. See the synchronous curve for the effect diagram. Figure 3 、 Figure 4 、 Figure 6 、 Figure 7 In the theoretical derivation and analysis, the role of the controller is used to meet the definition of achieving a certain synchronization. Here, the definition of Mittag-Leffle synchronization is achieved, which means that the drive system and the response system achieve Mittag-Leffle synchronization under this controller.

[0154] In summary, the synchronization control method provided in this embodiment can realize Mittag-Leffle synchronization of a fractional-order quaternion-valued memristor neural network under a state feedback controller.

[0155] 2. Numerical Simulation

[0156] In this embodiment, taking a two-dimensional fractional-order quaternion-valued memristor neural network system with multiple time delays as an example, the model of the drive-response system of the neural network system is determined as follows:

[0157]

[0158] Where α = 0.95, c1 = c2 = 6, τ1 = 0.5, τ2 = 0.8, I1 = I2 = 0, and the activation function f(·) is defined as According to the Lipschitz condition, we can get

[0159] The memristor connection weights of the quaternary values ​​satisfy:

[0160]

[0161]

[0162]

[0163]

[0164]

[0165]

[0166]

[0167]

[0168] According to the state feedback controller, in order to satisfy We set the control parameters η1 = η2 = 3.5, ξ1 = ξ2 = 1.46, and λ1 = λ2 = 1. The calculated values ​​are ρ = 24.2 and δ = 16.7, which satisfy the Mittag-Leffler synchronization requirement.

[0169] In summary, from the changing trend of the curve in the simulation results obtained from the simulation experiment, it can be found that the fractional-order quaternion memristor neural network provided in this embodiment can achieve Mittag-Leffle synchronization of the system under an open-loop controller.

[0170] Example 2

[0171] This embodiment is based on the first embodiment, that is, the drive-response system is implemented to achieve Mittag-Leffle synchronization, and is applied to image encryption.

[0172] like Figure 9 As shown in the figure, in a fractional-order quaternion-valued memristor neural network system, the drive system is used for encryption and the response system is used for decryption. The specific encryption process is as follows:

[0173] 1. Algorithm

[0174] a. Chaotic sequence generation

[0175] Step 1: Iterate the driving system and the response system in the fractional-order memristor neural network for N0+m×n times, discard the first N0 numbers in each sequence, and obtain eight state values: x1(h), x2(h), x4(h), x5(h), x6(h), x7(h), x8(h), h=1,2,...,m×n.

[0176] Step 2: Use the eight state values ​​obtained in step 1 to generate a new sequence k(ix,iy)∈[0,255],k′(ix,iy)∈[0,m×n], ix=1,2,...,8,iy=1,2,...,mn. The sequence value is calculated as follows:

[0177] k(ix,iy)=mod{floor((|x ix (iy)|-floor(|x ix (iy)|))×10 7),256},

[0178] k′(ix,iy)=mod{floor(mod(((|x ix (iy)|-floor(|x ix (iy)|))×10 15 ),10 8 )),m×n},

[0179] Where mod(·) represents the modulo operation, floor(·) represents rounding down, and a set of numbers is selected from k(ix,iy) as the chaotic sequence z σ ={z1,z2,...,z m×n}, select a set of numbers in k′(ix,iy) as the chaotic sequence w σ ={w1,w2,...,w m×n}.

[0180] Encryption process

[0181] b. Scrambling

[0182] Step 1: Convert the grayscale image matrix P m×n , dimensionality reduction processing is converted into a one-dimensional vector sequence P σ ={P1,P2,...,P m×n}.

[0183] Step 2: For the chaotic random sequence w σ ={w1,w2,...,w m×n} Arrange in ascending order to obtain index sequence A σ ={A1,A2,...,A m×n}.

[0184] [~,A i ]=sort(w i ).

[0185] Step 3: Use index sequence A as follows σ Descrambled sequence P σ , get the scrambled gray sequence B σ ={B1,B2,...,B m×n}.

[0186] B i =P Ai .

[0187] The scrambling process is completed here, and the diffusion process is performed later.

[0188] c. Diffusion

[0189] Step 1: The iterative chaotic random sequence zσ ={z1,z2,...,z m×n}Use m sequence s σ ={s1,s2,...,s m×n} is disturbed to obtain the chaotic sequence C σ ={C1,C2,...,C m×n};

[0190]

[0191] Step 2: Scramble the sequence B σ ={B1,B2,...,B m×n} and chaotic sequence C σ ={C1,C2,...,C m×n} Perform XOR operation to get D σ ={D1,D2,...,D m×n};

[0192]

[0193] Step 3: XOR the sequence D σ ={D1,D2,...,D m×n}Restore it to a matrix of image size m×n, and obtain the encrypted image E.

[0194] E=reshape(D σ ,m,n).

[0195] The decryption process is the reverse operation of the encryption process.

[0196] Decryption process

[0197] Similar to the chaotic sequence generation of the driving system, the chaotic sequence r for diffusion operation is generated by iteratively responding to the system. σ ={r1,r2,...,r m×n} and the chaotic sequence o for scrambling operation σ ={o1,o2,...,o m×n}.

[0198] d. Back diffusion

[0199] Step 1: Transform the ciphertext image matrix V m×n , dimensionality reduction processing is converted into a one-dimensional vector sequence V σ ={V1,V2,...,V m×n}.

[0200] Step 2: The chaotic random sequence r obtained by iterating the response system σ ={r1,r2,...,rm×n}Use m sequence s σ ={s1,s2,...,s m×n} to descramble and obtain the chaotic sequence G σ ={G1,G2,...,G m×n};

[0201]

[0202] Step 3: Sequence V obtained by ciphertext image σ ={V1,V2,...,V m×n} and chaotic sequence G σ ={G1,G2,...,G m×n} Perform XOR operation to get H σ ={H1,H2,...,H m×n};

[0203] e. Reverse chaos

[0204] Step 1: For the chaotic random sequence o σ ={o1,o2,...,o m×n} Arrange in ascending order to obtain index sequence J σ ={J1,J2,...,J m×n};

[0205] Step 2: Use the index sequence J as follows σ Descrambled sequence H m×n , get the scrambled gray sequence L σ ={L1,L2,...,L m×n};

[0206] [~,J σ ]=sort(o σ )

[0207] Step 3: Reverse the scrambled sequence L σ ={L1,L2,...,L m×n}Restore it to a matrix of image size m×n, and get the decrypted image Q.

[0208] Q = reshape(L σ ,m,n).

[0209] The encryption and decryption method of the present invention can be configured in the form of software, such as an independent APP designed or an embedded software that can be called at any time and applied in a computer terminal. The computer terminal includes a memory, a processor, and a computer program stored in the memory and running on the processor. The computer terminal can also be a smart phone, a tablet computer, a laptop computer, etc. that can execute a program. The processor can be a central processing unit (CPU), a controller, a microcontroller, a microprocessor, or other data processing chip in some embodiments. The processor is generally used to control the overall operation of a computer device. In the present embodiment, the processor is used to run the program code stored in the memory or process data. When the processor executes the program, the steps of the encryption and decryption method of the present invention can be implemented.

[0210] The encryption and decryption methods of the present invention can also be designed as a readable storage medium, such as a USB-shield, which stores computer program instructions. When the computer program instructions are read and executed by a processor, the encryption and decryption methods described above are executed. The USB-shield can be electronically plugged into a computer, which reads and executes the computer program instructions stored in the USB-shield.

[0211] 2. Security Analysis

[0212] In order to verify the effectiveness of the encryption algorithm, we use the drive system to encrypt and the response system to decrypt. We select a grayscale image of "parrot" with a pixel size of 768×512 as the original image for verification. The encryption and decryption process is shown in the figure below. Figure 9 As shown in the figure, the original image is scrambled by scrambling the pixel positions to create a scrambled image. The pixel values ​​are then diffused to create the encrypted ciphertext image. The ciphertext image is blurry and unclear, effectively hiding information. However, after decryption, the decrypted image is indistinguishable from the original image and remains clear, demonstrating the effectiveness of the encryption algorithm.

[0213] In order to more significantly study the encryption effect, the information contained in the encrypted image is analyzed in the following aspects.

[0214] (1) Histogram analysis

[0215] The histogram can intuitively describe the distribution of pixel values ​​in an image. If the histogram distribution is uneven, it means that the image is not effective in encryption and is easy to be deciphered. In order to obtain a secure encryption algorithm that is well resistant to statistical attacks, the pixel values ​​in the histogram of the ciphertext image must be uniformly distributed. Comparing the histogram of the original parrot image with the encrypted image can further evaluate the security of the encryption algorithm. Figure 10As shown in the comparison chart, the two groups of comparisons in the figure show that the pixels of the original image before encryption are roughly concentrated in the middle of the interval, showing a non-uniform distribution, while the pixel values ​​of the encrypted image are evenly distributed in the interval [0, 255], and the number of pixels at each point is generally balanced.

[0216] (2) Correlation analysis

[0217] Pixel correlation is a parameter that describes the relationship between pixel values ​​in an image. There is a certain correlation between the pixels of the original image, which increases with the increase of the correlation coefficient; there is no correlation between the pixels of the ciphertext image, or the correlation coefficient is close to zero. By comparing the correlation coefficients of 5000 pairs of adjacent pixels selected from the original image and the ciphertext image in different directions, the correlation coefficients of adjacent pixels are calculated from the horizontal, vertical and diagonal directions of the image. The results are as follows: Figure 11 As shown. Figure 11 As can be seen in (a), (b), and (c), the correlation between adjacent pixels in the original and encrypted images is linear, with a coefficient close to 1. The results in (d), (e), and (f) show that after encryption, the correlation between adjacent pixels in the original and encrypted images becomes very low, meaning that the pixel values ​​are randomly distributed, and the correlation in each direction is close to 0. This analysis demonstrates that the encryption algorithm can effectively randomize the distribution of pixels, thereby reducing the correlation between pixels and achieving better encryption results.

[0218] (3) Chosen Plaintext Attack Analysis

[0219] As we all know, a secure image encryption algorithm should be able to resist common attacks. Among many common attack types, chosen plaintext attacks are generally considered to be the most threatening attack method, so most attackers use chosen plaintext attacks to analyze image encryption.

[0220] Attackers typically select specific plaintext information and, using publicly available encryption algorithms, obtain the corresponding ciphertext. This allows them to analyze and obtain useful decryption rules, ultimately decrypting the original confidential image. Although attackers know the encryption algorithm, they are unable to extract useful information from the selected encrypted and decrypted image. Because all pixel values ​​in pure black and pure white images are 0 and 255, respectively, they are often considered special images for analysis. Figure 12 Ciphertext images for encrypting pure black and pure white images are presented. Taking an all-zero plaintext image as an example, after pixel value updates, the all-zero image becomes a non-zero image. Therefore, using an all-zero image prevents the permutation process from being ignored, as is the case with other encryption algorithms. Furthermore, during the diffusion process, attackers cannot use the all-zero image to determine the mathematical relationship between plaintext and ciphertext pixels, making the algorithm resistant to chosen-plaintext attacks.

[0221] (4) Analysis of Clipping Attacks

[0222] The purpose of cropping attack analysis is to analyze whether the encrypted image can be deciphered when the ciphertext image is incomplete. First, the ciphertext image obtained above is cropped. The first one is cropped by 6.25% ( Figure 13 (a)), the second one is cropped by 25% ( Figure 13 (b)), the third one is cropped by 50% ( Figure 13 (c)), the experimental results are based on Figure 13 The results show that the encrypted image can be decrypted and restored to the naked eye even when subjected to cropping attacks of varying degrees. Even when the encrypted image is cropped by as much as 50%, the outline of the decrypted image is still discernible to the naked eye. Therefore, the algorithm is robust against cropping attacks. It is worth noting that the clarity of the reconstructed plaintext image decreases significantly as the cropping ratio increases.

[0223] The above-described embodiments merely illustrate several implementations of the present invention, and while their descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the patent. It should be noted that a person skilled in the art would be able to make numerous variations and improvements without departing from the spirit of the present invention, all of which fall within the scope of protection of the present invention. Therefore, the scope of protection of the patent for this invention shall be determined by the appended claims.

Claims

1. A Mittag-Leffle synchronization control method based on a quaternary memristor neural network, which is used to synchronize a drive system and a response system within a fractional-order memristor neural network, characterized by: The synchronous control method comprises the following steps: First, define the error function e of the fractional-order memristor neural network that changes with time t p (t), let: e p (t)=y p (t)-x p (t) Where x p (t) represents the state variable of the driving system; y p (t) represents the state variable of the response system; Next, design the open-loop controller u p (t) is: Where η p ,ξ p ,λ p Both represent real numbers, τ β represents the βth delay from one neuron to another, β = 1, 2, ..., n; The model design of the drive system is as follows: Where, The symbol for fractional derivatives, 0<α<1, p represents the number of rows of neurons, q represents the number of columns of neurons, p=1,2,…,n, q=1,2,…,n, x p (t) represents the neuron state of the p-th row of neurons in the driving system changing with time t, c p represents the self-feedback coefficient of the p-th row of neurons, a pq (x q (t)) represents the memristor weight of a neuron in the p-th row and q-th column, x q (t) represents the state of the qth column of neurons, f q (y q (t)) represents the activation function of the qth column neuron; β represents the type of delay from one neuron to another neuron, b pq (x q (t)) represents another memristor weight of the connection between the p-th row and q-th column neurons, g q (x q (t-τ β )) represents the activation function of the neuron in the qth column with the βth delay, I p (t) represents the external interference of the p-th row of neurons changing with time t; The model design of the response system is: Where, The symbol for the fractional derivative, 0 < α < 1, p represents the number of rows of neurons in the response system, q represents the number of columns of neurons in the response system, p = 1, 2, ..., n, q = 1, 2, ..., n, y p (t) represents the neuron state of the p-th row of neurons in the response system changing with time t, c p represents the self-feedback coefficient of the p-th row of neurons, a pq (y q (t)) represents the memristor weight of a connection between the p-th row and q-th column neurons, y q (t) represents the state of the qth column of neurons, f q (y q (t)) represents the activation function of the qth column neuron; β represents the type of delay from one neuron to another neuron, b pq (y q (t)) represents another memristor weight of the connection between the p-th row and q-th column neurons, g q (y q (t-τ β )) represents the activation function of the neuron in the qth column with the βth delay, I p (t) represents the external interference of the p-th row of neurons changing with time t; u p (t) represents the controller applied to the neurons in the pth row; The model of the error function is: These are the results after differential inclusion and extreme value mapping respectively; When the driving system and the response system in the fractional-order memristor neural network are synchronized, the driving system is used to generate a chaotic sequence. The scrambling diffusion technique is used to perform a global scrambling operation and a diffusion operation with added disturbance on the image. Finally, the components of the image diffuse with each other to obtain an encrypted image. Decryption is performed by performing the inverse operation using the chaotic sequence of the response system.

2. The Mittag-Leffle synchronization control method based on a quaternary memristor neural network according to claim 1, wherein: a pq (x q (t)) and b pq (x q (t)) takes the following values: Where i, j, and k represent the three imaginary parts in the quaternion field.

3. An image encryption method based on fractional-order memristor neural network, characterized in that: The method uses the Mittag-Leffle synchronization control method based on a quaternary memristor neural network as described in claim 1 to control the drive system and the response system in the fractional-order memristor neural network to achieve synchronization. When the drive system and the response system in the fractional-order memristor neural network achieve synchronization, the drive system is used to generate a chaotic sequence, and a scrambling diffusion technique is used to perform a global scrambling operation and a diffusion operation with a disturbance on the image. Finally, the components of the image diffuse with each other to obtain an encrypted image. Decryption is performed by performing an inverse operation using the chaotic sequence of the response system. The scrambling is: Step 1: The grayscale image matrix P representing the plaintext image m×n , dimensionality reduction processing is converted into a one-dimensional vector sequence P σ ={P1,P2,...,P m×n }, where m and n represent the number of rows and columns of the original image matrix respectively; Step 2: For the chaotic random sequence w σ ={w1,w2,...,w m×n } Arrange in ascending order to obtain index sequence A σ ={A1,A2,...,A m×n }; [~,A σ ]=sort(w σ ) Step 3: Using index sequence A σ Descrambled sequence P σ , get the scrambled gray sequence B σ ={B1,B2,...,B m×n }; The diffusion is: Step 1: The iterative chaotic random sequence z σ ={z1,z2,...,z m×n }Use m sequence s σ ={s1,s2,...,s m×n } is disturbed to obtain the chaotic sequence C σ ={C1,C2,...,C m×n }; Step 2: Scramble the sequence B σ ={B1,B2,...,B m×n } and chaotic sequence C σ ={C1,C2,...,C m×n } Perform XOR operation to get D σ ={D1,D2,...,D m×n }; Step 3: XOR the sequence D σ ={D1,D2,...,D m×n }Restore it to a matrix of image size m×n, and obtain the encrypted image E.

4. The image encryption method based on fractional-order memristor neural network according to claim 3, characterized in that: The method for generating the chaotic random sequence comprises the following steps: Step 1: Iterate the driving system and the response system in the fractional-order memristor neural network for N0+m×n times, discard the first N0 numbers in each sequence, and obtain eight state values: x1(h), x2(h), x4(h), x5(h), x6(h), x7(h), x8(h), h=1,2,...,m×n; Step 2: Use the eight state values ​​obtained in step 1 to generate a new sequence k(ix,iy)∈[0,255], k′(ix,iy)∈[0,m×n],ix=1,2,...,8,iy=1,2,...,mn, the sequence value is calculated as follows: k(ix,iy)=mod{floor((|x ix (iy)|-floor(|x ix (iy)|))×10 7 ),256}, k′(ix,iy)=mod{floor(mod(((|x ix (iy)|-floor(|x ix (iy)|))×10 15 ),10 8 )),m×n}, Where mod(·) represents the modulo operation, floor(·) represents rounding down, and a set of numbers is selected from k(ix,iy) as the chaotic sequence z σ ={z1,z2,...,z m×n }, select a set of numbers in k′(ix,iy) as the chaotic sequence w σ ={w1,w2,...,w m×n }.

5. The image encryption method based on fractional-order memristor neural network according to claim 3, characterized in that: The synchronous control method of the drive system and the response system comprises the following steps: First, define the error function e of the fractional-order memristor neural network that changes with time t p (t), let: e p (t)=y p (t)-x p (t) Where x p (t) represents the state variable of the driving system; y p (t) represents the state variable of the response system; Next, design the open-loop controller u p (t) is: Where η p ,ξ p ,λ p Both represent real numbers, τ β represents the βth delay from one neuron to another, p = 1, 2, …, n.

6. The image encryption method based on fractional-order memristor neural network according to claim 5, characterized in that: The model of the drive system is: Where, The symbol for fractional derivatives, 0<α<1, p represents the number of rows of neurons, q represents the number of columns of neurons, p=1,2,…,n, q=1,2,…,n, x p (t) represents the neuron state of the p-th row of neurons in the driving system changing with time t, c p represents the self-feedback coefficient of the p-th row of neurons, a pq (x q (t)) represents the memristor weight of a neuron in the p-th row and q-th column, x q (t) represents the state of the qth column of neurons, f q (y q (t)) represents the activation function of the qth column neuron; β represents the type of delay from one neuron to another, b pq (x q (t)) represents another memristor weight of the connection between the p-th row and q-th column neurons, g q (x q (t-τ β )) represents the activation function of the neuron in the qth column with the βth delay, I p (t) represents the external interference of the p-th row of neurons changing with time t; The model of the response system is: Where, The symbol for fractional derivatives, 0<α<1, p represents the number of rows of neurons, q represents the number of columns of neurons, p=1,2,…,n, q=1,2,…,n, y p (t) represents the neuron state of the p-th row of neurons in the response system changing with time t, c p represents the self-feedback coefficient of the p-th row of neurons, a pq (y q (t)) represents the memristor weight of a connection between the p-th row and q-th column neurons, y q (t) represents the state of the qth column of neurons, f q (y q (t)) represents the activation function of the qth column neuron; β represents the type of delay from one neuron to another, b pq (y q (t)) represents another memristor weight of the connection between the p-th row and q-th column neurons, g q (y q (t-τ β )) represents the activation function of the neuron in the qth column with the βth delay, I p (t) represents the external interference of the p-th row of neurons changing with time t; u p (t) represents the controller applied to the neurons in the pth row; The model of the error function is: They are the results after differential inclusion and extreme value mapping respectively.

7. A method for image decryption based on a fractional-order memristor neural network, for decrypting an image encrypted using the image encryption method based on a fractional-order memristor neural network according to any one of claims 3 to 6, characterized in that: Utilize the chaotic sequence of the response system to perform inverse operations: inverse diffusion and inverse scrambling; Reverse diffusion: Step 1: Transform the ciphertext image matrix V m×n , dimensionality reduction processing is converted into a one-dimensional vector sequence V σ ={V1,V2,...,V m×n }; Step 2: The chaotic random sequence r obtained by iterating the response system σ ={r1,r2,...,r m×n }Use m sequence s σ ={s1,s2,...,s m×n } to descramble and obtain the chaotic sequence G σ ={G1,G2,...,G m×n }; Step 3: Sequence V obtained by ciphertext image σ ={V1,V2,...,V m×n } and chaotic sequence G σ ={G1,G2,...,G m×n } Perform XOR operation to get H σ ={H1,H2,...,H m×n }; Reverse scrambling: Step 1: For the chaotic random sequence o σ ={o1,o2,...,o m×n } Arrange in ascending order to obtain index sequence J σ ={J1,J2,...,J m×n }; Step 2: Use the index sequence J as follows σ Descrambled sequence H m×n , get the scrambled gray sequence L σ ={L1,L2,...,L m×n }; [~,J σ ]=sort(o σ ) Step 3: Reverse the scrambled sequence L σ ={L1,L2,...,L m×n }Restore it to a matrix of image size m×n, and get the decrypted image Q.

8. A fractional-order memristor neural network, characterized in that: It encrypts the image using the image encryption method based on the fractional-order memristor neural network as described in any one of claims 3 to 6, and decrypts the image using the image decryption method based on the fractional-order memristor neural network as described in claim 7.

9. A computer terminal comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the steps of the image encryption method based on the fractional-order memristor neural network as described in any one of claims 3 to 6 and the steps of the image decryption method based on the fractional-order memristor neural network as described in claim 7 are implemented.

10. A readable storage medium, characterized in that: The readable storage medium stores computer program instructions. When the computer program instructions are read and executed by a processor, the steps of the image encryption method based on the fractional-order memristor neural network as described in any one of claims 3 to 6 and the steps of the image decryption method based on the fractional-order memristor neural network as described in claim 7 are implemented.

Images