A hybrid encryption scheme based on optical scanning cryptography and Fibonacci–Lucas transformation

The whole digital world today enforces an extended demand for information security and confidentiality at various levels of communication. The current field of interest includes secret transfer of personal data, medical data, and radiograph records of patients, restricted defense and intelligence information of countries, data related to examinations, etc. Medical images are regarded as important and sensitive data in the medical informatics systems. The confidentiality of health records are of primary concern while dealing with telemedicine or tele-health platforms. Medical image security in both intranet and Internet faces severe threats. Except for the intranet environment, medical image transmission over wired or wireless networks has elevated in demand. Cryptographic methods and encryption standards are revolutionary methods of securing information that transforms the original information to an unreadable format. To retrieve the original information, knowledge about encryption keys and encryption methods is required. As an advancement of the primary image encryption, optical image encryption methods have been developed. The optical means of encryption has the advantage of high speed and many degrees of freedom. Coherent optical systems are used for implementing most investigated optical encryption techniques. Refregier and Javidi first proposed and implemented this optical image encryption.¹ The above optical encryption uses the Double Random Phase Encoding (DRPE). This is the most established and well implemented optical encryption methodology.^2–9 Rajput and Nishchal introduced another implementation of DRPE with fractional Fourier transform.¹⁰ Instead of using random phase mask, Zamran et al. proposed¹¹ a scheme that uses deterministic phase masks. Others also developed various optical image encryption schemes.^12–19 References 20 and 21 employ the technique of compressive sensing for image encryption. Multiple image encryption based on optical methods is also implemented.^22,23

The aforementioned optical encryption uses coherent techniques. However, there are only a few incoherent optical encryption techniques implemented until now. These offer better Signal-to-Noise ratio (S/N) compared with the coherent systems. In conventional incoherent optical systems, the intensity of the object is manipulated by real and positive Point Spread Functions (PSFs). This gives an extra bonus that limits the way one can access the information. Optical scanning holography is one such example.^24–27 Advances in OSH and OSC are implemented in Refs. 28–30. The OSC, by using biometric key, was proposed by Tang and Zhang.³¹ Applying this principle, OSC for multidepth objects is discussed in Ref. 32. More sophisticated digital encryptions are done with the help of chaotic maps, and other diffusion models are also implemented.^33–41 Recently, several research studies on encryption schemes based on optical, chaotic, or both have been implemented.^42–57 

A hybrid encryption scheme that uses incoherent holography followed by Fibonacci–Lucas transform⁵⁸ is proposed here. The first stage encryption uses OSC with Fused Biometric Array (FBA) key. The second high-security level is provided by the application of a post-processing step on the encrypted hologram. Fibonacci–Lucas transformation provides the second stage encryption. The technique finds applications in securing large sized e-documents or medical documents safe.

This paper is arranged as follows. Section II explains the basic theory behind the cryptographic system design. The succeeding subsections of Sec. II deal with the Optical Scanning Cryptography with enhanced security. Section III contains the results of the proposed method. For effective evaluation of the proposed methodology, various figure of merits like Correlation Coefficient (CC) analysis, Structural Similarity Index Measure (SSIM), maximum deviation analysis, etc. are taken into account. The proposed system is tested against differential attacks also.

Due to the recent progress in the development of optical components and their improved and reliable performance, and the effectiveness of chaotic maps in scrambling the information randomly, a two-layer encryption–decryption system using these concepts is designed.

The section deals with the basic theory of optical scanning cryp-tography. Figure 1 shows the optical setup of encryption and decry-ption. Here, the traditional pupil set is used. Figure 2 represents the flowchart for the simulation of the optical setup of OSC.

A terahertz optical source, such as laser or visible light, acts as the light source. A beam splitter divides the light into two, and one of its frequencies is slightly shifted and passed to two pupils. These two pupils, p₁(x, y) and q₁(x, y), are illuminated by the laser beam of angular frequencies Ω and Ω + ω. The beam combiner combines two beams. The combined beam passes through a lens, and it is the scanning beam, which is used for 2D scanning of the object or image to be encrypted. The amplitude of object to be encrypted is denoted by U₀(x, y, z_c). Here, z_c (coding distance) represents the distance of object from the back focal plane of the lens. The X–Y scanner performs 2D scanning over the object.

The photodetector collects all the light transmitted from the object. The subsequent step is the electronic processing. In electronic processing stage, the current from the photodetector is given to a narrow Band Pass Filter (BPF) tuned at frequency ω, which delivers a heterodyne current i_ω(x, y, z_c). This current is demodulated by an electronic multiplier and a Low Pass Filter (LPF), which forms a lock-in amplifier. By multiplying the incoming signal i_ω (x, y, z_c) by cos(ωt) and sin(ωt) and through low-pass filtering, we obtain two signals, i_cos(x, y, z_c) and i_sin(x, y, z_c), respectively, which can be added in a complex way with the aid of a computer to give a final encrypted image, i(x, y, z_c),

The optical transfer function of the encryption stage at coding distance z_c is

where k₀ is the wave number. Here, only the intensity of the input pattern is processed and hence called incoherent system. OTF is determined by pupil function p₁(x, y) and q₁(x, y). Set q₁(x, y) as a pinhole and p₁(x, y) as a delta function. The derived OTF is

For decryption, the optical system is the same and the laser beams are now scanning pinhole. This helps in retrieving the key for the transmission section. Through electronic processing circuitry, the output of the photodetector will then be processed. The process can be repeated but by replacing z_c with z_d. Here, choose p₁(x, y) as a pinhole and keep q₁(x, y) as is,

This is the output generated at the decryption stage, where the decryption key has been inserted into the stage. The information is now stored in the digital computer to be used later to decrypt the information.

Fibonacci–Lucas transformation⁵⁸ is a 2D transformation that is specially meant for the encryption of images. The Lucas series is a Fibonacci series’ special case.

Several special cases of the Fibonacci series can be constructed by changing the seed values. By choosing an appropriate seed value for the Fibonacci series and selecting the same seed value for the Lucas series, a new transformation can be achieved. The Fibonacci–Lucas transformation is described as follows:

Here, F_n is given by

Here, L_n is given by

where N is the size of the input and n is the seed or initial point of Fibonacci or Lucas transformation. This is similar to the modified Arnold transform. FLT is periodic in nature with a maximum possible periodicity of N² − 1. Since this can produce different scrambling patterns with different periodicities, FLT will provide a more secure encryption compared to basic Arnold transform and modified Arnold transform.

Let m_i represent elements of a plain text in which i varies from 1 to 64. Table I represents elements of the plain text. Table II shows the elements of the transformed text using the FLT.

An optical cryptographic system based on optical scanning holography is proposed. In this system, the encryption stage consists of two levels. In the first stage, the encryption using incoherent holography (OSC) is performed at a coding distance of z_c. The second level of encryption consists of Fibonacci–Lucas transformation. The applied Fibonacci–Lucas transformation converts the pixels of the sine hologram image and cosine hologram image in the order given in Table II. The traditional pupils or any other pupils can be for this experiment are a pinhole and a unity function. The key used for encryption can also be varied.

The simulation experiment is conducted on a two-dimensional grayscale object. The two keys used for this Optical Scanning Holography (OSC) based encryption are a random phase key and a Fused Biometric Array (FBA) key.⁵⁷ 

Figures 3 and 4 show the biometric keys used for the proposed system.⁵⁸ Figure 5 shows the fused biometric array, which is created by fusing Figs. 3 and 4. The FBA is an array of iris images of the sender arranged in a specific order and then combined with the fingerprint of the sender. Each element of iris array corresponds to various eye movements of the same person (sender). The array used in the system consists of nine elements. In order to achieve PSF engineering, the pupils used for optical scanning are 2D rectangular function and 2D Gaussian function. The change in pupil functions in the OSC system finally results in a different pattern of light distribution for the encryption. From the optical encryption section, the encrypted hologram image is digitally encrypted once again. The second encryption is done separately for the real and imaginary parts of the OSC encrypted image. After the second level of encryption, the real and imaginary parts are combined to form a complex hologram image. The flow diagram of the encryption system is shown in Fig. 6. The optical stage in this experiment is conducted by taking into account the mathematical equivalents of components used in the optical system. The appropriate optical system transfer function for OSC (while taking the new pupil set into consideration) is implemented. Then, the second level of encryption is experimentally conducted by the mathematical modeling of the FLT. The optical decryption is done with the same fused biometric array and random phase key. The pupil set used is the same as that of the encryption side. The actual result will be reproduced only at a decoding distance, which is equal to the coding distance, exactly the same seed point for FLT that is used in the encryption. Figure 7 shows the implemented block diagram representation for the simulation experiment of the PSF engineered optical scanning cryptosystem. The amplitude distribution corresponding to the new pupil set is shown in this block.

In the decryption side, the inverse transformation is applied to the encrypted hologram to get back the original complex hologram image. After that, an optical decryption is done with the same fused biometric array and random phase key. The actual result will be obtained only at a decoding distance, which is equal to the coding distance. The flowchart for decryption is shown in Fig. 8.

The experiment is conducted with two sets of pupils. First, the experiment is carried out with a traditional pupil set, i.e., the pinhole and the unity. Since this experiment has resulted in poor encryption in the first stage, a new set of pupils is used for the implementation of the OSC system. p₁(x, y) is chosen as a Gaussian function, and q₁(x, y) is chosen as a rectangular function. The encryption and decryption keys are the same as in Sec. II D. The theoretical expressions for the used pupil pair and the corresponding OTF are shown as follows:

where γ is a constant in Eq. (10). With these pupils, the transfer function becomes

where k_x=$k0xf$⁠, k_y=$k0yf$ and k₀ is the wave number, and G(k_x, k_y) and R(k_x, k_y) are the spectral domain representations of p₁(x, y) and q₁(x, y), respectively. While decryption, the following condition is to satisfied:

The encrypted hologram is once again transformed with the help of Eqs. (14) and (15),

Here, H_T is the transformed hologram and H is the encrypted hologram.

The following figures show the simulation results of the proposed system. Simulations are done with Matlab R2016a. These are done with the help of a personal computer (i3 processor and 4 GB RAM). Many Matlab experiments have been carried out to test the proposed technique. A total of 10 images is used in experiments. Visual results for an ultra sound medical image are shown in this paper, and the results for all other images are tabulated. Figure 9 shows the sample ultrasound image taken from the database⁵⁶ for doing the simulation experiment. Figures 10 and 11 show the results obtained from the proposed technique with the traditional pupil set. These figures help in the qualitative evaluation of the technique.

Figures 12 and 13 represent the encrypted hologram images obtained for the first stage (engineered pupil) and the proposed hybrid scheme, respectively. Figures 14 and 15 show the decrypted results for correct key and wrong key (mismatched decoding distance for OSC), respectively. Qualitative analysis shows that the PSF engineering in the proposed method outweighs the traditional pupil based OSC. This can be justified by the tabulated results in the tables.

The sample data images are tested for this proposed crypto system. The parameters used for the verification of the quality of encryption are structural similarity index (SSIM), correlation (CC) between the original and encrypted images, and Maximum Absolute Deviation (MAD), a parameter that indicates the absolute deviation between pixel values of original and encrypted images. Next two parameters considered for the quantitative analysis of the proposed method are Maximum Deviation Analysis (Max Dev), a parameter that shows the area under graph of the absolute difference or deviation between the histogram curves of the original and encrypted images, and Irregular Deviation (Irr. Dev) Analysis, a measure that is calculated on each individual pixel value and its input image before getting the histogram. It does not preserve any information about the positions of the pixels. Tables III and IV indicate the values of different quality metrics calculated between the original image and the encrypted hologram image when the first stage of the proposed hybrid method uses traditional pupil pair and point spread function engineered pupil pair, respectively. SSIM and CC for all the sample sets are remarkably low, which indicates that the signal dependency for the test and reference image is very weak. The metric maximum deviation is more, and irregular deviation is less, which is an essential requirement of good encryption. The analysis reveals that the modified pupil with double encryption gives steady values so that this method could be more advisable for implementations. Another set of results between the encrypted and decrypted images for the proposed technique with unity and delta function pupil is shown in Table V. Tables VI and VII show a quantitative comparison between the proposed PSF engineered method and traditional OSC based encryption. The tables clearly indicate a visible hike in the SSIM (66.6%), MAD (8.67%) for the double encryption using PSFE-OSC, and FLT compared to the encryption based on PSFE-OSC only. Considering the quantitative analysis of single encryption based on traditional optical scanning cryptography only and proposed encryption system, the performance measures are superior in terms of SSIM (23.4%), CC (53.94%), and MAD (7.5%). It is clear that the proposed hybrid method provides good quality results.

Table VIII represents the effect of differential attack analysis on this proposed system. The opto-digital encryption system is efficient in terms of correlation coefficient (CC) also (71%). The vertical coefficient (VCC) outweighs the existing works by about 88.27%. The diagonal correlation coefficient (DCC) is efficient by about 54.47% as compared with the mentioned existing works. The maximum deviation parameter is performing well about 50.49% in comparison with the shown existing research studies. These comparisons are based on Refs. 42–57 [Tables IX–X].

The number of arrangements possible with the iris array in the proposed system is ∼362 880. A single order change in the array will lead to a wrong key, thereby resulting in a different light distribution from the object side.

An optical scanning cryptographic system, which uses double encryption and fused biometric array key, is proposed here. The second level of encryption is performed using a new chaotic like series called Fibonacci–Lucas transformation. The change of seed points of FLT allows many degrees of freedom for the keys. The number of arrangements in the biometric array is the next option for the key space. While implementing with an optical setup, each of the arrangements produces separate light distributions. This ensures the decryption of correct output with exactly the same order of biometric array that is used for encryption. Even a single change in position will not reproduce the input. Two variants of double encryption are proposed. The first cryptosystem is implemented with a traditional pupil set of OSH with FBA key. The second cryptosystem uses Gaussian and rectangular pupils with FBA key. Analysis reveals that the proposed cryptosystem with modified pupil based gives better performance or encryption–decryption system, resulting in a steady distribution of parameters. The proposed double encryption cryptosystem is compared with the traditional cryptosystem based on OSC. According to the tabulated results, the proposed cryptosystem’s performance is impressive.

To the best of our knowledge, this is the first time a post-encryption using a periodic chaotic like transformation is performed onto encrypted optical scanning hologram. This is a novel optical elucidation for the intruder problems that can affect medical image sharing. Current technology allows the sharing of these images using the cloud. In telemedicine, the current internal security gaps are lack of authentication and effectual encryption. By combining the proposed methodology with the existing digital cryptographic techniques, a system with more resistance to key based attacks can be achieved; simulation studies have been performed.

We would like to acknowledge the support extended by Indian Institute of Space Science and Technology and Center for Development of Imaging Technology, Kerala, India; Group at the Center for Biometrics Security Research (CBSR); National Laboratory of Pattern Recognition (NLPR); and Institute of Automation, Chinese Academy of Sciences (CASIA). The ultrasound image used in this experiment has been taken from the database of The SP Lab research group, which is a part of the Brno University of Technology, Czech Republic.

The article receives no external funding.

The authors declare no conflicts of interest.

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