CN106570818A - Optical image encryption method based on LCT and Logistic chaos - Google Patents

Abstract

The invention relates to image information security and optical information processing, and aims to effectively resist known-plaintext attack and chosen-plaintext attack, make key management and transmission more convenient, further guarantee the security, and achieve good ability to resist violent attack, statistic attack, noise attack and cropping attack. An optical image encryption method based on LCT and Logistic chaos of the invention comprises the following steps: (1) construction of a 2D Logistic chaos: a 2D Logistic chaos system is a 2D extension of a 1D Logistic chaos system; (2) generation of a chaos key: two random phase masks as main keys are respectively generated by 2D Logistic chaos systems controlled by different chaos parameters; and (3) image encryption and decryption based on linear canonical transform LCT: a decrypted image is obtained. The method of the invention is mainly used in image information security and optical information processing occasions.

Description

Optical image encryption method based on LCT and Logistic chaos

Technical field

The present invention relates to image information safety and optical information processing technical field, more particularly to it is a kind of based on the change of linear canonical Change the optical image encryption method of (LCT) and two dimension Logistic chaotic.

Background technology

Digital picture has as one of current most popular multimedia form in the field such as politics, economic, military, education And be widely applied.In Internet technology highly developed today, how to protect digital picture exempt from distort, bootlegging and biography Broadcast with important practical significance.Research to image encryption technology has become one of the focus in current information security field.

Optical information processing technique with its high processing rate, high degree of parallelism, can quickly realize that convolution and related operation etc. are excellent Point, in image encryption research field the great interest (see document [1]) of people is caused.In optical image encryption technology, most Representative is the Double random phase technology (see document [2]) of the propositions such as Javidi.The technology opens optical picture As the frontier of encryption research, large quantities of optical encryption new methods and new technique are born (see Review literature based on the technology [3]).Additionally, as a kind of Fourier transformation of broad sense, fractional fourier transform, fresnel transform etc., linear canonical transform (LCT) can also be used in optical image encryption (see document [4]).

However, in the optical image encryption method based on Double random phase, mostly there are the following problems：

1) key is the random phase mask of picture size, therefore, key management and transmission are inconvenient (see document [5])；

2) because the inconvenience of random phase mask updates, therefore, encryption system is easily attacked by chosen -plain attact and known-plaintext Hit (see document [6] and [7]).

List of references：

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

[2] P.R é fr é gier and B.Javidi, Optical image encryption based on input Planeand 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] A.Sahin, H.Ozaktas, D.Mendlovic, Optical implementations of twodimensional fractional Fourier transforms and linear canonical Transformswith arbitrary parameters.Applied Optics, 1998,37：2130-41

[5] S.Yuan, Y.Xin, M.Liu, S.Yao, and X.Sun, An improved method to enhance The security of double random-phaseencoding 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-randomphase 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- plaintextheuristic attack on the Fourier plane encryption Algorithm.Opt.Express, 2006,14：3181-3186.

The content of the invention

To overcome the deficiencies in the prior art, it is contemplated that proposing image encryption method, realization is effective against known-plaintext Attack and chosen -plain attact, and cause key management and transmission to become more convenient, safety is further ensured, tool There are good violence attack, statistical attack, attacked by noise and shearing attack ability.The technical solution used in the present invention is, base In the optical image encryption method of LCT and Logistic chaos, step is as follows：

1) construction of two dimension Logistic chaotic：Two dimension Logistic chaotic system is dimensional Logistic chaos system 2 D extension；

2) generation of chaotic key：Play two pieces of random phase masks of master key effect respectively by different chaos state modulators Two dimension Logistic chaotic system generate, the initial value and control parameter of chaos system is used as master key；Additionally, linear canonical becomes The geometric parameter of (LCT) system is changed as the auxiliary key in encryption process；

3) image encryption based on linear canonical transform LCT and decryption：(1) in ciphering process, image to be encrypted is first First modulated by first piece of chaos random phase masks, then carry out first time linear canonical transform (LCT1), the image after conversion Modulated by second piece of chaos random phase masks again, then carry out second linear canonical transform (LCT2)；(2) in decrypting process In, the image after encryption carries out first the inverse transformation (ILCT2) of second linear canonical transform, then random by second piece of chaos The complex conjugate modulation of phase mask, it is modulated after image carry out the inverse transformation (ILCT1) of first time linear canonical transform again, most Afterwards again by the complex conjugate modulation of first piece of chaos random phase masks.

An example of the invention is comprised the concrete steps that：

(1) construction of two dimension Logistic chaotic：

The mathematic(al) representation of the discrete form of dimensional Logistic chaos system is：

x_n+1=μ x_n(1-x_n) (1)

When control parameter μ ∈ (3.56,4] when, Logistic systems be in chaos state；x_nFor the initial of chaos system Value, x_n+1For the iteration output valve of chaos system；

It is two dimensional form that dimensional Logistic chaos system is expanded, and the mathematic(al) representation of its discrete form is：

When control parameter μ=4, α=0.89, β=0.89, during γ=0.1, two-dimentional Logistic systems are in chaos shape State；x_n, y_nThe respectively initial value of two dimension Logistic chaotic system；x_n+1And y_n+1The respectively output valve of chaos system；

(2) generation of chaotic key：

Two pieces of chaos random phase masks play master key effect in encryption method, linear canonical transform (LCT) system it is several What parameter plays auxiliary key effect.Generate this two pieces of chaos random phases using two dimension Logistic chaotic system with regard to how below Mask describes in detail.

The size for assuming the image to be encrypted is M × N number of pixel, then the size of two pieces of chaos random phase masks is also M × N number of pixel, for the two dimension Logistic chaotic system by two groups of different chaos state modulators so as to iteration (M × N)/2 time Afterwards, two groups of random number sequences X are obtained₁={ 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)/2With y "₁, y "₂..., y "_(M×N)/2For the output valve of chaos system, by this two groups Random number sequence is integrated into respectively form Z of two two-dimensional matrixs₁={ z '_{M, n}| m=1,2 ..., M；N=1,2 ..., N } and Z₂= {z″_{M, n}| m=1,2 ..., M；N=1,2 ..., N }, wherein z '_{M, n}With z "_{M, n}For the element of two-dimensional matrix, m, n are matrix element Coordinate.Two pieces of chaos random phase masks can be then obtained, its mathematic(al) representation is respectively C₁(x₁, y₁)=exp (j2 π z '_{M, n}) And C₂(x₂, y₂)=exp (j2 π z "_{M, n})；Wherein, (x₁, y₁) and (x₂, y₂) it is respectively position residing for two pieces of chaos random phase masks The coordinate put, j is imaginary unit, and π is pi；

(3) image encryption based on linear canonical transform (LCT) and decryption：

1) in ciphering process, image U to be encrypted₀(x₀, y₀) modulated by first piece of chaos random phase masks first, Then first time linear canonical transform LCT1 is carried out, the image after conversion is modulated again by second piece of chaos random phase masks, so After carry out second linear canonical transform LCT2, Jing modulate twice and twice convert after can be obtained by encrypted image U (x, y)：

Wherein, (x₀, y₀) and (x, y) be respectively the coordinate of original image and encrypted image present position；LCT_{α, beta, gamma}{ } table Show that transformation parameter is α, the linear canonical transform of beta, gamma, its form is as follows：

In formula (3), α₁, β₁, γ₁And α₂, β₂, γ₂There is respectively following form：

Wherein, λ is the wavelength of Object light wave, and f is the focal length of lens；d₁, d₂, d₃, d₄For the geometry of linear canonical transform system Parameter；

2) in decrypting process, image U (x, y) after encryption carries out first the inverse transformation of second linear canonical transform ILCT2, then by the complex conjugate modulation of second piece of chaos random phase masks, it is modulated after image carry out the first sublinear again The inverse transformation ILCT1 of contact transformation, finally again by the complex conjugate modulation of first piece of chaos random phase masks, after being decrypted Image

Wherein, * is complex conjugate operator.

Of the invention the characteristics of and beneficial effect are：

In the optical image encryption method that the present invention is provided, the use of chaotic key so that this encryption method can be effective Opposing known plain text attack and chosen -plain attact, and cause key management and transmission to become more convenient.Linear canonical transform (LCT) geometric parameter of system is used as the auxiliary key in encryption process so that the safety of this encryption method is entered The guarantee of one step.

Description of the drawings：

Fig. 1 encryption process schematic diagrams.In figure：

The ciphering process schematic diagram of a optical image encryption method that () provides for the present invention；

The decrypting process schematic diagram of b optical image encryption method that () provides for the present invention；

Fig. 2 encryption and decryption image comparison figures.In figure：

A () is original image to be encrypted；

B () is the image of this method encryption；

C () is decrypted image when all keys are correct；

Decrypted image comparison diagram under Fig. 3 different situations.In figure：

A () is the initial value x of the two dimension Logistic chaotic system for controlling second piece of random phase masks₂Mistake, other are close Decrypted image when key is correct；

B () is the initial value y of the two dimension Logistic chaotic system for controlling second piece of random phase masks₂Mistake, other are close Decrypted image when key is correct；

C () is control parameter μ of the two dimension Logistic chaotic system for controlling second piece of random phase masks₂Mistake, its Decrypted image when its key is correct；

D () is control parameter α of the two dimension Logistic chaotic system for controlling second piece of random phase masks₂Mistake, its Decrypted image when its key is correct；

E () is control parameter β of the two dimension Logistic chaotic system for controlling second piece of random phase masks₂Mistake, its Decrypted image when its key is correct；

F () is control parameter γ of the two dimension Logistic chaotic system for controlling second piece of random phase masks₂Mistake, its Decrypted image when its key is correct；

G () is the geometric parameter d of linear canonical transform (LCT) system₁Mistake, decryption figure when other keys are correct Picture；

H () is the geometric parameter d of linear canonical transform (LCT) system₂Mistake, decryption figure when other keys are correct Picture；

I () is the geometric parameter d of linear canonical transform (LCT) system₃Mistake, decryption figure when other keys are correct Picture；

J () is the geometric parameter d of linear canonical transform (LCT) system₄Mistake, decryption figure when other keys are correct Picture；

Decrypted image comparison diagram under Fig. 4 difference deletion conditions.

A () is that the image for obtaining is decrypted from the encryption figure of 12.5% information of disappearance；

B () is that the image for obtaining is decrypted from the encryption figure of 25% information of disappearance；

C () is that the image for obtaining is decrypted from the encryption figure of 50% information of disappearance；

Fig. 5 difference noise like decrypted image comparison diagrams.In figure：

A () is that the image for obtaining is decrypted from the encryption figure containing 10% Gaussian noise；

B () is that the image for obtaining is decrypted from the encryption figure containing 10% salt-pepper noise；

C () is that the image for obtaining is decrypted from the encryption figure containing 10% speckle noise；

In accompanying drawing, the list of parts representated by each label is as follows：

CRPM₁：First piece of chaos random phase masks；CRPM₂：Second piece of chaos random phase masks；CRPM₁*：First The complex conjugate of block chaos random phase masks；CRPM₂*：The complex conjugate of second piece of chaos random phase masks；L₁：Lens；L₂：Thoroughly Mirror.

Specific embodiment

The invention provides a kind of optical image encryption for being based on linear canonical transform (LCT) and two dimension Logistic chaotic Method.The optical image encryption method that the present invention is provided by two dimension Logistic chaotic construction, the generation of chaotic key is based on The image encryption and decryption composition of linear canonical transform (LCT).The use of chaotic key so that this encryption method can be supported effectively Anti- known plain text attack and chosen -plain attact, and cause key management and transmission to become more convenient.Linear canonical transform (LCT) geometric parameter of system is used as the auxiliary key in encryption process so that the safety of this encryption method is entered The guarantee of one step.Many experiments show that this encryption method has good violence attack, statistical attack, attacked by noise and cuts Cut attacking ability.It is described below：

1) construction of two dimension Logistic chaotic：Two dimension Logistic chaotic system is dimensional Logistic chaos system 2 D extension；Compared to dimensional Logistic chaos system, two dimension Logistic chaotic system has bigger parameter space, more Good pseudo-randomness and more random number sequences can be produced.

2) generation of chaotic key：Play two pieces of random phase masks of master key effect respectively by different chaos state modulators Two dimension Logistic chaotic system generate, the initial value and control parameter of chaos system is used as master key；Additionally, linear canonical becomes The geometric parameter of (LCT) system is changed as the auxiliary key in encryption process.Because key updating is convenient in encryption process, Therefore, this encryption method can be effective against known plain text attack and chosen -plain attact；Additionally, key management and transmission are also more For convenience.

3) image encryption based on linear canonical transform (LCT) and decryption：(1) in ciphering process, image to be encrypted Modulated by first piece of chaos random phase masks first, then carry out first time linear canonical transform (LCT1), the figure after conversion , by second piece of chaos random phase masks modulation, second linear canonical transform (LCT2) is then carried out as again；(2) decrypted Cheng Zhong, the image after encryption carries out first the inverse transformation (ILCT2) of second linear canonical transform, then by second piece of chaos with The complex conjugate modulation of machine phase mask, it is modulated after image carry out the inverse transformation (ILCT1) of first time linear canonical transform again, It is last to be modulated by the complex conjugate of first piece of chaos random phase masks again.

To make the object, technical solutions and advantages of the present invention clearer, below further is made to embodiment of the present invention Ground is described in detail.

Embodiment 1

Based on linear canonical transform (LCT) and the optical image encryption method of two dimension Logistic chaotic, it adds solution to one kind Close process schematic as shown in figure 1, encryption method by two dimension Logistic chaotic construction, the generation of chaotic key, based on line Property contact transformation (LCT) image encryption and decryption composition.

(1) construction of two dimension Logistic chaotic：

In the encryption method that the present invention is provided, two dimension Logistic chaotic system is the two of dimensional Logistic chaos system Dimension is expanded；Compared to dimensional Logistic chaos system, two dimension Logistic chaotic system has bigger parameter space, more preferably Pseudo-randomness and more random number sequences can be produced.

(2) generation of chaotic key：

In the encryption method that the present invention is provided, two pieces of random phase masks for playing master key effect are joined respectively by different chaos The two dimension Logistic chaotic system of numerical control is generated, and the initial value and control parameter of chaos system is used as master key；Additionally, linear The geometric parameter of contact transformation (LCT) system is used as the auxiliary key in encryption process.Because key is more in encryption process It is new convenient, therefore, this encryption method can be effective against known plain text attack and chosen -plain attact；Additionally, key management and Transmission is also more convenient.

(3) image encryption based on linear canonical transform (LCT) and decryption：

1) in ciphering process, image to be encrypted is modulated first by first piece of chaos random phase masks, is then carried out First time linear canonical transform (LCT1), the image after conversion is modulated again by second piece of chaos random phase masks, is then carried out Second linear canonical transform (LCT2)；2) in decrypting process, the image after encryption carries out first the change of the second sublinear canonical The inverse transformation (ILCT2) changed, then by the complex conjugate modulation of second piece of chaos random phase masks, it is modulated after image enter again The inverse transformation (ILCT1) of row first time linear canonical transform, finally again by the complex conjugate tune of first piece of chaos random phase masks System.

In sum, the use of chaotic key so that this encryption method can be effective against known plain text attack and selection Plaintext attack, and cause key management and transmission to become more convenient.The geometric parameter conduct of linear canonical transform (LCT) system Auxiliary key in encryption process so that the safety of this encryption method has obtained further guarantee.

Embodiment 2

The scheme in embodiment 1 is introduced in detail with reference to Fig. 1, design principle, it is described below：

Based on linear canonical transform (LCT) and the optical image encryption method of two dimension Logistic chaotic, it adds solution to one kind Close process schematic is as shown in Figure 1.Encryption method by two dimension Logistic chaotic construction, the generation of chaotic key, based on line Property contact transformation (LCT) image encryption and decryption composition.The specific embodiment of this three part is given respectively in detail below Description.

(1) construction of two dimension Logistic chaotic：

The mathematic(al) representation of the discrete form of dimensional Logistic chaos system is：

x_n+1=μ x_n(1-x_n) (1)

When control parameter μ ∈ (3.56,4] when, Logistic systems be in chaos state；x_nFor the initial of chaos system Value, x_n+1For the iteration output valve of chaos system.

It is two dimensional form that dimensional Logistic chaos system is expanded, and the mathematic(al) representation of its discrete form is：

When control parameter μ=4, α=0.89, β=0.89, during γ=0.1, two-dimentional Logistic systems are in chaos shape State；x_n, y_nThe respectively initial value of two dimension Logistic chaotic system；x_n+1And y_n+1The respectively output valve of chaos system.It is worth note Meaning, when above-mentioned control parameter takes other values, two-dimentional Logistic systems are likely in mixed degree state.

(2) generation of chaotic key：

Two pieces of chaos random phase masks play master key effect in encryption method, linear canonical transform (LCT) system it is several What parameter plays auxiliary key effect.Generate this two pieces of chaos random phases using two dimension Logistic chaotic system with regard to how below Mask describes in detail.

The size for assuming the image to be encrypted is M × N number of pixel, then the size of two pieces of chaos random phase masks is also M × N number of pixel, for the two dimension Logistic chaotic system by two groups of different chaos state modulators so as to iteration (M × N)/2 time Afterwards, two groups of random number sequences X are obtained₁={ 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)/2With y "₁, y "₂..., y "_(M×N)/2For the output valve of chaos system.By this two groups Random number sequence is integrated into respectively form Z of two two-dimensional matrixs₁={ z '_{M, n}| m=1,2 ..., M；N=1,2 ..., N } and Z₂= {z″_{M, n}| m=1,2 ..., M；N=1,2 ..., N }, wherein z '_{M, n}With z "_{M, n}For the element of two-dimensional matrix, m, n are matrix element Coordinate.Two pieces of chaos random phase masks can be then obtained, its mathematic(al) representation is respectively C₁(x₁, y₁)=exp (j2 π z '_{M, n}) And C₂(x₂, y₂)=exp (j2 π z "_{M, n})；Wherein, (x₁, y₁) and (x₂, y₂) it is respectively position residing for two pieces of chaos random phase masks The coordinate put, j is imaginary unit, and π is pi.Because chaos random phase mask is by the initial value and control ginseng of chaos system Number controlling, therefore, the master key of the initial value and control parameter of chaos system as encryption system.Due to master key and auxiliary Key is all some numerals, therefore, managing and transmit these numerals will become very convenient；Additionally, updating in encryption process These numerals will also become very convenient.

(3) image encryption based on linear canonical transform (LCT) and decryption：

1) in ciphering process, image U to be encrypted₀(x₀, y₀) modulated by first piece of chaos random phase masks first, Then first time linear canonical transform LCT1 is carried out, the image after conversion is modulated again by second piece of chaos random phase masks, so After carry out second linear canonical transform LCT2, Jing modulate twice and twice convert after can be obtained by encrypted image U (x, y)：

Wherein, (x₀, y₀) and (x, y) be respectively the coordinate of original image and encrypted image present position；LCT_{α, beta, gamma}{ } table Show that transformation parameter is α, the linear canonical transform of beta, gamma, its form is as follows：

In formula (3), α₁, β₁, γ₁And α₂, β₂, γ₂There is respectively following form：

Wherein, λ is the wavelength of Object light wave, and f is the focal length of lens；d₁, d₂, d₃, d₄For the geometry of linear canonical transform system Parameter.

2) in decrypting process, image U (x, y) after encryption carries out first the inverse transformation of second linear canonical transform ILCT2, then by the complex conjugate modulation of second piece of chaos random phase masks, it is modulated after image carry out the first sublinear again The inverse transformation ILCT1 of contact transformation, finally again by the complex conjugate modulation of first piece of chaos random phase masks, after being decrypted Image

Wherein, * is complex conjugate operator.

In sum, the use of chaotic key so that this encryption method can be effective against known plain text attack and selection Plaintext attack, and cause key management and transmission to become more convenient.The geometric parameter conduct of linear canonical transform (LCT) system Auxiliary key in encryption process so that the safety of this encryption method has obtained further guarantee.

Embodiment 3

Feasibility checking is carried out to the scheme in embodiment 1 and 2 with reference to specific accompanying drawing, it is described below：

After the encryption method for implementing offer using the present invention is encrypted to piece image (as shown in Fig. 2 (a)), obtain Shown in encrypted image such as Fig. 2 (b).

By Fig. 2 (b) as can be seen that any information of original image is all hidden.When all keys are correct, decrypt Image such as Fig. 2 (c) shown in.By Fig. 2 (c) as can be seen that original image can be reduced completely.Illustrate using the system to ash The encryption and decryption of degree image is successful.

Additionally, when some wrong cipher key during correct other keys, shown in decrypted result such as Fig. 3 (a) -3 (j).Thus It can be seen that, the safety of the system can be guaranteed.

Fig. 4 (a) -4 (c) is the decrypted image under encryption figure 12.5%, 25% and 50% information state of disappearance.Fig. 5 (a) -5 C () is encryption figure containing the decrypted image in the case of 10% Gaussian noise, salt-pepper noise and speckle noise.As can be seen here, even if Encrypted image lacks a part of information or to a certain extent by sound pollution, and the embodiment of the present invention remains able to decrypt necessarily The original image of quality, demonstrates the feasibility of the system, meets the various needs in practical application.

To the model of each device in addition to specified otherwise is done, the model of other devices is not limited the embodiment of the present invention, As long as the device of above-mentioned functions can be completed.

It will be appreciated by those skilled in the art that accompanying drawing is the schematic diagram of a preferred embodiment, the embodiments of the present invention Sequence number is for illustration only, does not represent the quality of embodiment.

The foregoing is only presently preferred embodiments of the present invention, not to limit the present invention, all spirit in the present invention and Within principle, any modification, equivalent substitution and improvements made etc. should be included within the scope of the present invention.

Claims (2)

1. a kind of optical image encryption method based on LCT and Logistic chaos, is characterized in that, step is as follows：

1) construction of two dimension Logistic chaotic：Two dimension Logistic chaotic system is the two dimension of dimensional Logistic chaos system Expand；

2) generation of chaotic key：Play two pieces of random phase masks of master key effect respectively by the two of different chaos state modulators Dimension Logistic chaos systems are generated, and the initial value and control parameter of chaos system is used as master key；Additionally, linear canonical transform (LCT) geometric parameter of system is used as the auxiliary key in encryption process；

3) image encryption based on linear canonical transform LCT and decryption：(1) in ciphering process, image to be encrypted first by The modulation of first piece of chaos random phase masks, then carries out first time linear canonical transform (LCT1), the image after conversion again by Second piece of chaos random phase masks modulation, then carries out second linear canonical transform (LCT2)；(2) in decrypting process, Image after encryption carries out first the inverse transformation (ILCT2) of second linear canonical transform, then by the random phase of second piece of chaos The complex conjugate modulation of bit mask, it is modulated after image carry out the inverse transformation (ILCT1) of first time linear canonical transform again, finally Again by the complex conjugate modulation of first piece of chaos random phase masks.

2. the optical image encryption method based on LCT and Logistic chaos as claimed in claim 1, is characterized in that, one Example is comprised the concrete steps that：

(1) construction of two dimension Logistic chaotic：

The mathematic(al) representation of the discrete form of dimensional Logistic chaos system is：

x_n+1=μ x_n(1-x_n) (1)

When control parameter μ ∈ (3.56,4] when, Logistic systems be in chaos state；x_nFor the initial value of chaos system, x_n+1 For the iteration output valve of chaos system；

It is two dimensional form that dimensional Logistic chaos system is expanded, and the mathematic(al) representation of its discrete form is：

$\left\{\begin{array}{c}{x}_{n+1}={\mathrm{\μ\αx}}_n(1-x_n)+{\mathrm{\γy}}_n\\ y_{n+1}={\mathrm{\μ\βy}}_n(1-y_n)+{\mathrm{\γx}}_n\end{array}\right.---\left(2\right)$

When control parameter μ=4, α=0.89, β=0.89, during γ=0.1, two-dimentional Logistic systems are in chaos state；x_n, y_nThe respectively initial value of two dimension Logistic chaotic system；x_n+1And y_n+1The respectively output valve of chaos system；

(2) generation of chaotic key：

Two pieces of chaos random phase masks play master key effect, the geometric parameters of linear canonical transform (LCT) system in encryption method It is several to play auxiliary key effect.Generate this two pieces of chaos random phase masks using two dimension Logistic chaotic system with regard to how below Describe in detail.

The size for assuming the image to be encrypted is M × N number of pixel, then the size of two pieces of chaos random phase masks is also M × N number of Pixel, for the two dimension Logistic chaotic system by two groups of different chaos state modulators so as to after iteration (M × N)/2 time, obtain To 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)/2With y "₁,y″₂,…,y″_(M×N)/2It is random by this two groups for the output valve of chaos system Number Sequence is integrated into respectively form Z of two two-dimensional matrixs₁={ z '_m,n| m=1,2 ..., M；N=1,2 ..., N } and Z₂= {z″_m,n| m=1,2 ..., M；N=1,2 ..., N }, wherein z '_m,nWith z "_m,nFor the element of two-dimensional matrix, m, n are matrix element Coordinate.Two pieces of chaos random phase masks can be then obtained, its mathematic(al) representation is respectively C₁(x₁,y₁)=exp (j2 π z '_m,n) And C₂(x₂,y₂)=exp (j2 π z "_m,n)；Wherein, (x₁,y₁) and (x₂,y₂) it is respectively position residing for two pieces of chaos random phase masks The coordinate put, j is imaginary unit, and π is pi；

(3) image encryption based on linear canonical transform (LCT) and decryption：

1) in ciphering process, image U to be encrypted₀(x₀,y₀) modulated by first piece of chaos random phase masks first, then First time linear canonical transform LCT1 is carried out, the image after conversion is modulated again by second piece of chaos random phase masks, Ran Houjin Second linear canonical transform LCT2 of row, Jing after modulating twice and convert twice encrypted image U (x, y) is can be obtained by：

$U(x,y)={\mathrm{LCT}}_{{\mathrm{\α}}_2,{\mathrm{\β}}_2,{\mathrm{\γ}}_2}\left\{{\mathrm{LCT}}_{{\mathrm{\α}}_1,{\mathrm{\β}}_1,{\mathrm{\γ}}_1}\right\{U_0(x_0,y_0)C_1(x_1,y_1)\left\}{C}_2(x_2,y_2)\right\}---\left(3\right)$

Wherein, (x₀,y₀) and (x, y) be respectively the coordinate of original image and encrypted image present position；LCT_α,β,γ{ } represents and becomes It is α to change parameter, and the linear canonical transform of beta, gamma, its form is as follows：

$\begin{array}{c}U\left(x,y\right)={\mathrm{LCT}}_{\mathrm{\α},\mathrm{\β},\mathrm{\γ}}\left\{U_0\left(x_0,y_0\right)\right\}\left(x,y\right)\\ =\sqrt{\mathrm{\β}}\·\exp \left(\frac{-j\mathrm{\π}}{4}\right)\underset{-\mathrm{\∞}}{\overset{+\mathrm{\∞}}{\∫}}\underset{-\mathrm{\∞}}{\overset{+\mathrm{\∞}}{\∫}}{U}_0\left(x_0,y_0\right)\exp \left\{\mathrm{\α}\left(x_0^2+y_0^2\right)-2\mathrm{\β}\left(x_{0}x+y_{0}y\right)+\mathrm{\γ}\left(x^2+y^2\right)\right\}{\mathrm{dx}}_0{\mathrm{dy}}_0\end{array}---\left(4\right)$

In formula (3), α₁,β₁,γ₁And α₂,β₂,γ₂There is respectively following form：

${\mathrm{\α}}_1=\frac{d_1-f}{\mathrm{\λ}\[f(d_1+d_2)-d_1{d}_2\]}$

${\mathrm{\β}}_1=\frac{f}{\mathrm{\λ}\[f(d_1+d_2)-d_1{d}_2\]}$

${\mathrm{\γ}}_1=\frac{d_2-f}{\mathrm{\λ}\[f(d_1+d_2)-d_1{d}_2\]}$

${\mathrm{\α}}_2=\frac{d_3-f}{\mathrm{\λ}\[f(d_3+d_4)-d_3{d}_4\]}$

${\mathrm{\β}}_2=\frac{f}{\mathrm{\λ}\[f(d_3+d_4)-d_3{d}_4\]}$

${\mathrm{\γ}}_2=\frac{d_4-f}{\mathrm{\λ}\[f(d_3+d_4)-d_3{d}_4\]}---\left(5\right)$

Wherein, λ is the wavelength of Object light wave, and f is the focal length of lens；d₁,d₂,d₃,d₄For the geometric parameter of linear canonical transform system；

2) in decrypting process, image U (x, y) after encryption carries out first the inverse transformation ILCT2 of second linear canonical transform, Then by the complex conjugate modulation of second piece of chaos random phase masks, it is modulated after image carry out the change of the first sublinear canonical again The inverse transformation ILCT1 for changing, finally again by the complex conjugate modulation of first piece of chaos random phase masks, the image after being decrypted

Wherein, * is complex conjugate operator.