Jin-Fen Liu(劉進(jìn)芬), Yue Dong(董玥), Le Wang(王樂(lè)), and Sheng-Mei Zhao(趙生妹),?

1Institute of Signal Processing and Transmission,Nanjing University of Posts and Telecommunications,Nanjing 210003,China

2NanJing Tech University Pujiang Institute,Nanjing 211222,China

Keywords: optical encryption,ghost imaging,spread spectrum,correlated imaging

1.Introduction

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 Pittmanet 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, Clementeet 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, Tanhaet 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, Kanget 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,Zhenget 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 orderN,and randomly selectsM(M

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.

2.Theory

2.1.Theoretical method

whereIk(x,y) is the intensity distribution of thekth speckle pattern, and the direct sequence code?k(t), which is an orthogonal walsh hadamard code,should satisfy

whereTis 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 asBk(t),

whereT(x,y) is the transmission function of the object,Ais the illuminated region byPk(x,y,t).This operation is repeatedMtimes forMdifferent temporal-spatial speckle patternPk(x,y,t),and obtainsMbucket detection signals{Bk(t)}1.Then a summation of each bucket detection signal,∑1Bk(t),which is called ciphertext,is transmitted in a public channel.

whereBk=Ik(x,y)T(x,y)dxdy.

Fig.1.Schematic diagram of the SSGI-OE scheme.

It is shown that thekth bucket detection signal corresponding to thekth 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]

2.2.Security analysis

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 isMand the length of each direct sequence code isN,then the probability of obtaining the entire secure key for successfully reconstructing the original image,denoted asr,can be written as

From Eq.(9)we can see that,whenNis larger,the probability of successris 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.

3.Results and discussion

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 thekth speckle patternIk(x,y) is firstly modulated by thekth direct sequence code?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 signalsBk(t), which are recorded via an analogue-to-digital converter(NI USB-6351).Then Alice sends the speckle patterns{Ik(x,y)}1and the ciphertext∑1Bk(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 signalto-noise ratio (PSNR) are used as the evaluations, which are defined as[26]

where ?T(x,y) andT(x,y) refer to the reconstructed and the original image respectively,(x,y)is Cartesian coordinate,Lis the pixels of the image,and maxValis the maximum possible pixel value of the image.

Fig.2.The experimental setup for the proposed SSGI-OE scheme system.LED: light emitting diode.DMD: digital mirror device.L: lens.PD:photodetector.DAQ:data acquisition card.

Fig.3.Comparison of the numerical simulations and experimental results by SSGI-OE and CGI-OE with Hadamard speckle patterns and random speckle patterns.

We first demonstrate the feasibility of the proposed SSGIOE 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 orderN=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 SSGIOE scheme(M=1024,N=2048)can be calculated by

whereMis the length of the secure key,Nis the order of the generated Hadamard matrix,and log2Nrefers 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

whereMis the number of speckle patterns,andLis 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 whenM=1024,N=2048, the number of bits sent from Alice to Bob by using SSGI-OE scheme is about 1.1264×104bits,while the secure key by using the CGI-OE(M=1024)scheme is about 1.048×106bits, 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.

Table 1.The number of bits of the secure key sent from Alice to Bob by using the SSGI-OE and CGI-OE schemes.

Fig.4.Comparison of simulated and experimental results under different eavesdropping ratios.

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

whereM′refers to the length of the eavesdropping secure key,andMis 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 isN=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 isN=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 GIOE.The size of the row numbers is absolutely smaller than that of the speckle patterns.

Fig.5.The MSE and PSNR of the recovered images against the eavesdropping ratio: (a)the simulated results,(b)the experimental results.

Fig.6.Simulated and experimental results by the proposed SSGI-OE scheme under different eavesdropping ratios at the extreme circumstance.

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 matrixHof orderNis in natural order,which can be defined as

whereSirefers to theith row vector of the hadamard matrixH,and walhdenotes the natural ordered Walsh functions.Here,walois defined as the sequency-ordered Walsh functions,then

where〈i〉stands for the bit-reversed representation ofiandb(〈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 isM=1024,and the length of the direct sequence code isN=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 isM′=1024×0.15=154 and the length of each direct sequence code isN=2048,then the probabilityris

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

4.Conclusion

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 orderNare 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 SSGIOE(M=1024,N=2048)scheme is only 0.0107 times over the CGI-OE scheme.It is indicated that our proposed SSGIOE 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.

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant Nos.61871234 and 62001249), the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No.KYCX20_0729), the Natural Science Research Project of Higher Education of Jiangsu Province(Grant No.20KJB510030),the Qing Lan Project of Jiangsu Province(Su Teacher’s Letter[2022]No.29),the Research project of NanJing Tech University Pujiang Institute(Grant No.njpj2022-1-25), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.