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A new asymmetric optical double image encryption algorithm is proposed, which combines phase truncation and singular value decomposition. The plain text is encrypted with two-stage phase keys to obtain a uniformly distributed cipher text and two new decryption keys. These keys are generated during the encryption process and are different from encryption keys. It realizes asymmetric encryption and improves the security of the system. The unscrambling keys in the encryption operation are mainly related to plain text. At the same time, the system is more resistant to selective plain text attacks; it also improves the sensitivity of decryption keys. With the application of phase truncation, the key space expanded and the security of the cryptographic system is enhanced. The efficacy of the system is calculated by evaluating the estimated error between the input and retrieved images. The proposed technique provides innumerable security keys and is robust against various potential attacks. Numerical simulations verify the effectiveness and security of the proposed technique.

With the rapid development of modern online transactions, protecting information security has become increasingly difficult. Unauthorized users may access the data, so information hiding techniques are required to conceal the multimedia data from unintended users. To overcome this problem, many encoding methods have been developed in the field of optical security, biometrics, and holographic storage. Parameters such as phase, amplitude, wavelength, frequency, and polarization have multiple degrees of freedom, thus increasing the key space. Therefore, optical techniques are the prime requirement, where a given data will be transmitted secretly on it, and any attacker in the middle cannot obtain the data. In the previous years, many encoding techniques were developed [

In all these techniques, the indistinguishable RPM acts as a key during the decryption process. Similarly, due to the inherent nature of attacks, symmetrical schemes are used to deal with attacks, such as chosen cipher attack (CCA), chosen plain attack (CPA), and known plain attack (KPA) [

In this communication, we have introduced an unpredictable method for narrating duplex picture encryption using two-dimensional LCT and DPM besides esteem deterioration. The proposed LCT has comparable properties, such as numerous well-known scientific changes. We know that the classical DRPE experiences issue of key space and optical hub arrangement. To overcome these issues and to extend the key space, we favor utilizing a deterministic phase mask (DPM) [

In this article, the security of the cryptosystem depends on the linear canonical transform domain. The rest of the manuscript is organized as follows: Section

The deterministic phase mask (DPM) is produced by characterizing the arrangement

Generation of deterministic phase mask (DPM) for

For simplicity, we display the example when the order of encryption

LCT is generated by ABCD transform, generalized Fresnel transform, and amplified fractional Fourier transform. LCT may be directly changing course, with three parameters characterized as follows [

LCT_{(α, β, γ)} [] indicate that the LCT administrator has three genuine change parameters. These three parameters are independent of (_{(−α, −β, −γ)} [] denotes the inverse LCT. If

The optical implementation of an LCT using a single lens is shown in Figure

Representation of single lens LCT.

The LCT parameters

The SVD may be a numerical method utilized to diagonalizable matrices. It breaks down a

The matrix

If the multiplication arrangement has been changed to [UVS], [VUS], and [VSU], you cannot recompose the image. Finally, if the multiplication order is USV, the image can be restored. Because of multiplication, order plays an important role, so if the door opener gets any components from any channel, he will not be able to determine the first image.

PTFT can be a preparation of Fourier transform, but it has phase truncation, which means that because it only retains the amplitude (modulus part) of the Fourier spectrum, the phase part of the spectrum is truncated. Let

The phase truncation (PT) and the phase reservation (PR) operations can be expressed, respectively, as follows:

The input image _{1}, _{1}, _{1}. Here, _{1}) and the phase truncation (PT) value, bonded with another phase mask DPM (for the set _{2}, _{2}, _{2} order then further take linear canonical transformed which give the encrypted image, which is further amplitude-truncated to generate a second decryption key (DK_{2}). Also, at the conclusion, SVD is connected; steps are appearing underneath. Finally, the encrypted image

Flow chart for the encryption scheme.

The decryption process is shown in Figure _{1} again perform LCT.

Flow chart for decryption scheme.

Steps for decryption is as follows:

An optoelectronic experimental setup of the proposed encryption scheme has appeared in Figure _{2}, which is additionally connected with the computer and after that performing an inverse LCT. The resultant spectrum is recorded by the charged coupled device (CCD) camera and stored in the computer system. The phase truncated part may be made by CCD. The amplitude-truncated part may be done by phase-shifting interferometry. In the decryption process, the digitally acquired image _{1} irradiated with a laser source, and then subjected linear canonical transform through, then information is displayed on SLM_{2} is associated with computer and intensity of the decrypted image is recorded in the output plane.

Proposed optoelectronic encryption setup.

A series of computer simulations were performed using MATLAB (version R2020a) software to verify the efficiency of the proposed approach. The size of all the plain text images and target images selected were 256 × 256. In this scheme, two grayscale images tree and baboon Figures

(a, b) Input images of 256 × 256 pixels; (c, d) encrypted images; (e, f) decrypted images.

In this section, mean squared error (MSE) and a peak signal-noise ratio (PSNR) are used as the convergence criterion in the iterative process. If

In equation (^{−25} and 2.5223 × 10^{−25}. The minimum value of MSE demonstrates a superior likeness to the tried image.

PSNR measures the modification between the input image

The numerical esteem of PSNR obtained for our suggested algorithm for the tree and baboon image is 92.42 dB and 91.48 dB, respectively.

The relative error is computed between plain image and decoding image using mathematical expression represented in the following equation:

Among them,

Image encryption technique is sensitive to the initial values of the secret key. To obtain a sensitivity analysis of the image encryption technique, take incorrect parameters. The correct parameters are

(a, b) Retrieved images of tree, baboon for wrong two LCT orders; (c, d) decrypted images using another two-wrong parameter; (e, f) wrong values of DPM and RPM.

(a) MSE plots and the number of iterations of first LCT order; (b) MSE plot against the number of iterations for single image; (c) MSE plot and the number of iterations of second LCT order.

In this section, the correlation coefficient (CC) of two adjacent pixels in the original image and its encrypted image is examined. The CC is calculated by the following relations:

Calculation of CC value along horizontal, diagonal, and vertical of original and cipher images.

Image | Original image | Cipher image | ||||
---|---|---|---|---|---|---|

Horizontal | Diagonal | Vertical | Horizontal | Diagonal | Vertical | |

Lena | 0.9562 | 0.9509 | 0.9817 | 0.0093 | 0.0121 | 0.0086 |

Baboon | 0.9077 | 0.7566 | 0.8179 | 0.0077 | 0.0092 | 0.0157 |

OPT | 0.9497 | 0.9354 | 0.9695 | 0.0106 | 0.0020 | 0.0092 |

Optical image processing | 0.8994 | 0.8586 | 0.9465 | 0.0006 | 0.0038 | 0.0048 |

Plots of correlation distribution for randomly chosen 15,000-pixel pairs; (a, d, g, j, m) of input images; (b, e, h, k, n) correlation distribution of input images; (c, f, i, l, o) correlation distribution of encrypted images.

To demonstrate the ability to resist statistical attacks of the proposed image encryption algorithm, different kinds of statistical analysis methods are being utilized.

In order to obtain an effective and safe optical image encryption scheme, it should be able to encrypt different input images into an encryption form with similar histograms. The histograms of the tree, baboon, and their corresponding encrypted images are shown in Figures

Histogram plots (a) of an input tree and (b) input image of a baboon; (c, d) encrypted images of tree and baboon.

Entropy (

The chi-square value is calculated by the following equation [

where

The entropy calculations of grayscale and text images.

S. No | Image | Type/size | Entropy of input image | Entropy of encrypted image | Entropy of decrypted image |
---|---|---|---|---|---|

1 | Baboon | JEPG/256 | 7.0587 | 8.000 | 5.3239 |

2 | Tree | JEPG/256 | 6.9181 | 8.000 | 6.2558 |

3 | Optical image processing | JEPG/256 | 6.7272 | 7.647 | 6.7272 |

4 | OPT | JEPG/256 | 0 | 7.627 | 0 |

5 | Lena | JEPG/256 | 7.9216 | 7.8350 | 7.9216 |

Variance is the quantitative measure of histogram analysis. In addition, histogram variances are mainly used to quantitatively examine the uniformity of an image. Lower variance means higher uniformity of an image, alternatively the better security of the particular algorithm. The mathematical expression for calculating the variance is as follows [

Histogram variance results of 5 images using the proposed cryptosystem.

S. No | Image | Variance of original image | Variance of encrypted image | Ref. [ | Ref. [ | Ref. [ |
---|---|---|---|---|---|---|

1. | Lena | 38842.58 | 245.0547 | 244.31 | 276.39 | 260.70 |

2. | Tree | 161272.20 | 233.272 | — | — | — |

3. | Optical image processing | 21762.61 | 222.02 | — | — | — |

4. | OPT | 31541.41 | 230.593 | — | — | — |

5. | Baboon | 628013.38 | 228.59 | — | — | — |

The result shows our encryption scheme is better and more efficient, giving better results than others.

The chi-square test is the degree of deviation between the actual observation value and the theoretical inference value of the statistical sample. The larger the chi-square value, the less conformable and on the contrary, the more it is consistent. If the two values are completely equal, the chi-square value is 0, indicating that the theoretical value is completely consistent.where

Table

Results of the

S. No | Images | Result | |
---|---|---|---|

1 | Lena (512 × 512) | 232.1440 | Accept |

2 | Baboon (512 × 512) | 230.1243 | Accept |

3 | Tree (512 × 512) | 234.2467 | Accept |

4 | OPT (512 × 512) | 231.1053 | Accept |

In an occlusion attack, some parts of the encrypted image are blocked, which will cause the encrypted image to be blurred. This leads to blurred decrypted images depending on the size of the blocked parts. Different cases have been evaluated by taking the different sizes of the filter in the encrypted image. When the encrypted image is occluded or blocked, it impacts the quality of the recovered images. The occlusion is considered by changing the encrypted image by 10%, 25%, 50%, and 75%. As the occlusion percentage increases, the quality of the restored image gradually decreases. But still, the recovered images are visible till 25%. Figure

Occlusion results to the grayscale images of varying degrees of occlusion. (a) For 10% occlusion of the encrypted image. (b) For 25% occlusion of the encrypted image. (c) For 50% occlusion of the encrypted image. (d) For 75% occlusion of the encrypted image; (e) MSE plot with the percentage of the occluded area; (f) CC plot with the percentage of occluded area.

The encrypted image on the stage of image processing and image transmission is susceptible to different kinds of noise. These noises have a great influence on the quality of the decryption images. In this work, we have added Gaussian noise to the encoded image. The noise interferes with the ciphered images by relation [

Retrieved image for (a)

MSE plot of images tree and baboon with varying noise factor k.

The security of a cryptographic system depends on its resistance to four basic attacks. These basic attacks are cipher text only attack, known plain text attack, chosen plain text attack, and chosen cipher text attack. Among them, chosen plain text attack is the most powerful attack. If a cryptosystem is secure against this attack, it is secure against the other three attacks. The proposed scheme has eight keys; one from a deterministic phase mask, one from RPM, and six from linear canonical transform parameters, and the scheme is highly sensitive to all these parameters. If a small change is made in these parameters, the results would be completely different. So, the proposed scheme is secure enough against chosen plain text attacks and hence against other classical attacks too. The analysis proves that the presented system is resistant to several attacks which threaten the authenticity of any cryptosystem. Hence it is a much more secure and powerful yet simple cryptosystem.

In CPA, the attacker has the plain image and scheme. With respect to these, he will try the cipher image. Normally, DRPE is highly vulnerable to CPA. If an attacker chooses the Dirac delta function, which is shown in the below equation:

Dirac delta function is to be considered a single nonzero pixel at the centre of the image and all the other values are zero. In order to perform chosen plaintext analysis, created Dirac delta function is considered as plain image and cipher image calculation is given in the equation.

From the above equation, the second secret key is easily obtained by

(a) 3D plot of Dirac delta function; (b) Dirac delta function; (c) decrypted image of DRPE with CPA; (d) DRPE encrypted image with CPA; (e) encrypted image with DPM phase key.

The speed of the scheme measures the performance and the time taken by the scheme for the execution. It is important that the encryption and decryption schemes are fast enough to meet real time requirements. The scheme needs to be faster to reach the level of real time applications. The scheme has been tested against the speed by executing the scheme on a personal computer with configuration Intel (R) Core (TM) i3-2328 CPU @ 2.20 GHz–2.71 GHz, 2 GB RAM running Windows 10 on MATLAB R2020a 5 (9.6..0.1174912) 64 bit (win64), LM: 40664749. The total time taken for both encryption and decryption is 0.66 s. This time is due to the introduction of DPM. After introducing the DPM, the time taken is still very small and proves the scheme to be fast and efficient.

The proposed scheme is quantitatively compared with various recent schemes in terms of entropy, execution time, and key space. Table

Quantitative comparison analysis.

Parameters | Ref [ | Ref. [ | Ref. [ | Ref. [ | Present scheme |
---|---|---|---|---|---|

Element in key space | 9 | 6 (RPM + SPM) | 6 | 6 | DPM + RPM+6 (LCT order) |

Entropy of the encrypted image | 7.452 | 7.841 | 7.530 | 7.991 | 7.942 |

Execution time | 3.80553 s | 2.8371 s | — | — | 1.515843 s |

The Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI): Let encrypted images before and after one pixel change in the image be

In this contribution, it demonstrates that the simulation LCT encrypting system is capable of information security with noise-free recovery. The experimental results show that the linear canonical transform order can be considered as an extra security key. It is found that a small variation in order will lead to a large change in CC, MSE, and PSNR values. The use of DPM increases the key space also as extra security of the scheme. The proposed scheme is asymmetric and also uses the SVD operation that increases the security of the algorithm. The scheme is tested for various attacks such as occlusion and noise and it was found the scheme is not vulnerable. Numerical simulations are performed to demonstrate the feasibility and validity of this method. Key sensitivity has been analyzed by MSE curves under different decryption keys. Several possible attacks such as KPA and CPA have been considered and results demonstrate that the proposed encryption system has higher security.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors wish to thank the management of The NorthCap University, Gurugram, India, for their encouragement in supporting various research facilities.