Chao Han and Yuzhen Shen

Abstract—In order to increase the capacity of encrypted information and reduce the loss of information transmission,a three-dimensional(3D)scene encryption algorithm based on the phase iteration of the angular spectrum domain is proposed in this paper.The algorithm, which adopts the layer-oriented method,generates the computer generated hologram by encoding the three-dimensional scene. Then the computer generated hologram is encoded into three pure phase functions by adopting the phase iterative algorithm based on angular spectrum domain,and the encryption process is completed. The three-dimensional scene encryption can improve the capacity of the information,and the three-phase iterative algorithm can guarantee the security of the encryption information.The numerical simulation results show that the algorithm proposed in this paper realized the encryption and decryption of three-dimensional scenes. At the same time, it can ensure the safety of the encrypted information and increase the capacity of the encrypted information.

I.Introduction

IMAGES have become an important part of people’s information exchange and transmission with their vivid expression.The rapid development of network technology has brought convenience to information transmission.At the same time, the security of information has become an issue of people’s concern.The emergence of optical encryption technology has aroused great interest in the field of information processing,and due to the high-speed parallel processing capabilities of optical systems,optical information security techniques have been widely developed in recent years.

Since Javidi and Refregier first proposed the double random phase encoding(DRPE)algorithm[1],optical encryption technology has received more and more attention[2]–[5].Matoba and Javidi proposed an optically encrypted storage system based on Fresnel-transform(FrT)domain of the double random phase encoding,which introduced the location parameters as the third dimensional key and expanded the key space[6].Liuet al.proposed a DPRE algorithm based on fractional Fourier domain[7],[8],which further extended the application of DPRE algorithm.Since the encryption scheme of DPRE algorithm is vulnerable to attack techniques[9]–[12],Chenetal.[13] proposed a random phase encoding algorithm with a rear-mounted phase mask to improve the security of it.However,compared with the previous algorithm,it only adds a random phase in the rear focal plane of the second lenses,so the security risks still exist.In order to further improve the security of DPRE algorithm,the scholars have proposed a series of new algorithms based on the DPRE encryption scheme[14]–[16].An iterative encryption algorithm was first proposed by Wanget al.[14],encoded the original image into phase-only masks(POM s)and increased the security of the encrypted information.Liet al. proposed an algorithm by improving[14],the algorithm reduced the computation complexity by adopting the computer generated POM(s)[15].In order to expand the capacity of encrypted information,Situ and Zhang first proposed a wavelengthmultiplexed multi-image encryption algorithm,which encrypted each original image through the double-phase encoding algorithm and then superimposes them to generate the final encrypted image.This algorithm guaranteed the security of encrypted information and also increased the capacity of encrypted information[17].Subsequently,the researchers proposed a variety of multi-image and 3D scene encryption schemes[18]–[21].Chenet al.[22]proposed an asymmetric multi-image encryption scheme based on compressed sensing and feature fusion.However,the encryption and decryption process of the scheme is complicated,and it is difficult to realize this process by optical methods.Zhaoet al.proposed a 3D display algorithm which used an angular-spectrum layer-oriented method to encode a 3D scene to generate computer generated hologram(CGH).The algorithm can realize 3D scene display[23].Konget al.[24]proposed an encryption algorithm in the Fourier domain by using the layer-oriented method of [23].It adopted a layer-oriented method to generate a CGH from 3D scenes and used a vector stochastic decomposition algorithm(VSDA)to decompose the CGH into two pure phase functions, which can realize the encryption and display of the 3D scenes and increase the capacity of encrypted information.Subsequently, the updated experiments based on[24]is proposed,where the algorithm solves the inherent silhouette problem,the holographic encryption system based on interleaved CGHs is more practical and simple and expands the application range of the holographic encryption algorithm[25].

Summarizing the above algorithms,in order to improve the accuracy of information transmission and expand the capacity of encrypted information,a 3D scene encryption algorithm that combines accurate angular-spectrum diffraction theory with three-phase iterative algorithm is proposed in this paper.The algorithm first adopts the layer-oriented method to encode the 3D scene into a CGH,and then the CGH is encoded to generate three pure phase functions according to the phase iterative algorithm. Numerical simulation results show that the proposed algorithm can achieve the encryption and decryption of 3D scenes,ensure the safety of information and increase the capacity of encrypted information.

II.Encryption and Decryption Algorithm

III.Numerical Simulation and Analysis

Fig.2.The Algorithm of[3]:(a)Original image;(b)POM 1 phase distribution;(c) POM 2 phase distribution;(d)Decrypted image.

A monochrome unit plane wave with a wavelength of λ=532 nm is used in the encryption and decryption process,the diffraction distance parametersZ1,Z2,Z3are 50mm,70mm,and 80mm,respectively.In this paper,we propose an iterative encryption algorithm based on[3].First,the cameraman image will be used in 2D image numerical simulation analysis,and the image size is 256×256 pixels.Under the same simulation environment,the numerical simulation of the algorithm in[3]and the one in this paper is shown in Figs.2 and 3, respectively.Fig.2(a)represents the original image,Figs.2(b)and 2(c)respectively represent the phase distributions of POM 1 and POM 2.The decryption images of the algorithm in[3]are shown in Fig.2(d).For the algorithm in this paper,Figs.3(a)and 3(e)respectively represent the original image and the decrypted image,Figs.3(b),3(c),and 3(d)respectively represent the phase distribution of POM 1,POM 2,and POM 3.

In quantitative analysis of reconstructed image quality,the correlation coefficient is used as the evaluation criterion.The correlation coefficients of the algorithm of this paper and[3]are shown in Fig.4.At the initial iteration,the coefficients of the proposed algorithm in this paper are already larger than the ones in [3].In the subsequent phase iteration,although the correlation coefficients of the algorithm in this paper have a phased decrease,in the overall curve the correlation coefficients obtained by the algorithm in this paper are always higher than the correlation coefficients in [3].Therefore,it can be concluded that the proposed algorithm can improve the quality of reconstructed images under the same number of iterations.

Fig.3.The algorithm of this paper:(a)Original image;(b)POM 1 phase distribution;(c)POM 2 phase distribution;(d)POM 3 phase distribution;(e)Decrypted image.

Fig.4.The number of iterations and correlation curves between [3]and the algorithm of this paper.

The numerical simulation results of the 3D scenes verified that the algorithm can realize 3D scene encryption and decryption and expand the capacity of the encrypted information.The“plane”is selected as the 3D scene in the numerical simulation,its depth and location images are shown in Figs.5(a)and 5(b),respectively.The sizes of depth image and location image of 3D scenes are both 256×256 pixels.The capacity of encryption information is 131 072 pixels.Four quantization levels are used for the initial random phase in the paper.An angular spectrum diffraction algorithm is used to encode the“plane” into a CGH based on the layer-oriented method,the CGH is shown in Fig.5(c).Then the CGH is encoded into the three phase masks(POM 1,POM 2,and POM 3) by the phase iterative algorithm.Figs.6(a),6(b),and 6(c)respectively show the phase distributions on the POM 1,POM 2,and POM 3.Fig.7 shows the reconstructed images of two correct secret keys.Figs.7(a),7(b),and 7(c)are reconstructed images with decryption distancesd1=?173.33mm ,d2=?166.7 mm,andd3=?160 mm,respectively.

Fig.5.(a)Depth image of the plane;(b)Location image of the plane;(c)CGH of the plane.

Fig.6.Phase masks of encoding CGH of the plane:(a)Phase distribution of POM 1;(b)Phase distribution of POM 2;(c)Phase distribution of POM 3.

The size of the key space is an important indicator of encryption algorithm;according to[26],the size of key-space of this paper is3×4256×256×2. And the size of key space increases as the size of the encrypted image increases.

Fig.7.Reconstructed images with two secret keys:(a)d1=?173.33 mm;(b) d 2=?166.7 mm; (c)d 3=?160 mm.

Fig.8.(a)Depth image of the train;(b)Location image of the train;(c)CGH of the train.

Fig.9.Phase masks of encoding CGH of the train:(a)Phase distribution of POM 1;(b)Phase distribution of POM 2;(c)Phase distribution of POM 3.

Fig.10.Reconstructed images with two secret keys:(a) d 1=?160 mm;(b)d2=?166.7mm; (c)d 3=?173.33 mm.

In addition,in order to verify the feasibility of the algorithm,the “train”is selected as the 3D model,and the numerical simulation of the“train” is performed.Figs8(a)and 8(b)respectively represent the depth and location images of the “train”,and Fig.8(c)shows the CGH of the“train”.Fig.9 shows the phase distribution of three phase masks(POM 1,POM 2,and POM 3).Fig.10 shows the reconstructed images of the “train”with two correct secret keys,and Figs10(a),10(b),and 10(c)represent the decryption distances ofd1=?160 mm,d2=?166.7 mm,andd3=?173.33 mm,respectively.

IV.Security Analysis of the Known Plaintext Attacks

In order to strengthen the proposed idea,known plaintext attack(KPA)is carried out.Assume that the phase φ3(u3,v3)is the known-plaintext,and phases φ2(u2,v2) and φ1(u1,v1)are regarded as the secret keys.In the case of known encryption algorithm,the attack analysis of the proposed scheme is as follows.

1)When only the known-plaintext is obtained and the secret keys of phases φ2(u2,v2) and φ1(u1,v1)are unknown.The estimated secret keys or random secret keys are used for decryption(the random phase is adopted in this paper),and the decryption results are shown in Fig.11.

2)The plaintext φ3(u3,v3)and one of the correct secret keys φ2(u2,v2)( or secret key φ1(u1,v1))are known,and the other key 1(or key 2)is replaced by a random phase,decryption is shown in Fig.12(or Fig.13).

Fig.11.The reconstructed images of the“plane” with the only the plaintext POM 3:(a)d 1=?173.33 mm; (b) d 2=?166.7 mm; (c)d 3=?160 mm.

Fig.12.The reconstructed images of the known secret key φ2(u2,v2):(a)d1=?173.33 mm; (b)d 2=?166.7 mm; (c) d 3=?160 mm.

According to the above numerical simulation results,it can be said that the three-phase iterative algorithm based on the angular spectral domain has high security performance.In the case of illegal decryption,if the two secret keys cannot be completely obtained,the wrong result will occur in the decryption process,which ensures the security of original object information transmission.

V.Conclusion

The 3D scene encryption algorithm based on the phase iteration algorithm of the angular spectral domain is proposed in this paper.The layer-oriented method is used in the encryption algorithm to encode 3D scene into a CGH.Then the CGH is encoded into three pure phase functions by using accurate angular spectrum transformation and the three-phase iterative algorithm in the encryption process.The numerical simulation of 3D scenes results show that the encryption algorithm can realize the encryption and decryption of 3D scenes,and not only increases the capacity of the encrypted information, but also ensures the safety of the encrypted information.

Fig.13.The reconstructed images of the known secret key φ1(u1,v1):(a)d1=?173.33 mm; (b) d 2=?166.7 mm; (c)d 3=?160 mm.