CN103955883A - Asymmetric double-image encryption method based on fractional Fourier domain phase recovery procedure - Google Patents

Abstract

An asymmetric dual-image encryption method based on a phase recovery process in the fractional Fourier domain, including pure phase extraction, phase modulation, and fractional Fourier transform steps. The present invention applies the asymmetric encryption method based on the fractional Fourier domain phase recovery process to double image encryption, the encryption process is different from the decryption process, and the encryption key is different from the decryption key, which solves the problem that the traditional symmetric encryption algorithm is easy to attack and encrypt The key has the same disadvantage as the decryption key; through the attack test, it is proved that the present invention not only has strong resistance to brute force attack, but also has strong resistance to noise and other specific attacks. At the same time, the key space of the present invention is large, which solves the problem of insufficient key space of traditional encryption algorithms. The decryption process of the invention can be realized by optical method, the system is simple and the operation is convenient.

Description

Asymmetric Dual-Image Encryption Method Based on Phase Recovery Process in Fractional Fourier Domain

technical field

The invention belongs to the technical field of virtual optical information encryption methods, and relates to an asymmetric double-image encryption method based on a fractional Fourier domain phase recovery process.

Background technique

With the rapid popularization of the Internet, images, as a very efficient information carrier in contemporary society, have been widely used in various fields, and image encryption has become an important field in the field of information security. Since Refregier and Javidi proposed the classic double random phase encryption (DRPE) technology, in the past ten years, many encryption and authentication based on Fourier transform domain, fractional Fourier transform domain, Fresnel domain, GT domain systems have been proposed. Moreover, Alfalou and Brosseau point out: these techniques can also be used as compression operations. Although most of the published DRPE-based optical encryption systems have excellent parallel and multi-dimensional processing capabilities for signal processing. But we should point out that all these strategies fall under the category of symmetric cryptosystems, where the encryption key is simultaneously used as the decryption key. Several research investigations have shown that these strategies are vulnerable due to their inherently linear properties in mathematics and optical transformations. To resist these attacks, Qin and Peng proposed an asymmetric cryptosystem based on phase truncated Fourier transform (PTFT). The encryption key of this strategy is different from the decryption key, which is avoided by using a nonlinear Linearity properties of cryptosystems.

Recently, since Situ Guohai and Zhao Daomu proposed multi-image encryption technology, multi-image encryption technology based on multiplex technology has received more and more attention in the field of information security. Alfalou and Mansour proposed an encryption scheme with two security layers. The first layer uses the phase recovery process to multiplex and simultaneously encrypt the target image, and the second layer uses a double random phase system to encrypt the image. In subsequent work, Alfalou et al. used discrete cosine transform to simultaneously compress and encrypt multiple images. Wang Xiaogang and Zhao Daomu proposed an all-phase image encryption scheme based on the superposition principle and hologram, which encrypted the real-valued original image into a pure phase function (POF). Deng Xiaopeng and Zhao Daomu proposed a multi-image encryption method using Fourier domain phase recovery process and phase modulation, which completely avoids the influence of crosstalk noise. Hwang Hone-Ene et al proposed a color image encryption scheme based on the improved Gerchberg–Saxton algorithm (MGSA) in the Fresnel domain, which can significantly reduce the interference of crosstalk noise on image information.

As a special case of multi-image encryption, dual-image encryption has also attracted great attention in optical encryption systems. Li Huijuan and Wang Yurong proposed double-image encryption based on iterative Gyrator transformation, using different sets of Gyrator transformation angles to simultaneously encrypt two original images into one ciphertext image. The work done by Li Huijuan and others is to encrypt two images to the real part and imaginary part of a complex function respectively. Wang Xiaogang and Zhao Daomu proposed to encrypt two images to be encrypted into one bright image based on phase recovery algorithm and PTFT. In this method, the encryption key is different from the decryption key. However, Wang Xiaogang and Zhao Daomu devise a special attack that uses a two-step iterative amplitude recovery method. Under this attack, when the encryption key is used as the public key, the encrypted information will be revealed. Subsequently, Wang Xiaogang and Zhao Daomu proposed another double-image encryption technology that is highly resistant to this attack. Later, Li Huijuan and Wang Yurong proposed a dual-image encryption technology based on discrete fractional random transformation and chaotic mapping, which can improve the efficiency of encryption, storage, and conversion. Xiao Di and others proposed dual-image optical encryption based on discrete Chirikov standard mapping. In this method, the two original images are used as the amplitude and phase of the complex function respectively. After using the Chirikov standard mapping to scramble the complex function, the discrete fraction based on chaos The final ciphertext is obtained under the effect of random transformation and two-dimensional chaotic random mask. Although the above algorithm simplifies the encryption process to a certain extent, there are still problems such as low security, small key space, and slow convergence speed.

Contents of the invention

The purpose of the present invention is to propose an asymmetric double-image encryption method based on the phase recovery process in the fractional Fourier domain to solve the problems of low security and vulnerability to attack in the prior art.

The technical scheme adopted in the present invention is an asymmetric double-image encryption method based on the fractional Fourier domain phase recovery process, including pure phase extraction, phase modulation, and fractional Fourier transform steps. Specifically include the following steps:

The first step, pure phase extraction: there are two original grayscale images, use the fractional Fourier domain phase recovery process to extract the pure phase of the i (i=1,2) grayscale image fi ( _i =1,2) Phase function exp(jξ _i,1 )(i=1,2); in the process of extracting the pure phase function of the original grayscale image f _i (i=1,2) using the fractional Fourier domain phase recovery process, the concomitant generation Phase template function φ _i,1 ,φ _i,2 ,ξ _i,2 (i=1,2);

The second step, phase modulation: in the temporary image generation module, a complex matrix H _i (i=1,2) is generated through the amplitude image g and the corresponding two phase functions ξ _i,1 , ξ _i,2

H _i ＝F ^β2 {F ^β1 [gexp(jξ _i,1 )]exp(jξ _i,2 )} (1)

For phase modulation, two H _i are combined into a matrix H by convolution operation

H=H ₁ *H ₂ (2) j in formula (1) and (2) is the symbol of imaginary part, exp{ } is exponent operation, g is phase image, F ^α represents fractional Fourier transform, ξ _{i, 1.} ξi _,2 is the phase function, H is the modulation result, and * means the convolution operation;

The third step, fractional Fourier transform: implement α _3rd order fractional Fourier transform on the modulation result H obtained in the second step to obtain calculate Amplitude of matrix That is, the ciphertext C _final , and the decryption key φ _i,d is also generated at the same time:

${\mathrm{<span\; class="notranslate">\; \&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; f</span>}}^{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span><span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; f</span>}}^{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; |</span>}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

Among them, |H| represents the amplitude of the combined matrix, j is the symbol of the imaginary part, exp{·} is the exponent operation, and arg{·} represents the phase of the matrix.

The fractional Fourier domain phase recovery process uses three phase template functions, φ _i,1 ,φ _i,2 ,ξ _i,1 , (i=1,2) are the three phase template functions of the process;

The decryption process of the above encryption method is specifically to multiply the final ciphertext C _final by the phase function exp(jφ _i,d ) to generate a complex matrix right Implement the -(α ₂ +α ₃ ) order inverse fractional Fourier transform to obtain the modulated result H _i , multiply H _i and the phase matrix exp(jφ _i,2 ) to obtain h _i , implement -α ₁ on h _i Order inverse fractional Fourier transform to get then extract The amplitude of the decrypted image f _i , namely:

f _i ＝|F ^-α1 {F ^-(α2+α3) [C _final exp(jφ _i,d )]}expj(φ _i,2 )| (4)

The decryption device used in the above decryption method includes two spatial light modulators and two lenses, and the two spatial light modulators and the two lenses are arranged at intervals; the two spatial light modulators are spatial light modulator PM1 and spatial light modulator PM2 , the two lenses are lens L1 and lens L2; lens L1 is arranged between spatial light modulator PM1 and spatial light modulator PM2, lens L2 is arranged between spatial light modulator PM2 and the decrypted image, spatial light modulator PM1 and The spatial light modulator PM2 is connected with the electronic controller, and the electronic controller is connected with the decrypted image through a computer.

Set the spatial light modulator PM1 and the spatial light modulator PM2 as exp(jφ _i,d ) and exp(jφ _i,2 ) respectively, the decryption process takes the ciphertext image C _final as the incident light input, and uses the spatial light modulator PM1 Modulate the ciphertext image C _final , and the modulated image passes through the lens L1 to realize -(α ₂ +α ₃ ) order fractional Fourier transform; use the spatial light modulator PM2 to modulate -(α ₂ +α ₃ ) order fractional Fourier transform As a result of the transformation, the modulated image undergoes -α _1st -order fractional Fourier transformation through the lens L2, and the original plaintext image f _i can be obtained.

The above-mentioned deciphering method can also be realized using a photoelectric hybrid device, which includes two masks and two lenses, and the two masks and the two lenses are arranged at intervals; the two masks are mask exp(jφ _i,d ) and mask exp(jφ _i,2 ), the two lenses are lens L1 and lens L2; lens L1 is set between mask exp(jφ _i,d ) and mask exp(jφ _i,2 ), lens L2 is set Between the mask exp(jφ _i,2 ) and the decrypted image, the mask exp(jφ _i,d ) and the mask exp(jφ _i,2 ) are connected to the electronic controller, and the electronic controller communicates with the CCD through the computer The image sensor is connected, and the decrypted image can be obtained from the CCD.

The present invention has following beneficial effects:

1. The encryption process of the present invention is different from the decryption process, and the encryption key is different from the decryption key, which avoids the disadvantage that the encryption key of the asymmetric encryption algorithm is the same as the decryption key and is vulnerable to attacks, and improves security.

2. Through the attack test, it is proved that the present invention not only has strong resistance to brute force attack, but also has strong resistance to noise and other specific attacks. Illegal users cannot obtain any valuable information by attacking images.

3. The encryption method of the present invention solves the problem of small key space in the existing encryption system and improves the key space. Through statistical analysis, for the key φ _i,d , the key space is approximately S ₁ ≈116 ^256×256 ; the key space of φ _i,2 is approximately S ₁ ≈6 ^256×256 . Since the key space of the encryption system is S ₁ ×S ₂ , the present invention has a sufficiently large key space against brute force attacks.

4. The encryption method of the present invention improves the convergence speed at the same time. The convergence times of the existing single-channel encryption method are about 400 times, while the convergence times of the encryption method of the present invention are about 40 times, and the convergence speed is obviously improved.

5. The decryption process of the present invention can be realized by optical method, the system is simple and the operation is convenient.

Description of drawings

FIG. 1 is a schematic diagram of an asymmetric double-image encryption method based on a phase recovery process in the fractional Fourier domain of the present invention.

Fig. 2 is a diagram of the encryption process of the asymmetric double-image encryption method based on the fractional Fourier domain phase recovery process of the present invention.

Fig. 3 is a diagram of the decryption process of the asymmetric double-image encryption method based on the phase recovery process in the fractional Fourier domain of the present invention.

Fig. 4 is a schematic structural diagram of an asymmetric double-image decryption method device based on a phase recovery process in the fractional Fourier domain of the present invention.

Fig. 5 is the original image "Zelda" encrypted by the asymmetric double-image decryption method based on the fractional Fourier domain phase recovery process of the present invention.

Fig. 6 is the original image "Peppers" encrypted by the asymmetric double-image decryption method based on the fractional Fourier domain phase recovery process of the present invention.

Fig. 7 is a ciphertext image encrypted by using the asymmetric double-image encryption method based on the fractional Fourier domain phase recovery process of the present invention.

Fig. 8 is the decryption key φ _i,2 of the image "Zelda" encrypted by the asymmetric double-image decryption method based on the fractional Fourier domain phase recovery process of the present invention.

Fig. 9 is the decryption key φ _i,d of the image "Zelda" encrypted by the asymmetric double-image decryption method based on the fractional Fourier domain phase recovery process of the present invention.

Fig. 10 is the decryption key φ _i,2 of the image "Peppers" encrypted by the asymmetric double-image decryption method based on the fractional Fourier domain phase recovery process of the present invention.

Fig. 11 is the decryption key φ _i,d of the image "Peppers" encrypted by the asymmetric double-image decryption method based on the fractional Fourier domain phase recovery process of the present invention.

Fig. 12 is a decrypted image of the image "Zelda" encrypted by the asymmetric double-image decryption method based on the fractional Fourier domain phase recovery process of the present invention.

Fig. 13 is a decrypted image of the image "Peppers" encrypted by the asymmetric double-image decryption method based on the fractional Fourier domain phase recovery process of the present invention.

Detailed ways

The present invention will be described in detail below in conjunction with the accompanying drawings and specific embodiments.

An asymmetric dual-image encryption method based on the phase recovery process in the fractional Fourier domain, including pure phase extraction, phase modulation, and fractional Fourier transform; the specific steps are as follows:

The first step, pure phase extraction: there are two original grayscale images, use the fractional Fourier domain phase recovery process to extract the pure phase of the i (i=1,2) grayscale image fi ( _i =1,2) Phase function ξ _i,1 (i=1,2); in the process of extracting the pure phase function of the original grayscale image f _i (i=1,2) using the fractional Fourier domain phase recovery process, the phase template function is concomitantly generated φ _i,1 ,φ _i,2 ,ξ _i,2 (i=1,2);

The second step, phase modulation: in the temporary image generation module, a complex matrix H _i (i=1,2) is generated through the amplitude image g and the corresponding two phase functions ξ _i,1 , ξ _i,2

H _i ＝F ^β2 {F ^β1 [gexp(jξ _i,1 )]exp(jξ _i,2 )} (1)

For phase modulation, two H _i are combined into a matrix H by convolution operation

H＝H ₁ *H ₂ (2)

In formulas (1) and (2), j is the symbol of the imaginary part, exp{ } is the exponential operation, g is the phase image, F ^α is the fractional Fourier transform, ξ _i,1 and ξ _i,2 are the phase functions, H is the modulation result, * means the convolution operation;

The third step, fractional Fourier transform: implement α _3rd order fractional Fourier transform on the modulation result H obtained in the second step to obtain calculate the amplitude of the matrix | That is, the ciphertext C _final , and the decryption key φ _i,d is also generated at the same time:

${\mathrm{<span\; class="notranslate">\; \&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; \{</span>{\mathrm{<span\; class="notranslate">\; f</span>}}^{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span><span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; f</span>}}^{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; |</span>}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

Among them, |H| represents the amplitude of the combined matrix, j is the symbol of the imaginary part, exp{·} is the exponent operation, and arg{·} represents the phase of the matrix.

The fractional Fourier domain phase recovery process uses three phase template functions, φ _i,1 ,φ _i,2 ,ξ _i,1 , (i=1,2) are the three phase template functions of the process.

The decryption process of the above encryption method is specifically to multiply the final ciphertext C _final by the phase function exp(jφ _i,d ) to generate a complex matrix right Implement the -(α ₂ +α ₃ ) order inverse fractional Fourier transform to obtain the modulated result H _i , multiply H _i and the phase matrix exp(jφ _i,2 ) to obtain h _i , implement -α ₁ on h _i Order inverse fractional Fourier transform to get then extract The amplitude of the decrypted image f _i , namely:

f _i ＝|F ^-α1 {F ^-(α2+α3) [C _final exp(jφ _i,d )]}expj(φ _i,2 )| (4)

The working principle of the encryption method of the present invention is as follows: firstly, an asymmetric double-image encryption method based on a fractional Fourier domain phase recovery process is used to extract the pure phase function of each image in two original grayscale images. Then, the two images are combined into one image using convolution operation. Finally, perform a α _3rd -order fractional Fourier transform on the modulated result, and extract the amplitude of the transformed result to obtain the final ciphertext. At the same time, the decryption key is also generated during the encryption process.

Refer to Fig. 1 for the principle of the encryption method of the present invention, where ( _xi , y _i ) and (x _o , y _o ) denote input and output coordinates, respectively. And transform the core as follows:

K(x _i ,y _i ; x _o ,y _o )＝A _φ exp{iπ[(x _i ² +y _i ² +x _o ² +y _o ² )cotφ _α -2(x _i x _o +y _i y _o )cscφ _α ]}

(5)

${\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; \&phi;</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i\&pi;sgn</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; i</span>}{\mathrm{<span\; class="notranslate">\; \&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}<span\; class="notranslate">\; ]</span>}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}<span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

φ _α = απ/2 (7)

Referring to Fig. 2 for the encryption process of the present invention, firstly, the pure phase function of each of the two original grayscale images is extracted. The specific process: use the fractional Fourier domain phase recovery process to extract the pure phase function ξ i,1 of the i-th grayscale image _{f i ,} φ _i,1 , φ _i,2 , ξ _i,2 are to extract the i-th original The phase template function concomitantly generated during the pure phase function process of the grayscale image, where i=1,2. α ₁ , α ₂ , α ₃ , β ₁ , β ₂ are the fractional indices of the iterative phase recovery process. Secondly, for the obtained two H _i (i=1, 2), calculate H by convolution operation. Next, the relevant fractional indices α ₁ , α ₂ , α ₃ , β ₁ , β ₂ are set. Implement α _3rd order fractional Fourier transform on the result H obtained in the second step to get Calculate the amplitude of the matrix That is, the ciphertext C _final , and the decryption key φ _i,d is also generated simultaneously according to the formula.

Decryption is the reverse process of encryption. Referring to Fig. 3 for the decryption process of the present invention, specifically, the final ciphertext C _final is multiplied by the phase function exp(jφ _i,d ) to generate a complex matrix right Implement the -(α ₂ +α ₃ ) order inverse fractional Fourier transform to obtain the modulated result H _i , multiply H _i and the phase matrix exp(jφ _i,2 ) to obtain h _i , implement -α ₁ on h _i Order inverse fractional Fourier transform to get then extract The amplitude of the decrypted image f _i ,

f _i ＝|F ^-α1 {F ^-(α2+α3) [C _final exp(jφ _i,d )]}expj(φ _i,2 )| (4)

FIG. 4 is a schematic structural diagram of the device used in the asymmetric double-image decryption method based on the phase recovery process in the fractional Fourier domain of the present invention. The device is similar to DRPE's 4f imaging system, including two spatial light modulators and two lenses, and the two spatial light modulators and two lenses are arranged at intervals. In the embodiment of the present invention, the two spatial light modulators are a spatial light modulator PM1 and a spatial light modulator PM2, and the two lenses are a lens L1 and a lens L2. The lens L1 is arranged between the spatial light modulator PM1 and the spatial light modulator PM2, the lens L2 is arranged between the spatial light modulator PM2 and the decrypted image, PM1 and PM2 are connected with an electronic controller, and the electronic controller communicates with the image sensor through a computer CCD connection. Set PM1 and PM2 as exp(jφ _i,d ) and exp(jφ _i,2 ) respectively, the decryption process takes the ciphertext image C _final as the incident light input, uses the spatial light modulator PM1 to modulate the ciphertext image C _final , and modulates The image of the -(α ₂ +α ₃ ) order fractional Fourier transform is realized through the lens L1; the spatial light modulator PM2 is used to modulate the result of the -(α ₂ +α ₃ ) order fractional Fourier transform, and the modulated image Through the lens L2 to realize -α _1st order fractional Fourier transform, the decrypted plaintext image f _i can be obtained on the CCD.

The above encryption method can also be implemented with a photoelectric hybrid device, which is similar to the 4f imaging system of DRPE. In the encryption process, only PM1 and PM2 are replaced by phase masks exp(jφ _i,d ) and exp(jφ _i,2 ) .

In the present invention, the correlation coefficient (CC) or mean square error (MSE) is used as the phase recovery process iteration end criterion, when CC is greater than a preset value close to 1 or the mean square error MSE value is less than a preset value close to 0 value, the iteration terminates, and the calculation formulas of CC and MSE are as follows:

$\mathrm{<span\; class="notranslate">\; CC</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; g</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; g</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; g</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; g</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; \}</span>}{\sqrt{\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; g</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; g</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \}</span>}\sqrt{\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; g</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; g</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \}</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; MSE</span>}<span\; class="notranslate">\; =</span>\frac{\underset{\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\underset{\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; g</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; g</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

Among them, g is the plaintext image, g ^k is the approximate plaintext image obtained by iteration, M and N are the width and height of the plaintext image respectively, and E[·] is the expected value calculation symbol. Assuming that the last number of iterations is K, the optimal phase function is:

φ ₁ ＝φ ₁ ^K , φ ₂ ＝φ ₂ ^K , ξ ₁ ＝ξ ₁ ^K - ¹ , ξ ₂ ＝ξ ₂ ^K (10)

The ciphertext image obtained after encrypting the original images of "Zelda" and "Peppers" in Figure 5 and Figure 6 is shown in Figure 7. The ciphertext image presents a fixed white noise distribution and only contains intensity information, so the image cannot be illegal Users provide any valuable information. It can be seen that the encryption degree of the encryption method of the present invention is very high.

The decryption key φ _i,2 of the image "Zelda" is shown in Figure 8, and it can be seen that the decryption key is similar to a white noise distribution.

The decryption key φ _i,d of the image "Zelda" is shown in Figure 9, and it can be seen that the decryption key is similar to a white noise distribution.

The decryption key φ _i,2 of the image "Peppers" is shown in Figure 10, and it can be seen that the decryption key is similar to a white noise distribution.

The decryption key φ _i,d of the image "Peppers" is shown in Fig. 11. It can be seen that the decryption key is similar to a white noise distribution.

See Figure 12 for the decrypted image "Zelda". Comparing the original image "Zelda" and the decrypted "Zelda" image in Figure 5, it can be seen that there is almost no error between the decrypted image and the original image.

See Figure 13 for the decrypted image "Peppers". Comparing the original image "Peppers" in Figure 6 with the decrypted "Peppers" image, it can be seen that there is almost no error between the decrypted image and the original image.

The encryption method of the invention solves the problem of small key space in the existing encryption system and improves the key space. According to statistics, for the key φ _i,d , the key space is approximately S ₁ ≈116 ^256×256 ; the key space of φ _i,2 is approximately S ₁ ≈6 ^256×256 . Since the key space of the encryption system is S ₁ ×S ₂ , the present invention has a sufficiently large key space against brute force attacks.

The encryption key of the present invention is different from the decryption key, avoids the disadvantage that the encryption key of the asymmetric encryption algorithm is the same as the decryption key and is easily attacked, and improves the security.

Through the attack test, it is proved that the present invention not only has strong resistance to brute force attack, but also has strong resistance to noise and other specific attacks. Illegal users cannot obtain any valuable information by attacking images.

The decryption process of the invention can be realized by optical method, the system is simple and the operation is convenient.

Claims (7)

1. the asymmetric double image encryption method based on fractional Fourier domain phase place rejuvenation, is characterized in that, comprises pure phase extraction, phase-modulation, fractional fourier transform step.

2. the asymmetric double image encryption method based on fractional Fourier domain phase place rejuvenation as claimed in claim 1, is characterized in that ciphering process is different from decrypting process, and encryption key is different from decruption key, specifically comprises the steps:

The first step, pure phase extraction: have two width original-gray image, use fractional Fourier domain phase place rejuvenation to extract i width gray level image f _ithe pure phase bit function ξ of (i=1,2) _{i, 1}(i=1,2); Using fractional Fourier domain phase place rejuvenation extraction original-gray image f _iin the pure phase bit function process of (i=1,2), supervene phase mask function phi _{i, 1}, φ _{i, 2}, ξ _{i, 2}(i=1,2);

Second step, phase-modulation: in intermediate images generation module, a complex matrix H _iby amplitude image as g and corresponding two phase function ξ _{i, 1}, ξ _{i, 2}produce

H _i＝F ^β2{F ^β1[gexp(jξ _i,1)]exp(jξ _i,2)} (1)

Carry out phase-modulation, two H _iby convolution algorithm, be combined into a matrix H

H＝H ₁*H ₂ (2)

In formula (1), (2), j is the imaginary part of symbol, and exp{} is exponent arithmetic, and g is phase image, F ^αrepresent fractional fourier transform, ξ _{i, 1}, ξ _{i, 2}be phase function, H is modulation result, and * represents convolution algorithm;

The 3rd step, fractional fourier transform: the result H that second step is obtained implements α ₃rank fractional fourier transform obtains be ciphertext C _final, decruption key φ _i,dalso generated simultaneously:

Wherein, | H| represents the amplitude of combinatorial matrix, and j is the imaginary part of symbol, and exp{} is exponent arithmetic, the phase place of arg{} representing matrix.

3. the asymmetric double image encryption method based on fractional Fourier domain phase place rejuvenation as claimed in claim 2, is characterized in that: described fractional Fourier domain phase place rejuvenation is used three phase place masterplate functions, φ _{i, 1}, φ _{i, 2}, ξ _{i, 1}(i=1,2) are three phase mask functions of this process.

4. the asymmetric double image encryption method based on fractional Fourier domain phase place rejuvenation as described in claim 1-3 any one, is characterized in that, its decrypting process is specially: by final ciphertext C _finalbe multiplied by phase function exp (j φ _i,d) generation complex matrix right enforcement-(α ₂+ α ₃) result H after rank are modulated against fractional fourier transform _i, by H _iwith phasing matrix exp (j φ _{i, 2}) multiplying each other obtains h _i, to h _ienforcement-α ₁rank obtain against fractional fourier transform then extract amplitude as deciphering image f _i, that is:

f _i＝|F ^-α1{F ^-(α2+α3)[C _finalexp(jφ _i,d)]}expj(φ _i,2)| (4)

Wherein, j is the imaginary part of symbol, and exp{} is exponent arithmetic, C _finalrepresent final ciphertext of encrypting, F ^αrepresent fractional fourier transform.

5. the asymmetric double image decryption method based on fractional Fourier domain phase place rejuvenation as claimed in claim 4, it is characterized in that: decryption device used comprises two spatial light modulators and two lens, two spatial light modulators and two lens are spaced; Two spatial light modulators are spatial light modulator PM1 and spatial light modulator PM2, and two lens are lens L1 and lens L2; Lens L1 is arranged between spatial light modulator PM1 and spatial light modulator PM2, lens L2 is arranged on spatial light modulator PM2 between deciphering image, spatial light modulator PM1 is connected with electronic controller with spatial light modulator PM2, electronic controller is connected with ccd image sensor by computing machine, by obtaining deciphering image on CCD.

6. the asymmetric double image decryption method based on fractional Fourier domain phase place rejuvenation as claimed in claim 5, is characterized in that: spatial light modulator PM1 and spatial light modulator PM2 are set to respectively to exp (j φ _i,d) and exp (j φ _{i, 2}), decrypting process is with ciphertext image C _finalas incident light input, usage space photomodulator PM1 modulation ciphertext image C _final, the image of modulation sees through lens L1 realization-(α ₂+ α ₃) rank fractional fourier transform; Usage space photomodulator PM2 modulation-(α ₂+ α ₃) result of rank fractional fourier transform, the image after modulation sees through lens L2 realization-α ₁rank fractional fourier transform, can obtain the plaintext image f decrypting _i.

7. the asymmetric double image decryption method based on fractional Fourier domain phase place rejuvenation as claimed in claim 4, it is characterized in that: decryption device is PMD, PMD comprises two masks and two lens, and two masks and two lens are spaced; Two masks are mask exp (j φ _i,d) and mask exp (j φ _{i, 2}), two lens are lens L1 and lens L2; Lens L1 is arranged on mask exp (j φ _i,d) and mask exp (j φ _{i, 2}) between, lens L2 is arranged on mask mask exp (j φ _{i, 2}) and decipher between image mask exp (j φ _i,d) and mask exp (j φ _{i, 2}) be connected with electronic controller, electronic controller is connected with ccd image sensor by computing machine, by obtaining deciphering image on CCD.