CN112765623B - Optical multi-image authentication and encryption method based on phase recovery algorithm - Google Patents
Optical multi-image authentication and encryption method based on phase recovery algorithm Download PDFInfo
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Abstract
The invention discloses an optical multi-image authentication and encryption method based on a phase recovery algorithm, which comprises the following steps of S1: obtaining a sparse phase by using a binary image by utilizing a Fresnel domain phase recovery algorithm, and then dividing the sparse phase into phases P for image authentication 1 And phase P 2 (ii) a S2, multi-image encryption: phase P of sparse phase 1 As a fixed constraint, the phase P of the sparse phase is 2 As sparse constraint, encrypting the binary images to enable sparse phases to be embedded into a key, and encrypting each binary image to obtain two pure phase masks, wherein each pure phase mask comprises sparse phase data; s3, decryption: and finishing decryption after verifying the sparse phase data. The invention can encrypt a plurality of images, improves the encryption capacity and ensures the quality and the safety of the decrypted images.
Description
Technical Field
The invention relates to the technical field of information security, in particular to an optical multi-image authentication and encryption method based on a phase recovery algorithm.
Background
Information encryption technology based on optical principles has attracted much attention from researchers with its unique advantages over the past few decades. Refregion and Javidi of connecticut university first proposed a Dual Random Phase Encryption (DRPE) technique, which has attracted researchers' great interest in the field of optical image encryption and has been extended to the fresnel domain, the fractional fourier domain, and the Gyrator domain.
Since the DRPE system belongs to a linear system, the dependency between plaintext, secret key and ciphertext is very simple, and it has been found that it is vulnerable to attack. In order to improve security, Qin proposed in 2010 an encryption system based on phase truncated fourier transform. However, the cryptographic system has also been found to be vulnerable to Phase Recovery Algorithm (PRA) based attacks, and the resulting phase-only masks (POMs) present a profile problem. To solve these problems, the recently proposed asymmetric image encryption based on the Yang-Gu hybrid phase recovery algorithm, which involves a two-step iterative encryption process, requires high computational cost to obtain a high-quality decrypted image. In 2008, Zhang proposed a method for encrypting an image into two phase-only masks based on the principle of optical interference. However, it has also been found that there are profile problems with the interference-based encryption schemes. Subsequently, some new methods are proposed to further improve the security of the phase-only mask. Meanwhile, some image encryption methods are applied to multi-image encryption in conjunction with spatial multiplexing techniques. For example, Alfalou et al perform a specific spectral multiplexing, enabling simultaneous encryption and compression of multiple images, but require additional filtering during the demultiplexing process. In recent years, a highly secure optical authentication method has been proposed based on an optical or digital image processing technique such as DRPE. In 2011, Perez-Cabre proposes a cryptographic system based on photon counting imaging, and is widely applied to information authentication in recent years. Another approach is to use a sparse representation algorithm to generate a phase-only mask for authentication.
The optical image encryption based on the phase recovery algorithm adopts a numerical calculation mode to encrypt an image into a plurality of different phase masks, an optical means is adopted for decryption during decryption, the hardware cost required by the optical decryption is high, the flexibility is low, and the requirement on the spatial arrangement precision of a phase template is very high, so that the optical image encryption is still greatly limited in practical application at present; the optical multi-image encryption by utilizing the multiplexing technology has the problems of small encryption capacity, poor decryption quality and the like.
Disclosure of Invention
The invention aims to provide an optical multi-image authentication and encryption method based on a phase recovery algorithm. The invention can encrypt a plurality of images, improves the encryption capacity and ensures the quality and the safety of the decrypted images.
In order to solve the technical problems, the technical scheme provided by the invention is as follows: the optical multi-image authentication and encryption method based on the phase recovery algorithm comprises the following steps:
s1, sparse phase acquisition: obtaining a sparse phase by using a binary image by utilizing a Fresnel domain phase recovery algorithm, and then dividing the sparse phase into phases P for image authentication 1 And phase P 2 ；
S2, multi-image encryption: phase P of sparse phase 1 As a fixed constraint, the phase P of the sparse phase is 2 As sparse constraint, encrypting the binary images to enable sparse phases to be embedded into a key, and encrypting each binary image to obtain two pure phase masks, wherein each pure phase mask comprises sparse phase data;
s3, decryption: and finishing decryption after verifying the sparse phase data.
In the above optical multi-image authentication and encryption method based on the phase recovery algorithm, in step S1, three binary images g are used 1 ，g 2 And g 3 Generating a sparse phase for authentication, comprising the steps of:
(i) for the m (m is 1,2,3) th binary image g m Performing Fresnel domain double random phase encoding, wherein the corresponding amplitude constraint of the output surface is as follows:
wherein U is m The amplitude of the mth binary image after the double random phase coding of the Fresnel domain; FrT represents the Fresnel transformation; PT { } represents taking amplitude operation, namely phase information of complex amplitude is removed; z is a radical of 1 And z 2 Respectively represent diffraction distances; g m (m ═ 1,2,3) stands for generating the sparse phase S m Binary image of R 1 And R 2 Are two random phases generated by a computer, each specifically denoted as exp [ i γ ] y 1 (x,y)]And exp [ i γ ] 2 (u,v)]Where (x, y) and (u, v) denote the coordinates of the spatial domain and the Fresnel domain, respectively, γ 1 (x, y) and γ 2 (u, v) represents two in the interval [0, 2 π]Random matrices with uniform probability distribution and statistical independence thereon;
then, a phase function of the output surface is obtained by utilizing a Fresnel domain phase recovery algorithm
Represents a distribution in [0, 2 π]Wherein the superscript n represents the nth iteration and the subscript m represents the mth binary image;
(ii) when the nth iteration operation is performed, the amplitude function corresponding to the random phase with the initial value generated by the computer is performed
And R 1 Is taken as the distance z 1 Fresnel transformation of (1), then multiplication by R 2 As a distance z 2 Obtaining a complex amplitude function by Fresnel diffraction
Namely, it is
Wherein
Is a computer-generated random amplitude having a value in the interval 0,1]And continuously updating in the iterative process;
(iii) then the amplitude U is measured m And phase
Is taken as the distance z 2 Inverse Fresnel transformation of (1), then multiplying by R 2 Has a complex conjugate as a distance z 1 Inverse Fresnel transformation of (2) to obtain a new complex amplitude function
Represents a distribution in [0, 2 π]Of phase (i) i
FrT therein -1 Representing the inverse Fresnel transformation, and the conjugate operation, the amplitude of which is the next iteration
Is used for replacing
(iv) (iv) repeating (ii) and (iii) up to g m And
the value of the correlation coefficient between the two reaches a set threshold value, the iteration stops, and the mathematical expression of the correlation coefficient is
Wherein E represents a mathematical expectation;
assuming the iteration stops at the Nth time, the sparse phase S m From phase function
Random extraction is carried out, and then the sparse phase S is extracted m The method is divided into two parts:
namely S 1 +S 2 Is kept constant for S 1 +S 2 Embedding random phase R pixel by pixel at position with middle pixel value as zero 3 To obtain a phase P 1 Wherein the phase function S is used to avoid crosstalk 1 ，S 2 Should not overlap with each other; s 3 Non-zero pixels remain unchanged, for S 3 Embedding random phase R pixel by pixel at position with middle pixel value as zero 4 To obtain a phase P 2 (ii) a In the formula, R 3 And R 4 Representing an independently distributed random phase,
representing data embedding.
In the foregoing optical multi-image authentication and encryption method based on the phase recovery algorithm, the specific process of step S2 is as follows:
assuming that K images need to be encrypted, the K (K is 1,2,3, … K) th image is f k ；
(a) Image f by using sparse constraint-based phase recovery algorithm k Encrypting, and performing j iteration operation to obtain phase P 1 As a fixed constraint, phase P 2 As a result of the sparsity constraint,
for phase masking, in the first iteration
Generating a desired phase in a jth iteration
Then the phase P is adjusted 1 And
added up to make a distance z 1 Fresnel transformation of the amplitude
And phase
Namely, it is
Wherein PR represents taking phase operation;
(b) random phase generated by computer
For the first iteration only, the required phase can be generated in the j-th iteration
Will amplitude of vibration
And phase
Multiplied and then made a distance z 2 The fresnel diffraction of (a) is performed,
continuously updated in the next iteration; obtaining amplitude after transformation
And phase
Namely, it is
(c) To-be-encrypted graphImage f k And phase
Multiply by and make a distance z 2 Inverse Fresnel diffraction, and transforming to obtain amplitude
And phase
Namely, it is
Wherein in (j +1) iterations, the phase
Is used for updating
(d) Will phase
Multiplied by the amplitude
Then proceeding for a distance z 1 Inverse Fresnel transformation of (2), calculating the resulting complex amplitude
Comprises the following steps:
(e) updating phase
Is calculated by the following formula, namely
(f) Repeating steps (a) - (e) until
And f k When the value of the correlation coefficient reaches a preset threshold value, the iteration is stopped;
assuming that the iteration stops at the J-th time, the resulting phase P 2k Is composed of
Decryption key utilization for kth pictures
And
is obtained by calculation, i.e.
For a plurality of images, all phases P are multiplexed 2k Adding to obtain the ciphertext and the key, i.e.
Wherein, phase P' 2 For authentication and decryption, amplitude E is used as the ciphertext.
In the foregoing optical multi-image authentication and encryption method based on the phase recovery algorithm, the specific process of step S3 is as follows:
due to the phase P 1 And P' 2 The phase information containing the binary image ciphertext needs to be decrypted for P 1 And P' 2 Authentication is performed with a corresponding decrypted amplitude distribution of
By using a non-linear correlation algorithm, the amplitudes g' are respectively compared with the original image g 1 And g 2 Comparison, g' with the original image g 3 Comparison, i.e.
NC1＝|IFT{|FT(g')·[FT(g 1 )] * | ω-1 ·FT(g')·[FT(g 1 )] * }| 2 ；
NC2＝|IFT{|FT(g')·[FT(g 2 )] * | ω-1 ·FT(g')·[FT(g 2 )] * }| 2 ；
NC3＝|IFT{|FT(g″)·[FT(g 3 )] * | ω-1 ·FT(g″)·[FT(g 3 )] * }| 2 ；
Wherein FT (-) represents a Fourier transform, IFT (-) represents an inverse Fourier transform, and ω represents a nonlinear intensity coefficient;
phase P of passing authentication 1 And P' 2 By using the correct decryption key D k Setting the same diffraction distance z as the encryption process 1 And z 2 Obtaining a corresponding decrypted image, i.e.
Compared with the prior art, the method has the advantages that the Fresnel domain phase recovery algorithm is utilized, the binary image is used for obtaining the sparse phase, then the sparse phase is divided into two phases for image authentication, one phase serves as fixed constraint, the other phase serves as sparse constraint, the binary image is encrypted, the sparse phase is embedded into the key, two pure phase masks are obtained after each binary image is encrypted, the pure phase masks contain sparse phase data, therefore, the key needs to be authenticated before decryption, and the safety of the system is improved. Furthermore, the invention adopts the multiplexing technology to realize the encryption of a plurality of images, improves the encryption capacity and ensures the quality of the decrypted images, and the iterative algorithm of the invention has high convergence speed, and the decryption key is calculated by the numerical value in the iterative process, thereby further improving the security of the image encryption.
Drawings
FIG. 1 is a schematic diagram for generating a verifiable phase mask.
Fig. 2 is a flow chart of an encryption process.
Fig. 3 is a schematic diagram of the decryption process.
Fig. 4(a) - (c) are binary maps g for generating sparse phases m ；(d)-(f)φ m The phase distribution of (a); (g) - (i) sparse phase S m ；(d)P 1 And P 2 The phase profile of (a).
In FIG. 5, (a) - (d) are images f to be encrypted k ；(e)-(f)P 1 And P' 2 And (4) phase distribution.
The correlation coefficient curves after 50 iterations of (a) - (d) in fig. 6.
The amplitude distributions of (a) - (b) g' and g "in FIG. 7; (c) - (d) g' with g, respectively 1 And g 2 Comparing the authentication results; (e) g' and g 3 And comparing the authentication results.
Decryption Key D in FIGS. 8(a) - (D) k (ii) a (e) - (h) a decrypted image obtained using the correct decryption key.
FIG. 9 is an authentication system security analysis. (a) Is different from g m A binary map of (2); (b) - (c) authentication of graph (a)And (4) obtaining the result.
Fig. 10 is a graph showing the effect of varying the diffraction distance on the decryption result. (a) z is a radical of formula 1 Changing the decryption result after 0.001 m; (b) z is a radical of 1 Changing the decryption result after 0.002 m; (c) z is a radical of 2 Changing the decryption result after 0.01 m; (d) z is a radical of 2 Changing the decryption result after 0.02 m; (e) z is a radical of 1 Change by 0.001m, z 2 Changing the decryption result after 0.01 m; (f) z is a radical of 1 Change by 0.002m, z 2 The decryption result after 0.02m is changed.
Fig. 11 is a ciphertext clipping security analysis. (a) - (e) the ciphertext is cropped by 16% and the corresponding decrypted image; (f) - (j) the ciphertext is cropped by 25% and the corresponding decrypted image. (k) Graph of mean correlation coefficient versus shear ratio.
Fig. 12 is a ciphertext plus noise security analysis. (a) - (e) ciphertext adding gaussian noise with standard deviation of 0.5 and decrypting the image; (f) - (j) the ciphertext adding gaussian noise with a standard deviation of 1 and the decrypted image; (k) a plot of the average correlation coefficient as a function of the gaussian noise standard deviation.
Fig. 13 is a multiplexing capability analysis. (a) The decryption result of the 16 gray-scale images;
fig. 14 is a graph of the average correlation coefficient as a function of the number of encrypted images.
Detailed Description
The present invention will be further described with reference to the following examples and drawings, but the present invention is not limited thereto.
Example (b): the optical multi-image authentication and encryption method based on the phase recovery algorithm comprises the following steps:
s1, sparse phase acquisition: obtaining a sparse phase by using a binary image by utilizing a Fresnel domain phase recovery algorithm, and then dividing the sparse phase into phases P for image authentication 1 And phase P 2 ；
The specific process of step S1 is as follows:
(i) as shown in fig. 1, two independently distributed random phases R 1 、R 2 Are respectively arranged in a space domain and a Fresnel domain when the m-th binary image g m When Fresnel domain double random phase coding is carried out as input, output thereofThe corresponding amplitudes of the faces are:
wherein U is m The amplitude of the mth binary image after the double random phase coding of the Fresnel domain; FrT represents the Fresnel transformation; PT { } represents taking amplitude operation, namely phase information of complex amplitude is removed; z is a radical of 1 And z 2 Respectively represent diffraction distances; g m Representation for generating sparse phase S m Binary image of R 1 And R 2 Is two random phases generated by a computer, and is respectively expressed as exp [ i gamma [ [ gamma ]) 1 (x,y)]And exp [ i γ ] 2 (u,v)]Where (x, y) and (u, v) denote the coordinates of the spatial domain and the Fresnel domain, respectively, γ 1 (x, y) and γ 2 (u, v) represents two in the interval [0, 2 π]Random matrices with uniform probability distribution and statistical independence thereon;
then, a phase function of the output surface is obtained by utilizing a Fresnel domain phase recovery algorithm
Represents a distribution in [0, 2 π]Wherein the superscript n represents the nth iteration and the subscript m represents the mth binary image;
(ii) when the nth iteration operation is performed, the amplitude function corresponding to the random phase with the initial value generated by the computer is performed
And R 1 Is taken as the distance z 1 Fresnel transformation of (1), then multiplication by R 2 As a distance z 2 Obtaining a complex amplitude function by Fresnel diffraction
Namely, it is
Wherein
Is a computer generated random amplitude having a value in the interval 0,1]And continuously updating in the iterative process;
(iii) then the amplitude U is measured m And phase
Is taken as the distance z 2 Inverse Fresnel transformation of (1), then multiplying by R 2 Has a complex conjugate as a distance z 1 Inverse Fresnel transformation of the signal to obtain a new complex amplitude function
Represents a distribution in [0, 2 π]Of phase (i) i
FrT therein -1 Representing the inverse Fresnel transformation, representing the conjugate operation, in the next iteration, the amplitude
Is used for replacing
(iv) (iv) repeating (ii) and (iii) up to g m And
the value of the correlation coefficient between the two reaches a set threshold value, the iteration stops, and the mathematical expression of the correlation coefficient is
Wherein E represents a mathematical expectation;
assuming the iteration stops at the Nth time, the sparse phase S m From phase function
Random extraction is carried out, and then the sparse phase S is extracted m The method is divided into two parts:
namely S 1 +S 2 Is kept constant for S 1 +S 2 Embedding random phase R pixel by pixel at position with middle pixel value as zero 3 To obtain a phase P 1 Wherein the phase function S is used to avoid crosstalk 1 ，S 2 Should not overlap with each other; s 3 Non-zero pixels remain unchanged, for S 3 Embedding random phase R pixel by pixel at position with middle pixel value as zero 4 To obtain a phase P 2 (ii) a In the formula, R 3 And R 4 Representing an independently distributed random phase,
representing data embedding.
S2, multi-image encryption: phase P of sparse phase 1 As a fixed constraint, the phase P of the sparse phase is 2 As sparse constraint, encrypting the binary images to enable sparse phases to be embedded into a key, and encrypting each binary image to obtain two pure phase masks containing sparse phase data; the specific process is as follows:
assuming that K images need to be encrypted, the firstK (K is 1,2,3, … K) is f k ；
(a) As shown in fig. 2, a phase recovery algorithm based on sparse constraint is adopted for the image f k Encrypting, and performing j iteration operation to obtain phase P 1 As a fixed constraint, phase P 2 As a result of the sparsity constraint,
for phase masking, in the first iteration
Generating a desired phase in a jth iteration
Then the phase P is adjusted 1 And
added up to make a distance z 1 Fresnel transformation of the amplitude
And phase
Namely, it is
Wherein PR represents taking phase operation;
(b) random phase generated by computer
For the first iteration only, the required phase can be generated in the j-th iteration
Will amplitude of vibration
And phase
Multiplied and then made a distance z 2 The fresnel diffraction of (a) is performed,
continuously updated in the next iteration; obtaining amplitude after transformation
And phase
Namely, it is
(c) Image f to be encrypted k And phase
Multiply by and make a distance z 2 Inverse Fresnel diffraction, transforming to obtain amplitude
And phase
Namely, it is
Wherein in (j +1) iterations, the phase
Is used for updating
(d) Will phase
Multiplied by the amplitude
Then proceeding for a distance z 1 Inverse Fresnel transformation of (2), calculating the resulting complex amplitude
Comprises the following steps:
(e) updating phase
Is calculated by the following formula, namely
(f) Repeating steps (a) - (e) until
And f k The value of the correlation coefficient between the two reaches the pre-valueSetting a threshold value first, and stopping iteration;
assuming that the iteration stops at the J-th time, the resulting phase P 2k Is composed of
Decryption key utilization for kth pictures
And
is obtained by calculation, i.e.
For a plurality of images, all phases P are multiplexed 2k Adding to obtain the ciphertext and the key, i.e.
Wherein, phase P' 2 For authentication and decryption, amplitude E is used as the ciphertext.
S3, decryption: after verification of the sparse phase data, decryption is completed, as shown in FIG. 3, with phase mask P 1 And P' 2 Ciphertext E and P 'are placed on the input surface' 2 Placing the two coherent light beams on the same input surface, irradiating each input surface by coherent light, modulating the two coherent light beams by a Fresnel domain decryption key after the two coherent light beams are interfered by a beam splitter, and obtaining a distance z 2 After the Fresnel is diffracted, recording the recovered image by the CCD; the specific process is as follows:
due to the phase P 1 And P' 2 IncludedThe phase information of the binary image ciphertext needs to be decrypted for P 1 And P' 2 Authentication is performed with a corresponding decrypted amplitude distribution of
By using a non-linear correlation algorithm, the amplitudes g' are respectively compared with the original image g 1 And g 2 Comparison, g' with the original image g 3 Comparison, i.e.
NC1＝|IFT{|FT(g')·[FT(g 1 )] * | ω-1 ·FT(g')·[FT(g 1 )] * }| 2
NC2＝|IFT{|FT(g')·[FT(g 2 )] * | ω-1 ·FT(g')·[FT(g 2 )] * }| 2
NC3＝|IFT{|FT(g″)·[FT(g 3 )] * | ω-1 ·FT(g″)·[FT(g 3 )] * }| 2 ；
Wherein FT (-) represents a Fourier transform, IFT (-) represents an inverse Fourier transform, and ω represents a nonlinear intensity coefficient;
phase P of passing authentication 1 And P' 2 By using the correct decryption key D k Setting the same diffraction distance z as the encryption process 1 And z 2 Obtaining a corresponding decrypted image, i.e.
The invention will be further explained with reference to specific embodiments and the accompanying drawings.
First, three binary maps with a size of 256 × 256 are selected for generating the sparse phase, as shown in fig. 4(a) - (c), and in the numerical simulation process, the wavelength λ is equal to632.8nm, diffraction distance z 1 ＝0.2m，z 2 0.25 m. Phase distribution phi obtained after iterative operation m As shown in FIGS. 4(d) - (f), for phi respectively m Thinning is carried out, the proportion of extracted pixels is respectively 14%, 16% and 10%, and the obtained sparse phase S m As shown in fig. 4(g) - (i), respectively. Will sparsely phase S m Dividing the phase into two parts, and embedding a random phase into pixel points with zero pixel values pixel by pixel to obtain a phase P 1 And P 2 As shown in fig. 4(j) and 4(k), respectively.
Performing image encryption operation according to fig. 2, selecting four images shown in fig. 5(a) - (d) as images to be encrypted, setting the maximum iteration number J to 50 in an experiment, respectively performing 50 iterations on the four images to be encrypted, wherein correlation coefficients all reach more than 0.98, and the obtained curves of the correlation coefficient values changing along with the iteration number are shown in fig. 6(a) - (d), and obtaining a phase P by using a multiplexing technology 1 And P' 2 As shown in fig. 5(e) and 5 (f).
Before decryption using fig. 3, the phase P needs to be aligned 1 And P' 2 Performing authentication by firstly using Fresnel domain dual random phase coding system to perform phase P 1 And P' 2 As shown in fig. 7(a) and 7(b), it can be seen that no information can be obtained by the naked eye, and then the authentication results obtained by comparing the non-linear correlation coefficient with the original figure are shown in fig. 7(c) - (e). After the phase authentication is passed, the ciphertext is respectively decrypted by using the decryption keys shown in fig. 8(a) - (d), and the decrypted images are obtained as shown in fig. 8(e) - (h), and as can be seen from fig. 8(e) - (h), the decrypted images basically have little difference from the original images, which illustrates that the quality of the decrypted images is ensured by the invention.
The security of the invention is examined below. Firstly, the safety is analyzed by using a frame different from g m The binary image of (a) is shown in fig. 9(a), and the two are compared with the decryption amplitudes g' and g ″ respectively by using nonlinear correlation coefficients, and the obtained authentication results are shown in fig. 9(b) and 9(c), and it can be seen that the obtained authentication results have no obvious peak and the authentication fails. Next, by changing the diffraction distance, the decryption system was investigatedFor the sake of convenience, the system security uses only one encryption map "Boat" as the test image, and first determines the diffraction distance z 1 The resulting decrypted images are shown in fig. 10(a) and 10(b) with a 0.001m and 0.002m change, respectively; then the diffraction distance z 2 The resulting decrypted images are shown in fig. 10(c) and fig. 10(d) with a 0.01m and 0.02m change, respectively; finally, the diffraction distance z is simultaneously varied 1 And z 2 A value of (1), i.e. Δ z 1 ＝0.001m,Δz 2 0.01m and Δ z 1 ＝0.002m,Δz 2 The resulting decrypted image is shown in fig. 10(e) and 10(f), and it can be seen that the diffraction distance z is 0.02m 1 The effect on the decryption result is large, and even an error of 1mm causes the quality of the decrypted image to be poor. The experimental results show that the encryption method provided by the invention has high security.
Further, applicants also tested robustness. The robustness of the invented system is tested by cutting the ciphertext to 16% as shown in fig. 11(a) and then decrypting, the resulting decrypted image is shown in fig. 11(b) - (e), and the correlation coefficients are 0.7319,0.7019,0.7202and 0.7449, respectively. The ciphertext is cropped by 25% as shown in fig. 11(f) and then decrypted, and the resulting decrypted image is shown in fig. 11(g) - (j), in which the correlation coefficients are 0.6071,0.6392,0.6217and 0.6399, respectively. By changing the cut-off ratio to the ciphertext, a curve of the average correlation coefficient as a function of the cut-off ratio is drawn as shown in fig. 11 (k). Next, the robustness of the present invention is tested by adding noise to the ciphertext. First, gaussian noise with a standard deviation of 0.5 is added to the ciphertext as shown in fig. 12(a), the resulting decrypted images are shown in fig. 12(b) - (e), and the correlation coefficients are 0.8312,0.7751,0.8251, and 0.8913, respectively; then, gaussian noise with a standard deviation of 1 is added to the ciphertext as shown in fig. 12(f), and the decrypted images are shown in fig. 12(g) - (j), with correlation coefficients of 0.6530,0.7352,0.7739, and 0.7627, respectively; by changing the standard deviation of the applied Gaussian noise, a curve of the average correlation coefficient changing along with the standard deviation of the Gaussian noise is drawn as shown in FIG. 12(k), and as can be seen from FIG. 12, the method has good robustness.
Finally, testing the multiplexing capability of the present invention, first selecting 16 256 × 256 gray-scale images as test images, the obtained decrypted images are shown in fig. 13(a), the average correlation coefficient value of these decrypted images is 0.8563, and when the average correlation coefficient is set to 0.7 as the standard for evaluating the quality of the decrypted images, the proposed encryption system can encrypt at least 30 gray-scale images. By changing the number of the encrypted images, a curve of the average correlation coefficient changing with the number of the encrypted images is drawn as shown in fig. 14, and the system of the invention has high multiplexing capability as can be seen from fig. 14.
Claims (4)
1. The optical multi-image authentication and encryption method based on the phase recovery algorithm is characterized in that: the method comprises the following steps:
s1, sparse phase acquisition: obtaining a sparse phase by using a binary image by utilizing a Fresnel domain phase recovery algorithm, and then dividing the sparse phase into phases P for image authentication 1 And phase P 2 ；
S2, multi-image encryption: phase P of sparse phase 1 As a fixed constraint, the phase P of the sparse phase is 2 As sparse constraint, encrypting the binary images to enable sparse phases to be embedded into a key, and encrypting each binary image to obtain two pure phase masks, wherein each pure phase mask comprises sparse phase data;
s3, decryption: and finishing decryption after verifying the sparse phase data.
2. The phase recovery algorithm based optical multi-image authentication and encryption method according to claim 1, wherein: in the step S1, three binary images g are used 1 ，g 2 And g 3 Generating a sparse phase for authentication, comprising the steps of:
(i) for the m (m is 1,2,3) th binary image g m Performing Fresnel domain double random phase encoding, wherein the corresponding amplitude constraint of the output surface is as follows:
wherein U is m Is the firstAmplitude of the m binary images after Fresnel domain double random phase coding; FrT represents the Fresnel transformation; PT { } represents taking amplitude operation, namely phase information of complex amplitude is removed; z is a radical of 1 And z 2 Respectively represent diffraction distances; g m (m-1, 2,3) stands for generating a sparse phase S m Binary image of R 1 And R 2 Is two random phases generated by a computer, and is respectively expressed as exp [ i gamma [ [ gamma ]) 1 (x,y)]And exp [ i γ ] 2 (u,v)]Where (x, y) and (u, v) denote the coordinates of the spatial domain and the Fresnel domain, respectively, γ 1 (x, y) and γ 2 (u, v) represents two in the interval [0, 2 π]Random matrices with uniform probability distribution and statistical independence thereon;
then, a phase function of the output surface is obtained by utilizing a Fresnel domain phase recovery algorithm
Represents a distribution in [0, 2 π]Wherein the superscript n represents the nth iteration and the subscript m represents the mth binary image;
(ii) when the nth iteration operation is performed, the amplitude function corresponding to the random phase with the initial value generated by the computer is performed
And R 1 Is taken as the distance z 1 Fresnel transformation of (1), then multiplication by R 2 As a distance z 2 Obtaining a complex amplitude function by Fresnel diffraction
Namely, it is
Wherein
Is a computer-generated random amplitude having a value in the interval 0,1]And continuously updating in the iterative process;
(iii) then the amplitude U is measured m And phase
Is taken as the distance z 2 Inverse Fresnel transformation of (2), then multiplying by R 2 Has a complex conjugate distance z 1 Inverse Fresnel transformation of (2) to obtain a new complex amplitude function
Represents a distribution in [0, 2 π]Of phase (i) i
FrT therein -1 Representing the inverse Fresnel transformation, and the conjugate operation, the amplitude of which is the next iteration
Is used for replacing
(iv) (iv) repeating (ii) and (iii) up to g m And
the value of the correlation coefficient between the two reaches a set threshold value, the iteration stops, and the mathematical expression of the correlation coefficient is
Wherein E represents a mathematical expectation;
assuming the iteration stops at the Nth time, the sparse phase S m From phase function
Random extraction is carried out, and then the sparse phase S is extracted m The method is divided into two parts:
namely S 1 +S 2 Is kept constant for S 1 +S 2 Embedding random phase R pixel by pixel at position with middle pixel value as zero 3 To obtain a phase P 1 Wherein the phase function S is used to avoid crosstalk 1 ，S 2 Should not overlap with each other; s 3 Non-zero pixels remain unchanged, for S 3 Embedding random phase R pixel by pixel at position with middle pixel value as zero 4 To obtain a phase P 2 (ii) a In the formula, R 3 And R 4 Representing an independently distributed random phase,
representing data embedding.
3. The phase recovery algorithm based optical multi-image authentication and encryption method according to claim 2, wherein: the specific process of step S2 is as follows:
assuming that K images need to be encrypted, the K (K is 1,2,3, … K) th image is f k ；
(a) Image f by using sparse constraint-based phase recovery algorithm k To carry outEncrypting, when performing the j-th iteration, the phase P 1 As a fixed constraint, phase P 2 As a result of the sparsity constraint,
for phase masking, in the first iteration
Generating a desired phase in a jth iteration
Then the phase P is adjusted 1 And
added up to make a distance z 1 Fresnel transformation of the amplitude
And phase
Namely, it is
Wherein PR represents taking phase operation;
(b) random phase generated by computer
For the first iteration only, the required phase can be generated in the j-th iteration
Will amplitude of vibration
And phase
Multiplied and then made a distance z 2 The fresnel diffraction of (a) is performed,
continuously updated in the next iteration; obtaining amplitude after transformation
And phase
Namely, it is
(c) Image f to be encrypted k And phase
Multiply by and make a distance z 2 Inverse Fresnel diffraction, transforming to obtain amplitude
And phase
Namely that
Wherein in (j +1) iterations, the phase
Is used for updating
(d) Will phase
Multiplied by the amplitude
Then proceeding for a distance z 1 Inverse Fresnel transformation of (2), calculating the resulting complex amplitude
Comprises the following steps:
(e) updating phase
Is calculated by the following formula, namely
(f) Repeating steps (a) - (e) until
And f k When the value of the correlation coefficient reaches a preset threshold value, the iteration is stopped;
assuming that the iteration stops at the J-th time, the resulting phase P 2k Is composed of
Decryption key utilization for kth pictures
And
is obtained by calculation, i.e.
For a plurality of images, all phases P are multiplexed 2k Adding to obtain the ciphertext and the key, i.e.
Wherein, phase P' 2 For authentication and decryption, amplitude E is used as the ciphertext.
4. The phase recovery algorithm based optical multi-image authentication and encryption method according to claim 3, wherein: the specific process of step S3 is as follows:
due to the phase P 1 And P' 2 The phase information containing the binary image ciphertext needs to be decrypted for P 1 And P' 2 Authentication is performed with a corresponding decrypted amplitude distribution of
By using a non-linear correlation algorithm, the amplitudes g' are respectively compared with the original image g 1 And g 2 Comparison, g' with the original image g 3 Comparison, i.e.
NC1＝|IFT{|FT(g')·[FT(g 1 )] * | ω-1 ·FT(g')·[FT(g 1 )] * }| 2 ；
NC2＝′IFT{|FT(g')·[FT(g 2 )] * | ω-1 ·FT(g')·[FT(g 2 )] * }| 2 ；
NC3＝|IFT{|FT(g″)·[FT(g 3 )] * | ω-1 ·FT(g″)·[FT(g 3 )] * }| 2 ；
Wherein FT (-) represents a Fourier transform, IFT (-) represents an inverse Fourier transform, and ω represents a nonlinear intensity coefficient;
phase P of passing authentication 1 And P' 2 By using the correct decryption key D k Setting the same diffraction distance z as the encryption process 1 And z 2 Obtaining a corresponding decrypted image, i.e.
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