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Abstract
Optical holographic encryption (OHE) has been extensively researched in the field of information security due to its parallel and multi-dimensional characteristics. However, although some progress in OHE has been made in recent years, inherent security flaws resulting from the robust nature of holograms persist. In this study, we propose a multilevel holographic encryption method based on the Tiger Amulet (TA) concept. Compared with the normal OHE, our method employs two ciphertexts. It strategically utilizes the low-level plaintext as intentional deceptive content to confound the potential eavesdroppers. Furthermore, we ingeniously exploit the hologram’s robustness in reverse, thereby establishing an additional protection mechanism to enhance the security of the middle-level plaintext. Leveraging the TA concept, the high-level plaintext can only be decrypted when two matched ciphertexts are combined and collimated. The TA based decryption mechanism enhances the security and sensitivity deciphering high-level plaintext. Benefiting from the security mechanisms above, our proposed method demonstrates promising applicability across diverse scenarios and holds the potential to redefine the landscape of multilevel OHE design.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the digital era, the exchange of information is on the rise due to the advancements in information technology. However, this surge in information also amplifies the risk of information leakage. Consequently, information security has emerged as a paramount concern over the past decades. Various encryption technologies [1–4] have been proposed to enhance the security of information during its transmission and storage. Among these encryption technologies, optical holographic encryption (OHE) stands out as a novel encryption approach. Capitalizing on its characteristics of high speed, parallelism, and multidimensionality [5,6], OHE has drawn considerable attention and extensive research. Leveraging the multidimensional nature of light, distinct physical parameters, including polarization [7], wavelength [8], phase [9,10], amplitude [10], and orbital angular momentum (OAM) [11], can be used to record and store ciphertext within OHE. This sets OHE apart from commonly used electronic encryption technologies, offering increased capacity and more degrees of freedom for key design.
Advancements in OHE have led to three primary implementation schemes: employing liquid crystals (LCs), utilizing metasurfaces, or combining metasurfaces and LCs. In LC-based OHE, the modulated basic parameter of the light field is usually the phase [12–14] and polarization [15]. LCs are widely utilized in holographic encryption due to their cost-effectiveness, expansive area coverage, and dynamic switching capability. Commercial LC devices, such as spatial light modulators (SLMs), offer reprogrammability as an advantage. However, limitations, such as micron-sized pixels, modest modulation capability, limited resolution, narrow field of view, twin imaging, narrow bandwidth, and high-order diffraction, impede LC’s progression in holographic encryption. Metasurface employment enables manipulation of fundamental light field parameters within an ultrashort distance. Metasurfaces enable modulation of individual parameters, such as amplitude [16], phase [17,18], polarization [7,19] and OAM [20], and even simultaneous modulation of multiple parameters [21,22]. Although meta-holograms can achieve high resolution and desired diffraction orders due to their subwavelength period structure, metasurface manufacturing costs are higher than LCs, and their overall size is smaller. Additionally, once a metasurface’s structure is determined, its modulation ability becomes fixed, limiting adjustability. Recent findings reveal anisotropic LC’s modulation potential of the Pancharatnam–Berry phase [23,24], leading to the proposal of an LC-metasurface hybrid scheme for OHE [25]. The hybrid scheme addresses the downsides of both technologies. This amalgamation achieves dynamic holographic encryption, assuring quality in decrypted hologram.
Except for the implementation schemes, other progresses in OHE are made to improve its performance. For instance, chaotic theory is used to improve the holographic encryption schemes [26–28]. Biometric information is combined with OHE to improve the security [27,29]. Other transform domains, such as gyrator domain [30,31], fractional Fourier domain [32], and Fresnel domain [33], are used in OHE to achieve multiple image encryption.
Despite substantial progress in holographic [34], OHE’s security remains vulnerable to the robustness of holograms. For example, even if the decrypted wavelength deviates from the target wavelength, resulting in the phase offsets of hologram within a certain range, the plaintext can still be extracted from an inaccurate decrypted hologram. And the leakage of partial key is enough for the eavesdropper to get the hidden information from the ciphertext.
In this study, we propose and demonstrate a multilevel holographic encryption method based on the Tiger Amulet (TA) concept [35] to improve the security of OHE. TA is a certificate for deploying troop in ancient China. The most important feature of TA is that, it is split into two parts and allocated to separate individuals in usual. When the TA is separated, each part holds no validity. Only when two parts of the TA are combined into an entirety, can the TA have validity. To implement TA functionality in OHE, we develop a phase extraction Gerchberg–Saxton (PEGS) algorithm to form four kinds of functional zones on ciphertexts: camouflage, protection, senior and TA zones. The camouflage zone generates deceptive information akin to low-level plaintext. The senior zone generates middle-level plaintext, whereas the protection zone forms protection mechanism for the middle-level plaintext. The TA zone remains inactive alone, only unlocking the high-level plaintext upon combination and alignment with senior zones.
Compared with normal OHE, the proposed method has three main advantages: achieving multilevel encryption, introducing deceptive information (low-level plaintext) and protection mechanism to enhance middle-level plaintext security, and enhancing sensitivity and security through the TA-based decryption of high-level plaintext.
The rest of this article is organized as follows. Section 2 outlines the proposed method’s principles. Section 3 presents numerical simulations and analyses. Section 4 delves into the optical experiment. Section 5 provides the main results and conclusion.
2. Principle
The working principle of our scheme, rooted in TA philosophy, is shown in Fig. 1.
In Fig. 1(a), our scheme employs two ciphertexts. When these two ciphertexts are decrypted independently without keys, a maple leaf and a meaningless rectangular pattern will be obtained simultaneously. The maple leaf serves as deceptive data to baffle eavesdroppers, and the rectangular pattern functions as a protection mechanism for middle-level plaintext. However, as shown in Fig. 1(b), decryption with accurate middle-level keys prompts the repositioning of the protection mechanism. Subsequently, a paper and a trophy can be obtained from these two ciphertexts. Most importantly, as shown in Fig. 1(c), a clear high-level plaintext can be decrypted from the collimated ciphertexts with the help of the high-level key upon concatenating the two ciphertexts. In addition, the high-level plaintext will become indiscernible in the case of noncollimated ciphertexts.
The flowchart for the realization of multilevel holographic encryption is shown in Fig. 2. From Fig. 2, the execution of the proposed method can be segmented into three steps: 1) generation of multilevel keys; 2) generation of phase extraction masks (PEMs); and 3) generation of ciphertexts.
2.1 Generation of multilevel keys
Within our method, the middle-level key functions as a switch governing distinct functional zones. A binary mask, comprising units containing solely 1s and 0s, is chosen as the key. When a unit holds a value of 1, rays can traverse it, whereas a value of 0 restrains them. In the process of generating middle-level keys, the unit values on the key can be set tailored to users’ specifications. For example, a middle-level key can be generated by utilizing a user’s fingerprint feature [36] to ensure its uniqueness and security. In this study, middle-level keys are generated by randomly assigning elements of 1 or 0 to the mask units. During the assignment, the probabilities for 0 and 1 are all 50% to maximize the key space.
Owing to the decryption characteristics of high-level plaintext, protection zones and camouflage zones introduces noise in the object plane when combining two ciphertexts. Hence, the high-level key must possess the capability to deactivate invalid functional zones during high-level plaintext decryption. For the ultimate authority, the high-level key is generated through an exclusive OR (XOR) operation between two middle-level keys:
where the symbol ${\oplus} $ denotes the XOR operation, and $K$ represents the key in our method. For the superscript of $K$, $H$ and $M$ denote the high-level and middle-level of keys respectively. The subscript corresponds to the respective ciphertext’s serial number. The schematic depicting keys in our method is shown in Fig. 3.After all the keys are obtained, the next step involves generating PEMs for the PEGS algorithm to establish different functional zones within the ciphertexts.
2.2 Generation of phase extraction masks
In our proposed method, PEMs are created by combining unit distributions from two middle-level keys. Combining two middle-level keys produces four kinds of unit combinations: (0,0), (0,1), (1,0), and (1,1). The illustrative depiction of combining middle-level keys is shown in Fig. 4.
In Fig. 4, the symbol ${\otimes} $ denotes the combination operation and the results of unit combination are represented by marking the units with different colors.
Based on the fact that middle-level keys can enable deceptive information but deactivate protection mechanisms, utilizing the combined result in Fig. 4, PEMs for the protection zone and camouflage zone are assembled using the unit combinations of (0,0) and (1,1), respectively. These PEMs can be expressed as:
where the symbol $\bar {} $ represents the NOT logic operation, and symbol ${\bullet} $ stands for the AND logic operation.$PZ$ and $CZ$ donate the masks for the protection and camouflage zones, respectively, with the subscript corresponding to the ciphertext’s serial number.Furthermore, as the middle-level keys exclusively active their corresponding middle-level plaintexts and TA zones cooperate with senior zones, PEMs for senior zones corresponding to $K_1^M$ and $K_2^M$ comprise the unit combinations of (1,0) and (0,1), respectively. Therefore, the PEMs for senior zones can be expressed as:
where $SZ$ represents the mask for the senior zone. Utilizing the cooperation relationship between the senior and TA zones in two ciphertexts, the PEM for the TA zone in one ciphertext should coincide with that of the senior zone in the other ciphertext. Hence, PEMs of the TA zone in each ciphertext can be represented as:Following the acquisition of the PEM for the TA zone in each ciphertext, the comprehensive PEM of the TA zone can be expressed as:
where the symbol ${\cup} $ donates the union operation.Utilizing Eqs. (2)–(8) and the middle-level keys in Fig. 1, all the PEMs are shown in Fig. 5.
After obtaining all the PEMs, the PEGS algorithm can be conducted to generate the ciphertexts in our method.
2.3 Generation of ciphertexts
For normal holograms, the holographic reconstruction from the hologram plane to the object plane can be represented by the Fourier transform:
where $I({x,y} )$ signifies the intensity in the object plane, $\mathrm{{\cal F}}$ represents the 2D Fourier transform operation, $A({u,v} )$ and $\varphi ({u,v} )$ are the distributions of light field’s amplitude and phase in the hologram plane, respectively. Based on Eq. (9), the Gerchberg–Saxton (GS) algorithm [37] can be used to calculate the desired phase by iteratively using the target image as a constraint. However, the traditional GS algorithm is not suitable for our method. To obtain the desired ciphertexts in our method, we develop a PEGS algorithm based on traditional GS algorithm. For the sake of simplifying ciphertext generation, the amplitude distribution is set to 1, and the phase distribution serves as the ciphertext in our method. The comprehensive flowchart of the PEGS algorithm is shown in Fig. 6.The PEGS algorithm’s commencement point is a random phase and the target image. An iterative process ensues until the total number of iterations $n$ reaches the preset number $N$. During the iteration process, the target image acts a constraint. Fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) are performed to calculate the amplitude $I$ and phase $\varphi $ in the object plane and the hologram plane, respectively. When the parameter's subscript is 2, it pertains to the object plane, otherwise, it pertains to the hologram plane. Different from the traditional GS algorithm, a PEM is employed to extract the phase in the hologram plane. The successful reconstruction of the target image from the extracted phase depends on the high robustness of hologram. Guided by the target image and the PEM within the algorithm, the output ${\varphi _o}$ signifies the phase distribution for a specific functional zone within the ciphertext. For example, if the deceptive information and the PEM of the camouflage zone are used, then the output phase will be the ciphertext for the camouflage zone.
When $TZ$ is used in PEGS, the corresponding target image is the high-level plaintext. Because the TA zone should cooperate with two senior zones across in two ciphertexts, the output phase ${\varphi _{TZ}}$ must be further segmented and assigned to the two ciphertexts. Thus, the phases for the TA zone in two ciphertexts are defined as:
where the subscript represents the location of phase distribution within the ciphertext.And about the protection mechanism, it is a strategy to transform the middle-level plaintext into an unrecognizable figure. In our method, the protection mechanism is established by using the hologram of protection zone to eliminate middle-level plaintext features. The required protection image can be generated by extracting the middle-level plaintext from a rectangular pattern, as shown in Fig. 7.
In Fig. 7, the symbol $\mathrm{\ \ominus }$ donates the extraction operation. Benefiting from the protection mechanism, the decryption results without a correct key will be the protected image and protection image. Their superposition will form a meaningless rectangular pattern in the object plane. When the correct key deactivates the protection mechanism, the middle-level plaintext can be obtained. Hence, the protection mechanism enhances the middle-level plaintext’s security.
Utilizing the PEGS algorithm, phases ${\varphi _{P{Z_i}}}$, ${\varphi _{C{Z_i}}}$, ${\varphi _{S{Z_i}}}$,and ${\varphi _{T{Z_i}}}$ in protection, camouflage, senior, and TA zones for two ciphertexts can be obtained. After obtaining all the phase distributions for the functional zones, the ciphertexts of our method can be expressed as:
2.4 Decryption of ciphertexts
Based on Eq. (9) and the design of our method, low-level plaintexts decrypted without keys can be expressed as:
where $P$ represents the plaintext. The subscript and superscript of $P$ indicate the serial number and level of the plaintext respectively.If the middle-level key is used, then the decrypted plaintext is represented as:
Moreover, if the high-level key is used to decrypt the combined ciphertexts, the decrypted plaintext is represented as:
3. Numerical simulation and analysis
In this section, numerical simulation and analysis are conducted to demonstrate the performance of the proposed method.
3.1 Numerical simulation of the multilevel encryption scheme
For the numerical simulation, grayscale images of 7, 9, 8 and 3, shown in Figs. 8(a)–(d), respectively, are chosen as two middle-level plaintexts, deceptive information, and high-level plaintext. Two binary middle-level keys with a size of 128 × 128 pixels are generated randomly. In the middle-level key, the quantities of 0 and 1 are equal. The two middle-level keys and the high-level key generated by XOR operation of the middle-level keys are shown in Figs. 8(e)–(g), respectively.
Fig. 8. Plaintexts and keys. (a) and (b) represent two middle-level plaintexts; (c) signifies the deceptive information; (d) corresponds to the high-level plaintext; (e) and (f) donate middle-level keys corresponding to (a) and (b); (g) is the high-level key.
Utilizing the PEMs described in Section 2.2 and the PEGS algorithm in Section 2.3, the obtained ciphertexts, which are the phase distribution of the holograms, are shown in Fig. 9.
As depicted in Fig. 9, no discernible information associated with the plaintexts is observed. In accordance with the decryption principle introduced in Section 2.4, the eavesdropper, without decryption keys for the ciphertext, can only decrypt the deceptive information. The obtained deceptive information from the two ciphertexts is shown in Figs. 10(a) and (b). However, the protection mechanisms will be eliminated when the plaintexts are decrypted using the correct middle-level keys by the authenticated receiver, revealing the middle-level plaintexts in the object plane. The decrypted results for the correct middle-level keys are shown in Figs. 10(c) and (d). Upon combining and decrypting the two ciphertexts with the high-level key, the high-level plaintext, which is shown in Fig. 10(e), can be successfully decrypted.
Fig. 10. Decryption results. (a) and (b) represent the decryption results of middle-level plaintext without keys; (c) and (d) denote the decryption results of middle-level plaintext with correct middle-level keys; and (e) signifies the decryption results of high-level plaintext.
For quantitatively assessing the image quality of the decrypted results, the correlation coefficient (CC) between the original plaintext and the decrypted plaintext is calculated by:
Table 1. Correlation strength in relation to CC values
From Table 1, a higher CC value indicates a stronger correlation strength between the ciphertext and the potential decryption, leading to an increased likelihood of successful decryption. When CC value is lower than 0.6, the correlation strength between the ciphertext and the decrypted ciphertext will be lower than strong. In this study, a threshold CC value of 0.6 is used to evaluate the decrypted results.
Utilizing Eq. (16), the calculated results of CC between the decrypted plaintexts in Fig. 10 and the original plaintexts are listed in Table 2.
Table 2. CC values between the decrypted plaintexts and the original plaintexts in the simulation
As shown in Table 2, the CC values between the deceptive information “8” and the decrypted results in Figs. 10(a)–(e) are 0.8436, 0.8753, 0.9304, 0.9351, and 0.0768 respectively. In Figs. 10(a)–(d), the CC results indicate the successful recovery of deceptive information from each ciphertext. The decrypted result is sufficiently effective in confusing the eavesdroppers. Conversely, the very low CC value in Fig. 10(e) signifies the complete removal of deceptive information from the decrypted result. Both the middle-level plaintexts have been successfully recovered from the ciphertexts using the correct middle-level keys by comparing the CC results of the middle-level plaintexts. Although the CC values between the middle-level plaintexts and the decrypted results in Figs. 10(a) and (b) are 0.5646 and 0.5734, the features of both middle-level plaintexts have been eliminated in Figs. 10(a) and (b). Hence, the protection mechanism effectively renders middle-level plaintexts unrecognizable. The reason for the almost strong correlation between the middle-level plaintexts and the decrypted results in Figs. 10(a) and (b) is that the protection mechanism conceals only the middle-level plaintexts in the object plane. Moreover, the CC results of the middle-level plaintexts in Fig. 10(e) show that the perfect coupling of senior zones and TA zones eliminates all middle-level plaintexts. The CC results of the high-level plaintext show that a single ciphertext from our method is insufficient for decrypting the high-level plaintext. However, it can be almost fully recovered when the high-level plaintext is decrypted from the collimated ciphertext.
From the simulation results in this section, the proposed method can achieve multilevel encryption. However, some high-frequency information of the plaintexts will lose when the PEGS algorithm is conducted. If the grayscale plaintexts are too complex, the decryption results will inevitably degrade. Hence, the proposed method is more suitable for encrypt simple grayscale figures and binary figures to ensure its efficiency.
3.2 Numerical simulation on the influence of keys
The first case to be simulated is the influence of incorrect middle-level keys. Two incorrect middle-level keys are generated randomly to decrypt the ciphertexts in Fig. 9. The corresponding results are shown in Figs. 11(a) and (b).
Fig. 11. Decryption results using incorrect keys. (a) and (b) represent the decryption results of middle-level plaintexts with incorrect keys.
In Figs. 11(a) and (b), the middle-level plaintexts remain well-protected by the protection mechanisms. The CC values of the deceptive information are 0.6603 and 0.6219 respectively. This implies that the eavesdropper can only extract the deceptive information from the ciphertexts when the middle-level keys are incorrect.
For the high-level plaintext, cooperation between two ciphertexts is crucial for decryption. Hence, each ciphertext can be regarded as half of the whole decryption key. When two ciphertexts are coupled, cases of high-level plaintext decryption can emerge: 1) incorrect ciphertexts match; 2) misalignment between two ciphertexts. For incorrect match, we suppose that one of the ciphertexts is coupled with another incorrect ciphertext. For misalignment, we simulate a one-pixel shift in both the horizontal and vertical directions between two ciphertexts. The decryption results for incorrect match and misalignment are shown in Figs. 12(a) and (b), respectively.
Fig. 12. Decryption results for incorrect match and misalignment between two ciphertexts. (a) represents the decryption result for an incorrect match; (b) represents the decryption result for misalignment.
Here, ${C_1}$ in Fig. 9 is used as the correct ciphertext while ${C_2}$ in Fig. 9 is rotated 90 degrees clockwise to form the incorrect ciphertext. From Figs. 12(a) and (b), no discernible information associated with the high-level plaintext “3” can be observed.
Another case simulated is the leakage of key. To simulate various percentages of key leakage, we use an extraction mask with varying duty cycles, as shown in Fig. 13, to extract the unit distribution from the middle-level keys. When the unit distribution on the middle-level key is extracted, the units located in the extraction zone are retained, whereas those outside the extraction zone are set to 1.
In Fig. 13, the side length of the square extraction zone is $b$ pixels and the total side length of the key is $a$ pixels. Hence, the duty cycle of the extraction mask can be represented as:
As the leakage percentage increases, the protection mechanism gradually weakens, and the middle-level plaintexts become more prominent. The CC between the protection mechanism in the decrypted plaintext and the original protection mechanism can be used as an indicator to describe the influence of key leakage. The CC-leakage percentage curve is shown in Fig. 14.
From Fig. 14, the CC values of the protection mechanism remain above 0.6 when the leakage percentage is 75%. However, the CC values drop to 0.6065 and 0.5846 at a leakage percentage of 80%. Simultaneously, the details of the middle-level plaintexts can be recognized, indicating that the protection mechanism has fully failed. Hence, if the leakage percentage of the key does not exceed 80%, then the protection mechanism in our method can keep the middle-level plaintexts safe.
For high-level plaintext, if the eavesdropper is unaware of our method’s scheme, then decrypting the high-level plaintext becomes impossible. Even if the eavesdropper is aware of the scheme, the requirement for two ciphertexts and their precise location matching greatly increases the difficulty of high-level plaintext decryption.
3.3 Attack resistance analysis
Brute force attack is the most basic attack method for cryptanalysis. The practical encryption system is required to have a large key space to resist brute force attacks. In the proposed scheme, the probabilities for 0 and 1 in the binary mask are all 50%. If the key length is $N \times N$, the size of the key space can be expressed as:
where the symbol $C$ is the mathematical operator of combination and the symbol $\lfloor{} \rfloor $ donates the rounding down operation. Based on Eq. (18), the key space will achieve about $1.0 \times {10^{29}}$ even if the key length is only $10 \times 10$. Because the key space is too large to be simulated by the exhaustive method, to simplify the analysis, we analyze the key space theoretically based on the conclusion of key leakage in Section 3.2.According to the conclusion of key leakage, we can draw the equivalent conclusion that when the correct rate of an illegal key exceeds 80%, the protection mechanism will be disabled and the decrypted middle-level plaintext will be obtained. Thus, when the key length is $N \times N$, the number of illegal keys in the key space is:
And then, the proportion of illegal keys in the whole key space can be represented as:
Utilizing Eq. (20), the illegal key proportion-key length curve is shown in Fig. 15.
From Fig. 15, the trend of illegal key is that its proportion in key space decreases dramatically with the increase of key length. When the key lengths are $10 \times 10$ and $128 \times 128$, the proportions of illegal key are ${10^{ - 8.1543}} \approx 7.0 \times {10^{ - 9}}$ and ${10^{ - 1371.7}} \approx 1.9 \times {10^{ - 1372}}$ respectively. Hence, the key space of the proposed scheme is large enough to ensure the security of the plaintext.
Despite the hologram’s exceptional robustness to noise, the combination of two ciphertexts can intensify noise contamination in the high-level plaintext. Hence, assessing the robustness of the high-level plaintext against noise attack is crucial to ensure the efficacy of the proposed method. Gaussian random noises with a mean value of 0 and varying signal-to-noise ratios (SNRs) are added to two ciphertexts in the simulation. Subsequently, the middle-level plaintexts and high-level plaintext are decrypted using these noise-contaminated ciphertexts. The CC values of the decrypted plaintexts are calculated to generate the CC-SNR curve, which is shown in Fig. 16.
From Fig. 16, we observe that when the SNR surpasses 3 dB, the CC values between the decrypted plaintexts and the original plaintexts remain above 0.6. This indicates that the valid information in the decrypted plaintexts can be identified easily. However, the CC of the high-level plaintext drops to only 0.4146 as the SNR decreases to 2 dB, although the CC values of the middle-level plaintexts still exceed 0.6. This implies that the high-level plaintext has lost its distinguishing features and is no longer recognizable. Hence, the proposed method possesses the ability to resist noise with an SNR higher than 3 dB.
For the convenience to transmit the ciphertext, the pure-phase ciphertext is usually transformed into the grayscale figure. However, the frequency of each gray level in the grayscale ciphertext may exposure the information in plaintext. To test the resistance of the proposed scheme against statistical attack, ciphertexts in Fig. 9 are transformed into grayscale figures and then histogram analysis is performed. Figure 17 shows the histograms of the deceptive information, two middle-level plaintexts, one high-level plaintext, and two grayscale ciphertexts respectively.
According to the results in Fig. 17, all the plaintexts have an obvious bar for grayscale value of 0. And the bars of non-zero grayscale values, which form the effective information of the plaintexts, are too small to be seen in the histograms. In the histogram of the ciphertexts, the distribution of grayscale value is close to the uniform distribution. Because the histogram of the ciphertext is significantly different from that of the plaintext, the ciphertext will not provide any information about the distribution of the plaintext. Hence, the proposed scheme is resistant to the statistical attack.
Except for the statistical attack, occlusion attack can also be implemented. To test the robustness of the proposed scheme against data loss attack, we use a variable square to remove the data block from the center of two ciphertexts. And then, two middle-level plaintexts and the high-level plaintext are decrypted from the remaining ciphertexts with the correct keys. The CC values of the decrypted plaintexts are calculated to generate the CC- Occlusion curve, which is shown in Fig. 18.
In Fig. 18, three ciphertexts with the occlusion percentage of 10%, 43%, and 76% are displaced respectively at the bottom. The CC values between the decrypted plaintexts and the original plaintexts remain above 0.6 when the occlusion percentage is lower than 76%. Moreover, Fig. 18 shows that, the robustness of the high-level plaintext against occlusion attack is better than that of the middle-level plaintext. The reason why high-level plaintext has stronger robustness is that, the cooperation of TA zones and senior zones will form more effective phases for high-level plaintext decryption when two ciphertexts are concatenated. From the quantitative results, the proposed scheme can resist the occlusion attack when the occlusion percentage does not exceed 76%.
4. Experiment
In the experiment, the setup, as shown in Fig. 19, is employed.
In Fig. 19, the light source produces collimated linearly polarized light with a wavelength of 632.8 nm. Two SLMs (HOLOEYE, LC 2012 with a resolution of 1024${\times} $768 pixels, and a pixel size of 36 µm) controlled by computers 1 and 2 respectively are used to load the ciphertexts. SLM1 and SLM2 are combined using the fixture when decrypting the high-level plaintext. When decrypting a single ciphertext, SLM1 is removed. Behind the SLMs, a Fourier lens (GCO-0202 M) with a focal length of 300 mm is used to construct the holograms. A CMOS camera (SP-20000C-USB with a resolution of 5120${\times} $3480 pixels, and a pixel size of 6.4 µm) is placed at a distance of 300 mm behind the Fourier lens to record the decrypted plaintexts.
In order to simplify the middle-level keys in the experiment, the middle-level keys and high-level key used are shown in Figs. 20(a)–(c), respectively. The deceptive information, high-level plaintext, two middle-level plaintexts, and their corresponding protection mechanisms are illustrated in Figs. 20(d)–(i). The combined results of the middle-level plaintext and the protection mechanism are shown in Figs. 20(j)–(k).
Fig. 20. Experimental material. (a) and (b) represent two middle-level keys; (c) depicts the high-level key; (d) is the deceptive information “I”; (e) is the high-level plaintext “A”; (f) and (g) donate the middle-level plaintexts “B” and “L”; (h) and (i) signify the protection mechanisms corresponding to “B” and “L” respectively; and (j) and (k) depict the protective effects of the two protection mechanisms.
As SLMs represent modulated phase using grayscale values ranging from 0 to 255, the output ciphertexts obtained by the PEGS algorithm and the materials in Fig. 20 are presented as two grayscale images in Figs. 21(a) and (b).
Utilizing the setup in Fig. 19 and the ciphertexts in Fig. 21, the decrypted results from ${C_1}$ and ${C_2}$ without middle-level keys are listed in Figs. 22(a) and (b). When the middle-level keys are used, the corresponding results are shown in Figs. 22(c) and (d). Upon combining two ciphertexts for the high-level plaintext, the results obtained from the noncollimated ciphertexts and the collimated ciphertexts with a high-level key are shown in Figs. 22(e) and (f), respectively.
Fig. 22. Decrypted results. (a) and (b) show the deceptive information and the protected middle-level plaintexts; (c) and (d) represent the decrypted middle-level plaintexts and the deceptive information; (e) and (f) show the decryption results of high-level plaintexts from the noncollimated ciphertexts and the collimated ciphertexts.
In Figs. 22(a) and (b), only the deceptive information “I” is recognizable whereas the middle-level plaintexts, protected by the designed mechanism, are unrecognizable. In Figs. 22(c) and (d), the protection mechanisms are disabled when the middle-level keys are used for decryption, allowing two distinct middle-level plaintexts “B” and “L” to be observed in the object plane. The different results for the high-level plaintext in Figs. 22(e) and (f) show that only through combing and collimating two ciphertexts, can the high-level plaintext “A” be recovered. Moreover, different from the simulation results, two middle-level plaintexts are also discernible in the object plane when decrypting the high-level plaintext. This phenomenon arises due to imperfect collimation of the two ciphertexts, leading to phase leakage. The hologram’s exceptional robustness enables the leaked phases to construct their corresponding hologram.
Relevant plaintexts from the decrypted results are extracted to verify the effectiveness of the proposed encryption method quantitatively. Subsequently, CC is again used as an indicator to evaluate the correlation between the original plaintexts and the decryption results. The CC calculation results are listed in Table 3.
Table 3. CCs between the decrypted plaintexts and the original plaintexts in experiment
As shown in Table 3, the CC values for the deceptive information “I” are all higher than 0.8 when decrypting a single ciphertext. This indicates that the deceptive information is constructed well enough to mislead the eavesdroppers. Upon combining and decrypting two ciphertexts with a high-level key, the correlation strength for the deceptive information is at the level of very weak. Furthermore, the CC values of the protected middle-level plaintexts in the experiment are similar to those obtained in the simulation. However, middle-level plaintexts can be well recovered with the CC values surpassing 0.9 when the correct middle-level keys are used. In addition, the CC values for the middle-level plaintexts reconstructed from the leaked phases indicate the leaked phases are adequate to decrypt the middle-level plaintexts. For the high-level plaintext “A”, its CC surpasses 0.8 only when two ciphertexts are combined and collimated. Based on the results in Fig. 22 and the data in Table 3, the effectiveness and feasibility of multilevel holographic encryption based on TA concept have been verified.
5. Conclusion
In this study, we proposed and demonstrated a multilevel holographic encryption method based on the concept of TA. The proposed method employs four types of functional zones as the basic unit to create ciphertexts through a PEGS algorithm. Utilizing the layout of the four functional zones, the low-level plaintext can be decrypted directly. The middle-level plaintext requires the corresponding key for decryption, and the high-level plaintext can only be recovered through the combination and collimation of two matched ciphertexts. Compared with normal OHE, deceptive information and additional protection mechanisms are introduced to enhance the security of the method. The incorporation of the TA concept into this method serves as a form of physical protection, heightening the sensitivity and security of the high-level plaintext. In addition, more ciphertexts can be split using the proposed method to form more levels of plaintexts and further enhance the security of the highest-level plaintext.
Considering the distinct advantages of multilevel encryption, the incorporation of deceptive information mechanisms, additional protection mechanisms for high-level plaintext, and TA-based decryption method for the high-level plaintext, our study holds substantial potential for applications in image storage, information concealment, holographic encryption/displays, and related domains.
Funding
Key Laboratory of Optical System Advanced Manufacturing Technology, Chinese Academy of Sciences (2022KLOMT02-01).
Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable suggestions.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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