CN106651736A - Optical image encryption method based on Gyrator transform and coupled chaos - Google Patents

Abstract

The invention relates to the technical fields of image information security and optical information processing, and aims to effectively resist known plaintext attacks and chosen plaintext attacks, and make key management and transmission more convenient. The technical scheme that the present invention adopts is, Gyrator transformation and coupled chaotic optical image encryption method, the steps are as follows: 1) the structure of coupled Logistic chaos: two one-dimensional Logistic chaotic maps are linked together by a coupling parameter; 2) chaotic key The generation of two random phase masks that act as the master key are respectively generated by the coupled Logistic chaotic system controlled by different chaotic parameters; 3) Image encryption and decryption based on Gyrator transformation: (1) During the encryption process, the to-be-encrypted The image is first modulated by the first chaotic random phase mask and (2) by the complex conjugate of the first chaotic random phase mask. The present invention is mainly applied to image information security occasions.

Description

Image Encryption Method Based on Gyrator Transform and Coupling Chaotic Optics

technical field

The invention relates to the technical fields of image information security and optical information processing, in particular to an optical image encryption method based on Gyrator transformation and coupled Logistic chaos.

Background technique

As one of the most popular multimedia forms at present, digital images are widely used in the fields of politics, economy, military affairs, education and so on. In today's highly developed Internet technology, how to protect digital images from tampering, illegal copying and dissemination has important practical significance. The research on image encryption technology has become one of the hotspots in the field of information security.

Optical information processing technology has aroused great interest in the field of image encryption research because of its high processing speed, high parallelism, and fast realization of convolution and correlation operations (see literature [1]). In the optical image encryption technology, the most representative is the double random phase encoding technology proposed by Javidi et al. (see literature [2]). This technology has opened up a new field of optical image encryption research, and a large number of new optical encryption methods and technologies have been born based on this technology (see review literature [3]). In addition, as a generalized Fourier transform, Gyrator transform can also be used in optical image encryption (see literature [4]).

However, most of the optical image encryption methods based on double random phase encoding have the following problems:

1) The key is a random phase mask of the image size, therefore, key management and transmission are inconvenient (see literature [5]);

2) Since the random phase mask is inconvenient to update, the encryption system is vulnerable to chosen plaintext attack and known plaintext attack (see literature [6] and [7]).

references:

[1] O. Matoba, T. Nomura, E. Perez-Cabre, M. Millan, and B. Javidi, Optical techniques for information security, Proceedings of IEEE 2009, 97: 1128-1148

[2]P.Réfrégier and B.Javidi,Optical image encryption based on inputplane and Fourier plane random encoding,Opt.Lett.,1995,20:767-769

[3] S. Liu, C. Guo, and J. T. Sheridan, A review of optical image encryption techniques, Optics & Laser Technology, 2014, 57:327-342

[4] J. Rodrigo, T. Alieva, M. Calvo, Gyrator transform: properties and applications, Opt. Express, 2007, 15: 2190-2203

[5] S.Yuan, Y.Xin, M.Liu, S.Yao, and X.Sun, An improved method to enhance the security of double random-phase encoding in the Fresnel domain, Optics&Laser Technology, 2012,44:51-56

[6] X. Peng, H. Wei, and P. Zhang, Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain, Opt. Lett., 2006, 31:3261-3263

[7] U. Gopinathan, D.S. Monaghan, T.J. Naughton, and J.T. Sheridan, A known-plain text heuristic attack on the Fourier plane encryption algorithm. Opt. Express, 2006, 14:3181-3186.

Contents of the invention

In order to overcome the deficiencies of the prior art, the present invention aims at effectively resisting known plaintext attacks and chosen plaintext attacks, and makes key management and transmission more convenient. The technical solution adopted in the present invention is, Gyrator transformation and coupling chaotic optical image encryption method, the steps are as follows:

1) Construction of coupled Logistic chaos: link two one-dimensional Logistic chaos maps together through a coupling parameter;

2) Generation of chaotic key: the two random phase masks used as the master key are respectively generated by the coupled Logistic chaotic system controlled by different chaotic parameters, and the initial value and control parameters of the chaotic system are used as the master key; in addition, the Gyrator transform The angle is used as an auxiliary key in the process of encryption and decryption;

3) Image encryption and decryption based on Gyrator transformation: (1) During the encryption process, the image to be encrypted is first modulated by the first chaotic random phase mask, and then undergoes Gyrator transformation with an angle of a ₁ , and the transformed image is then It is modulated by the second chaotic random phase mask, and then undergoes Gyrator transformation with an angle of a ₂ ; (2) In the decryption process, the encrypted image first undergoes a Gyrator inverse transformation with an angle of a ₂ , and then is transformed by the second chaotic The complex conjugate modulation of the random phase mask, the modulated image is subjected to the Gyrator inverse transformation with an angle of a ₁ , and finally is modulated by the complex conjugate of the first chaotic random phase mask.

Concrete steps in an embodiment are:

(1) Construction of coupled Logistic chaos:

The mathematical expression of the discrete form of the one-dimensional Logistic chaotic system is:

x _n+1 ＝μx _n (1-x _n ) (1)

Among them, x _n is the initial value of the chaotic system, x _n+1 is the iterative output value of the chaotic system; and when the control parameter μ∈(3.56,4], the Logistic system is in a chaotic state;

Introduce a coupling parameter ε, which satisfies ε∈(-2,2), then the mathematical expression of the discrete form of the coupled Logistic chaotic system is:

x _n+1 ＝μx _n (1-x _n )+ε(y _n -x _n ) (2)

y _n+1 ＝μy _n (1-y _n )+ε(x _n -y _n ) (3)

Among them, x _n and y _n are the initial values of the coupled Logistic chaotic system respectively, and x _n+1 and y _n+1 are the iterative output values of the coupled Logistic chaotic system respectively. Similarly, when the control parameter μ∈(3.56,4] , the coupled Logistic system is in a chaotic state;

(2) Generation of chaotic key:

In the encryption method, two chaotic random phase masks act as the master key, and the Gyrator transformation angle acts as the auxiliary key. Assuming that the size of the image to be encrypted is M×N pixels, the size of the two chaotic random phase masks is also M×N pixels. For the coupled Logistic chaotic system controlled by two sets of different chaotic parameters, after iterating (M×N)/2 times, two sets of random number sequences X ₁ ={x′ ₁ ,x′ ₂ ,…,x′ _{( M×N)/2} }, Y ₁ ={y′ ₁ ,y′ ₂ ,…,y′ _(M×N)/2 } and X ₂ ={x″ ₁ ,x″ ₂ ,…,x″ _{( M×N)/2} }, Y ₂ ={y″ ₁ ,y″ ₂ ,…,y″ _(M×N)/2 }; where, y′ ₁ ,y′ ₂ ,…,y′ _(M×N)/2 , x″ ₁ ,x″ ₂ ,…,x″ _(M×N)/2 and y″ ₁ ,y″ ₂ ,…, y″ _(M×N)/2 are the iterative output values of the coupled chaotic system respectively, and these two sets of random number sequences are respectively integrated into two two-dimensional matrices Z ₁ ={z′ _i,j |i=1, 2,...,M; j=1,2,...,N} and Z ₂ ={z″ _i,j |i=1,2,...,M; j=1,2,...,N}, where z ′ _{i, j} and z″ _{i, j} are the elements of the two-dimensional matrix, and the subscripts i, j are the coordinates of the matrix elements, then two chaotic random phase masks can be obtained, and their mathematical expressions are respectively C ₁ (x ₁ ,y ₁ )=exp(j2πz′ _i,j ) and C ₂ (x ₂ ,y ₂ )=exp(j2πz″ _i,j ); among them, (x ₁ ,y ₁ ) and (x ₂ ,y ₂ ) are the coordinates of the positions of the two random phase masks, j is the imaginary unit, and π is the pi. Since the chaotic random phase mask is controlled by the initial value and control parameters of the chaotic system, the initial value of the chaotic system and control parameters as the master key of the encryption system;

(3) Image encryption and decryption based on Gyrator transformation:

1) During the encryption process, the image U ₀ (x ₀ , y ₀ ) to be encrypted is firstly modulated by the first chaotic random phase mask, and then undergoes Gyrator transformation with an angle of a ₁ , and the transformed image is then modulated by the second Block chaotic random phase mask modulation, and then perform Gyrator transformation with an angle of a _2. After two modulations and two transformations, the encrypted image U(x′,y′) can be obtained:

U(x′,y′)=GT _a2 {GT _a1 {U ₀ (x ₀ ,y ₀ )C ₁ (x ₁ ,y ₁ )}C ₂ (x ₂ ,y ₂ )} (4)

Among them, (x′,y′) is the position coordinate of the output surface; GT _a { } represents the Gyrator transformation with angle a, and its form is as follows:

Among them, U ₀ (x ₀ ,y ₀ ) and U(x,y) represent the input image and the transformed image respectively; (x ₀ ,y ₀ ) and (x,y) represent the input image and the transformed image respectively The coordinates of the position; sin represents the sine function, cos represents the cosine function;

2) During the decryption process, the encrypted image U(x′,y′) is first subjected to Gyrator inverse transformation with an angle of a ₂ , and then modulated by the complex conjugate of the second chaotic random phase mask, the modulated The image is then subjected to Gyrator inverse transformation with an angle of a ₁ , and finally modulated by the complex conjugate of the first chaotic random phase mask to obtain the decrypted image:

Among them, * represents the complex conjugate operator.

Features and beneficial effects of the present invention are:

In the optical image encryption method provided by the present invention, the use of the chaotic key enables the encryption method to effectively resist known plaintext attacks and chosen plaintext attacks, and makes key management and transmission more convenient. The Gyrator transformation angle is used as an auxiliary key in the process of encryption and decryption, which further guarantees the security of this encryption method.

Description of drawings:

Fig. 1 is a schematic diagram of the encryption and decryption process of the optical image encryption method provided by the present invention. In the picture:

(a) a schematic diagram of the encryption process of the optical image encryption method provided by the present invention;

(b) a schematic diagram of the decryption process of the optical image encryption method provided by the present invention;

Figure 2 Comparison of encrypted and decrypted images. In the picture:

(a) is the original image to be encrypted;

(b) the image encrypted for this method;

(c) is the decrypted image when all keys are correct.

Figure 3 Comparison of decrypted images when there is an error. In the picture:

(a) The initial value x2 of the coupled Logistic chaotic system controlling the second random phase mask is wrong, and the decrypted image when the other keys are all correct;

(b) The initial value y2 of the coupled Logistic chaotic system for controlling the second random phase mask is wrong, and the decrypted image when other keys are all correct;

(c) The decrypted image when the control parameter μ2 of the coupled Logistic chaotic system controlling the second random phase mask is wrong and the other keys are all correct;

(d) is the decrypted image when the Gyrator transformation angle a1 is wrong and other keys are correct;

(e) is the decrypted image when the Gyrator transformation angle a2 is wrong and other keys are correct;

Figure 4 is a comparison chart of missing decrypted images.

(a) is the image decrypted from the encrypted image with 12.5% missing information;

(b) is the image decrypted from the encrypted image with 25% missing information;

(c) is the image decrypted from the encrypted image with 50% missing information;

Figure 5 Comparison of images under different Gaussian noises.

(a) is an image decrypted from an encrypted image containing 10% Gaussian noise;

(b) is an image decrypted from an encrypted image containing 10% salt and pepper noise;

(c) is an image decrypted from an encrypted image containing 10% speckle noise;

In the accompanying drawings, the list of parts represented by each label is as follows:

CRPM ₁ : first chaotic random phase mask; CRPM ₂ : second chaotic random phase mask; CRPM ₁ *: complex conjugate of first chaotic random phase mask; CRPM ₂ *: second chaotic random phase mask Complex conjugate of phase mask; GT: Gyrator transform; IGT: inverse Gyrator transform.

detailed description

The invention provides an optical image encryption method based on Gyrator transformation and coupled Logistic chaos. The optical image encryption method provided by the invention consists of the structure of coupling Logistic chaos, the generation of chaotic keys, and the image encryption and decryption based on Gyrator transformation. The use of chaotic keys makes this encryption method effective against known-plaintext attacks and chosen-plaintext attacks, and makes key management and transmission more convenient. The Gyrator transformation angle is used as an auxiliary key in the process of encryption and decryption, which further guarantees the security of this encryption method. A large number of experiments show that this encryption method has a good ability to resist brute force attack, statistical attack, noise attack and shear attack. See the description below for details:

1) Construction of coupled Logistic chaos: The coupled Logistic chaotic system links two one-dimensional Logistic chaotic maps together through a coupling parameter; compared with a single one-dimensional Logistic chaotic system, the coupled Logistic chaotic system has a larger parameter space, Better pseudo-randomness, and more random number sequences can be generated.

2) Generation of chaotic key: the two random phase masks used as the master key are respectively generated by the coupled Logistic chaotic system controlled by different chaotic parameters, and the initial value and control parameters of the chaotic system are used as the master key; in addition, the Gyrator transform The angle is used as an auxiliary key in the process of encryption and decryption. Since the key is easily updated during encryption and decryption, the encryption method can effectively resist known plaintext attacks and chosen plaintext attacks; in addition, key management and transmission are also more convenient.

3) Image encryption and decryption based on Gyrator transformation: (1) During the encryption process, the image to be encrypted is first modulated by the first chaotic random phase mask, and then undergoes Gyrator transformation with an angle of a ₁ , and the transformed image is then It is modulated by the second chaotic random phase mask, and then undergoes Gyrator transformation with an angle of a ₂ ; (2) In the decryption process, the encrypted image first undergoes a Gyrator inverse transformation with an angle of a ₂ , and then is transformed by the second chaotic The complex conjugate modulation of the random phase mask, the modulated image is subjected to the Gyrator inverse transformation with an angle of a ₁ , and finally is modulated by the complex conjugate of the first chaotic random phase mask.

In order to make the purpose, technical solution and advantages of the present invention clearer, the implementation manners of the present invention will be further described in detail below.

Example 1

An optical image encryption method based on Gyrator transform and coupled Logistic chaos. The schematic diagram of the encryption and decryption process is shown in Figure 1. The encryption method consists of the construction of coupled Logistic chaos, the generation of chaotic keys, and the image encryption and decryption based on Gyrator transform. .

(1) Construction of coupled Logistic chaos:

In the encryption method provided by the present invention, the coupled Logistic chaotic system connects two one-dimensional Logistic chaotic maps through a coupling parameter; compared with a single one-dimensional Logistic chaotic system, the coupled Logistic chaotic system has a larger parameter space, Better pseudo-randomness, and more random number sequences can be generated.

(2) Generation of chaotic key:

In the encryption method provided by the present invention, the two random phase masks that play the role of the master key are generated by the coupled Logistic chaotic system controlled by different chaotic parameters respectively, and the initial value and control parameters of the chaotic system are used as the master key; in addition, the Gyrator transform The angle is used as an auxiliary key in the process of encryption and decryption. Since the key is easily updated during encryption and decryption, the encryption method can effectively resist known plaintext attacks and chosen plaintext attacks; in addition, key management and transmission are also more convenient.

(3) Image encryption and decryption based on Gyrator transformation:

1) During the encryption process, the image to be encrypted is firstly modulated by the first chaotic random phase mask, and then undergoes Gyrator transformation with an angle of a ₁ , and the transformed image is then modulated by the second chaotic random phase mask, and then Perform Gyrator transformation with an angle of a ₂ ; 2) In the decryption process, the encrypted image first undergoes a Gyrator inverse transformation with an angle of a ₂ , and then is modulated by the complex conjugate of the second chaotic random phase mask, after modulation The image of is subjected to Gyrator inverse transformation with an angle of a ₁ , and finally modulated by the complex conjugate of the first chaotic random phase mask.

In summary, the use of chaotic keys makes this encryption method effective against known plaintext attacks and chosen plaintext attacks, and makes key management and transmission more convenient. The Gyrator transformation angle is used as an auxiliary key in the process of encryption and decryption, which further guarantees the security of this encryption method.

Example 2

Below in conjunction with Fig. 1, design principle, the scheme in embodiment 1 is introduced in detail, see the following description for details:

An optical image encryption method based on Gyrator transform and coupled Logistic chaos, the schematic diagram of the encryption and decryption process is shown in Figure 1. The encryption method consists of the construction of coupled Logistic chaos, the generation of chaotic keys, and the image encryption and decryption based on Gyrator transform. The specific implementation manners of these three parts will be described in detail below.

(1) Construction of coupled Logistic chaos:

The mathematical expression of the discrete form of the one-dimensional Logistic chaotic system is:

x _n+1 ＝μx _n (1-x _n ) (1)

Among them, x _n is the initial value of the chaotic system, x _n+1 is the iterative output value of the chaotic system; and when the control parameter μ∈(3.56,4], the Logistic system is in a chaotic state.

Introduce a coupling parameter ε, which satisfies ε∈(-2,2), then the mathematical expression of the discrete form of the coupled Logistic chaotic system is:

x _n+1 ＝μx _n (1-x _n )+ε(y _n -x _n ) (2)

y _n+1 ＝μy _n (1-y _n )+ε(x _n -y _n ) (3)

Among them, x _n and y _n are the initial values of the coupled Logistic chaotic system, respectively, and x _n+1 and y _n+1 are the iterative output values of the coupled Logistic chaotic system, respectively. Similarly, when the control parameter μ∈(3.56,4], the coupled Logistic system is in a chaotic state.

(2) Generation of chaotic key:

In the encryption method, two chaotic random phase masks act as the master key, and the Gyrator transformation angle acts as the auxiliary key. The following is a detailed introduction on how to use the coupled Logistic chaotic system to generate these two chaotic random phase masks.

Assuming that the size of the image to be encrypted is M×N pixels, the size of the two chaotic random phase masks is also M×N pixels. For the coupled Logistic chaotic system controlled by two sets of different chaotic parameters, after iterating (M×N)/2 times, two sets of random number sequences X ₁ ={x′ ₁ ,x′ ₂ ,…,x′ _{( M×N)/2} }, Y ₁ ={y′ ₁ ,y′ ₂ ,…,y′ _(M×N)/2 } and X ₂ ={x″ ₁ ,x″ ₂ ,…,x″ _{( M×N)/2} }, Y ₂ ={y″ ₁ ,y″ ₂ ,…,y″ _(M×N)/2 }; among them, x′ ₁ ,x′ ₂ ,…,x′ _{(M× N)/2} ，y′ ₁ ,y′ ₂ ,…,y′ _(M×N)/2 , x″ ₁ ,x″ ₂ ,…,x″ _(M×N)/2 and y″ ₁ ,y ″ ₂ ,…,y″ _(M×N)/2 are the iterative output values of the coupled chaotic system, respectively. These two sets of random number sequences are respectively integrated into two two-dimensional matrices Z ₁ ={z′ _i,j |i=1,2,...,M; j=1,2,...,N} and Z ₂ ＝{z″ _i,j |i=1,2,…,M; j=1,2,…,N}, where z′ _i,j and z″ _i,j are elements of two-dimensional matrix, i, j is the coordinate of the matrix element. Then two chaotic random phase masks can be obtained, and their mathematical expressions are respectively C ₁ (x ₁ ,y ₁ )=exp(j2πz′ _i,j ) and C ₂ (x ₂ ,y ₂ )=exp(j2πz″ _i,j ); where (x ₁ ,y ₁ ) and (x ₂ ,y ₂ ) are the coordinates of the two random phase masks, j is the imaginary unit, and π is the circumference ratio. Due to the chaotic random phase mask The membrane is controlled by the initial value and control parameters of the chaotic system. Therefore, the initial value and control parameters of the chaotic system are used as the master key of the encryption system. Since the master key and the auxiliary key are some numbers, the management and It will be very convenient to transmit these numbers; in addition, it will be very convenient to update these numbers during the encryption and decryption process.

(3) Image encryption and decryption based on Gyrator transformation:

1) During the encryption process, the image U ₀ (x ₀ , y ₀ ) to be encrypted is firstly modulated by the first chaotic random phase mask, and then undergoes Gyrator transformation with an angle of a ₁ , and the transformed image is then modulated by the second Block chaotic random phase mask modulation, and then perform Gyrator transformation with an angle of a _2. After two modulations and two transformations, the encrypted image U(x′,y′) can be obtained:

U(x′,y′)=GT _a2 {GT _a1 {U ₀ (x ₀ ,y ₀ )C ₁ (x ₁ ,y ₁ )}C ₂ (x ₂ ,y ₂ )} (4)

Among them, (x′,y′) is the position coordinate of the output surface; GT _a { } represents the Gyrator transformation with angle a, and its form is as follows:

Among them, U ₀ (x ₀ ,y ₀ ) and U(x,y) represent the input image and the transformed image respectively; (x ₀ ,y ₀ ) and (x,y) represent the input image and the transformed image respectively The coordinates of the position; sin represents the sine function, and cos represents the cosine function.

2) During the decryption process, the encrypted image U(x′,y′) is first subjected to Gyrator inverse transformation with an angle of a ₂ , and then modulated by the complex conjugate of the second chaotic random phase mask, the modulated The image is then subjected to Gyrator inverse transformation with an angle of a ₁ , and finally modulated by the complex conjugate of the first chaotic random phase mask to obtain the decrypted image:

Among them, * represents the complex conjugate operator.

In summary, the use of chaotic keys makes this encryption method effective against known plaintext attacks and chosen plaintext attacks, and makes key management and transmission more convenient. The Gyrator transformation angle is used as an auxiliary key in the process of encryption and decryption, which further guarantees the security of this encryption method.

Example 3

Below in conjunction with specific accompanying drawing, the scheme in embodiment 1 and 2 is carried out feasibility verification, see the following description for details:

After an image (as shown in Figure 2(a)) is encrypted by using the encryption method provided by the implementation of the present invention, the obtained encrypted image is shown in Figure 2(b).

As can be seen from Figure 2(b), any information of the original image is hidden. When all the keys are correct, the decrypted image is shown in Figure 2(c). As can be seen from Figure 2(c), the original image can be completely restored. It shows that the encryption and decryption of grayscale images using this system is successful.

In addition, when a certain key is wrong and other keys are correct, the decryption results are shown in Figure 3(a)-3(e). It can be seen that the security of the system can be guaranteed.

Figures 4(a)-4(c) are the decrypted images when the encrypted image is missing 12.5%, 25% and 50% of the information. Figures 5(a)-5(c) are the decrypted images when the encrypted image contains 10% Gaussian noise, salt and pepper noise and speckle noise. It can be seen that even if the encrypted image lacks some information or is polluted by noise to a certain extent, the embodiment of the present invention can still decrypt the original image with a certain quality, which verifies the feasibility of the system and meets various needs in practical applications .

In the embodiments of the present invention, unless otherwise specified, the models of the devices are not limited, as long as they can complete the above functions.

Those skilled in the art can understand that the accompanying drawing is only a schematic diagram of a preferred embodiment, and the serial numbers of the above-mentioned embodiments of the present invention are for description only, and do not represent the advantages and disadvantages of the embodiments.

The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included in the protection of the present invention. within range.

Claims (2)

1. A Gyrator transformation and coupling chaotic optical image encryption method is characterized by comprising the following steps:

1) construction of coupling Logistic chaos: connecting two one-dimensional Logistic chaotic mappings together through a coupling parameter;

2) generating a chaotic key: two random phase masks which play a role of a master key are respectively generated by coupling Logistic chaotic systems controlled by different chaotic parameters, and an initial value and a control parameter of the chaotic system are used as the master key; in addition, the Gyrator transforms the angle to be used as an auxiliary key in the encryption and decryption process;

3) image encryption and decryption based on Gyrator transform: (1) in the encryption process, an image to be encrypted is firstly modulated by a first chaotic random phase mask, and then the angle a is carried out₁The transformed image is modulated by a second chaotic random phase mask and then the angle is a₂Gyrator transformation of (1); (2) in the decryption process, the encrypted image is firstly processed with an angle a₂Inverse transformation of Gyrator, complex conjugate modulation by the second chaotic random phase mask, and angle a of the modulated image₁And (4) performing inverse transformation on the Gyrator, and finally performing complex conjugate modulation on the Gyrator by using a first chaotic random phase mask.

2. The Gyrator transform and coupled chaotic optical image encryption method as claimed in claim 1, wherein in one embodiment, the specific steps are as follows:

(1) construction of coupling Logistic chaos:

the mathematical expression of the discrete form of the one-dimensional Logistic chaotic system is as follows:

x_n+1＝μx_n(1-x_n) (1)

wherein x is_nIs an initial value of the chaotic system, x_n+1Is an iterative output value of the chaotic system, and when the control parameter mu ∈ (3.56, 4)]When the system is in a chaotic state, the Logistic system is in a chaotic state;

introducing a coupling parameter, wherein the element belongs to (-2,2), and then the mathematical expression of the discrete form of the coupling Logistic chaotic system is as follows:

x_n+1＝μx_n(1-x_n)+(y_n-x_n) (2)

y_n+1＝μy_n(1-y_n)+(x_n-y_n) (3)

wherein x is_nAnd y_nRespectively, an initial value, x, of the coupled Logistic chaotic system_n+1And y_n+1Respectively, the iterative output values of the coupled Logistic chaotic system, and similarly, when the control parameter mu ∈ (3.56, 4)]When the coupling Logistic system is in a chaotic state;

(2) generating a chaotic key:

assuming that the size of an image to be encrypted is M × N pixels, the sizes of the two chaotic random phase masks are M × N pixels, and for a coupled Logistic chaotic system controlled by two groups of different chaotic parameters, iterating the coupled Logistic chaotic system for 2 times (M × N), obtaining two groups of random number sequences X₁＝{x′₁,x′₂,…,x′_(M×N)/2}，Y₁＝{y′₁,y′₂,…,y′_(M×N)/2And X₂＝{x″₁,x″₂,…,x″_(M×N)/2}，Y₂＝{y″₁,y″₂,…,y″_(M×N)/2}; wherein, x'₁,x′₂,…,x′_(M×N)/2，y′₁,y′₂,…,y′_(M×N)/2，x″₁,x″₂,…,x″_(M×N)/2And y ″)₁,y″₂,…,y″_(M×N)/2Respectively, the two groups of random number sequences are respectively integrated into two-dimensional matrix forms Z₁＝{z′_i,j1,2, …, M; j ═ 1,2, …, N } and Z₂＝{z″_i,j1,2, …, M; j ═ 1,2, …, N }, where z'_i,jAnd z'_i,jTwo chaotic random phase masks can be obtained by using subscript i, j as the coordinate of matrix element as the element of two-dimensional matrix, and the mathematical expressions are respectively C₁(x₁,y₁)＝exp(j2πz′_i,j) And C₂(x₂,y₂)＝exp(j2πz′_i,j) (ii) a Wherein (x)₁,y₁) And (x)₂,y₂) The coordinates of the positions of the two random phase masks are respectively, j is an imaginary number unit, and pi is a circumference ratio;

(3) image encryption and decryption based on Gyrator transform:

1) in the encryption process, it is addedDense image U₀(x₀,y₀) Firstly modulated by a first chaotic random phase mask, and then the angle is a₁The transformed image is modulated by a second chaotic random phase mask and then the angle is a₂After two times of modulation and two times of transformation, the encrypted image U (x ', y') can be obtained:

U(x′,y′)＝GT_a2{GT_a1{U₀(x₀,y₀)C₁(x₁,y₁)}C₂(x₂,y₂)} (4)

wherein, (x ', y') is the position coordinate at the output face; GT system_a{. denotes the Gyrator transform for angle a, which is of the form:

$\begin{array}{c}U(x,y)={\mathrm{GT}}_a\left\{U_0\right(x_0,y_0\left)\right\}(x,y)\\ =\frac{1}{\left|\sin a\right|}\underset{-\mathrm{\∞}}{\overset{+\mathrm{\∞}}{\∫}}\underset{-\mathrm{\∞}}{\overset{+\mathrm{\∞}}{\∫}}{U}_0(x_0,y_0)\exp \left\{i2\mathrm{\π}\frac{(xy+x_0{y}_0)\cos a-(x_{0}y+{\mathrm{xy}}_0)}{\sin a}\right\}{\mathrm{dx}}_0{\mathrm{dy}}_0\end{array}---\left(5\right)$

wherein, U₀(x₀,y₀) And U (x, y) represents the input image and the transformed image, respectively; (x)₀,y₀) And (x, y) coordinates representing the positions of the input image and the transformed image, respectively; sin represents a sine function and cos represents a cosine function;

2) in the decryption process, the encrypted image U (x ', y') is first processed with an angle a₂Inverse transformation of Gyrator, complex conjugate modulation by the second chaotic random phase mask, and angle a of the modulated image₁Inverse transformation of the Gyrator, and finally modulation by complex conjugate of the first chaotic random phase mask to obtain a decrypted image:

wherein denotes a complex conjugate operator.