Image Cryptosystem in Optical Gyrator Transform Domain Using Audio Keys

Abstract

Two remote sensing image encryption algorithms based on the randomness of audio channel sequences are proposed and their robustness is verified by many attack experiments. The first encryption algorithm uses the randomness of audio to encrypt image pixels in RMD. Compared with traditional image encryption algorithms, this algorithm has high randomness and security and can resist stronger password attacks. However, the encryption algorithm takes a long time. Considering that information sometimes needs to be transmitted urgently, a second encryption algorithm is proposed. By combining vocal tract and random phase to form new public and private keys, the number of computation amount and dislocations is reduced. The second algorithm is short in time but low in security.

1. Introduction

At present, we are in an era of rapid development of information technology. Networks can transmit information through the Internet at high speed and realize communication and exchange functions. With the development and application of remote-sensing technology, remote-sensing images play an increasingly important role in meteorology, hydrology, geology, and so on. Therefore, remote-sensing images are often vulnerable to accidental or malicious attacks, which compromise the integrity of data. Therefore, more and more attention is being paid to image information protection in remote sensing image transmission and storage. In the image encryption method, the security of the image can be improved effectively by using optical technology to design a targeted image encryption algorithm. The optical hiding method is of great significance in encrypting returning satellite remote sensing images. By sampling the digital image and then introducing the optical information encryption of human physiology, such as speech, its extensive and unique characteristics make the encrypted voice encryption related to user identity can improve the information security system further. At present, the encryption algorithms of grayscale image, color image, and hyperspectral image are mainly realized by chaos model and optical transformation. Arnold transform, Fresnel diffraction, and gyroscope transform are commonly used [1].

In this paper, the phase in audio features is encrypted by the Fourier transform based on the gyrator domain, and the image is encrypted and dislocated by key perspective, which further improves the algorithm’s anti-attack ability.

1.1. Optical Image Encryption

The optical image encryption technique has many advantages such as fast encryption operation, large key capacity, and more information available for encoding, so it has attracted a lot of attention from scholars. In 1995, Refregier et al. combined the input signal into optical and image scrambling techniques and implemented them numerically to convert the input signal into smooth white noise and obtained a stable encryption method [2]. They proposed a highly representative dual phase random encoding (DPRE) based on a 4f system. The 4f system is widely used in linear optical processing systems, through the 2f distance between the two lenses, object distance f, and phase distance f, so it is a 4f system [3]. Inspired by DRPE [4,5], some scholars propose methods to encode wavelength, phase, amplitude, etc., in the image, to improve the encryption dimension or to improve the fractional Fourier transform and the higher order transformation, so as to improve the information security of the image effectively. Asymmetric optical cryptography and its feasibility are realized in further research using masks (random phase diffusers) to transform and increase key space. To address the vulnerability of DRPE, mainstream encryption techniques typically go through three operations: transformation, random phase mask, and preprocessed optical transformation. The preprocessing Ssoperations include singular value decomposition, Arnold transform, random phase mask and so on. Optical transformation usually adopts Gyrator transform, Fresnel transform, Fourier transform, etc. [6].

In recent years, while optical encryption methods are becoming more mature, researchers have paid more attention to their robustness and anti-attack capability. In the last century, Refregier et al. introduced random phase coding to protect images against forgery [7], making it easier to hide image information and playing an initial role in protection. Then, some scholars introduced phase reconstruction iterative algorithm, so that asymmetric encryption system get encrypted holograms [8,9] and cannot recover holograms when we do not know the public and private keys. In 2013, Rajput et al. applied this encryption method to grayscale images [10]: some features of the image were used as private keys and the linear characteristics of the original encryption system were eliminated by the truncation technique. In 2020, Yuan Guo et al. solved the shortcoming of weak resistance to optically encrypted selective plaintext attacks [11], increasing the efficiency and sensitivity of key transmissions, effectively defending against various attacks, and enhancing security. In 2021, Zhu et al. proposed security analysis and improvement of image encryption cryptosystems based on planar extraction and multiple chaos [12]. In 2022, Peng et al. proposed convolutional neural networks generate pseudo-keys through iteration to achieve the purpose of forgery attacks [13].

In recent years, DNA sequences and fingerprints are often used to replace pixels, increasing resistance to attacks by damaging pixel values [14,15]. Because biometrics have proprietary features such as strong parallelism and high information density. Therefore, researchers use these features as a key vehicle for information encryption, such as DNA encoding and computation using complementary addition and subtraction operations of base pairs, which gives it the advantage of high storage capacity and high computational speed [16].

Status of Gyrator Transform Domain Optical Encryption

Common optical system transformations include the Fresnel transform, polarization transform, rotary transform, dual random phase encoding, and fractional Fourier transform. The direction of the optical phase coding technique extends from Fourier transform domain to the fractional Fourier transform domain and Fresnel transform domain. The optical dimension of information expands from the initial phase to the amplitude or polarization information. The performance characteristics of these common optical transforms are shown in

Table 1. Performance comparison of different encryption methods.

Transformation	Correlation between Adjacent Pixels	Anti-Noise Attack	Correlation between Adjacent Pixels and Resistance to Shear Attacks	Optical Device Realization
Fourier Transformation (FT)	×	×	×	√
Fast Fourier Transform (FFT)	√	√	√	×
Fourier Series Transform (FST)	√	×	×	√
Gyrator Transformation (GT)	√	√	√	√

×: negetive. √: positive.

, the Gyrator domain transform has higher performance than other transforms.

Both the fractional Fourier transform and the Gyrator transform are the general forms of conventional transformation. In 1993, Lohmann et al. combined optical devices with the Fourier transform [17] and led future generations in designing an optical device based on the Gyrator transform. In 2016, Lili Yao et al. implemented an asymmetric encrypted Gyrator transform [18], splitting the image component into two phase plates and using Gyrator transform to generate two private keys in the encryption process. In 2015, Cai Ning et al. studied image encryption based on Fourier transform and GT transform [19], adding two random phases as transformation angles and increasing key space. The order variables of the Gyrator transform are used as additional keys for rotation and transformation. Encryption method is more intuitive and secure because only a complete key can restore the plaintext information [20].

This paper presents an optical remote sensing color image encryption method based on Gyrator with acoustic characteristics. The two-phase channel of the audio dual-channel is used as the encryption key. The Gyrator transform is superior to other optical transformations in terms of anti-noise attack, encryption performance and realization ability of optical equipment.

2. Optical Image Encryption Security Analysis

2.1. Basic Concepts of Information Security

Cryptography is the basic discipline of information security, and the common ways are mainly asymmetric encryption and symmetric encryption. There are five basic components in the encryption model as follows, and the relationship is shown in Figure 1.

2.2. Common Evaluation Functions for Image Restoration

The experiments to detect whether the encryption and decryption methods are qualified are judged by some key quantitative data to compare the degree of image information restoration. The MSE mean square error function is introduced in Equation (1) to calculate. MSE is used to determine the difference between the decrypted image and the original image. Where I_d and I₀ mark the decrypted image and the original image, respectively. M and N are the number of ranks of the matrix. In fractional order in a very small range to decrypt properly. Mathematically, the mean square error between the original image and the decrypted image is represented [21].

$$MSE=\frac{1}{MN}{\displaystyle \sum }_{m,n}|I_d\left(m,n\right)-I_0\left(m,n\right){|}^2$$

(1)

Meanwhile, we also use PSNR (peak signal-to-noise ratio), defined as Equation (2) in dB, the larger the absolute value the closer the information of the two images. Where MAX_I is the maximum possible pixel value of the image.

$$PSNR=10\cdot {\log }_{10}\left(\frac{MA{X}_I{}^2}{MS}\right)=20\cdot {\log }_{10}\left(\frac{MA{X}_I}{\sqrt{MSE}}\right)$$

(2)

According to the formula, the smaller the MSE, the larger the PSNR and the better the image quality (i.e., the closer to the original image). Greater than 40 dB means the image is excellent and extremely close to the real image. 30 dB–40 dB means the image is good, with partial distortion but can be adopted. Anything below 20 dB is considered a failure in image restoration; photographs are unrecognizable and remote sensing images with such results are not normally used [22].

2.3. Cryptographic Analysis

No encryption technology is absolutely secure, and an attacker with enough time and resources can always find a way to break the encryption algorithm. From Figure 2, we can find that there are four common attacks by attackers in the process of information transmission, and their ultimate goal is to obtain the key. The more information is known, the easier and stronger the attack is. The strongest attack method, CCA, can obtain a decrypted image of part of the specified ciphertext and is the only method that has access to the decrypted ciphertext. In real life, the most common attacks are known plaintext attack (KPA) and selective plaintext attack (CPA).

Known plaintext attack KPA: The attacker already has a partial plaintext and ciphertext pair and tries to infer the complete plaintext, key, and corresponding encryption algorithm. In the GT transform, it corresponds to the enemy’s known rotation angle α (one of the keys) and the dislocation method to try to unscramble the image.

Select plaintext attack CPA: The attacker already knows the encryption algorithm and can select the ciphertext corresponding to the part of the plaintext. For example, an attacker can infiltrate the encryption system disguised as Bob to create a plaintext (Lena) and get the cryptogram, but the key needs to be inferred by itself. Only algorithms with good diffusivity and dislocation can resist such attacks.

3. Optical Gyrator Transform Domain-Based Audio Key Encryption Algorithm

3.1. Development of the Gyrator Transform Domain

Optical Gyrator is an image processing approach for Gyrator spatial phase plane rotation proposed by Alieva et al. in 2007, providing a new approach to phase transformation [23]. The Gyrator domain transformation at different angles can be achieved by an optical system consisting of a combination of cylindrical lenses. In simulated experiments, GT has properties related to shift, scaling, and Parseval’s theorem, and is one of the three mainstream image encryption algorithms alongside Fresnel diffraction and Arnold transform. In 2010, Liu et al. studied the fast Gyrator transform [24] and rewrote the GT formula using convolution operation to improve the computational efficiency by introducing a fast Fourier transform into the calculation. Some scholars have carried out a lot of research on optical image encryption based on the fractional Fourier transform and Gyrator transform [23,25]. Compared with traditional optical encryption techniques, Gyrator domain optical encryption has higher security and stronger anti-attack capability.

3.2. Gyrator Definition

The mathematical definition of the two-dimensional Gyrator transform is given by the following equation:

$$F(u,v)=G^{\alpha }[f(x,y)](u,v)=\frac{1}{\left|\sin \alpha \right|}{\displaystyle \iint f(x,y)\exp [i2\pi \frac{(xy+uv)\cos \alpha -xv-yu}{\sin \alpha }]dxdy}$$

(3)

The Gyrator has the same properties as the fractional Fourier transform such as linear additivity and energy conservation. In the discrete Gyrator we use a method of convolution operations to implement the program algorithm.

3.2.1. Optical Implementation of Gyrator [26]

The experimental setup for performing the Gyrator transform is shown in Figure 3: a phase modulation setup consisting of three lenses. Among them, lenses L1 and L3 are identical and have the same focal length f (i.e., the distance z between adjacent lenses) and angular parameters. The focal length of lens L2 is z/2.

Figure 4 Generalized lens composed of two identical cylindrical lenses. Where L1 (L3) and L2 correspond to the angular parameters of the above figure and, respectively, the transformation order of the Gyrator can be realized by two angular parameters. The expression of the phase modulation function is given by.

In Equation (4), f is the focal length of the generalized lens and λ is the wavelength of the incident light wave. Where ${\mathsf{\phi }}_1$ = −$\mathsf{\phi },\mathsf{\phi }$₂ = $\mathsf{\phi }$ − π/2, when the lens angle parameter of the generalized lens is 0, π/2 or π, it is equivalent to the orthogonal combination of two column lenses in the experiment, and the generalized lens becomes an ordinary spherical lens, which becomes a 4f system of two Fourier cascade transformations.

$$t(x,y)=\exp (-i\pi \frac{x^2+y^2-2xy\sin 2\varphi }{\lambda f})$$

(4)

Figure 5 shows the Gyrator encryption and decryption process, GL1, GL2 are generalized lens in GT transform. The encrypted image is f(x,y), which is first multiplied by a random phase C1, and then multiplied by another random phase C2 for the second order transformation in order to finally obtain the output image g(x′,y′) as the image complex amplitude. The decryption method is to perform the −α inverse transform in turn, and then multiply by the complex conjugate phase for two reductions to get the original image f(x,y).

3.2.2. Numerical Implementation of Gyrator

In the above equation, the GT operation requires a larger computational effort. Chen et al. [27] can accelerate information processing by computing the analog convolution of discrete GT. The fast Fourier transform was introduced into the convolution calculation to improve the expression of the Gyrator transform. For second order functions, the GT transform of order ɑ can be obtained in an alternative form defined as follows:

$${\mathrm{q}}_{\mathrm{g}}(x_0,y_0)=G^{\alpha }\left[q(x_i,y_i)\right](x_0,y_0)={\displaystyle \iint q\left(x_i,y_i\right)}K\left(x_i,y_i;x_0,y_0\right)d{x}_{i}d{y}_i$$

(5)

where q_g is the GT output function, G^ɑ represents the GT transform, and ɑ is the transform order (rotation angle). $(x_i,y_i)$ is the input plane coordinate (spatial frequency) and $(x_0,y_0)$ is the output plane coordinate (spatial position), which is equivalent to the fractional spectral plane.

$K\left(x_i,y_i;x_0,y_0\right)$ is the integral transform kernel function of GT, defined as

$$K_{\alpha }\left(x_i,y_i,x_0,y_0\right)=\frac{1}{\left|\sin \alpha \right|}\exp \left(i2\pi \frac{\left(x_0{y}_0+x_i\right)\cos \alpha -\left(x_i{y}_0+x_0{y}_i\right)}{\sin \alpha }\right)$$

(6)

When the following angular equivalent transformations are used:

$$\cot \alpha =-\tan \frac{\alpha }{2}+\frac{1}{\left|\sin \alpha \right|}$$

(7)

Equation (6) can be rewritten as

$$K\left(x_i,y_i,x_0,y_0\right)=\frac{\exp \left[-2i\pi \left(x_i{y}_i+x_0{y}_0\right)\tan \frac{\alpha }{2}\right]}{\left|\sin \alpha \right|}\times \exp \left[2i\pi \frac{\left(x_i-x_0\right)\left(y_i-y_0\right)}{\sin \alpha }\right]$$

(8)

Combining Equations (5) and (8) can be equated in GT as the convolution equation:

$$q_g\left(x_0,y_0\right)=p\left(x_0,y_0\right)\left\{\left[q\left(x_i,y_i\right)p_1\left(x_i,y_i\right)\right]\otimes p_2\left(x_i,y_i\right)\right\}$$

(9)

$\otimes$ representing the sign of the convolution, $P_1$ and $P_2$ are two pure phase functions:

$$\begin{gathered} P_1\left(x,y\right)=\exp \left(-2i\pi xy\tan \frac{\alpha }{2}\right) \\ {P}_2\left(x,y\right)=\frac{\exp [-2i\pi xy\csc \alpha ]}{\left|\sin \alpha \right|} \end{gathered}$$

(10)

According to the convolution property, Equation (9) can be rewritten as follows:

$$q_B=p_1{f}^{-1}\left[f\left(p_{1}q\right)p_2\right]$$

(11)

where $f$ and $f^{-1}$ represent the computation of Fourier transform and inverse transform. The FFT computation can compute the GT transform in discrete cases and successfully increase the computation speed.

In addition, the analytical formula for the phase function $p_2(u,v)$ is

$$p_2(u,v)=f[p_2(x_i,y_i)](u,v)=\exp (-2i\pi uv\sin \alpha )$$

(12)

Equation (11) becomes:

$$q_B=p_1{f}^{-1}\left[f\left(p_{1}q\right)p_2\right]$$

(13)

Assuming that the number of samples in the X-direction Y-direction is equivalent to $N$ and $M$, respectively, the sample points are defined as follows:

$$\{\begin{array}{l}{x}_i(k)=x_0(k)=u(k)=\frac{k-(N-1)/2}{\sqrt{N}}\\ y_i(j)=y_0(j)=v(j)=\frac{j-(M-1)/2}{\sqrt{M}}\end{array}$$

(14)

Therefore, the GT transform can be implemented by two FFT calculations, and the equations of the fast Fourier transform can be obtained from Equations (13) and (14).

3.3. Selecting Sound Encryption

The introduction of random modes of hyperchaotic systems and rotational domains for image decomposition dislocation (RMD) is a simple and efficient method to implement and is often used in image encryption systems for pixel-value dislocation [28]. In order to improve the security of the encryption algorithm, we use RMD to combine and randomize the intermediate hyperchaotic data, and the result can be used as the private key and public key. RMD is an improved version of the asymmetric encryption scheme based on equivalent mode decomposition (EMD), with greatly reduced constraints.

Chen et al. demonstrated in 2016 [26] the sensitivity of the initial conditions of chaotic systems, which are very sensitive to color images, with small changes in initial values leading to large differences. The initial value of the chaos setting is changed to x₀ = y₀ = z₀ = −10 and u₀ = −10.0001.

The security of cryptosystems can be greatly improved by introducing hyperchaotic systems into cryptosystems due to their high randomness and extreme sensitivity to initial conditions. In addition, some hyperchaotic systems are used in some cryptosystems to improve security since the initial values of chaotic systems are highly sensitive and can disrupt linear relationships [29,30]. The encryption process is shown in Figure 6, where the resulting chaotic data is used as the public key of the encryption system and encoded in a two-dimensional format. Intermediate variables are introduced into the Gyrator transform for propagation [31,32]. The results of the Gyrator transformation are classified into P1 and P2 by RMD, and pixel dislocation is performed using two thresholds, V1 and V2, to obtain ciphers and private keys [33]. Both thresholds can be used as additional keys to the encryption scheme to improve security, but the key is lost during decryption. The validity of the private key and key will be tested in subsequent chapters.

3.3.1. Encryption Process

The security of the encryption system is further improved by linking the key with the user’s identity through voice mode. The final encrypted image is obtained by sampling voice mode, constructing a random phase, using the chaos model to interrupt the phase, and completing the computation and interruption of the hyperchaotic system on a computer. In order to ensure the security of cipher text, two asymmetric encryption methods are proposed. The full encryption method is shown in Figure 7. The first encryption method takes the audio signal as the final disruption sequence, which reflects the randomness and uncertainty of the audio signal. The image is divided into bands and scrambled separately. Then, using Gyrator transform, the sound sequence is added with RMD scrambling. Finally, the image is scrambled again to get the encrypted image. The second encryption method, as shown in Figure 8, changes the random phase by sampling the audio signal and then performing an RMD transformation. Similar to Method I, the audio signal does not participate in the RMD operation, only as the last scrambling sequence parameters, to get the encrypted image of Method II. In this paper, we will test and compare the two encryption methods to see if the effect is significant.

• Method I

Figure 7. Method I.

Figure 7. Method I.

• Method II

Figure 8. Method II.

Figure 8. Method II.

3.3.2. Decryption Process

Since the proposed cryptosystem is asymmetric, complete decryption cannot be achieved without the correct public and private keys. In the first step of decryption, the private key is obtained by reverse pixel scrambling and RMD computation, respectively. Then, during decryption, the correct private key and α are introduced and decoded into an RGB format image.

Equation (15) uses phase key P₁, transform domain alpha to obtain the encrypted image I_E, I_oc to GT to obtain the image. The operation is a dislocation method. $I_E$ is the last encrypted image available.

$$I_E=I_{oc}\cdot operation(P_1)$$

(15)

The private key $P_2$ is used in Equation (16) to perform the Gyrator inversion with the encrypted image to obtain the decrypted image $I_D$.

$$I_D=IGT(I_E\cdot P_2)$$

(16)

4. Implementation and Experimental Validation

4.1. Experimental Setting

To verify that the audio can be encrypted in the Gyrator domain for remote sensing images, numerical simulations are performed using MATLAB R2020b software. In the simulation experiment, the initial parameters are set to order α as 0.8 and 256 × 256 × 189 hyperspectral remote sensing images are imported. Remote sensing images are encrypted and decrypted in the Gyrator domain using audio signals. The experiments were conducted on the system Windows 11 Professional.

Graphics card: NVIDIAGeForceGTX-1650i 16 GB

Version built: MATLAB R2020b, memory 16 GB

The audio is a certain popular song of about 3 min in length. The encrypted map is hyperspectral remote sensing image I256_8bit.mat, three bands are selected to synthesize the pseudo-color image for encryption, in order to ensure that the experiment has a certain degree of contrast, the control experiments are selected to synthesize the bands 11, 52 and 113.

Based on the knowledge in Section 2, we now judge images with PSNR greater than 30 dB as successful restoration and can identify the original information.

4.2. Simulation Results

Aiming at the characteristics of remote sensing images, the algorithm in this paper supports the encryption of multiple image forms and can encrypt single-band grayscale images and multi-band hyperspectral images. The use of sound for encryption makes the images noise-like, with good encryption effect and high applicability. The difference between the encrypted image and the original figure is obvious, and the original image information can be completely masked. The encrypted graph has certain masking and visual insensitivity.

As the key of image decryption, the space of the key of α and the random phase P₁, P₂ is very large, and the sum of the Gyrator transform has an infinite number of values. Therefore, the image encryption performance based on the Gyrator transform is very good [28].

The running time of Method I is 7.707928 s. The running time of Method II is 2.843909 s. It can be seen that the chaos model has a much longer dislocation operation time. As shown in Figure 9 and Figure 10. The three images from left to right are the original image, the encrypted image, and the decrypted image. The encrypted image is completely blurred and no original image information can be obtained by visual inspection, it can be seen that both methods can complete the encryption. When decrypting, we can get almost the same image as the original image and the value of PSNR is above 200 dB, which can also prove the high restoration of the image from the value, and both methods can decrypt better.

Figure 11 and Figure 12 show the curves plotted with the PSNR values (PSNR curves) corresponding to the decrypted images at different order transformations for the 141 decryptions completed by the two algorithms, respectively (The X-axis is the transformation order, and the Y-axis is the PSNR value). In the calculation, the rotation order ranges from 0.5 to 1.2 with a step size of 0.005. both algorithms only have a clear spike at 0.8, while the restoration of the decrypted images solved by other values is not high enough, so the order can be used as an additional key to protect the image. The more pronounced spikes prove good key sensitivity, and the amount of information restored by 0.8 is much greater than the restoration of 0.795 and 0.805.

The histogram of the plaintext (Figure 13) shows a clear statistical pattern (The X-axis represents the brightness value, and the Y-axis represents the number of pixels of each type). In order to prevent attackers from analyzing and deciphering the cryptographic code, we transform the histogram of the encrypted image into a completely different grayscale distribution. At this time, the feature information of the image is little, which ensures the security of the ciphertext in transmission, proving that both methods are stable and secure.

4.3. Performance and Evaluation Tests of the Gyrator Transform

4.3.1. Noise Attack Test

In the experiment, the image is normalized before adding the homemade Gaussian white noise, and the noise density is added from 0 to 0.05 in steps of 0.1. The results of noise attacks for the two encryption methods at each step are shown in Figure 14a,b, respectively, and the Gaussian white noise is gradually added for decryption. It can be found in the Figure 14b that its anti-noise effect has been stable and high-quality decryption has been completed every time.

4.3.2. Known Plaintext Attack Test

Assuming that part of the image is intercepted by the attacker, the algorithm gets two plaintext images and encrypted the corresponding ciphertext. A known plaintext attack is carried out using a phase recovery algorithm on each component, with 1000 rounds of attacks lasting about 20 min. In programming implementation, the core lies to calculate the current phase, then compare with the phase obtained in the previous iteration, obtain the direction of gradient convergence, and converge to a certain value after several iterations. The decryption phase is introduced into the GT algorithm and the final decryption map is obtained. The final attack results of both algorithms are shown in Figure 15a,b, respectively. There are only a few pixel points on the left, all in random patterns with no image information, suggesting that Method I the plaintext map is absolutely protected. By using PSNR, Figure 15c is the restoration of Figure 15a to the existing two plaintext messages, and Figure 15d is from Figure 15b. From PSNR, the raw information is not identifiable, proving that the algorithm is effective against attacks. One of the pictures has a high PSNR which means the attacker gains most of the image information. Some of the information in the secret image is visible in the correct image, proving that the algorithm is not completely resistant to KPA attack patterns.

4.3.3. Chosen Plaintext Attack Test

Based on a known plaintext attack, a scheme with a known encryption algorithm is added and a chosen plaintext attack is performed using the impact function. Attacks by both algorithms took about 50 min and produce the results shown in Figure 16a,b, respectively. Figure 16c is the restoration of Figure 16a to the existing two plaintext messages, and Figure 16d is from Figure 16b. Method I does not capture image information and the algorithm protects the plaintext encryption path. However, the chosen plaintext attack Method II can obtain the outline of two secret images, which proves that the method did not completely resist the attack.

4.3.4. Shear Attack Test

In the optical experiments, shear attack by using an opaque baffle to block the light path, so that part of the light can not pass through the lens as shown in Figure 17, obtained a partially black image, part of the information loss led to the restored image with large noise. The maximum number of shear pixels per layer in the experiment is 20 × 20.

The reduction degree (PSNR) of Method I encryption is shown in Figure 18a, where the pixel edge length is 0–20, step size 1, and high quality of decrypted map can be used when edge length p ≤ 5. The rest of the tracing point data are shown in the figure. Method II is shown in Figure 18b. Here, even if cut to the maximum value, the PSNR obtained by decryption is still as high as 40.4 and can be used.

4.4. Method II Robust Analysis

In Section 4.3 experiments, Method II was found to be more robust to more noise and shear attacks than Method I. In order to improve the algorithm, we attack both methods more widely, find its threshold, and observe the change of its decryption diagram.

Figure 19 is the result of a noise-increasing attack, with a noise factor of 0.116 for Method II, which is about twice as high as for Method I.

Figure 20 shows the results of Method II in a larger cutting area, the cutting area can reach 18 × 18 without distortion, even if there are obvious inflection points in the early stage but as much as possible to ensure the original image information and the overall smoothness of the later stage, so that even if the cutting area becomes larger it can still restore the image.

4.5. Optical Image Encryption Testing System V1.0

This paper designs and develops a graphical GUI interface based on the MATLAB environment. An intuitive interface can simplify operation and make it more convenient in practical applications. Aiming at the two algorithms, a test system is designed to verify the encryption effect. Its several functions are as follows: selection image data, band fusion, encrypted image, decryption of image, and interference (attack) with encrypted image, as shown in Figure 21 for the entire system.

4.5.1. Software Module Design

The visualization interface developed is shown in Figure 22, and the different functional modules are divided into the following areas:

(1)

Data pre-processing module: The red box area can be based on the characteristics of a remote sensing image with multiple bands, and three bands can be selected to synthesize the pseudo-color image after importing the data to obtain the visual information of the image initially. If you want to encrypt a grayscale image, it can directly make the same number in three bands. After the initial image is obtained, the image is encrypted and decrypted directly in order to obtain the encryption and decryption effect more intuitively.

(2)

Load data module: The blue box area is selected for a certain mp3 audio sequence, pending encryption and decryption operation.

(3)

Information Display Module: The white box area is the result of image processing. It is divided into three parts. When the corresponding button is clicked, the wave fusion image, the encrypted result and the corresponding decryption map are displayed respectively.

(4)

Attack module: The yellow box area is the attack window and provides multiple attack algorithms and checks the robustness of the algorithm. There are four attack algorithms, namely noise attack, known plaintext attack, selected plaintext attack, and shear attack.

4.5.2. Simulation Experiments

Firstly, band selection and pseudo-color image synthesis are performed on the I256_8bit.mat datasets consisting of 189 bands, 12 bands, 51 bands, and 170 bands. Once the pseudo-color image is obtained and the audio selected, encryption and decryption can start, with the final result shown in Figure 23. Finally, the attack method can be selected and four attack methods are provided to detect the algorithm.

In the case of noise attacks, as shown in Figure 24a, a pop-up noise attack window is created by selecting noise attack in the status bar and clicking the Attack button, as shown in Figure 24b. The attack result graph under different noise intensities is obtained. image restoration can be achieved through initial visual interpretation or by interacting with image results through tracer points as shown in Figure 24c to find noise factors corresponding to PSNR > 40 for high-recovery images.

5. Conclusions

This paper uses the Gyrator domain transform to encrypt remote sensing images, and enriches the key by introducing acoustic information so that the algorithm is not easily destroyed, thus enriching the theoretical research on the problem. Two encryption methods have been proposed. Method I has strong randomness and can resist plaintext attacks. Method II is phase encryption. It is faster and more powerful. The phase dislocation makes the key more random, but it also reduces the robustness key and may affect the decryption of the image if noise and clipping are encountered during transmission. The PSNR function proves that both algorithms can encrypt and decrypt images.

According to the actual needs, an image encryption test system is designed in the domain of optical gyroscope, and the function and operation of the system are introduced and demonstrated.