Optical encryption scheme based on spread spectrum ghost imaging

With the rapid development of the informatization of human society, information security is becoming more and more important. To meet the sever challenge, researchers have proposed a variety of encryption techniques successively. They mainly include computer encryption,^[1] quantum encryption,^[2–5] and optical encryption.^[6,7] As a promising encryption method, optical encryption has a lot of advantages, such as high-speed operation and possibility of hiding data in multiple dimensions (like phase, wavelength, spatial frequency, or polarization), which recently attracts interests from more and more researchers.

In a different context, ghost imaging (GI),^[8–21] as an intriguing optical technique, has been receiving considerable current attention since Pittman et al. proposed an optical imaging by using two-photon quantum entanglement in 1995.^[8] In 2002, Bennink^[9] provided an experimental demonstration of ghost imaging using a classical source which opened a new avenue to obtain higher resolution images in optically harsh or noisy environments. In ghost imaging, there are two optical beams. One beam, called the signal beam, illuminates an object and then is detected by a bucket detector without any spatial resolution. The other beam, named as the reference beam, is detected by a high spatial resolution detector, such as a charge coupled device (CCD). The information of the object can be retrieved by correlating the intensities detected by the two detectors. In 2008, Shapiro proposed an architecture, named as computational ghost imaging (CGI) to simplify the system of ghost imaging, in which only the signal beam was needed and the reference beam was calculated offline.^[22] This offline computational ghost imaging scheme makes the application of ghost imaging in optical encryption field possible.^[23–38] For instance, Clemente et al. proposed a computational ghost imaging optical encryption (CGI-OE) scheme based on the concepts of CGI in 2010,^[23] in which the information of an object is encrypted into the intensities of signal light. In 2012, Tanha et al. proposed a gray and a color optical encryption scheme to improve the security and develop the application.^[24] Then we presented a quick response (QR) coded compressive ghost imaging optical encryption scheme (QR-CGI-OE), where the computational GI technique, QR code, and compressive sensing technique were adopted in the scheme.^[25] In addition, we proposed another optical encryption scheme based on ghost imaging system with disordered speckles to obtain a higher security with a small key in 2017.^[26] Later, according to the stealth effect of the phase object in the ghost imaging, Kang et al. proposed an optical encryption method based on compressive ghost imaging and public key cryptography in 2018,^[27] which solved the key distribution problem of ghost imaging encryption, and reduced the additional cost of establishing security channels. In 2019, a novel information encryption scheme^[28] was presented based on the customized data container, where the primary information can be recovered completely from the ciphertext encrypted with computational ghost imaging. In 2020, we also proposed an optical encryption scheme based on ghost imaging with fractional Fourier transform (FrFT), which could dramatically reduce the number of bits of the key transferred between authorized users comparing to the existing optical encryption schemes based on CGI.^[29] Recently, Zheng et al. proposed an inverse computational GI scheme,^[35] in which bucket signals are firstly selected and then random patterns are calculated correspondingly. This scheme provided an opportunity to combine with other cryptographies, and enriched the GI-based encryption process.

It can be believed that the optical encryption schemes based on CGI have dramatically reduced the number of the bits required to transmit the object information to a remote party because the encryption of the object image is not a complex valued matrix but simply an intensity vector. However, the prepared random speckle patterns, commonly selected as the secure key in most existing optical encryption schemes based on CGI, have to be transmitted to the authorized user in a private channel, and the amount of the secure key is large due to the recovery algorithm of computational GI technique. Hence, the corresponding key distribution becomes a severe task in the realizations of the existing optical encryption schemes based on CGI. It is a promising direction to develop a secure encryption with a little key distribution.

In this paper, we propose a novel secure optical encryption scheme, named as SSGI-OE, based on spread spectrum ghost imaging,^[39] which is a promising technique in optical security because it has a great potential in protection against eavesdropping. In the scheme, Alice firstly generates a large Hadamard matrix of order N, and randomly selects M (M < N) rows’ numbers of the generated Hadamard matrix as the secure key and distributes them to Bob in a private channel. Alice extracts M corresponding row vectors of the generated Hadamard matrix and regards them as the direct sequence codes. With the help of the spread spectrum ghost imaging system, the measurement results from the bucket detector, called ciphertext, are obtained. Later, Alice transmits M speckle patterns and the ciphertext to Bob by a public channel. The unauthorized user, Eve, cannot reconstruct the image of the object because the correlations between the measurement results and the speckle patterns are destroyed. On the contrary, the authorized user, Bob, could despread the measurement results by the received key, and retrieve the image of the object with the theory of SSGI. The feasibility and security of the proposed optical encryption scheme are discussed by simulations and experiments.

This paper is organized as follows. In Section 2, we present the schematic setup of the SSGI-OE scheme, and give the theoretical analysis of the proposed scheme. In Section 3, we provide the numerical and experimental results to demonstrate the proposed scheme. Finally, we draw our conclusion in Section 4.

Figure 1 shows the schematic diagram of the proposed SSGI-OE scheme, where {Rowk}k=1M are rows’ numbers randomly chosen from N order Hadamard matrix, and M is the number of the selected row vectors. These rows’ numbers are the secure key in the scheme, and transmitted to Bob by a private channel. Alice takes the resultant row vectors of the generated Hadamard matrix as the direct sequence codes {φk(t)}k=1M to modulate the speckle patterns {Ik(x,y)}k=1M. The kth direct sequence code is used to modulate the kth speckle pattern. N is the length of each direct sequence code. Then these speckle patterns are modulated by a spatial modulator, such as a spatial light modulator (SLM) or digital micro-mirror device (DMD), to obtain the temporal-spatial speckle patterns {Pk(x,y,t)}k=1M.

where I_k(x,y) is the intensity distribution of the kth speckle pattern, and the direct sequence code φ_k(t), which is an orthogonal walsh hadamard code, should satisfy

where T is the duration of a speckle pattern.

Later, these speckle patterns illuminate the object directly, and a bucket detector is used to collect the total light intensities transmitted through the object, denoted as B_k(t),

where T(x,y) is the transmission function of the object, A is the illuminated region by P_k(x,y,t). This operation is repeated M times for M different temporal-spatial speckle pattern P_k(x,y,t), and obtains M bucket detection signals {Bk(t)}k=1M. Then a summation of each bucket detection signal, ∑k=1MBk(t), which is called ciphertext, is transmitted in a public channel.

where B_k = ∫_AI_k(x,y)T(x,y)dxdy.

Afterwards, Alice sends the speckle patterns {Ik(x,y)}k=1M to Bob in a public channel. With the correct secure key {Rowk}k=1M, the direct sequence codes {φk(t)}k=1M can be extracted exactly. Using these correct direct sequence codes, the ciphertext can be decrypted,

It is shown that the kth bucket detection signal corresponding to the kth modulated speckle pattern from the ciphertext can be separated efficiently.

Hence, Bob can retrieve the image of the object with the second-order correlation algorithm,^[40]

where M refers to the number of the speckle patterns, and 〈B〉=1M∑k=1MBk.

On the other hand, we conduct a comprehensive security analysis of the proposed scheme. The unauthorized user cannot recover the image of the object. The great advantage of the direct sequence codes is that we can obtain a high security in optical encryption. Assume that the unauthorized user decrypts the ciphertext ∑k=1MBk(t) with the direct sequence codes {φe(t)}e=1M, which are extracted from the wrong secure key {Rowe}e=1M, he will get

It is shown that B_e, instead of B_k, is separated from the ciphertext, which means that the correlations between the resultant detection results {Be}e=1M and the speckle patterns {Ik(x,y)}k=1M are destroyed, so the unauthorized user cannot recover the imaging T^e(x,y) with the second-order correlation,

where 〈Be〉=1M∑e=1MBe, and T^e(x,y) is random data. No useful information can be obtained from the random data T^e(x,y).

In addition, we further analyze the possibility of obtaining the entire secure key in the proposed SSGI-OE scheme. Suppose that the length of the secure key is M and the length of each direct sequence code is N, then the probability of obtaining the entire secure key for successfully reconstructing the original image, denoted as r, can be written as

From Eq. (9) we can see that, when N is larger, the probability of success r is close to zero. In other words, the unauthorized user has only a slim chance to eavesdrop the entire secure key. It is theoretically deduced that our proposed scheme has a high security.

In this section, we testify the proposed SSGI-OE scheme by both numerical simulations and experiments. In addition, we discuss the vulnerability of the proposed scheme.

The experimental setup for the proposed SSGI-OE scheme is shown in Fig. 2. A light emitting diode (LED) is driven by a direct-current (DC) source (Gwinstek GPD-3303S) of 2.9 V to produce the speckle patterns, which are modulated by a digital mirror device (DMD, ViALUX V-7001) with a spatial distribution, where the kth speckle pattern I_k(x,y) is firstly modulated by the kth direct sequence code φ_k(t). Then these speckle patterns are projected onto the object successively. Note that the Hadamard speckle patterns are utilized in practical experiment due to the limitations of experimental equipments. Then a photodetector (Thorlabs PDA100A-EC with gain 30 dB) is employed to collect the total transmission light intensities through the object to generate the bucket detection signals B_k(t), which are recorded via an analogue-to-digital converter (NI USB-6351). Then Alice sends the speckle patterns {Ik(x,y)}k=1M and the ciphertext ∑k=1MBk(t) to the authorized user, Bob, in a public channel. Meanwhile, the secure key is sent to Bob in a private channel.

In order to compare the quality of the reconstructed image quantitatively, mean square error (MSE) and peak signal-to-noise ratio (PSNR) are used as the evaluations, which are defined as^[26]

where T^(x,y) and T(x,y) refer to the reconstructed and the original image respectively, (x,y) is Cartesian coordinate, L is the pixels of the image, and max Val is the maximum possible pixel value of the image.

We first demonstrate the feasibility of the proposed SSGI-OE scheme shown in Fig. 3, where the picture of ‘NUPT’ and the ‘Ghost’ image with the size of 32 × 32 pixels are selected as the objects in the simulation, and only the ‘Ghost’ is used for the experiment. Here, 1024 Hadamard speckle patterns are considered for SSGI-OE scheme and 1024 random speckle patterns are used for the CGI-OE scheme. To provide a fair comparison, we also present the CGI-OE scheme with Hadamard speckle patterns together. For the SSGI-OE scheme, a Hadamard matrix of order N = 2048 is generated first, and 1024 rows are selected randomly as the direct sequence codes, whose rows’ numbers are regarded as the secure key. Note that we use the same secure key in both simulations and experiments. For the CGI-OE scheme, the random speckle patterns (or the Hadamard speckle patterns) are taken as the secure key. The results show that the reconstructed images with the correct key have the complete information of the original ones, especially when the Hadamard speckle patterns are used the recovered images are reconstructed perfectly by both optical encryption schemes, whereas the recovery images with the error key have no useful information of the original object images. In addition, from the column of SSGI-OE with Hadamard patterns and the column of CGI-OE with random patterns, we can see that the images reconstructed by the proposed SSGI-OE scheme are much better than those by the CGI-OE scheme. Therefore, we carry out the simulations and experiments with Hadamard speckle patterns later. Moreover, from the column of SSGI-OE with Hadamard patterns and the column of CGI-OE with Hadamard patterns, we can see that both encryption schemes have similar results, which indicates that the proposed SSGI-OE scheme has a similar decryption performance to the CGI-OE scheme in this situation.

Moreover, the proposed SSGI-OE scheme has a much smaller key. For the 32 × 32 pixels image, the number of bits of the secure key sent from Alice to Bob by the proposed SSGI-OE scheme (M = 1024, N = 2048) can be calculated by

where M is the length of the secure key, N is the order of the generated Hadamard matrix, and log₂N refers to the transmitted bits of each row number in the secure key. Similarly, for the 32 × 32 pixels image, the secure key in the CGI-OE scheme can be computed by

where M is the number of speckle patterns, and L is the number of pixels of each speckle pattern.

The number of bits of the secure key sent from Alice to Bob is listed in Table 1. From Table 1 we can see that when M = 1024, N = 2048, the number of bits sent from Alice to Bob by using SSGI-OE scheme is about 1.1264 × 10⁴ bits, while the secure key by using the CGI-OE (M = 1024) scheme is about 1.048 × 10⁶ bits, which is almost 93.04 times over the SSGI-OE scheme on key distribution. In other words, the number of bits sent from Alice to Bob by using the SSGI-OE (M = 1024, N = 2048) scheme is only 0.0107 times over the CGI-OE (M = 1024) scheme. It is indicated that the proposed SSGI-OE considerably reduces the number of bits sent by Alice to Bob.

Next, we test the security of the proposed SSGI-OE scheme with the object ‘Ghost’ and compare with the results of the CGI-OE scheme. Assume that a potential eavesdropper, Eve, has an eavesdropping ratio (ER) of the secure key. The eavesdropping ratio (ER) is defined as

where M′ refers to the length of the eavesdropping secure key, and M is the length of the whole secure key. The value of ER is set from 5% to 40%. The reconstructed results are displayed in Fig. 4. Here, the size of the binary ‘Ghost’ mage is 32 × 32 pixels, and the length of the direct sequence code is N = 2048. The proposed SSGI-OE scheme adopts 1024 Hadamard speckle patterns, and the CGI-OE scheme uses 1024 random speckle patterns. The first two rows are simulated results, from which we can see that the encrypted information by both schemes cannot be retrieved correctly when ER is less than 40%, even though the outline of the images reconstructed by the SSGI-OE scheme are more or less distinguished. The last two experimental results also show that Eve could not recover the original image when ER is less than 40%, which indicates that the proposed SSGI-OE scheme has a high security.

Figure 5 further shows the MSE and PSNR curves against the eavesdropping ratio by using the proposed SSGI-OE scheme and the CGI-OE scheme. Figure 5(a) shows the simulated results. Figure 5(b) shows the experimental results. Here the object is a binary Ghost image with the size of 32 × 32 pixels, and the length of the direct sequence code is N = 2048. The proposed SSGI-OE scheme uses 1024 Hadamard speckle patterns, and the CGI-OE scheme employs 1024 random speckle patterns. Both simulations and experimental results show that even though the MSEs (PSNRs) value by using the proposed SSGI-OE scheme is smaller (larger) than those by using CGI-OE scheme, Eve could not obtain any information of the original image by using SSGI-OE scheme. The proposed scheme has a high security with a much smaller key. In addition, compared with the other normal encryption scheme based on ghost imaging, such as the scheme demonstrated in Ref. [24], our proposed scheme has a smaller key distribution with a similar security. Because the randomly selected row numbers are the key in our proposed scheme, while the speckle patterns are often used as the key in the normal GI-OE. The size of the row numbers is absolutely smaller than that of the speckle patterns.

Finally, we discuss the security of the proposed SSGI-OE scheme at the extreme circumstance. As we all know, different Hadamard speckle patterns have different contributions on the imaging, the speckle patterns with lower frequency usually have a bigger contributions of imaging, while the higher frequency speckle patterns have less contributions. These speckle patterns with different frequencies can be known from the sequency ordering algorithm.^[41] Generally, a Hadamard matrix H of order N is in natural order, which can be defined as

where S_i refers to the ith row vector of the hadamard matrix H, and wal_h denotes the natural ordered Walsh functions. Here, wal_o is defined as the sequency-ordered Walsh functions, then

where 〈i〉 stands for the bit-reversed representation of i and b (〈i〉) is the gray code to binary conversion of 〈i〉. The gray code is a reflective binary code wherein two successive values differ in only one bit. By the sequency ordering algorithm, the Walsh–Hadamard speckle patterns are ordered from lower frequency to higher frequency.

Suppose that the eavesdropper, Eve, she eavesdrops the secure key to decrypt the ciphertext and achieves a number of bucket values which are properly corresponding to the speckle patterns with lower frequency, she extracts the image of the object at a lower eavesdropping ratio. The results are displayed in Fig. 6. Here, the size of the binary ‘Ghost’ image is 32 × 32 pixels, the number of the speckle patterns is M = 1024, and the length of the direct sequence code is N = 2048. The value of ER is set from 5% to 25%. Both simulations and experimental results show that the encrypted information begins to be retrieved when ER is less than 15%. However, this situation happens with a poor probability. According to Eq. (9), the length of the 15% secure key is M′ = 1024 × 0.15 = 154 and the length of each direct sequence code is N = 2048, then the probability r is

From Eq. (17), we can see that the probability r is close to 0. It is indicated that the proposed SSGI-OE scheme has a high security.

In summary, we have proposed a novel optical encryption scheme based on the spread spectrum ghost imaging, named as SSGI-OE. In the scheme, the row numbers randomly selected from a Hadamard matrix with order N are regarded as the secure key, transmitted from Alice to Bob in a private channel, and the corresponding row vectors are used as the direct sequence codes in the SSGI system. Only the authorized user, Bob, could decrypt the ciphertext by the correct key, and recover the encrypted image with the theory of SSGI, whereas the unauthorized user, Eve, could not obtain any useful information of the encrypted image. The numerical simulations and experimental results have shown the feasibility and the security of the proposed SSGI-OE scheme. When the ER is less than 40%, the eavesdropper cannot acquire any information of the encrypted image. Meanwhile, the authorized user could recover completely the encrypted information with the secure key. Moreover, for the extreme circumstance, the eavesdropper begins to retrieve the information of the encrypted image when ER is less than 15%. However, this situation happens with a lower probability. For the 32 × 32 pixels image, the number of bits sent from Alice to Bob by using the SSGI-OE (M = 1024, N = 2048) scheme is only 0.0107 times over the CGI-OE scheme. It is indicated that our proposed SSGI-OE scheme has a higher security with a smaller key distribution. It has provided a practical method to complement optical encryption with the CGI system. Furthermore, the proposed scheme could promise widespread application of ghost imaging encryption in high security data transferring.