Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2023; 7(6): 673-682
Published online December 25, 2023 https://doi.org/10.3807/COPP.2023.7.6.673
Copyright © Optical Society of Korea.
Seok Hee Jeon^{1}, Sang Keun Gil^{2}
Corresponding author: ^{*}skgil@suwon.ac.kr, ORCID 0000-0002-3828-0939
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, a three-key triple data encryption algorithm (TDEA) of a digital cryptosystem based on phase-shifting interferometry is proposed. The encryption for plaintext and the decryption for the ciphertext of a complex digital hologram are performed by three independent keys called a wavelength key k_{1}(λ), a reference distance key k_{2}(d_{r}) and a holographic encryption key k_{3}(x, y), which are represented in the reference beam path of phase-shifting interferometry. The results of numerical simulations show that the minimum wavelength spacing between the neighboring independent wavelength keys is about δλ = 0.007 nm, and the minimum distance between the neighboring reference distance keys is about δd_{r} = 50 nm. For the proposed three-key TDEA, choosing the deviation of the key k_{1}(λ) as δλ = 0.4 nm and the deviation of the key k_{2}(d_{r}) as δd_{r} = 500 nm allows the number of independent keys k_{1}(λ) and k_{2}(d_{r}) to be calculated as N(k_{1}) = 80 for a range of 1,530–1,562 nm and N(d_{r}) = 20,000 for a range of 35–45 mm, respectively. The proposed method provides the feasibility of independent keys with many degrees of freedom, and then these flexible independent keys can provide the cryptosystem with very high security.
Keywords: Cryptosystem, Digital hologram, Fourier optics, Optical encryption, Phase-shifting interferometry
OCIS codes: (060.4785) Optical security and encryption; (070.1170) Analog optical signal processing; (070.4560) Data processing by optical means; (090.1995) Digital holography; (090.2880) Holographic interferometry
In recent years, information hacking by unauthorized persons has become a serious problem because hacking techniques are being developed with rapid computer processing capability. In order to prevent attackers from getting confidential information, a cryptosystem is introduced in a data communication network to hide information. As for the digital types of information encryption methods, the advanced encryption standard (AES) [1] and triple data encryption standard (3DES) [2] are the most popular standard block encryption algorithms. Among the block cipher algorithms that use a symmetric key to encrypt information, 3DES is an approach that extends the short key size of DES. However, no matter how much the key size of 3DES increases, this algorithm needs more processing time and the larger quantity of data, notwithstanding increased security. Cheng et al. [3] compared overall encryption efficiency in terms of speed for a given electronic hardware platform with the three standard block encryption schemes DES, 3DES and AES. Digital encryption techniques generally use electronic devices to cipher information, while optical methods to encrypt information have been researched due to high-speed parallel processing and two-dimensional large data handling advantages. Various methods for optical cryptosystems have been proposed [4–6], among which digital holography [7–9] and phase-shifting interferometry [8–15] are promising methods to mix with digital processing. In particular, a complex digital hologram function with amplitude and phase information is digitally calculated by interferograms detected on a charge-coupled device (CCD) in phase-shifting interferometry. Recently, researchers have presented optical double or triple-key encryption methods to enhance security strength and data processing volume. Jeon and Gil [16] proposed a triple DES algorithm and its optical implementation based on dual XOR logic operations, Ahouzi et al. [17] proposed an advanced algorithm using a triple random-phase encryption (TRPE) scheme in the Fourier transform domain that improves the security of optical encryption based on double random-phase encryption (DRPE), and Kumari et al. [18] also suggested a TRPE cryptosystem in the Fresnel domain. As for the multiple keys for optical encryption, Zhang and Wang [19] applied the parameters of the optical configuration to serve as additional keys for optical image encryption based on interference, which motivated us to introduce the concept of three independent keys for phase-shifting digital interferometry.
In this paper, we propose the three-key triple data encryption algorithm (TDEA) of a digital cryptosystem based on phase-shifting interferometry. The proposed method carries out digital encryption and decryption processes with three independent keys called a wavelength key, a reference distance key and a holographic encryption key. The independent wavelength keys are assumed to be determined by the tunable wavelength of a light source and the independent reference distance keys are determined by varying placement of the reference input from the Fourier transform lens in optical interferometry. In Section Ⅱ, 3DES is briefly reviewed, and the proposed three-key TDEA is described. In Section Ⅲ, an evaluation of the proposed method is verified by the results of numerical simulations. Conclusions are summarized in Section Ⅳ.
The DES has been the most widely used symmetric key block cryptography and was chosen as a standard by the American National Standard Institute (ANSI) in 1977. The algorithm uses a fixed-length 56-bit key to encrypt and decrypt a 64-bit block of data. However, its 56-bit key can no longer guarantee enough security in recent cryptanalytic attacks against block ciphers. Generally, increasing the key length makes the cryptosystem more secure. The problem of increasing the key length can be overcome by using double or triple-length keys. In 1998, a 3DES called triple data encryption algorithm (TDEA), in which DES is applied three times, was adopted as the standard ANSI X9.52 [2]. Figure 1 shows a block diagram of the 3DES encryption and decryption procedure. The 3DES algorithm consists of three DES keys (k_{1}, k_{2} and k_{3}) for the cryptosystem. There are two variations of 3DES. If three 56-bit keys k_{1}, k_{2} and k_{3} are independent DES keys, it is referred to as three-key 3DES and produces an effective key length of 168 bits. If two keys, k_{1} and k_{2}, are independent keys and k_{3} is the same as k_{1}, it is referred to as two-key 3DES and gives an effective key length of 112 bits. The resultant 3DES algorithm is much harder to break compared to a single DES. The encryption and decryption of the 3DES algorithm is as follows. Assume that k_{1}, k_{2} and k_{3} are three independent keys in the 3DES cryptosystem. The encryption and decryption processes are expressed as
where m(x, y) is a plaintext to be encrypted and c(x, y) is a ciphertext.
In this paper, a cryptosystem that performs three-key TDEA based on the phase-shifting interferometry principle is proposed. The concept of the proposed method is described by the optical configuration shown in Fig. 2, which is based on a Mach–Zehnder interferometer architecture. A tunable laser diode beam is collimated by a collimating lens (CL) and is linearly polarized by a polarizer (P1), and it is divided by a beam splitter (BS1) into two plane waves of the object and the reference beams traveling in different directions. When shutters S1 and S2 are open, the downward object beam passes through an input amplitude-type spatial light modulator (SLM1) and a random phase mask (RPM), while the rightward reference beam passes through an input phase-type spatial light modulator (SLM2) and a λ/4 plate. Two lenses (L1 and L2) form a Fourier transform of the input functions into a CCD. A random phase mask is adopted to improve the dynamic range of the spatial frequency in the spatial frequency plane on the CCD. The RPM function is represented as r(x, y) = exp[j2πq(x, y)], where q(x, y) is a randomly distributed function over the interval [0, 1]. A λ/4 plate makes the wave along the vertical axis (s-polarization axis) occur with no phase shift and the wave along the horizontal axis (p-polarization axis) occur with a phase shift of π/2 radians. This scheme provides two-step phase-shifting interferometry [20].
Firstly, a cryptosystem based on optoelectronic two-step phase-shifting interferometry, considering the wavelength of light and the distance between the input and the lens, is briefly described. Let m(x, y) be a plaintext to be encrypted and k(x, y) be a holographic encryption key in phase-shifting interferometry. The Fourier transform diffraction patterns of the object and reference beams form complex amplitude distributions at the output spatial frequency (u, v) plane, and are expressed as
where λ is the wavelength of the light source, k = 2π/λ, f is the focal length of the lens, d_{o} is the object distance between SLM1 and lens L1, and d_{r} is the reference distance between SLM2 and lens L2. The output complex distribution will be the exact Fourier transform of the input except for the phase factor outside the integral when the distance between the input and the lens is the focal length of the lens, that is d_{o} = f and d_{r} = f. It is denoted from Eqs. (3) and (4) that the phase factor in front of the Fourier transform of m(x, y)r(x, y) is P_{o}(u, v; λ, d_{o}) and the phase factor in front of the Fourier transform of k(x, y) is P_{r}(u, v; λ, d_{r}), respectively. It is interesting to note that even if the object distance d_{o} has any distance value, resulting in any phase factor P_{o}(u, v; λ, d_{o}), it does not affect the phase change contribution in Eq. (3) because the RPM function r(x, y) includes a random phase distribution, while the phase factor P_{r}(u, v; λ, d_{r}) with different reference distance d_{r} affects the phase change contribution in Eq. (4). If the object distance d_{o} is assumed to have the focal length f for convenience, and if the integral part in Eqs. (3) and (4), which represent Fourier transforms of the object and reference complex distributions in the spatial frequency domain, are expressed as M(u, v) = F{m(x, y)r(x, y)} = |M(u, v)| e ^{jϕM(u,v)} and K(u, v) = F{k(x, y)} = |K(u, v)| e ^{jϕK(u,v)}, respectively, then Eqs. (3) and (4) are rewritten by
With Eqs. (5) and (6), the intensity pattern recorded by the CCD is given by
where ^{δ} is a phase shift in the reference beam. Two interference intensities at the output plane can be achieved with digital two-step phase-shifting interferometry when a phase shift of π/2 occurs between the s-polarization axis and the p-polarization axis in the reference beam. The suitable polarization direction of an output polarizer (P2) in front of the CCD gives two interference intensities. Representing the complex amplitude distribution of the object and the reference beams with Eqs. (5)–(7) can be expressed as
where variables in spatial frequency coordinates are omitted and ∆φ(u, v; λ, d_{r}) = φ_{M}(u, v) − {φ_{Pr}(u, v; λ, d_{r}) + φ_{K}(u, v)} denotes the phase difference between the reference and the object beams. This intensity pattern recorded by the CCD shows a noise-like random distribution due to the randomness of Eq. (3). Additionally, only the intensity distribution of the object beam I_{o} = |M(u, v)|^{2} and only the intensity distribution of the reference beam I_{r} = |P_{r} (u, v; λ, d_{r})|^{2} |K(u, v)|^{2}, which are DC-terms in the interference intensity of Eq. (8), are recorded on the CCD by controlling shutters S1 and S2 in the optical setup. In this method, a complex digital hologram function generated from phase-shifting interferometry is a kind of ciphertext that also provides random phase and random amplitude distributions. If the complex digital hologram function is assumed to be H(u, v; λ, d_{r}) = A(u, v; λ, d_{r}) e^{j∆ϕ(u,v;λ,dr)}, the amplitude A(u, v; λ, d_{r}) and the phase ∆φ(u, v; λ, d_{r}) can be calculated by two intensities I_{1}(u, v; λ, d_{r}; π/2) and I_{2}(u, v; λ, d_{r}; 0) after removing the DC-term |M|^{2} + |P_{r}|^{2} |K|^{2} from Eq. (8) as
To decrypt m(x, y) of the plaintext, the object complex function M(u, v) should be retrieved from the ciphertext of the complex digital hologram function H(u, v; λ, d_{r}) by the complex distribution function U_{r}(u, v; λ, d_{r}) of Eq. (4) including the holographic encryption key k(x, y) used in phase-shifting interferometry. According to the hologram memory principle, the object wavefront is reconstructed by illuminating the reference wavefront to the hologram. Likewise, the object complex function M(u, v) can be reconstructed only with knowledge of the reference complex function U_{r}(u, v; λ, d_{r}) which is now acting as a decryption key. Consequently, the complex distribution M(u, v) and the plaintext m(x, y) are recovered as follows:
where F ^{−1}{∙} denotes an inverse Fourier transform.
From now on, we propose a new digital method of three-key TDEA to introduce three keys in the phase-shifting interferometry scheme, which easily improves security without adding optical components to the phase-shifting interferometry encryption system. From the proposed optical configuration shown in Fig. 2, let us consider the wavelength of laser diode light as an independent key k_{1}(λ) and the reference distance between SLM2 and lens L1 as another independent key k_{2}(d_{r}), respectively, while the holographic encryption key k(x, y) in phase-shifting interferometry is maintained as an independent key k_{3}(x, y). With this concept, the complex distribution function U_{r}(u, v; λ, d_{r}) of Eq. (4) is modified into U_{r}(u, v; k_{1}(λ), k_{2}(d_{r})) so that the interference intensity of Eq. (7) is expressed as
Since the object and the reference beams in Eq. (13) are changed by two keys, k_{1}(λ) and k_{2}(d_{r}), the two-step phase-shifting interferometry makes it so that the complex digital hologram function of the ciphertext is modified into
where the amplitude A and the phase ∆φ are rewritten by replacing λ and d_{r} in Eqs. (9) and (10) with k_{1}(λ) and k_{2}(d_{r}). This means that the complex hologram function is dependent on the wavelength and the reference distance in the cryptosystem, and therefore the ciphertext is made by using three independent keys, k_{1}(λ), k_{2}(d_{r}) and k_{3}(x, y). Now, to decrypt m(x, y) of the plaintext from the ciphertext H(u, v; k_{1}(λ), k_{2}(d_{r}), k_{3}), it is necessary to know all three keys, not only about k_{1}(λ) of the light wavelength but also about k_{2}(d_{r}) of the holographic encryption key k_{3}(x, y) location in the reference beam. The decryption process for the proposed three-key TDEA is accomplished as follows:
Flowcharts of the encryption and decryption processes for the proposed three-key TDEA are shown in Fig. 3, where ⊗ represents the inner product between pixels, FT and IFT denote Fourier transform and inverse Fourier transform, PSI denotes phase-shifting interferometry, SQ denotes a square function, and TH denotes a function to make binary data by a proper threshold. Although wavelength tuning of the light source and precise location control of the distance are required to implement independent keys k_{1}(λ) and k_{2}(d_{r}) optically, the effect of three-key encryption is so powerful that the cryptosystem can improve security.
In the proposed digital algorithm applied from the optical configuration as shown in Fig. 2, a pair of a wavelength key and a reference distance key {k_{1}(λ), k_{2}(d_{r})} can be used as a variable public key like a one-time password (OTP). If a specific wavelength key k_{1}(λ_{s}) is challenged to the host server, a specific reference distance key k_{2}(d_{rs}) is acknowledged to the user according to a predetermined specific key pair {k_{1}(λ_{s}), k_{2}(d_{rs})}. With these two specific key pairs, a user can encrypt private information with a holographic encryption key k_{3}(x, y) which is used independently as a private key of a user. Conceptually, the encryption and decryption processes of the proposed three-key TDEA are expressed as
In the encryption process for the proposed three-key TDEA, generating a ciphertext of a complex digital hologram can be performed optically or digitally. However, it is very difficult to control the wavelength λ of light and to align the reference distance d_{r} precisely in implementing the optical setup shown in Fig. 2. A practical method that ignores these limited optical problems is to apply the proposed algorithm to computer-oriented digital processing, which is more convenient than the optical technique. To demonstrate the validity of the proposed method, numerical simulations using MATLAB (R2021) are carried out to show the performance. In this paper, all the data size of inputs is 256 × 256 pixels. A binary image of a monkey, instead of encoded digital data from information, is used as an input plaintext m(x, y) for visual convenience, as shown in Fig. 4(a).
From the proposed new algorithm shown in Figs. 3 and 4, a specific wavelength of light is chosen as the first independent wavelength key k_{1}(λ) and a specific reference distance is chosen as the second independent distance key k_{2}(d_{r}) before processing the phase-shifting interferometry encryption. The third holographic encryption key k_{3}(x, y) used in phase-shifting interferometry is assumed to have a randomly generated pattern, as shown in Fig. 4(b). By using Eq. (14), a complex digital hologram function H(u, v; k_{1}(λ), k_{2}(d_{r}), k_{3}) is generated by the three keys. It is clear that a plaintext is encrypted in a ciphertext with noise-like random phase and random amplitude distributions, A(u, v; k_{1}(λ), k_{2}(d_{r}), k_{3}) and ∆ϕ(u, v; k_{1}(λ), k_{2}(d_{r}), k_{3}), respectively. Figures 4(c) and 4(d) show the randomly distributed amplitude and phase maps of the complex digital hologram of the ciphertext.
To verify the reliability of the proposed algorithm, the influence of three independent keys, k_{1}, k_{2} and k_{3}, on the decrypted image is evaluated. The mean square error (MSE) between the decrypted binary image d(x, y) and the original plaintext image m(x, y), representing a relative error between them, is introduced as
where p × q is the pixel size of the image data. If the decrypted image is retrieved without any error, the MSE is calculated as 0%. In a phase-shifting interferometry cryptosystem, the most important encryption key is the holographic encryption key k_{3}(x, y), which is determined by independent users selecting a 2-D random distribution function in the key. A detailed explanation about phase-shifting digital holographic optical encryption is given in the [16, 20]. In this paper, we omit the role of the third key k_{3}(x, y).
First, we examine the suitability of the wavelength key k_{1}(λ) as an independent encryption key. To know the influence of the key k_{1}(λ) on the decrypted image, the MSE according to fractional wavelength change is analyzed for three different light sources. The key k_{1}(λ) with three different designing center wavelengths λ_{c} is given by k_{1}(λ_{1} = 1,550 nm), k_{1}(λ_{2} = 1,300 nm) and k_{1}(λ_{3} = 670 nm), respectively. The deviation of the wavelength and the fractional wavelength change are defined as δλ = λ_{c} − λ and ∆λ = δλ/λ_{c}, respectively. Figure 5(a) shows an MSE graph with respect to fractional wavelength change. As shown in Fig. 5(a), the shorter wavelength with k_{1}(λ_{3} = 670 nm) allows larger MSE compared to the long wavelength with k_{1}(λ_{1} = 1,550 nm) for ∆λ < 0.5 × 10^{−3}% when the same ∆λ is given. However, the MSE reaches more than 90% for ∆λ > 0.5 × 10^{−3}% regardless of different wavelengths, which means that the decrypted image d(x, y) is quite a different from the original image m(x, y). The deviation of the wavelength of k_{1}(λ_{1} = 1,550 nm) is about δλ = 0.007 nm when ∆λ = 0.5 × 10^{−3}%, for example. It is interesting that a very small deviation of the light wavelength can separate a new independent wavelength key k_{1}(λ_{1} + δλ) with different wavelength from an independent wavelength key k_{1}(λ_{1}).
Second, to examine the influence of the distance key k_{2}(d_{r}) on the decrypted image as an independent encryption key, the MSE according to fractional reference distance change is analyzed for different distances. The key k_{2}(d_{r}) with different reference distances d_{r} is chosen as k_{2}(d_{r1} = 30 mm), k_{2}(d_{r2} = 40 mm), k_{2}(d_{r3} = 50 mm), k_{2}(d_{r5} = 60 mm) and k_{2}(d_{r6} = 70 mm), respectively. The deviation of the distance and the fractional distance change are defined as δd_{r} = d_{c} − d_{r} and ∆d_{r} = δd_{r}/d_{c}, respectively, where d_{c} is a designing center reference distance from the lens to the input location. Figure 5(b) shows an MSE graph with respect to fractional distance change. As shown in Fig. 5(b), the longer distance with k_{2}(d_{r6} = 70 mm) allows a larger MSE compared to the wavelength with k_{2}(d_{r1} = 30 mm) for ∆d_{r} < 1.5 × 10^{−4}% when the same ∆d_{r} is given. The deviation of the reference distance of k_{2}(d_{r2} = 40 mm) is about δd_{r} = 60 nm when ∆d_{r} = 1.5 × 10^{−4}%, for example. The decrypted image d(x, y) is not discriminated from the original image m(x, y) for ∆d_{r} > 1.5 × 10^{−4}% because the MSE reaches more than 90% regardless of different distances. It is also noted that the distance separating the next independent wavelength key k_{2}(d_{r1} + δd_{r}) with different reference distance from an independent distance key k_{2}(d_{r1}) is very small.
In addition, Fig. 6 shows the results of decryption for the plaintext of binary image m(x, y) when the correct wavelength key k_{1}(λ = 1,550 nm) and the correct reference distance key k_{2}(d_{r} = 40 mm) are given in the proposed three-key TDEA. Figures 6(a)–6(c) show the decrypted images with MSE = 14.1% in the case of δλ = 0.002 nm (∆λ = 0.13 × 10^{−3}% deviation error), MSE = 63.7% in the case of δλ = 0.004 nm (∆λ = 0.26 × 10^{−3}% deviation error), and MSE = 89.3% in the case of δλ = 0.007 nm (∆λ = 0.45 × 10^{−3}% deviation error), respectively. Figures 6(d)–6(f) show the decrypted images with MSE = 45.0% in the case of δd_{r} = 20 nm (∆d_{r} = 0.5 × 10^{−4}% deviation error), MSE = 74.1% in the case of δd_{r} = 30 nm (∆d_{r} = 0.75 × 10^{−4}% deviation error), and MSE = 91.3% in the case of δd_{r} = 50 nm (∆d_{r} = 1.25 × 10^{−4}% deviation error), respectively.
Third, let us evaluate the dependence of the MSE on the change of the wavelength key k_{1}(λ) and the distance key k_{2}(d_{r}). To examine the effectiveness of key k_{1}(λ) and key k_{2}(d_{r}), it is assumed that the specific parameters used for the correct encryption and decryption in simulation are as follows: The wavelength key k_{1}(λ = 1,550 nm), the reference distance key k_{2}(d_{r} = 40 mm), the object distance d_{o} = 50 mm, and focal length of the lens f = 50 mm shown in Fig. 2, and the correct holographic encryption key k_{3}(x, y) shown in Fig. 4(b). Figure 7(a) shows an MSE graph for wavelength deviation from the correct wavelength key k_{1}(λ = 1,550 nm) with the correct reference distance key k_{2}(d_{r} = 40 mm). Although a very small deviation occurs, the MSE increases steeply, and the original plaintext cannot be decrypted. Figure 7(b) shows a detailed part near the correct wavelength key k_{1}(λ = 1,550 nm). The solid blue line shows the MSE for a wavelength key k_{1}(λ = 1,550 nm), while the red dashed line shows the MSE for another wavelength key k_{1}(λ = 1,550.007 nm). If k_{1}(λ = 1,550 nm) is the correct wavelength key in the proposed TDEA cryptosystem, then k_{1}(λ = 1,550.007 nm) is the incorrect key in the same TDEA cryptosystem. From the solid blue line, about 90% of the MSE is shown at a wavelength of λ = 1,550.007 nm. This means that k_{1}(λ = 1,550 nm) and k_{1}(λ = 1,550.007 nm) are keys that are independent from each other. Figure 8(a) shows an MSE graph for distance deviation from the correct reference distance key k_{2}(d_{r} = 40 mm) with the correct wavelength key k_{1}(λ = 1,550 nm). A very small deviation of the reference distance makes a steep MSE increase, similarly to the deviation of the wavelength. Figure 8(b) shows the detailed part near the correct reference distance key k_{2}(d_{r} = 40 mm). The solid blue line shows the MSE for a distance key k_{2}(d_{r} = 40 mm), while the red dashed line shows the MSE for another distance key k_{2}(d_{r} = 40.00005 mm). If k_{2}(d_{r} = 40 mm) is the correct wavelength key in the proposed TDEA cryptosystem, then k_{2}(d_{r} = 40.00005 mm) is the incorrect key in the same TDEA cryptosystem. From the solid blue line, about 90% of the MSE is shown at a reference distance of d_{r} = 40.00005 mm. This means that k_{2}(d_{r} = 40 mm) and k_{2}(d_{r} = 40.00005 mm) are keys that are independent from each other. Also, it is meaningful to show the separability of the incorrect keys k_{1}(λ = 1,550.007 nm) and k_{2}(d_{r} = 40.00005 mm) in the role of the independent key. In Fig. 7(b), the magenta dashed line shows the MSE for a correct wavelength key k_{1}(λ = 1,550 nm) and an incorrect distance key k_{2}(d_{r} = 40.00005 mm), which is a different view of the incorrect distance key k_{2}(d_{r} = 40.00005 mm) shown in Fig. 8(b) with respect to wavelength. Similarly, the magenta dashed line in Fig. 8(b) shows the MSE for an incorrect wavelength key k_{1}(λ = 1,550.007 nm) and a correct distance key k_{2}(d_{r} = 40 mm), which is a different view of the incorrect wavelength key k_{1}(λ = 1,550.007 nm) shown in Fig. 7(b) with respect to reference distance. According to a similar study on optical image encryption based on interference in [19], the simulation results showed that the wavelength sensitivity is 2 × 10^{−5} nm and the distance sensitivity is 2 nm.
To verify the cipher resistance to differential attacks on the proposed method, a number of pixel changing rate (NPCR) test is performed on the ciphertext c(x, y). The NPCR N(c_{1}, c_{2}) is defined as
The first evaluation is an NPCR randomness test for the proposed three-key TDEA, where c_{1} and c_{2} are ciphertexts before and after one pixel change in a plaintext. N(c_{1}, c_{2}) is calculated as about 4.88% from 10,000 iterations, which gives 0.003% of the MSE. The second consideration is that c_{1} and c_{2} are ciphertexts obtained by deviation from the correct wavelength key k_{1}(λ = 1,550 nm) or the correct reference distance key k_{2}(d_{r} = 40 mm) in the optical configuration. When the wavelength deviation δλ is 0.007 nm, N(c_{1}, c_{2}) is calculated as about 95.4%. When the distance deviation δd_{r} is 50 nm, N(c_{1}, c_{2}) is calculated as about 94.5%. These NPCR results suggest that the proposed method provides strong cipher resistance to attacks.
Finally, we discuss the feasibility of the proposed method in secure block cryptography with symmetric key. According to the three-key TDEA standardized as ANSI X9.52, the key length is 168 bits for block encryption. However, the proposed three-key TDEA provides a digital cryptosystem that has three independent keys, a wavelength key k_{1}(λ), a distance key k_{2}(d_{r}) and a 2-D holographic encryption key k_{3}(x, y) of 256 × 256 bits. At first, k_{3}(x, y) is dependent on the displaying capability of the SLM in implementing the cryptosystem optically. However, the size of key k_{3}(x, y) does not matter if only the proposed cryptosystem is implemented digitally. The only limitation to determine key k_{3}(x, y) is that a longer key needs a longer processing time in encryption. In this paper, we choose key k_{3}(x, y) of size 256 × 256 bits, which is reasonable for the optical or digital design. Next, the number of independent wavelength keys k_{1}(λ) is determined by considering a range of the wavelength. In the results as shown in Fig. 5(a) and Fig. 7(b), the minimum wavelength deviation to separate the independent keys of k_{1}(λ) is about δλ = 0.007 nm. But we choose the deviation of the key k_{1}(λ) as δλ = 0.4 nm, which is sufficient enough to discriminate each independent key k_{1}(λ). Next, we consider a tunable range of k_{1}(λ) as 1,530–1,562 nm because a tunable laser diode source with such a wavelength range can be achieved with the commercial laser diodes used in optical technology. The reason why we select δλ = 0.4 nm in the range of 1,530–1,562 nm is that it is the wavelength spacing used in the dense wavelength division multiplexing (DWDM) optical communication technique. From the range of 1,530–1,562 nm, we achieve the total number of independent wavelength keys k_{1}(λ) as N(k_{1}) = (1,562 − 1,530) / 0.4 = 80. It is also possible to get N(k_{1}) = (1,562 − 1,530) / 0.01 = 3,200 increasingly if we select δλ = 0.01 nm, which has no problem in the proposed digital cryptosystem. Lastly, the number of independent reference distance keys k_{2}(d_{r}) is determined by considering a range of reference distance. In the results as shown in Fig. 5(b) and Fig. 8(b), the minimum distance deviation to separate the independent keys of k_{2}(d_{r}) is about δd_{r} = 50 nm, but we choose the deviation of the key k_{2}(d_{r}) as δd_{r} = 500 nm, which gives sufficient deviation to separate each independent key k_{2}(d_{r}). In this paper, we consider a tunable range of k_{2}(d_{r}) as 35–45 mm with a designing center distance of k_{2}(d_{r} = 40 mm). From the range of 35–45 mm, we achieve the total number of independent distance keys k_{2}(d_{r}) as N(k_{2}) = (45 − 35) / 0.0005 = 20,000. If we expand the range of k_{2}(d_{r}) as 25–55 mm, then N(k_{2}) = (55 − 25) / 0.0005 = 60,000, three times larger than in the case of 35–45 mm. Furthermore, if we choose the deviation of the key k_{2}(d_{r}) as δd_{r} = 100 nm, which is two times larger than the minimum distance deviation of the independent keys of k_{2}(d_{r}) in the proposed cryptosystem, the total number of independent distance keys k_{2}(d_{r}) as N(k_{2}) = (45 − 35) / 0.0001 = 100,000 can be achieved. This flexibility is an advantage of a digital processing cryptosystem compared to an optical technique. The conventional three-key TDEA standard has a key length of 168 bits. This means that 2^{168} attempts are needed to find the exact security key, which is very sufficient to protect against key attacks in general. However, the proposed method is assumed to have the 2-D holographic encryption key k_{3}(x, y) of size 256 × 256 bits, and in addition to that, the wavelength key k_{1}(λ) and the distance key k_{2}(d_{r}) are assumed to have N(k_{1}) = 80 and N(d_{r}) = 20,000, respectively. Thus, the proposed cryptosystem requires 2^{256×256} × 80 × 20,000 attempts to hack the correct block data, which provides much more robustness to the cryptosystem under attack, and more importantly, it can handle 256 × 256 bits block encryption of data instead of encrypting a 64-bit block of data in the conventional three-key TDEA.
We the propose three-key TDEA of a digital cryptosystem based on phase-shifting interferometry, where a ciphertext of an encrypted binary image is acquired as a complex digital Fourier hologram function with complex amplitude and phase distribution functions. The encryption process is performed with the use of three encryption keys called a wavelength key k_{1}(λ), a reference distance key k_{2}(d_{r}) and a holographic encryption key k_{3}(x, y), which are independently represented in the reference beam path of the phase-shifting interferometry optical architecture. Different wavelengths of the light source and different reference distances determine independent keys k_{1}(λ) and k_{2}(d_{r}), respectively, which provides the keys with many degrees of freedom, and these flexible independent keys can protect against attacks on cryptosystems. The decryption process is carried out by digital processing for the ciphertext of the complex digital hologram with the three keys used in encryption. For the proposed three-key TDEA, the minimum wavelength deviation between the neighboring wavelength keys of k_{1}(λ) and the minimum distance deviation between the neighboring distance keys of k_{2}(d_{r}) are achieved as about δλ = 0.007 nm and about δd_{r} = 50 nm, respectively. For the proposed method in this paper, by choosing the deviation of the key k_{1}(λ) as δλ = 0.4 nm and the deviation of the key k_{2}(d_{r}) as δd_{r} = 500 nm, the number of independent keys k_{1}(λ) and k_{2}(d_{r}) is calculated as N(k_{1}) = 80 for a range of 1,530–1,562 nm and N(d_{r}) = 20,000 for a range of 35–45 mm, respectively, so that 2^{256×256} × 80 × 20,000 attempts are needed to find the correct key. The results of numerical simulations verify the feasibility of the proposed method.
This work was supported by Incheon National University (International Cooperative) Research Grant in 2021.
Incheon National University (International Cooperative) Research Grant in 2021.
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.
Curr. Opt. Photon. 2023; 7(6): 673-682
Published online December 25, 2023 https://doi.org/10.3807/COPP.2023.7.6.673
Copyright © Optical Society of Korea.
Seok Hee Jeon^{1}, Sang Keun Gil^{2}
^{1}Department of Electronic Engineering, Incheon National University, Incheon 22012, Korea
^{2}Department of Electronic Engineering, The University of Suwon, Hwaseong 18323, Korea
Correspondence to:^{*}skgil@suwon.ac.kr, ORCID 0000-0002-3828-0939
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, a three-key triple data encryption algorithm (TDEA) of a digital cryptosystem based on phase-shifting interferometry is proposed. The encryption for plaintext and the decryption for the ciphertext of a complex digital hologram are performed by three independent keys called a wavelength key k_{1}(λ), a reference distance key k_{2}(d_{r}) and a holographic encryption key k_{3}(x, y), which are represented in the reference beam path of phase-shifting interferometry. The results of numerical simulations show that the minimum wavelength spacing between the neighboring independent wavelength keys is about δλ = 0.007 nm, and the minimum distance between the neighboring reference distance keys is about δd_{r} = 50 nm. For the proposed three-key TDEA, choosing the deviation of the key k_{1}(λ) as δλ = 0.4 nm and the deviation of the key k_{2}(d_{r}) as δd_{r} = 500 nm allows the number of independent keys k_{1}(λ) and k_{2}(d_{r}) to be calculated as N(k_{1}) = 80 for a range of 1,530–1,562 nm and N(d_{r}) = 20,000 for a range of 35–45 mm, respectively. The proposed method provides the feasibility of independent keys with many degrees of freedom, and then these flexible independent keys can provide the cryptosystem with very high security.
Keywords: Cryptosystem, Digital hologram, Fourier optics, Optical encryption, Phase-shifting interferometry
In recent years, information hacking by unauthorized persons has become a serious problem because hacking techniques are being developed with rapid computer processing capability. In order to prevent attackers from getting confidential information, a cryptosystem is introduced in a data communication network to hide information. As for the digital types of information encryption methods, the advanced encryption standard (AES) [1] and triple data encryption standard (3DES) [2] are the most popular standard block encryption algorithms. Among the block cipher algorithms that use a symmetric key to encrypt information, 3DES is an approach that extends the short key size of DES. However, no matter how much the key size of 3DES increases, this algorithm needs more processing time and the larger quantity of data, notwithstanding increased security. Cheng et al. [3] compared overall encryption efficiency in terms of speed for a given electronic hardware platform with the three standard block encryption schemes DES, 3DES and AES. Digital encryption techniques generally use electronic devices to cipher information, while optical methods to encrypt information have been researched due to high-speed parallel processing and two-dimensional large data handling advantages. Various methods for optical cryptosystems have been proposed [4–6], among which digital holography [7–9] and phase-shifting interferometry [8–15] are promising methods to mix with digital processing. In particular, a complex digital hologram function with amplitude and phase information is digitally calculated by interferograms detected on a charge-coupled device (CCD) in phase-shifting interferometry. Recently, researchers have presented optical double or triple-key encryption methods to enhance security strength and data processing volume. Jeon and Gil [16] proposed a triple DES algorithm and its optical implementation based on dual XOR logic operations, Ahouzi et al. [17] proposed an advanced algorithm using a triple random-phase encryption (TRPE) scheme in the Fourier transform domain that improves the security of optical encryption based on double random-phase encryption (DRPE), and Kumari et al. [18] also suggested a TRPE cryptosystem in the Fresnel domain. As for the multiple keys for optical encryption, Zhang and Wang [19] applied the parameters of the optical configuration to serve as additional keys for optical image encryption based on interference, which motivated us to introduce the concept of three independent keys for phase-shifting digital interferometry.
In this paper, we propose the three-key triple data encryption algorithm (TDEA) of a digital cryptosystem based on phase-shifting interferometry. The proposed method carries out digital encryption and decryption processes with three independent keys called a wavelength key, a reference distance key and a holographic encryption key. The independent wavelength keys are assumed to be determined by the tunable wavelength of a light source and the independent reference distance keys are determined by varying placement of the reference input from the Fourier transform lens in optical interferometry. In Section Ⅱ, 3DES is briefly reviewed, and the proposed three-key TDEA is described. In Section Ⅲ, an evaluation of the proposed method is verified by the results of numerical simulations. Conclusions are summarized in Section Ⅳ.
The DES has been the most widely used symmetric key block cryptography and was chosen as a standard by the American National Standard Institute (ANSI) in 1977. The algorithm uses a fixed-length 56-bit key to encrypt and decrypt a 64-bit block of data. However, its 56-bit key can no longer guarantee enough security in recent cryptanalytic attacks against block ciphers. Generally, increasing the key length makes the cryptosystem more secure. The problem of increasing the key length can be overcome by using double or triple-length keys. In 1998, a 3DES called triple data encryption algorithm (TDEA), in which DES is applied three times, was adopted as the standard ANSI X9.52 [2]. Figure 1 shows a block diagram of the 3DES encryption and decryption procedure. The 3DES algorithm consists of three DES keys (k_{1}, k_{2} and k_{3}) for the cryptosystem. There are two variations of 3DES. If three 56-bit keys k_{1}, k_{2} and k_{3} are independent DES keys, it is referred to as three-key 3DES and produces an effective key length of 168 bits. If two keys, k_{1} and k_{2}, are independent keys and k_{3} is the same as k_{1}, it is referred to as two-key 3DES and gives an effective key length of 112 bits. The resultant 3DES algorithm is much harder to break compared to a single DES. The encryption and decryption of the 3DES algorithm is as follows. Assume that k_{1}, k_{2} and k_{3} are three independent keys in the 3DES cryptosystem. The encryption and decryption processes are expressed as
where m(x, y) is a plaintext to be encrypted and c(x, y) is a ciphertext.
In this paper, a cryptosystem that performs three-key TDEA based on the phase-shifting interferometry principle is proposed. The concept of the proposed method is described by the optical configuration shown in Fig. 2, which is based on a Mach–Zehnder interferometer architecture. A tunable laser diode beam is collimated by a collimating lens (CL) and is linearly polarized by a polarizer (P1), and it is divided by a beam splitter (BS1) into two plane waves of the object and the reference beams traveling in different directions. When shutters S1 and S2 are open, the downward object beam passes through an input amplitude-type spatial light modulator (SLM1) and a random phase mask (RPM), while the rightward reference beam passes through an input phase-type spatial light modulator (SLM2) and a λ/4 plate. Two lenses (L1 and L2) form a Fourier transform of the input functions into a CCD. A random phase mask is adopted to improve the dynamic range of the spatial frequency in the spatial frequency plane on the CCD. The RPM function is represented as r(x, y) = exp[j2πq(x, y)], where q(x, y) is a randomly distributed function over the interval [0, 1]. A λ/4 plate makes the wave along the vertical axis (s-polarization axis) occur with no phase shift and the wave along the horizontal axis (p-polarization axis) occur with a phase shift of π/2 radians. This scheme provides two-step phase-shifting interferometry [20].
Firstly, a cryptosystem based on optoelectronic two-step phase-shifting interferometry, considering the wavelength of light and the distance between the input and the lens, is briefly described. Let m(x, y) be a plaintext to be encrypted and k(x, y) be a holographic encryption key in phase-shifting interferometry. The Fourier transform diffraction patterns of the object and reference beams form complex amplitude distributions at the output spatial frequency (u, v) plane, and are expressed as
where λ is the wavelength of the light source, k = 2π/λ, f is the focal length of the lens, d_{o} is the object distance between SLM1 and lens L1, and d_{r} is the reference distance between SLM2 and lens L2. The output complex distribution will be the exact Fourier transform of the input except for the phase factor outside the integral when the distance between the input and the lens is the focal length of the lens, that is d_{o} = f and d_{r} = f. It is denoted from Eqs. (3) and (4) that the phase factor in front of the Fourier transform of m(x, y)r(x, y) is P_{o}(u, v; λ, d_{o}) and the phase factor in front of the Fourier transform of k(x, y) is P_{r}(u, v; λ, d_{r}), respectively. It is interesting to note that even if the object distance d_{o} has any distance value, resulting in any phase factor P_{o}(u, v; λ, d_{o}), it does not affect the phase change contribution in Eq. (3) because the RPM function r(x, y) includes a random phase distribution, while the phase factor P_{r}(u, v; λ, d_{r}) with different reference distance d_{r} affects the phase change contribution in Eq. (4). If the object distance d_{o} is assumed to have the focal length f for convenience, and if the integral part in Eqs. (3) and (4), which represent Fourier transforms of the object and reference complex distributions in the spatial frequency domain, are expressed as M(u, v) = F{m(x, y)r(x, y)} = |M(u, v)| e ^{jϕM(u,v)} and K(u, v) = F{k(x, y)} = |K(u, v)| e ^{jϕK(u,v)}, respectively, then Eqs. (3) and (4) are rewritten by
With Eqs. (5) and (6), the intensity pattern recorded by the CCD is given by
where ^{δ} is a phase shift in the reference beam. Two interference intensities at the output plane can be achieved with digital two-step phase-shifting interferometry when a phase shift of π/2 occurs between the s-polarization axis and the p-polarization axis in the reference beam. The suitable polarization direction of an output polarizer (P2) in front of the CCD gives two interference intensities. Representing the complex amplitude distribution of the object and the reference beams with Eqs. (5)–(7) can be expressed as
where variables in spatial frequency coordinates are omitted and ∆φ(u, v; λ, d_{r}) = φ_{M}(u, v) − {φ_{Pr}(u, v; λ, d_{r}) + φ_{K}(u, v)} denotes the phase difference between the reference and the object beams. This intensity pattern recorded by the CCD shows a noise-like random distribution due to the randomness of Eq. (3). Additionally, only the intensity distribution of the object beam I_{o} = |M(u, v)|^{2} and only the intensity distribution of the reference beam I_{r} = |P_{r} (u, v; λ, d_{r})|^{2} |K(u, v)|^{2}, which are DC-terms in the interference intensity of Eq. (8), are recorded on the CCD by controlling shutters S1 and S2 in the optical setup. In this method, a complex digital hologram function generated from phase-shifting interferometry is a kind of ciphertext that also provides random phase and random amplitude distributions. If the complex digital hologram function is assumed to be H(u, v; λ, d_{r}) = A(u, v; λ, d_{r}) e^{j∆ϕ(u,v;λ,dr)}, the amplitude A(u, v; λ, d_{r}) and the phase ∆φ(u, v; λ, d_{r}) can be calculated by two intensities I_{1}(u, v; λ, d_{r}; π/2) and I_{2}(u, v; λ, d_{r}; 0) after removing the DC-term |M|^{2} + |P_{r}|^{2} |K|^{2} from Eq. (8) as
To decrypt m(x, y) of the plaintext, the object complex function M(u, v) should be retrieved from the ciphertext of the complex digital hologram function H(u, v; λ, d_{r}) by the complex distribution function U_{r}(u, v; λ, d_{r}) of Eq. (4) including the holographic encryption key k(x, y) used in phase-shifting interferometry. According to the hologram memory principle, the object wavefront is reconstructed by illuminating the reference wavefront to the hologram. Likewise, the object complex function M(u, v) can be reconstructed only with knowledge of the reference complex function U_{r}(u, v; λ, d_{r}) which is now acting as a decryption key. Consequently, the complex distribution M(u, v) and the plaintext m(x, y) are recovered as follows:
where F ^{−1}{∙} denotes an inverse Fourier transform.
From now on, we propose a new digital method of three-key TDEA to introduce three keys in the phase-shifting interferometry scheme, which easily improves security without adding optical components to the phase-shifting interferometry encryption system. From the proposed optical configuration shown in Fig. 2, let us consider the wavelength of laser diode light as an independent key k_{1}(λ) and the reference distance between SLM2 and lens L1 as another independent key k_{2}(d_{r}), respectively, while the holographic encryption key k(x, y) in phase-shifting interferometry is maintained as an independent key k_{3}(x, y). With this concept, the complex distribution function U_{r}(u, v; λ, d_{r}) of Eq. (4) is modified into U_{r}(u, v; k_{1}(λ), k_{2}(d_{r})) so that the interference intensity of Eq. (7) is expressed as
Since the object and the reference beams in Eq. (13) are changed by two keys, k_{1}(λ) and k_{2}(d_{r}), the two-step phase-shifting interferometry makes it so that the complex digital hologram function of the ciphertext is modified into
where the amplitude A and the phase ∆φ are rewritten by replacing λ and d_{r} in Eqs. (9) and (10) with k_{1}(λ) and k_{2}(d_{r}). This means that the complex hologram function is dependent on the wavelength and the reference distance in the cryptosystem, and therefore the ciphertext is made by using three independent keys, k_{1}(λ), k_{2}(d_{r}) and k_{3}(x, y). Now, to decrypt m(x, y) of the plaintext from the ciphertext H(u, v; k_{1}(λ), k_{2}(d_{r}), k_{3}), it is necessary to know all three keys, not only about k_{1}(λ) of the light wavelength but also about k_{2}(d_{r}) of the holographic encryption key k_{3}(x, y) location in the reference beam. The decryption process for the proposed three-key TDEA is accomplished as follows:
Flowcharts of the encryption and decryption processes for the proposed three-key TDEA are shown in Fig. 3, where ⊗ represents the inner product between pixels, FT and IFT denote Fourier transform and inverse Fourier transform, PSI denotes phase-shifting interferometry, SQ denotes a square function, and TH denotes a function to make binary data by a proper threshold. Although wavelength tuning of the light source and precise location control of the distance are required to implement independent keys k_{1}(λ) and k_{2}(d_{r}) optically, the effect of three-key encryption is so powerful that the cryptosystem can improve security.
In the proposed digital algorithm applied from the optical configuration as shown in Fig. 2, a pair of a wavelength key and a reference distance key {k_{1}(λ), k_{2}(d_{r})} can be used as a variable public key like a one-time password (OTP). If a specific wavelength key k_{1}(λ_{s}) is challenged to the host server, a specific reference distance key k_{2}(d_{rs}) is acknowledged to the user according to a predetermined specific key pair {k_{1}(λ_{s}), k_{2}(d_{rs})}. With these two specific key pairs, a user can encrypt private information with a holographic encryption key k_{3}(x, y) which is used independently as a private key of a user. Conceptually, the encryption and decryption processes of the proposed three-key TDEA are expressed as
In the encryption process for the proposed three-key TDEA, generating a ciphertext of a complex digital hologram can be performed optically or digitally. However, it is very difficult to control the wavelength λ of light and to align the reference distance d_{r} precisely in implementing the optical setup shown in Fig. 2. A practical method that ignores these limited optical problems is to apply the proposed algorithm to computer-oriented digital processing, which is more convenient than the optical technique. To demonstrate the validity of the proposed method, numerical simulations using MATLAB (R2021) are carried out to show the performance. In this paper, all the data size of inputs is 256 × 256 pixels. A binary image of a monkey, instead of encoded digital data from information, is used as an input plaintext m(x, y) for visual convenience, as shown in Fig. 4(a).
From the proposed new algorithm shown in Figs. 3 and 4, a specific wavelength of light is chosen as the first independent wavelength key k_{1}(λ) and a specific reference distance is chosen as the second independent distance key k_{2}(d_{r}) before processing the phase-shifting interferometry encryption. The third holographic encryption key k_{3}(x, y) used in phase-shifting interferometry is assumed to have a randomly generated pattern, as shown in Fig. 4(b). By using Eq. (14), a complex digital hologram function H(u, v; k_{1}(λ), k_{2}(d_{r}), k_{3}) is generated by the three keys. It is clear that a plaintext is encrypted in a ciphertext with noise-like random phase and random amplitude distributions, A(u, v; k_{1}(λ), k_{2}(d_{r}), k_{3}) and ∆ϕ(u, v; k_{1}(λ), k_{2}(d_{r}), k_{3}), respectively. Figures 4(c) and 4(d) show the randomly distributed amplitude and phase maps of the complex digital hologram of the ciphertext.
To verify the reliability of the proposed algorithm, the influence of three independent keys, k_{1}, k_{2} and k_{3}, on the decrypted image is evaluated. The mean square error (MSE) between the decrypted binary image d(x, y) and the original plaintext image m(x, y), representing a relative error between them, is introduced as
where p × q is the pixel size of the image data. If the decrypted image is retrieved without any error, the MSE is calculated as 0%. In a phase-shifting interferometry cryptosystem, the most important encryption key is the holographic encryption key k_{3}(x, y), which is determined by independent users selecting a 2-D random distribution function in the key. A detailed explanation about phase-shifting digital holographic optical encryption is given in the [16, 20]. In this paper, we omit the role of the third key k_{3}(x, y).
First, we examine the suitability of the wavelength key k_{1}(λ) as an independent encryption key. To know the influence of the key k_{1}(λ) on the decrypted image, the MSE according to fractional wavelength change is analyzed for three different light sources. The key k_{1}(λ) with three different designing center wavelengths λ_{c} is given by k_{1}(λ_{1} = 1,550 nm), k_{1}(λ_{2} = 1,300 nm) and k_{1}(λ_{3} = 670 nm), respectively. The deviation of the wavelength and the fractional wavelength change are defined as δλ = λ_{c} − λ and ∆λ = δλ/λ_{c}, respectively. Figure 5(a) shows an MSE graph with respect to fractional wavelength change. As shown in Fig. 5(a), the shorter wavelength with k_{1}(λ_{3} = 670 nm) allows larger MSE compared to the long wavelength with k_{1}(λ_{1} = 1,550 nm) for ∆λ < 0.5 × 10^{−3}% when the same ∆λ is given. However, the MSE reaches more than 90% for ∆λ > 0.5 × 10^{−3}% regardless of different wavelengths, which means that the decrypted image d(x, y) is quite a different from the original image m(x, y). The deviation of the wavelength of k_{1}(λ_{1} = 1,550 nm) is about δλ = 0.007 nm when ∆λ = 0.5 × 10^{−3}%, for example. It is interesting that a very small deviation of the light wavelength can separate a new independent wavelength key k_{1}(λ_{1} + δλ) with different wavelength from an independent wavelength key k_{1}(λ_{1}).
Second, to examine the influence of the distance key k_{2}(d_{r}) on the decrypted image as an independent encryption key, the MSE according to fractional reference distance change is analyzed for different distances. The key k_{2}(d_{r}) with different reference distances d_{r} is chosen as k_{2}(d_{r1} = 30 mm), k_{2}(d_{r2} = 40 mm), k_{2}(d_{r3} = 50 mm), k_{2}(d_{r5} = 60 mm) and k_{2}(d_{r6} = 70 mm), respectively. The deviation of the distance and the fractional distance change are defined as δd_{r} = d_{c} − d_{r} and ∆d_{r} = δd_{r}/d_{c}, respectively, where d_{c} is a designing center reference distance from the lens to the input location. Figure 5(b) shows an MSE graph with respect to fractional distance change. As shown in Fig. 5(b), the longer distance with k_{2}(d_{r6} = 70 mm) allows a larger MSE compared to the wavelength with k_{2}(d_{r1} = 30 mm) for ∆d_{r} < 1.5 × 10^{−4}% when the same ∆d_{r} is given. The deviation of the reference distance of k_{2}(d_{r2} = 40 mm) is about δd_{r} = 60 nm when ∆d_{r} = 1.5 × 10^{−4}%, for example. The decrypted image d(x, y) is not discriminated from the original image m(x, y) for ∆d_{r} > 1.5 × 10^{−4}% because the MSE reaches more than 90% regardless of different distances. It is also noted that the distance separating the next independent wavelength key k_{2}(d_{r1} + δd_{r}) with different reference distance from an independent distance key k_{2}(d_{r1}) is very small.
In addition, Fig. 6 shows the results of decryption for the plaintext of binary image m(x, y) when the correct wavelength key k_{1}(λ = 1,550 nm) and the correct reference distance key k_{2}(d_{r} = 40 mm) are given in the proposed three-key TDEA. Figures 6(a)–6(c) show the decrypted images with MSE = 14.1% in the case of δλ = 0.002 nm (∆λ = 0.13 × 10^{−3}% deviation error), MSE = 63.7% in the case of δλ = 0.004 nm (∆λ = 0.26 × 10^{−3}% deviation error), and MSE = 89.3% in the case of δλ = 0.007 nm (∆λ = 0.45 × 10^{−3}% deviation error), respectively. Figures 6(d)–6(f) show the decrypted images with MSE = 45.0% in the case of δd_{r} = 20 nm (∆d_{r} = 0.5 × 10^{−4}% deviation error), MSE = 74.1% in the case of δd_{r} = 30 nm (∆d_{r} = 0.75 × 10^{−4}% deviation error), and MSE = 91.3% in the case of δd_{r} = 50 nm (∆d_{r} = 1.25 × 10^{−4}% deviation error), respectively.
Third, let us evaluate the dependence of the MSE on the change of the wavelength key k_{1}(λ) and the distance key k_{2}(d_{r}). To examine the effectiveness of key k_{1}(λ) and key k_{2}(d_{r}), it is assumed that the specific parameters used for the correct encryption and decryption in simulation are as follows: The wavelength key k_{1}(λ = 1,550 nm), the reference distance key k_{2}(d_{r} = 40 mm), the object distance d_{o} = 50 mm, and focal length of the lens f = 50 mm shown in Fig. 2, and the correct holographic encryption key k_{3}(x, y) shown in Fig. 4(b). Figure 7(a) shows an MSE graph for wavelength deviation from the correct wavelength key k_{1}(λ = 1,550 nm) with the correct reference distance key k_{2}(d_{r} = 40 mm). Although a very small deviation occurs, the MSE increases steeply, and the original plaintext cannot be decrypted. Figure 7(b) shows a detailed part near the correct wavelength key k_{1}(λ = 1,550 nm). The solid blue line shows the MSE for a wavelength key k_{1}(λ = 1,550 nm), while the red dashed line shows the MSE for another wavelength key k_{1}(λ = 1,550.007 nm). If k_{1}(λ = 1,550 nm) is the correct wavelength key in the proposed TDEA cryptosystem, then k_{1}(λ = 1,550.007 nm) is the incorrect key in the same TDEA cryptosystem. From the solid blue line, about 90% of the MSE is shown at a wavelength of λ = 1,550.007 nm. This means that k_{1}(λ = 1,550 nm) and k_{1}(λ = 1,550.007 nm) are keys that are independent from each other. Figure 8(a) shows an MSE graph for distance deviation from the correct reference distance key k_{2}(d_{r} = 40 mm) with the correct wavelength key k_{1}(λ = 1,550 nm). A very small deviation of the reference distance makes a steep MSE increase, similarly to the deviation of the wavelength. Figure 8(b) shows the detailed part near the correct reference distance key k_{2}(d_{r} = 40 mm). The solid blue line shows the MSE for a distance key k_{2}(d_{r} = 40 mm), while the red dashed line shows the MSE for another distance key k_{2}(d_{r} = 40.00005 mm). If k_{2}(d_{r} = 40 mm) is the correct wavelength key in the proposed TDEA cryptosystem, then k_{2}(d_{r} = 40.00005 mm) is the incorrect key in the same TDEA cryptosystem. From the solid blue line, about 90% of the MSE is shown at a reference distance of d_{r} = 40.00005 mm. This means that k_{2}(d_{r} = 40 mm) and k_{2}(d_{r} = 40.00005 mm) are keys that are independent from each other. Also, it is meaningful to show the separability of the incorrect keys k_{1}(λ = 1,550.007 nm) and k_{2}(d_{r} = 40.00005 mm) in the role of the independent key. In Fig. 7(b), the magenta dashed line shows the MSE for a correct wavelength key k_{1}(λ = 1,550 nm) and an incorrect distance key k_{2}(d_{r} = 40.00005 mm), which is a different view of the incorrect distance key k_{2}(d_{r} = 40.00005 mm) shown in Fig. 8(b) with respect to wavelength. Similarly, the magenta dashed line in Fig. 8(b) shows the MSE for an incorrect wavelength key k_{1}(λ = 1,550.007 nm) and a correct distance key k_{2}(d_{r} = 40 mm), which is a different view of the incorrect wavelength key k_{1}(λ = 1,550.007 nm) shown in Fig. 7(b) with respect to reference distance. According to a similar study on optical image encryption based on interference in [19], the simulation results showed that the wavelength sensitivity is 2 × 10^{−5} nm and the distance sensitivity is 2 nm.
To verify the cipher resistance to differential attacks on the proposed method, a number of pixel changing rate (NPCR) test is performed on the ciphertext c(x, y). The NPCR N(c_{1}, c_{2}) is defined as
The first evaluation is an NPCR randomness test for the proposed three-key TDEA, where c_{1} and c_{2} are ciphertexts before and after one pixel change in a plaintext. N(c_{1}, c_{2}) is calculated as about 4.88% from 10,000 iterations, which gives 0.003% of the MSE. The second consideration is that c_{1} and c_{2} are ciphertexts obtained by deviation from the correct wavelength key k_{1}(λ = 1,550 nm) or the correct reference distance key k_{2}(d_{r} = 40 mm) in the optical configuration. When the wavelength deviation δλ is 0.007 nm, N(c_{1}, c_{2}) is calculated as about 95.4%. When the distance deviation δd_{r} is 50 nm, N(c_{1}, c_{2}) is calculated as about 94.5%. These NPCR results suggest that the proposed method provides strong cipher resistance to attacks.
Finally, we discuss the feasibility of the proposed method in secure block cryptography with symmetric key. According to the three-key TDEA standardized as ANSI X9.52, the key length is 168 bits for block encryption. However, the proposed three-key TDEA provides a digital cryptosystem that has three independent keys, a wavelength key k_{1}(λ), a distance key k_{2}(d_{r}) and a 2-D holographic encryption key k_{3}(x, y) of 256 × 256 bits. At first, k_{3}(x, y) is dependent on the displaying capability of the SLM in implementing the cryptosystem optically. However, the size of key k_{3}(x, y) does not matter if only the proposed cryptosystem is implemented digitally. The only limitation to determine key k_{3}(x, y) is that a longer key needs a longer processing time in encryption. In this paper, we choose key k_{3}(x, y) of size 256 × 256 bits, which is reasonable for the optical or digital design. Next, the number of independent wavelength keys k_{1}(λ) is determined by considering a range of the wavelength. In the results as shown in Fig. 5(a) and Fig. 7(b), the minimum wavelength deviation to separate the independent keys of k_{1}(λ) is about δλ = 0.007 nm. But we choose the deviation of the key k_{1}(λ) as δλ = 0.4 nm, which is sufficient enough to discriminate each independent key k_{1}(λ). Next, we consider a tunable range of k_{1}(λ) as 1,530–1,562 nm because a tunable laser diode source with such a wavelength range can be achieved with the commercial laser diodes used in optical technology. The reason why we select δλ = 0.4 nm in the range of 1,530–1,562 nm is that it is the wavelength spacing used in the dense wavelength division multiplexing (DWDM) optical communication technique. From the range of 1,530–1,562 nm, we achieve the total number of independent wavelength keys k_{1}(λ) as N(k_{1}) = (1,562 − 1,530) / 0.4 = 80. It is also possible to get N(k_{1}) = (1,562 − 1,530) / 0.01 = 3,200 increasingly if we select δλ = 0.01 nm, which has no problem in the proposed digital cryptosystem. Lastly, the number of independent reference distance keys k_{2}(d_{r}) is determined by considering a range of reference distance. In the results as shown in Fig. 5(b) and Fig. 8(b), the minimum distance deviation to separate the independent keys of k_{2}(d_{r}) is about δd_{r} = 50 nm, but we choose the deviation of the key k_{2}(d_{r}) as δd_{r} = 500 nm, which gives sufficient deviation to separate each independent key k_{2}(d_{r}). In this paper, we consider a tunable range of k_{2}(d_{r}) as 35–45 mm with a designing center distance of k_{2}(d_{r} = 40 mm). From the range of 35–45 mm, we achieve the total number of independent distance keys k_{2}(d_{r}) as N(k_{2}) = (45 − 35) / 0.0005 = 20,000. If we expand the range of k_{2}(d_{r}) as 25–55 mm, then N(k_{2}) = (55 − 25) / 0.0005 = 60,000, three times larger than in the case of 35–45 mm. Furthermore, if we choose the deviation of the key k_{2}(d_{r}) as δd_{r} = 100 nm, which is two times larger than the minimum distance deviation of the independent keys of k_{2}(d_{r}) in the proposed cryptosystem, the total number of independent distance keys k_{2}(d_{r}) as N(k_{2}) = (45 − 35) / 0.0001 = 100,000 can be achieved. This flexibility is an advantage of a digital processing cryptosystem compared to an optical technique. The conventional three-key TDEA standard has a key length of 168 bits. This means that 2^{168} attempts are needed to find the exact security key, which is very sufficient to protect against key attacks in general. However, the proposed method is assumed to have the 2-D holographic encryption key k_{3}(x, y) of size 256 × 256 bits, and in addition to that, the wavelength key k_{1}(λ) and the distance key k_{2}(d_{r}) are assumed to have N(k_{1}) = 80 and N(d_{r}) = 20,000, respectively. Thus, the proposed cryptosystem requires 2^{256×256} × 80 × 20,000 attempts to hack the correct block data, which provides much more robustness to the cryptosystem under attack, and more importantly, it can handle 256 × 256 bits block encryption of data instead of encrypting a 64-bit block of data in the conventional three-key TDEA.
We the propose three-key TDEA of a digital cryptosystem based on phase-shifting interferometry, where a ciphertext of an encrypted binary image is acquired as a complex digital Fourier hologram function with complex amplitude and phase distribution functions. The encryption process is performed with the use of three encryption keys called a wavelength key k_{1}(λ), a reference distance key k_{2}(d_{r}) and a holographic encryption key k_{3}(x, y), which are independently represented in the reference beam path of the phase-shifting interferometry optical architecture. Different wavelengths of the light source and different reference distances determine independent keys k_{1}(λ) and k_{2}(d_{r}), respectively, which provides the keys with many degrees of freedom, and these flexible independent keys can protect against attacks on cryptosystems. The decryption process is carried out by digital processing for the ciphertext of the complex digital hologram with the three keys used in encryption. For the proposed three-key TDEA, the minimum wavelength deviation between the neighboring wavelength keys of k_{1}(λ) and the minimum distance deviation between the neighboring distance keys of k_{2}(d_{r}) are achieved as about δλ = 0.007 nm and about δd_{r} = 50 nm, respectively. For the proposed method in this paper, by choosing the deviation of the key k_{1}(λ) as δλ = 0.4 nm and the deviation of the key k_{2}(d_{r}) as δd_{r} = 500 nm, the number of independent keys k_{1}(λ) and k_{2}(d_{r}) is calculated as N(k_{1}) = 80 for a range of 1,530–1,562 nm and N(d_{r}) = 20,000 for a range of 35–45 mm, respectively, so that 2^{256×256} × 80 × 20,000 attempts are needed to find the correct key. The results of numerical simulations verify the feasibility of the proposed method.
This work was supported by Incheon National University (International Cooperative) Research Grant in 2021.
Incheon National University (International Cooperative) Research Grant in 2021.
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.