CN115883056B - Anti-spoofing attack image encryption and decryption method based on chaotic system sampling synchronous communication - Google Patents
Anti-spoofing attack image encryption and decryption method based on chaotic system sampling synchronous communication Download PDFInfo
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Abstract
The invention belongs to the technical field of new generation information, and particularly discloses a spoofing attack resistant image encryption and decryption method based on sampling synchronous communication of a chaotic system. The anti-deception attack image encryption and decryption method considers that under the network deception attack environment, the safety control gain is designed to enable the driving-response chaotic system to realize synchronization, namely the decryption key is successfully generated, the encrypted image is further successfully decrypted, the established driving-response closed-loop system model under the deception attack environment and the synchronization criterion can be applied to the prior art, and the characteristic matrix is selectedA、BNonlinear termg(x(t) The method can be degenerated into a specific chaotic system, and the solved safety control gains are different by selecting different parameters, so that the result of the method has universality.
Description
Technical Field
The invention belongs to the technical field of new generation information, and particularly relates to a spoofing attack resistant image encryption and decryption method based on sampling synchronous communication of a chaotic system.
Background
With the development of new generation information technology and digital networks, advanced communication technology makes the network control system have the characteristics of low cost communication, easy installation and the like. Accordingly, the resulting digital communication methods, including event-triggered communication, sampling communication, and intermittent communication, have become hot topics in the field of artificial intelligence. These communication schemes have significant features and advantages. However, digital communication is generally open in an actual communication environment. Thus, data transmission in a wireless network environment may be affected by network attacks. Therefore, an external attacker can easily handle the wireless transmission channel without considering security protection, which poses a great threat to economy and national security. On the other hand, images are used as an important expression form of multimedia, and are widely used in military, commercial, medical, etc.
In the field of image encryption, a driving chaotic system is generally used for generating pseudo random numbers as an encryption key for image encryption, so that an original image generates a ciphertext image; different initial values and controller parameters are generally selected at the receiving end, so that the same pseudo-random number generated by the response chaotic system becomes a decryption key, and the received ciphertext image is changed into a decryption image. However, in a digital communication environment, the transmitted sampled data is often affected by noise and network attack, so that the effectiveness and security of the transmission of the picture information are damaged, and the receiving end cannot acquire the decrypted image. Therefore, there is great interest in academia and industry in how to guarantee the network security of images during transmission. Up to now, network attacks can be divided into two cases, namely denial of service attacks and spoofing attacks. The former typically prevents transmission of control information, while the latter replaces or injects erroneous control information. In general, spoofing attacks with bernoulli distribution characteristics are considered the most dangerous and harmful because of their behavior to destroy the original information. This would result in the receiving end not having access to the decrypted image if the corresponding network security problem is not considered.
Thus, if the impact of spoofing attacks is taken into account, the previous digital controller design approach is no longer applicable. It follows that the design methodology and analysis techniques prior to redesign and reevaluation is a highly desirable problem. For the chaotic system, the basis and key of using the chaotic signal for secret communication are how to realize the synchronization between the two chaotic systems, and the synchronization also involves the design of the controller. However, the prior art considers that synchronization and image encryption and decryption are implemented in an absolutely secure environment and conventional point-to-point control, that is, the influence of external environments such as network attacks is not considered in implementing the synchronization process (decryption key generation process). In particular, with the development of new generation information technology, sampling communication technology has become a current research hotspot. However, the sampled data transmitted in the digital communication environment is generally affected by noise and network attack, so that the effectiveness and security of the transmission of the picture information are damaged, and the receiving end cannot acquire the decrypted image. Therefore, the results obtained by the prior art are not applicable to the new generation information technology field. That is, in the context of spoofing attacks, the prior art cannot achieve synchronization as well as image encryption and decryption. At present, no research on a spoofing attack resistant image encryption and decryption method based on sampling synchronous communication of a chaotic system exists. Therefore, how to characterize the attack intensity of the spoofing attack and how to establish a synchronous mathematical model of the driving-responding chaotic encryption system in the environment with the spoofing attack has important significance.
Disclosure of Invention
The invention aims to provide a spoofing attack resistant image encryption and decryption method based on sampling synchronous communication of a chaotic system. The method mainly considers how to design the security control gain under the network spoofing attack environment so that the driving-responding chaotic system can realize synchronization, namely successfully generating a decryption key and then successfully decrypting the encrypted image.
In order to achieve the above purpose, the invention adopts the following technical scheme:
a spoofing attack resistant image encryption and decryption method based on sampling synchronous communication of a chaotic system comprises the following steps:
firstly, designing a safety sampling communication controller according to the characteristics of spoofing attack obeying Bernoulli distribution, wherein the designed safety sampling communication controller comprises safety control gain and setting of sampling period;
based on a designed safe sampling communication controller, establishing a driving-responding closed-loop system model under a spoofing attack environment;
designing a Lyapunov function, and obtaining sufficient conditions capable of ensuring the asymptotic stability of a closed-loop system according to a matrix theory, an inequality estimation technology and a stability theory, namely establishing a criterion for ensuring the driving-response chaotic system to realize complete synchronization;
step 3, an image encryption process;
first, the original image is read to obtainM×NIs a pixel matrix of (a);
giving an initial value of a driving chaotic system, and generating an encryption key by using the driving chaotic systemM×NThe pseudo-random number is used for encrypting the pixel matrix of the original image by using an encryption algorithm to obtain a ciphertext image;
step 4, image decryption process;
giving an initial value of the response chaotic system, and solving a safety control gain which ensures that the driving-response chaotic system can be synchronized based on the criterion which is established in the step 2 and ensures that the driving-response chaotic system is completely synchronized;
in the image decryption process, the safety control gain parameter is applied to the closed-loop system model of the driving-response chaotic system established in the step 1, so that the driving-response chaotic system is completely synchronized in a spoofing attack environment;
finally, the decryption key generated by the response chaotic system is utilizedM×NAnd (3) carrying out inverse processing on the ciphertext image in the step (3) to obtain a plaintext image, so as to realize the image encryption and decryption process under the spoofing attack environment.
The invention has the following advantages:
as described above, the present invention provides a method for encrypting and decrypting anti-fraud attack images based on synchronous communication of chaotic system sampling, which aims at the technical problem that the prior art can not realize synchronization and image encryption and decryption in the fraud attack environment, and the method considers that under the network fraud attack environment, the security control gain is designed to enable the drive-response chaotic system to realize synchronization, namely successfully generate decryption keys and then successfully decrypt encrypted images, and the established closed loop system model and synchronization criterion can be applied to the prior art, and the characteristic matrix is selectedA、BNonlinear termg(x(t) Can degrade to a particular chaotic system. By selecting different parameters, the solved safety control gains are also different, namely the result of the invention has more universality.
Drawings
Fig. 1 is a flowchart of an anti-spoofing attack image encryption and decryption method based on sampling synchronous communication of a chaotic system.
Fig. 2 is a diagram of networked system synchronization control under spoofing attack.
Fig. 3 is a schematic diagram of a Lena ciphertext image.
Fig. 4 is a histogram of the Lena original image.
Fig. 5 is a histogram of a Lena ciphertext image.
Fig. 6 is a horizontal correlation diagram of the Lena original image.
Fig. 7 is a horizontal correlation diagram of a Lena ciphertext image.
Fig. 8 is a schematic diagram of an attractor driving a chaotic system.
Fig. 9 is a spoofing attack sequence diagram.
Detailed Description
The invention is described in further detail below with reference to the attached drawings and detailed description:
as shown in fig. 1, the anti-spoofing attack image encryption and decryption method based on sampling synchronous communication of a chaotic system comprises the following steps:
and 1, establishing a driving-response chaotic system closed-loop system model under a spoofing attack environment.
The secure sampling communication controller is first designed based on features of a spoof attack that obeys bernoulli distribution, wherein the designed secure sampling communication controller includes a secure control gain and a sampling period (i.e., hereinafterh k ) Is set by the setting of (2).
Based on the designed safe sampling communication controller, a driving-responding closed-loop system model under the spoofing attack environment is established.
The step 1 specifically comprises the following steps:
in order to unify the representation mode of the chaotic system, the following kinetic equation for driving the chaotic system is considered:
wherein,,x(t)∈R n is a state vector driving the chaotic system,A、B∈R n n× is a real matrix.
g(x(t) Is a nonlinear function that monotonically non-decreases and Li Puxi z continuous, i.e., for a constantaAndb，a≠bwith a constantm j So that
It is true that the method is that,j=1,2,…,n。
g j (a)、g j (b) Representing a functiong(x(t) First of the above)jIndividual elements are ata、bThe value of the position.
R n Representation ofnThe x 1-dimensional column vector is used,R n n× representation ofn×nA matrix of dimensions.
When matrixA、B、g(x(t) When set to a specific parameter, the system can be expressed as a plurality of classical chaotic systems. For example:
g(x(t))=0.5(|x(t)+1|-|x(t)-1|)；
the chaotic system is driven, namely, the Chua's circuit chaotic system is represented by a formula (1).
For the chaotic system, the basis and key of using the chaotic signal for secret communication are how to realize the synchronization between the two chaotic systems. If the two chaotic systems have slight differences in any one of the initial value and the system parameter, the system track becomes completely different after a period of time. Therefore, it is important to ensure that the drive-response chaotic system is synchronized.
Correspondingly, the dynamic equation of the response chaotic system is as follows:
wherein,,y(t)∈R n is a state vector of the response system,u(t)∈R n a communication controller for secure sampling.
Definition error systeme(t)=y(t)－x(t) The following converts the synchronization problem of the drive-response chaotic system into an error systeme(t) Stability problems of (2); according to the formula (1) and the formula (2), the kinetic equation of the error system is:
wherein,,g(e(t))=g(y(t))－g(x(t))；
g(y(t) Representing a nonlinear function, different chaotic attractors are generated by using different nonlinear functions.
In an ideal communication environment, a secure sampling communication controlleru(t) Expressed as:
u(t)=Ke(t k ), t k ≤t＜t k+1 。
wherein,,k=0,1,2…,t 0 sample period=0h k =t k+1 －t k And satisfy 0 <)h 1 ≤h k ≤h 2 ，h 1 Andh 2 representing the upper and lower bounds of the sampling period,K∈R n n× for the safety control gain matrix.
FIG. 2 clearly shows the sampled datae(t k )∈R n The process being transferred from the sensor to the controller, i.e. error system state informatione(t) Firstly, the data are transmitted to a sensor, and the sensor detects the system state and samples the system state to obtain discrete sampled datae(t k ) The sampled data is transferred to the controller through the communication network and becomesKe(t k ) The control signal is then transmitted to the zero-order keeper via the communication network to maintain signal continuity and finally to the actuator, thus forming a closed loop control system. For example, in a circuit implementation, first a hardware circuit of a drive and response system is constructed, obtained with an add-subtract comparatore(t). Then, a voltage-current sensor is adopted for acquisitione(t k ) (error signals in the circuit, i.e., voltage and current), data is transmitted to the CPU (i.e., controller) for processing over the wireless network. Finally, the control signal is applied to the circuit of the drive system via an amplifying circuit (actuator).
As shown in fig. 2, the sensors, controllers, actuators, drive systems, response systems, etc. are all connected via a communication network. The present invention considers sampled data since the communication network is vulnerable to hackinge(t k )∈R n The process from the sensor to the secure sampling communication controller is subject to spoofing attack signals by hackersα(t)。
Attack signalα(t) Satisfy Bernoulli distributionα(t)∈{0,1}；α(t) The symbol 1 indicates that an attack is occurring,α(t) =0 indicates that an attack is not occurring and the expectation of the occurrence of a spoof attack signal is E {α(t)=1}=α。
In fraud attackα(t) Under the influence of (a) the originally sampled data packete(t k ) Is replaced byf(e(t k )). Thus, consider a spoofing attack scenario, a secure sampling communications controlleru(t) Re-expressed as:
u(t)=(1-α(t))Ke(t k )+α(t)Kf(e(t k )), t k ≤t＜t k+1 (4)
the error system (3) is therefore re-expressed as:
the formula (5) is a drive-response closed-loop system model established in a spoofing attack environment.
Since the energy and the action time of the spoofing attack signal are limited, there are:
||f(e(t k ))|| 2 ≤||Le(t k )|| 2 (6)
wherein,,Lrepresented as a constant matrix; (t k ) Is shown int k The sampled data of the time instant is taken,f(e(t k ) Represents data after having been subjected to a spoofing attack by a hacker.
And 2, establishing a criterion for ensuring that the driving-response chaotic system realizes complete synchronization.
And designing a Lyapunov function, and obtaining a sufficient condition capable of ensuring the asymptotic stability of the closed-loop system according to a matrix theory, an inequality estimation technology and a stability theory, namely establishing a criterion for ensuring the driving-response chaotic system to realize complete synchronization.
The step 2 specifically comprises the following steps:
a criterion capable of ensuring the driving-response chaotic system to realize complete synchronization is established by designing a Lyapunov function according to a matrix theory, an inequality estimation technology and a stability theory.
For a given constanth 1 、h 2 、λ、α、β 1 、β 2 ，0＜h 1 ≤h 2 Lambda, alpha is any constant, beta 1 >0，β 2 >0。
If presentn×nDimension matrixP>0、U>0，n×nDimension diagonal matrixΛ>0，2n×2nDimensional symmetry matrix
、/>
，n×nDimension arbitrary matrixW、R 1 、Z 1 、Z 2 、Z 3 So that the following linear matrix inequalities (7) - (8) are for anyh k ∈{h 1 ,h 2 All are true:
Δ(h k )＜0 (7)
Γ(h k )＜0 (8)
wherein, delta is%h k )=[Δ ij ] n n6×6 ，Γ(h k )=[Γ ij ] n n7×7 。
M、M 1 、Q 2 Representation ofn×nAn arbitrary matrix is maintained,Q 1 、Q 3 representation ofn×nA dimensional symmetric matrix.
[Δ ij ] n n6×6 The expression is as follows:
wherein the symmetric part of the matrix is represented.
Δ 11 =－(M+M T )/2－Q 1 +h k (Q 2 +Q 2 T )－Z 1 －Z 1 T －R 1 T A－AR 1 ；
Δ 12 =P+h k (M+M T )/2+h k Q 1 －Z 2 －λAR 1 －R 1 T ；
Δ 13 =M－M 1 －h k Q 2 T +Q 1 +Z 1 T －Z 3 +(1-α)W；
Δ 14 =β 1 GΛ+R 1 T B；
Wherein,,G=diag{m 1 ,m 2 , …,m n the diagonal matrix is represented by the },m j ，j=1,2,…,n；
Δ 15 =αW；
Δ 16 =h k Q 3 －Q 2 ；
Δ 22 =h k U－λR 1 －λR 1 T ；
Δ 23 =h k (－M+M 1 )－h k Q 1 +Z 2 T +λ(1-α)W；
Δ 24 =λR 1 T B；
Δ 25 =λαW；
Δ 26 =h k Q 2 ；
Δ 33 =M 1 +M 1 T －(M+M T )/2－Q 1 +Z 3 +Z 3 T +β 2 L T L；
Δ 36 =Q 2 ；
Δ 44 =－2β 1 Λ；
Δ 55 =－β 1 I；
wherein,,Irepresenting the identity matrix;
Δ 66 =－Q 3 ；
Δ 34 、Δ 35 、Δ 45 、Δ 46 、Δ 77 is thatn×nIs a zero matrix of (c).
[Γ ij ] n n7×7 The expression is as follows:
wherein the symmetric part of the matrix is represented;
Γ 11 =－(M+M T )/2－Q 1 －Z 1 －Z 1 T －R 1 T A－AR 1 ；
Γ 12 =P－Z 2 －λAR 1 －R 1 T ；
Γ 13 =M－M 1 +Q 1 +Z 1 T －Z 3 +(1-α)W；
Γ 14 =β 1 GΛ+R 1 T B；
Γ 15 =h k Z 1 T ；
Γ 16 =αW；
Γ 17 =－Q 2 ；
Γ 22 =－λR 1 －λR 1 T ；
Γ 23 =Z 2 T +λ(1-α)W；
Γ 24 =λR 1 T B；
Γ 25 =h k Z 2 T ；
Γ 26 =αW；
Γ 33 =M 1 +M 1 T －(M+M T )/2－Q 1 +Z 3 +Z 3 T +β 2 L T L；
Γ 35 =h k Z 3 T ；
Γ 37 =Q 2 ；
Γ 44 =－2β 1 Λ；
Γ 55 =－h k U；
Γ 66 =－β 2 I；
Γ 77 =－Q 3 ；
Γ 27 、Γ 34 、Γ 36 、Γ 45 、Γ 46 、Γ 47 、Γ 56 、Γ 57 、Γ 67 is thatn×nIs a zero matrix of (c).
The drive-response closed loop system model built in the spoofing attack environment, equation (5), is mean square stable in the sampled communication, i.e., the drive-response system can achieve full synchronization in the sampled communication in the spoofing attack environment.
The invention solves the network spoofing attack environment by solving the linear matrix inequality criterion, namely the formula (7) and the formula (8)The problem of sampling synchronization of the lower chaotic system is solved easily by means of an LMI tool kit in MATLAB, and an unknown matrix in a linear matrix inequality criterion 7P、U、Λ、M、M 1 、Q 1 、Q 2 、Q 3 、W、R 1 、Z 1 、Z 2 、Z 3 。
Wherein,,P、U、M、M 1 、Q 1 、Q 2 、Q 3 is a matrix introduced in constructing the lyapunov function,Λ、W、R 1 、Z 1 、Z 2 、Z 3 is a free matrix, beta, introduced to solve the linear matrix inequality 1 、β 2 Lambda is a parameter introduced to solve the linear matrix inequality, the parameterα、h 1 、h 2 Has actual physical meaning.
In addition, in order to illustrate that the error system can realize the asymptotic stability of the mean square under the sampling communication, the following proving process is also provided:
the following lyapunov function was constructed:
。
wherein,,V 1 (t)=e T (t)Pe(t)；
taking the derivative yields the desired availability:
from the Jensen inequality
Wherein,,
。
from the characteristics of the attack signal and the nonlinear term, it is possible to:
wherein beta is 1 >0，β 2 >0，Λ>0 is a diagonal matrix.
For any matrix, according to the Newton-Leibiniz formulaZ 1 、Z 2 、Z 3 ∈R n n× The method comprises the following steps:
for arbitrary constantsλMatrixR 1 ∈R n n× The method comprises the following steps:
from formulas (9) - (13):
wherein:
according to formulas (7) - (8) in the calibration, we get:
。
according to the discrete Lyapunov theory, it is possible to:
Is obtainable by a closed loop system (5):
based on the Ke Xishi watt inequality, it is further available that:
the method is as follows from the equation of the Gerr Wo Lang:
due to
Is a fixed constant, so:
this means that the error system can achieve asymptotic stability of mean square under sampling communication, i.e. the driving-responding chaotic system can achieve synchronization under sampling communication under the environment of spoofing attack.
And 3, image encryption process.
First, the original image is read to obtainM×NIs provided.
Giving an initial value of a driving chaotic system, and generating an encryption key by using the driving chaotic systemM×NAnd encrypting the pixel matrix of the original image by using the pseudo-random number and an encryption algorithm to obtain the ciphertext image.
And 4, image decryption.
Giving an initial value of the response chaotic system, and solving a safety control gain which ensures that the driving-response chaotic system can be synchronized based on the criterion which is established in the step 2 and ensures that the driving-response chaotic system is completely synchronized.
Specifically, a safety control gain matrixKThe expression is as follows:K=(R 1 T ) -1 W。
in the image decryption process, the safety control gain parameter is applied to the closed-loop system model of the driving-response chaotic system established in the step 1, so that the driving-response chaotic system is completely synchronized under the spoofing attack environment.
Finally, the decryption key generated by the response chaotic system is utilizedM×NAnd (3) carrying out inverse processing on the ciphertext image in the step (3) to obtain a plaintext image, so as to realize the image encryption and decryption process under the spoofing attack environment.
In addition, to demonstrate the effectiveness of the method of the present invention, the following experiments were also performed:
and 2, establishing a criterion for ensuring that the driving-response chaotic system realizes complete synchronization.
The detailed procedures of step 1 and step 2 are already given in detail in the first half, and are not repeated here.
Taking the Lena original image as an example, the pixel values of the Lena gray scale image are read and stored in a 256×256-dimensional pixel matrixXThe histogram and horizontal correlation analysis are shown in fig. 4 and 6, respectively.
As can be seen from fig. 4 and 6, the original image has a strong correlation.
Selecting driving chaotic system parameters:
initial value of system state
。
FIG. 8 is an attractor for driving a chaotic system, using a driving hybrid system to generate 256×256 pseudorandom numbers for encryption keys and storing in a 256×256 dimensional matrixSThen the exclusive OR algorithm is used for the pixel matrix of the original imageEncryption and matrix storageOI.e.O=X⊕SThe encrypted ciphertext image is shown in fig. 3, and the histogram and horizontal correlation analysis thereof are shown in fig. 5 and 7, respectively. From the statistical perspective, the correlation of the ciphertext image is very low, and the encryption effect is proved to be good.
In the case of the view of figure 8,x 1 (t) and x 2 (t) Is a state vectorx(t) Is a component of (a).
Step 4, giving an initial value of the driving chaotic system
。
Considering the actual spoofing attack environment, the energy intensity of the spoofing attack is as follows
Spoofing attacks can cause sampled information to becomef(e(t k ))=(tanh(0.5e 1 (t k )), tanh(0.5e 2 (t k )), tanh(0.5e 3 (t k ))) T 。
Then according to the established stability criterion, namely formula (7) -formula (8), selecting sampling periodh 1 =h 2 =0.1,λ=1,α=0.5, β 1 =β 2 Shown in fig. 9 is a spoofing attack =1α(t) Is a sequence of (a).
Then the linear matrix inequality in the solution of MATLAB is available:
based on the obtained safety control gain, the safety control gain parameter is added into a closed loop system of the driving-response chaotic system, so that the driving-response chaotic system is completely synchronized, namely, a decryption key is successfully generated, further, an encrypted image is successfully decrypted, and a new view angle is provided for the technical field of new generation information.
Then generating a decryption key 256×256 pseudo-random number from the response system, storing in a 256×256 dimension matrixS 1 And for the matrix in step 3OAnd performing inverse operation, and obtaining a plaintext image after decryption. The anti-spoofing attack image encryption and decryption method based on the chaotic system sampling synchronous communication can effectively hide the original image information by comparing the original image, the ciphertext image and the plaintext image, and the decrypted image restores the original image information. That is, when a spoofing attack is applied, the generated decryption key can still successfully decrypt the ciphertext image to obtain the plaintext image.
The foregoing description is, of course, merely illustrative of preferred embodiments of the present invention, and it should be understood that the present invention is not limited to the above-described embodiments, but is intended to cover all modifications, equivalents and alternatives falling within the spirit and scope of the present invention as defined by the appended claims.
Claims (2)
1. The anti-spoofing attack image encryption and decryption method based on the sampling synchronous communication of the chaotic system is characterized in that,
the method comprises the following steps:
step 1, establishing a driving-responding chaotic system closed-loop system model under a spoofing attack environment;
firstly, designing a safety sampling communication controller according to the characteristics of spoofing attack obeying Bernoulli distribution, wherein the designed safety sampling communication controller comprises safety control gain and setting of sampling period;
based on a designed safe sampling communication controller, establishing a closed-loop system model of a driving-responding chaotic system under a spoofing attack environment;
step 2, establishing a criterion for ensuring that the driving-response chaotic system realizes complete synchronization;
designing a Lyapunov function, and obtaining a sufficient condition for ensuring the asymptotic stability of the closed-loop system according to a matrix theory, an inequality estimation technology and a stability theory, namely establishing a criterion for ensuring the driving-response chaotic system to realize complete synchronization;
step 3, an image encryption process;
firstly, reading an original image to obtain an MxN pixel matrix;
giving an initial value of the driving chaotic system, generating an encryption key MxN pseudo-random number by using the driving chaotic system, and encrypting a pixel matrix of an original image by using an encryption algorithm to obtain a ciphertext image;
step 4, image decryption process;
giving an initial value of the response chaotic system, and then solving a safety control gain capable of realizing the synchronization of the driving-response chaotic system based on the criterion for ensuring the complete synchronization of the driving-response chaotic system established in the step 2;
in the image decryption process, the safety control gain parameter is applied to the closed-loop system model of the driving-response chaotic system established in the step 1, so that the driving-response chaotic system is completely synchronized in a spoofing attack environment;
finally, the ciphertext image in the step 3 is subjected to inverse processing by using a decryption key MxN pseudo-random number generated by a response chaotic system to obtain a plaintext image, so as to realize the image encryption and decryption process under the spoofing attack environment;
the step 1 specifically comprises the following steps:
in order to unify the representation mode of the chaotic system, the following kinetic equation for driving the chaotic system is considered:
wherein x (t) ∈R n Is a state vector for driving the chaotic system, A, B epsilon R n×n Is a real matrix;
g(x(t))∈R n is a nonlinear function that monotonically non-decreases and Li Puxi z is continuous, i.e. for constants a and b, a+.b, there is a constant m j So that
Hold, j=1, 2, …, n;
g j (a)、g j (b) Representing the value of the j-th element in the function g (x (t)) at a and b; r is R n Represents an n×1-dimensional column vector, R n×n A matrix representing n x n dimensions; correspondingly, the dynamic equation of the response chaotic system is as follows:
wherein y (t) ∈R n Is the state vector of the response system, u (t) εR n Representing a secure sampling communications controller;
defining an error system e (t) =y (t) -x (t), and converting the synchronization problem of the driving-response chaotic system into a stability problem of the error system e (t); according to the formula (1) and the formula (2), the kinetic equation of the error system is:
wherein g (e (t)) = g (y (t)) -g (x (t));
g (y (t)) represents a nonlinear function, and different chaotic attractors are generated by using different nonlinear functions;
in an ideal communication environment, the secure sampling communication controller u (t) is represented as:
u(t)＝Ke(t k ),t k ≤t＜t k+1 ；
wherein k=0, 1,2 …, t 0 =0, sampling period h k ＝t k+1 －t k And satisfy 0<h 1 ≤h k ≤h 2 ，h 1 And h 2 Represent the upper and lower bounds of the sampling period, K.epsilon.R n×n Is a safety control gain matrix;
consider the sampled data e (t k )∈R n A spoofing attack signal alpha (t) sent by a hacker is received in the process from the sensor to the secure sampling communication controller; attack signal α (t) satisfies Bernoulli distribution α (t) ∈ {0,1};
α (t) =1 indicates that an attack is occurring, α (t) =0 indicates that an attack is not occurring, and the expectation of occurrence of a spoof attack signal is E { α (t) =1 } =α;
under the influence of the spoofing attack a (t), the originally sampled data packet e (t) k ) Is replaced by f (e (t) k ) A) is provided; thus, considering a spoofing attack scenario, the secure sampling communications controller u (t) is again denoted:
u(t)＝(1－α(t))Ke(t k )+ α(t) K f(e(t k )), t k ≤t＜t k+1 (4)
the error system (3) is therefore re-expressed as:
the formula (5) is a closed-loop system model of the driving-response chaotic system established under the spoofing attack environment;
since the energy and the action time of the spoofing attack signal are limited, there are:
|| f(e(t k ))|| 2 ≤||Le(t k )|| 2 (6)
wherein L is represented as a constant matrix;
e(t k ) Indicated at t k Time sample data, f (e (t k ) Representing data after having been subjected to a spoofing attack by a hacker;
the step 2 specifically comprises the following steps:
the standard capable of ensuring the driving-responding chaotic system to realize complete synchronization is established according to a matrix theory, an inequality estimation technology and a stability theory by designing a Lyapunov function;
for a given constant h 1 、h 2 、λ、α、β 1 、β 2 ，0＜h 1 ≤h 2 Lambda, alpha is any constant, beta 1 >0，β 2 >0；
If there is an n x n dimensional matrix P>0、U>0, n x n dimension diagonal matrix Λ>0,2n×2n-dimensional symmetric matrix
n x n dimensional arbitrary matrix W, R 1 、Z 1 、Z 2 、Z 3 So that the following linear matrix inequalities (7) - (8) are given for any h k ∈{h 1 ,h 2 All are true:
Δ(h k )＜0 (7)
Γ(h k )＜0 (8)
wherein, delta (h k )＝[Δ ij ] 6n×6n ，Γ(h k )＝[Γ ij ] 7n×7n ；
M、M 1 、Q 2 Represents an n x n-dimensional arbitrary matrix, Q 1 、Q 3 Representing an n x n dimensional symmetric matrix;
[Δ ij ] 6n×6n the expression is as follows:
wherein, represents the symmetric part of the matrix;
Δ 11 ＝－(M+M T )/2－Q 1 +h k (Q 2 +Q 2 T )－Z 1 －Z 1 T －R 1 T A－AR 1 ；
Δ 12 ＝P+h k (M+M T )/2+h k Q 1 －Z 2 －λAR 1 －R 1 T ；
Δ 13 ＝M－M 1 －h k Q 2 T +Q 1 +Z 1 T －Z 3 +(1-α)W；
Δ 14 ＝β 1 GΛ+R 1 T B；
wherein g=diag { m } 1 ,m 2 ,…,m n The diagonal matrix, m j ，j＝1,2,…,n；
Δ 15 ＝αW；
Δ 16 ＝h k Q 3 －Q 2 ；
Δ 22 ＝h k U－λR 1 －λR 1 T ；
Δ 23 ＝h k (－M+M 1 )－h k Q 1 +Z 2 T +λ(1-α)W；
Δ 24 ＝λR 1 T B；
Δ 25 ＝λαW；
Δ 26 ＝h k Q 2 ；
Δ 33 ＝M 1 +M 1 T －(M+M T )/2－Q 1 +Z 3 +Z 3 T +β 2 L T L；
Δ 36 ＝Q 2 ；
Δ 44 ＝－2β 1 Λ；
Δ 55 ＝－β 1 I；
Wherein I represents an identity matrix;
Δ 66 ＝－Q 3 ；
Δ 34 、Δ 35 、Δ 45 、Δ 46 、Δ 77 a zero matrix of n x n; [ Γ ] ij ] 7n×7n The expression is as follows:
wherein, represents the symmetric part of the matrix;
Γ 11 ＝－(M+M T )/2－Q 1 －Z 1 －Z 1 T －R 1 T A－AR 1 ；
Γ 12 ＝P－Z 2 －λAR 1 －R 1 T ；
Γ 13 ＝M－M 1 +Q 1 +Z 1 T －Z 3 +(1-α)W；
Γ 14 ＝β 1 GΛ+R 1 T B；
Γ 15 ＝h k Z 1 T ；
Γ 16 ＝αW；
Γ 17 ＝－Q 2 ；
Γ 22 ＝－λR 1 －λR 1 T ；
Γ 23 ＝Z 2 T +λ(1-α)W；
Γ 24 ＝λR 1 T B；
Γ 25 ＝h k Z 2 T ；
Γ 26 ＝αW；
Γ 33 ＝M 1 +M 1 T －(M+M T )/2－Q 1 +Z 3 +Z 3 T +β 2 L T L；
Γ 35 ＝h k Z 3 T ；
Γ 37 ＝Q 2 ；
Γ 44 ＝－2β 1 Λ；
Γ 55 ＝－h k U；
Γ 66 ＝－β 2 I；
Γ 77 ＝－Q 3 ；
Γ 27 、Γ 34 、Γ 36 、Γ 45 、Γ 46 、Γ 47 、Γ 56 、Γ 57 、Γ 67 a zero matrix of n x n;
the closed-loop system model of the driving-response chaotic system established under the spoofing attack environment, namely the formula (5) is stable in mean square under the sampling communication, namely the driving-response chaotic system can realize complete synchronization under the sampling communication under the spoofing attack environment.
2. The spoofing attack resistant image encrypting and decrypting method according to claim 1, wherein the method comprises the steps of
In the step 4, the safety control gain matrix K is obtained by the following expression: k= (R 1 T ) -1 W。
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