Information encryption based on the customized data container under the framework of computational ghost imaging

Sui Liansheng; Du Cong; Xu Minjie; Tian Ailing; Asundi Anand

doi:10.1364/OE.27.016493

1. Introduction

With the rapid development of information technology in recent years, the processing methods for image encryption, authentication, data hiding and sharing also have obtained great progress in the field of optics. Since the double random phase encoding (DRPE) is proposed in 1995 [1], more and more researchers have realized the intrinsic advantages of optical methods, such as parallelism, light speed processing and multi-degree of freedom. Therefore, plenty of optical schemes have been proposed based on various optical principles and digital techniques, such as polarized light, photon-counting, diffractive imaging, compressive sensing, structured phase mask, interference, joint transform correlator, integral imaging, iterative phase retrieval, phase-only hologram, correlated imaging, and transport of intensity equation have been successfully developed [2–21]. The detailed description of the principle, summaries, and trends of more optical information processing methods can be found in the literature [22–25]. In addition, in order to improve the robustness against common attacks, the DRPE has been further extended into different transform domains such as Fresnel transform domain, fractional Fourier domain and gyrator transform domain [26–35], where additional optical parameters can be employed as secret keys to enhance the security level of the cryptosystems.

Due to the physical advantages such as needless of the imaging lens and complexity of optical parameters, the computational ghost imaging has gained more and more researches in the field of information security [36–40]. Wu et al. [41] investigated the multiplexing capacity for optical multiple-image encryption, where each original image is encrypted into a vector of measured intensities using computational ghost imaging. Then, the ciphertext is obtained by superposing all vectors of intensities. Li et al. [42] reported a multiple-image encryption based on compressive ghost imaging, where secret images are firstly transformed with discrete cosine transform and then integrated with coordinate sampling matrices. The combined image is used as the input to collect the measured intensities. In the correlated-photon secured imaging proposed by Chen [43], the pre-generated random intensity-only maps are used to retrieve the phase-only masks using the iterative phase retrieval algorithm. Jiang et al. [44] presented an information security scheme based on computational temporal ghost imaging. A set of independent 2D random binary patterns considered as the secret keys are weighted with the data stream formed with information to be encrypted, and the ciphertext is obtained by summing these weighted secret keys. Zhang and Zhao [45] suggested a ghost imaging system with disordered speckles to achieve higher security. Sui et al. [46] proposed an optical image hiding under the framework of computational ghost image based on an expansion strategy, where many phase-only masks are used to collect the measured intensities to enhance the resistance against noise and occlusion attacks.

Since Barrera et al. [47,48] for the first time proposed the concept of information container, the feasibility of optical encryption schemes based on the quick response (QR) code have been demonstrated experimentally and digitally. Because of obvious advantages such as fast readability, large storage capacity, and high error tolerance capability, the QR code has become the more and more attractive topic in the field of information security. For example, Zhao et al. [49] proposed a high performance optical encryption, where the QR-coded image carrying original information is encrypted with the help of computational ghost imaging. In the decryption process, the compressive sensing technique is applied to reveal the information by QR decoding. To achieve the goal of large noise tolerance, Zea et al. [50] introduced the first tailor-made container in the optical cryptosystems, which can enhance the ability to encrypt more information while experimental requirements are reduced. Based on the concept of customized data container, Qin et al. [51] presented an information encryption scheme in ghost imaging including two levels. The primary information is transformed into the binary signal to control the generation of the data container in the first level. Then, the container is encrypted with the conventional ghost imaging. Also, Qin et al. [52] used the customized data container in the diffractive imaging based encryption scheme, which reduces the affection of the speckle noise and makes it possible to reveal the hidden information in the real applications.

Although optical information processing methods based on computational ghost imaging have been widely proposed for image encryption, authentication and hiding, there are still two major drawbacks in the process of ghost imaging. One is the low quality of the reconstructed object. Because randomly generated phase-only masks used to record the measured intensities contain overlap information between each other, the reconstructed information usually has noisy distribution even plenty of phase masks are used. Meanwhile, the huge number of phase masks are considered as the secret keys, which is very inconvenient for key management such as storage and transmission. These factors are inadaptable for applications requiring high visual quality. Another is the low security due to the inherent linearity of computational ghost imaging. The related schemes are vulnerable to some attacks such as chosen-plaintext attack. To cope with these drawbacks simultaneously, an information encryption is proposed based on the customized data container under the framework of computational ghost imaging in this paper. In the process of ghost imaging, the phase-only masks are retrieved from the 2D patterns formed with the rows of the Hadamard matrix using the modified phase retrieval algorithm, which assures that only a low number of phase masks are used to collect the measured intensities. To break the linearity of computational ghost imaging, two random sequences are generated based on the logistic map. One is used to obtain an interim data container, with the help of which the XOR operation is applied to the data container carrying the primary information. Another is used to scramble the XOR encoding result. The parameters such as the conditions of the logistic map can be considered as the secret keys to effectively enhance the security of the cryptosystem.

The rest of this article is organized as follows. In Section 2, the theoretical principles of the customized data container and phase-only masks generated using the modified phase retrieval algorithm are introduced. Afterward, the processes of encryption and decryption based on computational ghost imaging is described in detail. In Section 3, a set of numerical simulations and security analysis are performed. Finally, the conclusion is given in Section 4.

2. Scheme principle

2.1 The customized data container

Since the customized data container has been designed in the field of optical information security [50], it has attracted more and more attention due to its obvious advantages such as larger noise tolerance and encryption capacity than quick response code [51,52]. For a customized data container in terms of a binary image, a square block constituted by $3 \times 3$ pixels is used to represent the information of one bit. With respect to this rule, a customized data container shown in Fig. 1(a) is designed to carry the primary information in the proposed information encryption scheme. The size of this container Is $128 \times 128$ pixels, and there are $16 \times 16$ white square blocks in it. These blocks separate 4 pixels from each other along the horizontal and vertical directions. Basically, if a block is set to white color, namely the intensity values of 9 pixels in this block are set to 255, it will denote 1. Otherwise, it will denote 0 if it is set to black, where the intensity values are set to 0. As shown in Fig. 1(a), all blocks are set to white color before the primary information is loaded into it. It should be noted that a total of 256 bits of information can be encoded in this container.

Fig. 1 (a) The customized data container with 256 bits of information, (b) the container encoded with the primary information and (c) the 8 bits of information calculated for the character “X”.

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For the security cryptosystem based on the customized data container, the primary information constituted by characters are considered as the object to be encrypted, which is first transformed into a bit stream with certain criterions. Afterward, each bit is read sequentially from the bit stream to render the color for the corresponding block. Synchronously, the blocks are scanned from left to right and from top to bottom in the data container. A character is represented with 8 bits, and then 8 consecutive blocks are required for encoding it. Because the designed container contains 256 bits of information, it can include up to 32 characters during the process of encryption. When the message that includes characters such as “XUT: computational ghost imaging” is regarded as the primary information, the customized data container will be changed as shown in Fig. 1(b). Usually, the characters are converted into the bit stream according to ASCII standard [51], namely, a character is directly transformed into 8 bits of information according to its decimal value. For example, the decimal value of the character “X” is 88, from which the 8 bits of information can be calculated as “01011000”. The encoding principle can be further illustrated in Fig. 1(c).

2.2 Phase-only masks calculated based on the Hadamard matrix

Besides obvious merits such as needless of the imaging lens, the object to be protected can be encrypted into a vector of intensity values rather than the complex-valued matrix in the optical security field, which make the computational ghost imaging widely studied. However, plenty of phase-only masks should be randomly generated to collect of measured intensities for obtaining the reconstructed object with high quality, which is usually considered as the important secret keys shared with the sender and receiver. As result, it is very inconvenient for storage and transmission of these masks, which hinders wide applications utilizing the computational ghost imaging. To cope with this problem, a spatially orthogonal Hadamard matrix is used to retrieve phase-only masks in the proposed information encryption scheme. The redundancy between these masks can be decreased, which can reconstruct the object with high-quality only using less measured intensities.

Suppose the size of the original object to be encrypted is $N \times N$ pixels, which is the same size as the phase-only masks. First of all, the Hadamard matrix with order $2^{k}$ is constructed, where the condition as $N \times N = 2^{k}$ should be satisfied. The Hadamard matrix with any order is not only square and symmetric but also an orthogonal matrix composed of + 1 and - 1 elements. The second order Hadamard matrix can be defined as

H_{2} = [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}] .

Based on this basic matrix, the Hadamard matrix with order

2^{k}

can be calculated with the recursive procedure, which is mathematically described as

H_{2^{k}} = [\begin{matrix} H_{2^{k - 1}} & H_{2^{k - 1}} \\ H_{2^{k - 1}} & - H_{2^{k - 1}} \end{matrix}] .

After the required Hadamard matrix has been obtained, each row is rearranged into a 2D pattern with

N \times N

pixels. Thus, there are

2^{k}

patterns to be generated, from which

2^{k}

phase-only masks will be retrieved using an iterative phase retrieval algorithm.

Suppose one of these 2D patterns is denoted as $h_{i} (μ, υ), i = 1, 2, 3, \dots, 2^{k},$ from which the phase-only mask $\exp (j φ_{i} (x, y))$ is retrieved. Based on the modified Gerchberg-Saxton algorithm, the phase-only mask and 2D pattern are located in the input and output plane, respectively. Given an initial phase-only mask $\exp (j φ_{i}^{(0)} (x, y)),$ where the phase function $φ_{i}^{(0)} (x, y)$ is randomly distributed in the range $[0, 2 π],$ the iterative phase retrieval process can be described in detail as follows:

(1) In the $n th$ round, the phase-only mask is propagated to the output plane, from which the complex amplitude can be obtained as $E_{i}^{(n)} (μ, υ) = F r T_{λ, z} {\exp (j φ_{i}^{(n)} (x, y))},$
where the function $F r T_{λ, z} {\cdot}$ stands for the Fresnel transform with regard to the wavelength $λ$ and the axial distance $z .$
(2) After the amplitude is substituted with the support constraint, namely the square root of the 2D pattern, the above result in the output plane is updated as ${\hat{E}}_{i}^{(n)} (μ, υ) = \sqrt{h_{i} (μ, υ)} \exp (j \arg (E_{i}^{(n)} (μ, υ))),$
where the function $\arg (\cdot)$ is used to calculate the phase distribution. Then, this new wavefront propagates back to the input plane, and the corresponding process can be mathematically described as
$G_{i}^{(n)} (x, y) = F r T_{λ, - z} {{\hat{E}}_{i}^{(n)} (μ, υ)},$
where the function $F r T_{λ, - z} {\cdot}$ stands for the inverse Fresnel transform.
(3) Before starting the next round, the phase-only mask in the input plane is updated by extracting the phase function of the inverse propagation result, which can be expressed as $\exp (j φ_{i}^{(n + 1)} (x, y)) = \exp {j \arg (G_{i}^{(n)} (x, y))} .$
(4) Above steps constitute a cycle, which will not terminate until the correlation coefficient (CC) between the intensity ${| E_{i}^{(n)} (μ, υ) |}^{2}$ and the 2D pattern $h_{i} (μ, υ)$ achieves the required threshold. The CC value can be calculated as $CC = \frac{E [[{| U |}^{2} - E [{| U |}^{2}]][h - E [h]]]}{\sqrt{E [[{| U |}^{2} - E [{| U |}^{2}]]^{2}]} \sqrt{E [{[h - E [h]]}^{2}]}} .$
For brevity, the coordinates and subscripts are omitted. To assure the best estimation in the output plane, the threshold is usually set very close to 1. Once the iteration process terminated, the last updated result calculated with Eq. (6) is saved as the retrieved phase-only mask.

In addition, it should be noted that the elements with value −1 in these 2D patterns should be set to 0 before the iteration processes start, which is not favorable to reduce the redundancy between the retrieved phase-only masks.

2.3 Image encryption based on computational ghost imaging

With the help of computational ghost imaging technique, the customized data container carrying the primary information will be encoded into a series of intensity values in the proposed information encryption scheme. The schematic setup for illustrating the encoded process is depicted in Fig. 2. After collimated by a lens, the wave from the laser illuminates the spatial light modulator (SLM), which is controlled by computer to load plenty of phase-only masks sequentially. Thereafter, the beam pass through the transparent area of the customized data container at an axial distance from the SLM, where this distance equals to that used in the iterative process of retrieving the phase-only masks from 2D patterns. When the phase-only mask $\exp (j φ_{i} (x, y))$ is loaded into the SLM, the measured intensity denoted as $B_{i}$ is recorded by the bucket detector without spatial resolution, which can be expressed as

B_{i} = \iint d μ d υ I_{i} (μ, υ) T (μ, υ),

where the function

T (μ, υ)

is the transmission function according to the customized data container, and

I_{i} (μ, υ) = {| E_{i} (μ, υ) |}^{2}

is the related speckle pattern. It should be pointed out that

E_{i} (μ, υ)

is the Fresnel transform result of the mask

exp (j φ_{i} (x, y))

as described with Eq. (3). Because the phase-only masks are retrieved from 2D patterns generated based on the Hadamard matrix, the redundancy between these speckle patterns can be reduced greatly, which will be helpful to improve the quality of the reconstructed data container even less measured intensities are employed.

Fig. 2 The schematic setup for illustrating the encoded process: SLM, spatial light modulator; BD, bucket detector; POMs, phase-only masks.

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Although the encoded process is straightforward, it should be aware that there is a serious problem need to be solved, that is, how to avoid security risk originating from the linearity of computational ghost imaging. In order to cope with this question, before the customized data container is encoded, it is processed firstly based on the random sequence generated with logistic map. As a very simple one-dimensional nonlinear chaos function, logistic map has been increasingly applied to image encryption. A tiny variation of the conditions such as initial value and bifurcate parameter can yield dramatically different random iterative values. Meanwhile, these conditions can be considered as the secret keys to further enhance the security level due to their high sensitivity. Given the primary information to be encrypted, the detailed encryption process based on computational ghost imaging and logistic map is described as follows:

(1) According to the primary information, the customized data container carrying the related characters is constructed based on the aforementioned formation rule. Thus, a binary image representing this data container can be obtained, which is denoted as $f$ and has $N \times N$ pixels.
(2) Given the conditions of logistic map such as the bifurcate parameter $p$ and the initial value $x (0),$ the random sequence is generated to scramble the obtained data container. The nonlinear iterative form of logistic map can be mathematically described as $x (n + 1) = p \times x (n) \times (1 - x (n)),$
where $x (n) \in (0, 1)$ denotes the iterative value, and $p \in [3 .5699456,4]$ . Initially, a random sequence is generated using with the given conditions and an integer $P$ , which is denoted as $X = {x (j) | j = 1,2, \dots, 2 N N + P} .$ To increase randomness, the previous $P$ values are discarded to obtain two new random sequences, which are expressed as
$X^{'} = {x (j) | j = 1,2, \dots, N N},$ $Y^{'} = {x (j) | j = N N + 1, N N + 2, \dots, 2 N N} .$
(3) An initial customized data container similar to Fig. 1(a) is constructed, which also contains 256 square blocks. With the help of a predefined threshold $γ \in (0, 1)$ and the random sequence $Y^{'},$ the color of its blocks can be changed. Suppose the previous 256 random values of the sequence $Y^{'}$ are selected. If the $k th$ value is larger than the given threshold, the $k th$ block of this data container is set to white. Otherwise, it is set to black. Thus, a random data container is generated and denoted as $g .$ By performing the operation as $f XOR g,$ the XOR encoding result can be obtained and denoted as $f^{'} .$
(4) After sorting $X^{'}$ in the ascending or descending order, another sequence denoted as $X^{″} = {x (w (j)) | j = 1,2, \dots, N N}$ is generated, where the symbol $w (\cdot)$ is the address code and means that the $j th$ value in the sequence $X^{'}$ is mapped to the $w (j) th$ value in $X^{″} .$ Meanwhile, the XOR encoding result $f^{'}$ is reshaped into a one-dimensional sequence denoted as $F = {f (j) | j = 1,2, \dots, N N} .$ With the help of the sequence $X^{″},$ a new sequence $F^{'}$ is obtained, where the $j th$ value in $F$ is placed in the position $w (j)$ in $F^{'} .$ Finally, the sequence $F^{'}$ is reshaped into the two-dimensional matrix to form the scrambled data container $f^{″} .$
(5) Using Eqs. (1) and (2), the Hadamard matrix with order $2^{k}$ is constructed. Each row of this matrix is reshaped into a 2D pattern, from which a phase-only mask is retrieved using the iterative phase retrieval algorithm.
(6) Placing the scrambled data container $f^{″}$ into the object plane in Fig. 2, the whole or partial phase-only masks are loaded into SLM in turn. Thus, a series of measured intensities can be recorded by the bucket detector, which will be used as the ciphertext in the proposed scheme.

Although the encryption process is somewhat complicated, the decryption process is simple, which can be implemented digitally. Suppose there are $K$ measured intensities recorded by the bucket detector, the scrambled data container can be reconstructed by correlation between speckle patterns and measured intensities, which can be mathematically described as

f^{″} = \frac{1}{K} \sum_{i = 1}^{K} (B_{i} - 〈 B_{i} 〉) (I_{i} (μ, υ) - 〈 I_{i} (μ, υ) 〉),

where the calculation

〈 \cdot 〉

denotes the ensemble average. Based on two random sequences generated with the same conditions and the given integer in the process of encryption, the scrambled data container is inversely transformed to the XOR encoding result

f^{'},

and the interim data container

g

also is generated. Finally, the original customized data container can be obtained by performing the XOR operation between

f^{'}

and

g,

from which the primary information can be extracted. Because the color of blocks in the decrypted data container is not completely white or black, the bit of information represented with a block can be determined by evaluating the average intensity value of 9 pixels. According to a preset threshold in the range

[0, 255],

if the average is larger, it is identified as 1. Otherwise, it is identified as 0. This procedure is essential for recovering the primary information when only partial phase-only masks are used to collect the measured intensities. This judgment process is also referred to as threshold-judging recognition [51]. In addition, the conditions of logistic map and the integer applied in the process of scrambling the customized data container can be used as the secret keys, which not only enhances the security level due to their high sensitivity but also makes the management of secret keys very convenient.

3. Results and analysis

To demonstrate the feasibility and effectiveness of the proposed optical information encryption scheme, a series of numerical simulations are performed based on the computational ghost imaging setup shown in Fig. 2. It should be noted that the unwanted effects like speckle noise are not considered, which widely exists in the actual encryption applications [53–57]. The designed customized data container carrying the primary information is shown in Fig. 1(b). The wavelength of the collimated wave is 632.8 $nm$ and the axial distance between the SLM and the object plane is 50 mm. The bifurcate parameter and the initial value of logistic map are set to 3.99995 and 0.45, respectively. In the generation of random sequence values, the previous 2000 iterative values are discarded for increasing randomness. The random data container generated based on logistic map is shown in Fig. 3(a). By performing the XOR operation between Fig. 1(b) and Fig. 3(a), the XOR encoding result is obtained as shown in Fig. 3(b). After the XOR encoding result is scrambled based on the random sequence, the scrambled data container is shown in Fig. 3(c). Thus, the regular structure of the original customized data container is completely destroyed, which can break through the linearity of the cryptosystem and prevent an invalid user from accessing the primary information directly. Because the size of the data container shown in Fig. 1(b) is $128 \times 128$ pixels, the order of the Hadamard matrix is $2^{14} .$ There are $2^{14}$ phase-only masks which can be generated from the rows of the Hadamard matrix. Figure 3(d) shows a 2D pattern formed with one row of the Hadamard matrix. Figure 3(e) shows the retrieved phase-only mask using the iterative phase retrieval algorithm.

Fig. 3 (a) The random data container, (b) the XOR encoding result, (c) the scrambled data container, (d) a 2D pattern formed with one row of the Hadamard matrix, and (e) the corresponding phase-only mask.

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When 100%, 80%, 40%, 20%, 15% and 10% of the total masks are sequentially loaded into the SLM to record the measured intensities, the reconstructed XOR encoding result is displayed in Figs. 4(a)-4(f), respectively. It can be seen that the quality of the reconstructed results degrades gradually with the decrease in the number of phase masks. The CC values between Figs. 4(a)-4(f) and Fig. 3(b) are 0.9528, 0.8585, 0.6165, 0.4369, 0.3831 and 0.3152, respectively. After the XOR operation between Figs. 4(a)-4(f) and Fig. 3(a) is respectively performed, the customized data container can be reconstructed to extract the primary information. For the reconstructed results shown in Figs. 4(a)-4(e), the recovered data containers are the same as the original one shown in Fig. 1(b). The primary information can be extracted correctly when the threshold used in the threshold-judging recognition is set to 110. However, no matter how the threshold is changed, the primary information cannot be recovered entirely from Fig. 4(f). The recovered information is listed in Table 1 when the threshold is set to 110, 90, 70, 50 and 30, respectively. It should be explained that the recovered character will be set to “?” if its decimal value is not in the range $[32, 126]$ , because the character with other decimal value is usually considered as the format control character. Anyway, the primary information can be extracted completely by using much less phase-only masks in the proposed encryption scheme.

Fig. 4 (a)-(f) The reconstructed XOR encoding result when100%, 80%, 40%, 20%, 15% and 10% of the total masks are used to collect the measured intensities.

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Table 1. The recovered primary information when different thresholds are used.

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In the subsequent security analysis, it is considered that only 15% of phase-only masks are used to collect the measured intensities. In the proposed scheme, the conditions of logistic map can be considered as the secret keys to enhance the security. When the bifurcate parameter is altered by $\pm 1 \times 10^{-15},$ the reconstructed XOR encoding results are displayed in Figs. 5(a) and 5(b), respectively. The corresponding CC values between them and Fig. 3(b) are −0.0003 and 0.0119, respectively. After performing the XOR operation with random data container shown in Fig. 3(a) and applying the threshold-judging recognition with different thresholds, the recovered information is listed in Table 2. It can be seen that almost no valid characters are recognized when the threshold is set to 110. As the threshold decreases, more characters are gradually discerned. However, the original information cannot be perspective from these recovered results. The relation curve between CC values and the deviation of this parameter is displayed in Fig. 5(c). When this parameter has very tiny change, namely its deviation is very close to 0, the CC value will be larger than 0.35, which guarantees that the primary characters can be extracted from the reconstructed data container.

Fig. 5 (a)-(b) The reconstructed data container with the tiny changes of the bifurcate parameter, and (c) the relation curve between CC values and the deviation of the bifurcate parameter.

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Table 2. The recovered characters extracted with different thresholds when the bifurcate parameter has tiny changes.

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When the initial value of logistic map is changed by $\pm 1 \times 10^{-16},$ the reconstructed XOR encoding results are displayed in Figs. 6(a) and 6(b), respectively, which all have noisy distribution. The corresponding CC values between them and Fig. 3(b) are −0.0035 and 0.0002, respectively. After performing the XOR operation with random data container shown in Fig. 3(a) and applying the threshold-judging recognition with different thresholds, the recovered information is listed in Table 3. Similarly, more valid characters are discerned with the threshold decreasing. Nevertheless, the original information still cannot be recognized from these reconstructed data containers. The relation curve between CC values and the deviation of the initial value is depicted in Fig. 6(c), from which it can be seen that the CC value will fall below 0.35 sharply when the initial value has a very small variation. Therefore, the original information cannot be revealed from the corresponding data container even with a lower threshold.

Fig. 6 (a)-(b) The reconstructed data container with the tiny variation of the initial value, and (c) the relation curve between CC values and the deviation of the initial value.

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Table 3. The recovered characters extracted with different thresholds when the initial value has tiny changes.

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In order to enhance the randomness of the sequence generated based on the logistic map, an integer is used to indicates how many previous random values are discarded. Actually, this integer can also be considered as the security key in the proposed information encryption scheme. When this integer is altered by $\pm 1,$ the reconstructed XOR encoding results are displayed in Figs. 7(a) and 7(b), respectively. These reconstructed results have better visual quality, from which some white blocks can be observed faintly. The corresponding correlation coefficients between them and Fig. 3(b) reach high values such as 0.1020 and 0.1628, respectively. After performing the XOR operation with random data container shown in Fig. 3(a) and applying the threshold-judging recognition with different thresholds, the recovered information is listed in Table 4, where the wrong primary information is produced even the threshold reaches 110. The relation curve between CC values and the deviation of this integer varied from −20 to 20 is shown in Fig. 7(c), where the CC value will fall below 0.35 sharply once this integer is not correct.

Fig. 7 (a)-(b) The reconstructed data container with the tiny variation of the integer, and (c) the relation curve between CC values and the deviation of the integer.

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Table 4. The recovered characters extracted with different thresholds when the integer has tiny changes.

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Due to its inherent linearity, the image encryption scheme based on computational ghost imaging can be easily cracked by the chosen-plaintext attack, which means that the attacker can retrieve the phase-only masks loaded into spatial light modulator by freely choosing the object to be encoded in the object beam. In addition, the encryption process is carried out using the scrambling operation before DRPE in the general encryption model with the architecture scrambling-then-DRPE. Due to the linearity of DRPE, the vulnerability of the model to chosen-plaintext attack has been demonstrated [58], which indicates that the scrambling operation usually cannot compensate for the security problem. Although the proposed encryption scheme is not based on DRPE, the scrambling operation is used to change the structure of the customized data container, which potentially affords the risk of being attacked.

To cope with these serious security risks, the customized data container carrying the primary information is firstly changed by using the XOR operation with the random interim data container, and then the XOR encoding result is scrambled based on the random chaotic sequence. Finally, the scrambled data container is encrypted into the ciphertext values using computational ghost imaging. The XOR operation can hinder the attacker from directly accessing the object beam, and the linearity of the cryptosystem can be broken through thoroughly. Even the attacker can obtain a larger number of image pairs like Fig. 1(b) and Fig. 3(c) by using the chosen-plaintext attack, it is still very difficult to retrieve the parameters used in the cryptosystem. So, it can be concluded that the proposed scheme is totally immune to the chosen-plaintext attack. Meanwhile, the parameters of the cryptosystem such as the conditions of the logistic map and the integer used to discard the previous values of random sequence are considered as the secret keys. These parameters are remarkably sensitive to tiny variation, which not only enlarges the key space but also enhances the robustness against the brute force attack.

4. Conclusion

To conclude, an optical information encryption scheme is proposed based on computational ghost imaging and logistic map. The primary information composed of displayable characters is encoded into a customized data container. Before being encrypted into a vector of intensity values as the ciphertext, the customized data container is changed by the XOR operation and scrambled based on two random sequences generated with logistic map, where the inherent linearity of computational ghost imaging can be broken. Especially, a number of phase-only masks retrieved from the rows of the Hadamard matrix are used to collect the measured intensities of the customized data container. Thus, the primary information can be efficiently extracted from the reconstructed data container using more less measured intensities. Also, the proposed information encryption scheme has high security level due to high sensitivity of the secret keys such as the conditions of logistic map. Numerical results have demonstrated the efficiency and performance of the proposed method.

Funding

Key Laboratory Science Research Plan of Education Department of Shaanxi Province (16JS079).

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Threshold	Primary information recovered from Fig. 4(f)
110	XWT: compu?ational glost imaging
90	X?T: comtu?ati/nal glgst imaoing
70	\?T: cootu?ati/nal glgst imaoi.g
50	\??*”cO?tu?hTi/?Al!glgst imaoi.e
30	\??*”cO?t5?hDa/?Al!glgst#ikaoi.e

Threshold	Primary information recovered from Fig. 5(a)	Primary information recovered from Fig. 5(b)
110	= ?????????? = ????g?g??hy????????g	???x?8???}?f}??c??m???m???????v?
90	-?????????? = ????g?g??ly????????g	???x?????}?f\|??c??m???}???????v?
70	-?????????? = ??'?g~e??lx???????Xg	???xL????}??\|??c??i???}???????v?
50	%????]????? = ????g~EC??X?G?<???Zg	???\|l??G?]??\|??s??)??? = ???????w?
30	?????]??????????U~TC????^? = ??lZc	??;\|l??C????\??0??9???????????g?

Threshold	Primary information recovered from Fig. 6(a)	Primary information recovered from Fig. 6(b)
110	????????X7?}????76?????6?m???s??	f??0??g??????????l??~???m???>?x?
90	????????X?0\|????w6???????m???s??	f??0??g??????????,??~???m???>?x?
70	[???????\?0\|?????6???????m???3??	f??0??g??????????.??~???m???>?x?
50	????????\?0\|??>??4???????l???7_?	f??0??g?????????????}???q???6?x?
30	????????M?0t??>??????????l???7??	f??0??????o???a????? = ???q?4?V?h?

Threshold	Primary information recovered from Fig. 7(a)	Primary information recovered from Fig. 7(b)
110	???sX???G?2C??????xn??qL??{????? `	???????.??????>)k??K???cvg?{?2h?
90	???sX???Ww2C??????xn??uM??{?????	????>??.??????:)k??K???cve?m?2h?
70	???cT???Ww0C??????j.???M??;{????	????:??/?vWpu?;)k??K???cru?m?2`?
50	Y??b??^?U7 Cr??pt?J*X??]???{????	????:C?/1vWpuh; + k??K????ru?m?2??
30	Y??”?U^oU??AR??pp??:Y??S???y???S	?VF?:i?/1Tupah + + j??K????qy?mr;??

Threshold	Primary information recovered from Fig. 4(f)
110	XWT: compu?ational glost imaging
90	X?T: comtu?ati/nal glgst imaoing
70	\?T: cootu?ati/nal glgst imaoi.g
50	\??*”cO?tu?hTi/?Al!glgst imaoi.e
30	\??*”cO?t5?hDa/?Al!glgst#ikaoi.e

Information encryption based on the customized data container under the framework of computational ghost imaging

Abstract

1. Introduction

2. Scheme principle

2.1 The customized data container

2.2 Phase-only masks calculated based on the Hadamard matrix

2.3 Image encryption based on computational ghost imaging

3. Results and analysis

4. Conclusion

Funding

References

Cited By

Figures (7)

Tables (4)

Equations (12)

Optics Express