We demonstrate that all parameters of optical lightwave can be simultaneously designed as keys in security system. This multi-dimensional property of key can significantly enlarge the key space and further enhance the security level of the system. The single-shot off-axis digital holography with orthogonal polarized reference waves is employed to perform polarization state recording on object wave. Two pieces of polarization holograms are calculated and fabricated to be arranged in reference arms to generate random amplitude and phase distribution respectively. When reconstruction, original information which is represented with QR code can be retrieved using Fresnel diffraction with decryption keys and read out noise-free. Numerical simulation results for this cryptosystem are presented. An analysis on the key sensitivity and fault tolerance properties are also provided.
© 2014 Optical Society of America
The nature of light wave infuses new energy and possibility into security systems. Two obvious advantages can be observed when securing information with light. The first one is the high speed parallel processing capability. Due to the rapid growth of data size in modern society, this ability is essential and valuable. The other one can be recognized as the multiple parameters feature of light wave which give rise to more flexible key design than digital methods. Since the pioneering work known as double random phase encoding (DRPE)  was proposed, a series of relevant pursuits are conducted. Two aspects of these works can be distinguished. Some aims to enhance the security level of DRPE by introducing additional encryption parameters such as fractional domain DRPE, pixel scrambling DRPE, etc [2,3]. Others focus on attacking and hacking DRPE based cryptosystem, for instance, phase retrieval algorithm based known plaintext attack  and various attacks on the dual polarization encryption method , to name a few of them. All the researches regarding the two aspects promote the development of optical security technique from opposite sides.
Despite the rapid advance in optical security technique, there still exist several problems. More digital image processing operations are adopted and integrated into newly developed optical cryptosystems. Iterative phase retrieval algorithm  is applied to achieve image encryption in three dimensional space. Pixel pre-scrambling operation is also designed as part of DRPE system to further randomize the plaintext . These digital encryption approaches can effectively raise up the complexity of equivalent mathematical algorithm of cryptosystem. However, this also unavoidably hinders the global processing speed of systems. Hence, we expect for the all-optical information encryption architecture with high security level.
Another problem observed in literature is about the key space (KS) and key dimension (KD). Even if a lot of analyses on various kinds of attack have been reported and some strategies have been suggested to enhance the attack resistibility [8,9], less work about the expansion of KS and KD is proposed. Monaghan et al present depth analysis on KS of DRPE system and find that the KS depends on quantization levels of phase . In our previous work, KS of DRPE system is investigated and the result indicates that it is also determined by the position distribution of one phase value for binary phase mask . These studies disclose that KS of DRPE is not as large as thought in essence. And we notice that the KS may become a severe problem with the development of computing technology such as super computer and quantum computing. In this situation, we expect for the KS and KD of optical cryptosystem to be as large as possible to resist brute-force attack.
Most reported optical cryptosystems treat phase as main key such as DRPE and coherent diffractive imaging encryption . Diffraction distance is designed as key in Fresnel domain DRPE  and space-based optical encryption method . And it must be emphasized that the distance key in the latter one is two dimensional (2D) but one dimensional (1D) for the former one, hence, the key space is significantly larger for the latter. Together with the random phase key, we can claim that the key dimension of above proposal to be two, that is, diffraction distance and phase. Amplitude part of optical light wave can also be designed as key in cryptosystem although it may lead to the energy loss. Frequency domain random amplitude modulation is proposed to enhance the security level of DRPE technique . Multiple image encryption based on Fourier transform hologram is reported in which random amplitude key is located in reference arm . Polarization is an independent parameter of light wave which is designed as key. A dual encryption scheme is developed with an array of linear micro-polarizers using Mueller matrix calculus . Other approaches include double random polarization encoding , image encryption based on interference of polarized light  etc. It can be observed that the polarization key in the mentioned proposals is all two dimensional (2D) in transverse space. Apart from these polarization encryption schemes, polarization can be utilized to achieve multiplexing encryption . Others optical parameters could be used as extra encryption keys to multiplex data, thus improving the robustness of cryptosystem . However, similar to the above mentioned diffraction distance key in 1D or 2D, the optical parameters used for multiplexing are generally in one dimensional. The main and most important encryption keys are still the random phase key in 2D because the key space provided by the 1D and 2D keys are andrespectively for an image with M pixels and key with N quantization levels. This is incomparable in quantity for the two conditions. The significance of introducing various multiplexing keys lies mainly on increasing the data capacity and separating different channels for a specific cryptosystem. Besides these works, we propose a vector wave encryption scheme recently and success to set the key dimension to be two, which is, phase and polarization  both in 2D. In spite of the above works, all parameters of light wave and system structure parameter such as distance have never been simultaneously designed as keys in optical security system. In this work, we accomplish this goal by proposing an optical cryptosystem with four-dimensional keys in amplitude, phase, and polarization all in 2D and diffraction distance in 1D.
The challenge of implementing this four-dimensional keys encryption system mainly lies in the generation of 2D random amplitude, phase and polarization keys. Especially the generations of 2D random polarization key, this is still a research topic involved in generation of inhomogeneous vector waves. As we know, the spatial lateral resolution for vector wave is hard to be improved. Thus the validity of the proposed system is limited. To relieve this limitation, we propose to introduce the quick response (QR) code to represent original secret message. It is worth noting that the proposed algorithm exhibits a good practicability when the plaintext is represented with a QR code. The QR code is detected as a 2-dimensional digital image by an image sensor and analyzed by a programmed processor. One of the best features QR codes exhibit is a level of error correction which directly depend on the storage capacity. Thanks to these levels of correction contained in the generating algorithm, information can be noise-free retrieved despite of the pollution or information loss. A lot of works regarding the usage of QR code have been reported including the joint transform correlator based experimental QR code data recovery , single-intensity recording technique with QR code and phase retrieval algorithm  and holographic encryption with QR code , etc [26,27]. Original information can be noise-free recovered in these encryption methods and our proposal with the help of QR code. The polarization modulation or transformation can be implemented with a parallel aligned spatial light modulator (PAL-SLM) and original information can be effectively read out with a QR code which has been proved in our previous work  in which an asymmetric polarization cryptosystem is experimentally verified.
Hereinafter, we demonstrate this all-optical cryptosystem with four-dimensional keys. To the best of our knowledge, this is for the first time all parameters of optical wave are simultaneously designed as keys in a security system. This goal is achieved based on the single shot polarization digital holography technique. Two pieces of digital calculated and fabricated polarization holograms are introduced to achieve orthogonal polarized light with random amplitude and phase distribution respectively to interfere with object wave with scrambled polarization state. Original information can be read out from the decrypted QR code using Fresnel diffraction calculation and decryption keys during the reconstruction process.
2. Design for optical cryptosystem with four-dimensional keys
To make full use of all parameters of optical lightwave, our basic principle is to record vector wave with scrambled state of polarization (SOP) structure in object path. In the meanwhile, in order to achieve recording in single shot without any mechanical movement, the orthogonal polarized reference waves configuration is utilized. During generation of the two reference wave, specially tailored polarized light with random amplitude or phase distribution is obtained from two digital polarization holograms with defined magnitude and orientation of each pixel’s birefringence.
We first show the SOP and diffraction distance keys design in the object arm. The whole hologram recording also known as the encryption scheme is illustrated in Fig. 1. To clarify the encryption and decryption process, a block diagram is also presented in Fig. 2. The optical elements located in object arm are constituted by LP3, SLM1, QWP3, SLM2 and QWP4 (see the figure caption for the full names). Two SLMs are both reflective type parallel-aligned phase-only liquid crystal device (PAL-LCD) which is used to modulate the SOP of input wave. The orientations of light transmission for linear polarizers are marked with arrows. Also the fast axes of two SLMs which are orientated at with respect to x axis are indicated in the figure. The orientated fast axis can be easily implemented with an extra half wave plate placed in front of SLM  without any physical rotation of SLM itself. SLM1 is employed to convert the linearly polarized light into elliptically polarized light . QWP3, SLM2 and QWP4 are combined to rotate the principle axis direction for elliptical polarized light . Lens 3 and Lens 4 are set into a 4-f projection system to make alignment and calibration between the back surface of SLM1 and front surface of SLM2. This arrangement in object arm allows generation of arbitrary SOP structure with defined ellipticity and azimuth angle determined by the pattern displayed on SLMs. The SOP shape right behind the back surface of SLM2 can be deduced using Jones matrix calculation as :
The gray level pattern displayed on SLM1 is known as , which is normalized into 0~1. is a random amplitude pattern uniformly distribution between 0~1. is the original image. denotes a horizontally polarized light generated with LP3. The gray level random pattern displayed on SLM2 is denoted as which is uniformly distributed between 0~1 and taken as the polarization encryption key (the first dimensional key). The 4-f imaging system impinges SOP shape modulated by SLM1 onto SLM2 to scramble the spatial SOP structure. After this operation, a vector wave with random SOP structure is obtained behind SLM2. Then, the SOP distribution on the CCD surface can be represented by Fresnel transforming the SOP right behind SLM2 with a diffraction distance d which is taken as the second dimensional key. We notice that this vectorial diffraction process from SLM2 to CCD can be simplified with Jones matrix method together with Fourier transform under the condition of fully polarized and coherent illumination . Hence, SOP on CCD surface can be calculated as:
In Eq. (2), denotes Fresnel transform operator with Fresnel diffraction distance. denotes the wave propagation factor. denotes wavelength of illumination source. is the coordinate in CCD plane. The Fresnel diffraction distance is d. Fresnel diffraction can diffuse the information of plaintext effectively which gives rise to a random SOP pattern on CCD surface.
Then, we show random amplitude and phase keys design in reference arms. In the object arm design process, the polarization and diffraction distance keys are accomplished. The other two dimensions include random amplitude and phase keys. In order to generate orthogonal polarized wavefront, the digitally calculated and optically fabricated polarization holograms (DPH) with the azobenzene polymer film which is spin coated onto an aluminum mirror is introduced . This form of digital polarization hologram permits both the amplitude and the phase of a diffracted beam to be independently controlled. To fabricate the holograms, a birefringence with defined retardance and orientation of the fast axis is recorded into a photopolymer film. The holograms are selectively read out by choosing the polarization state of the read beam. The polarization hologram can be understood as spatially varying retarder plates. Every pixel of this hologram can be described in terms of a Jones matrix for a general linear retarder :
In Eq. (3), denotes the angle between fast axis of local retarder and x axis. A and B are complex transmission coefficients of the fast and slow axis of the retarder and are given by,, where is the retardation at given point in the hologram plane. When illuminated by left-circularly polarized plane wave, the transmitted field can be described by:
In right hand side of Eq. (4), the former term is a left circular polarized light with defined amplitude shape while the latter term is a right circular polarized light with both defined amplitude and phase shape which can be engineered by adjusting two variables of the hologram.
The method discussed in  is utilized to compute and for each pixel and fabricate this kind of holograms. Also, the readout scheme is also employed to obtain and observe diffraction patterns with either random amplitude or random phase distribution. We omit the detailed fabrication and computing process for this polarization hologram for simplicity here. In Fig. 1, QWP1, DPH1 and LP1 are combined to generate a pure random phase encryption key with horizontal polarized direction while QWP2, DPH2 and LP2 are for a pure random amplitude key with vertical polarized direction. Lens 1 and Lens 2 enables projection Fourier transform of optical field produced by DPH. Hence, by adjusting the diffraction distance, we acquire the random pattern required for encryption on CCD plane. M1 and M2 are arranged to shift the beam for a combination and reach of two reference waves and object wave on CCD.
During the interference procedure, off-axis geometry is used . Horizontal and vertical polarized reference waves reach CCD plane from upside and left side respectively with the same incident angle of . Then, we can represent two orthogonal references wave as:
In Eqs. (5)-(6), the , and can be varied across the polarization holograms plane with specific resolution and sample interval based on the writing configuration . Here, the three parameters are designed as random amplitude key for and phase key for . The incident angle can be determined by:
The digital recording, that is, the encryption process can be expressed as interference of three beams with specifically designed SOP and complex amplitude structure:
In Eq. (8), is taken as the final encrypted image i.e. the cyphertext. The three first terms form the zero order diffraction while the fourth and fifth terms produce two real image and last two terms form virtual images. For example, the last two terms are:
Due to the orthogonal polarized property of reference wave, only corresponding object wave components are recorded in respective terms.
With respect to the reconstruction algorithm, we perform it with the Fresnel transform based method. Observing Eq. (9) and Eq. (10), we can figure out that the reciprocal random amplitude function should be generated and used for concealing the noise superimposed in vertical polarized component of object wave while the conjugate random phase pattern should be used to decrypt horizontal polarized component of object wave. Then, an inverse Fresnel diffraction with right distance d should be performed, after the polarization key is compensated, the original information can be recovered. This procedure is implemented by digital method in the computer. Decryption reference waves are digitized as:
In which, and are the sampling intervals in the hologram plane (pixel size of the detector). For the recovery of horizontal component and vertical component of SOP in object wave on SLM2’s rear plane, Eq. (13) and Eq. (14) are utilized respectively:
In our simulation process, the pixel total number N = 512. Wavelength is set to be 632.8nm. Due to the off-axis geometry of recording, five diffraction patterns on rhs of Eq. (8) can be separated in space.
The elimination on polarization scrambling key can be achieved by generating an reverse rotation matrix, the original image can be calculated as:
Here in Eq. (16), the retrieved original information can be easily obtained by extracting the angle cosine function, after eliminating random number , original image is completely retrieved.
3. Simulation results
Considering the resource limitation in our lab and the disability of fabricating the polarization holograms, we present a proof-of-concept study and conduct a numerical simulation analysis on the proposed cryptosystem under the platform of MATLAB 2008a.
It is worth noting that the proposed system performs well when using quick response (QR) code to represent plaintext information. QR code is detected as a 2-dimensional digital image by an image sensor and analyzed by a programmed processor. The useful feature QR codes exhibit is a level of error correction which depends on the storage capacity. Thanks to these levels of correction contained in the generating algorithm, information can be noise-free retrieved despite of the pollution. The error correction ability can be classified into four levels as L, M, Q and H in which the minimum error percentage is 7%, 15%, 25% and 30% respectively. When the error percentage occurred in the polluted QR codes exceeds the minimum values for corresponding level, the information contained in the QR code cannot be read out. The version can be sorted into 40 types according to the data capacity. The larger the number of type, the larger data capacity is. Note that all the QR codes used below are generated with software named “Psytec QR code editor”. And the scanning is performed with an app on the smart phone called “ScanLife”. The two apps are both widely accessible for everyone.
Hereinafter, original image shown in Fig. 3(a) with pixel number is coded into QR code with version 6 type M shown in Fig. 3(b). After polarization scrambling induced by random pattern on SLM2, the resulted random SOP distribution can be represented with horizontal component shown in Fig. 3(c) and vertical component shown in Fig. 3(d). Then, the random pattern on CCD plane in object arm can be calculated with Fresnel diffraction integral. For both horizontal and vertical components, Fig. 3(e) and Fig. 3(f) can be observed. Finally, the cyphertext recorded after interference is illustrated in Fig. 3(g).
Some simulation conditions and system parameters can be found here. The amplitude and phase keys are quantized into four steps due to the fabricating technique limitation of polarization hologram. The SOP key is quantized in 256 level induced by the gray levels displayed on LCD. Wavelength of light source is set to be 632.8 nm. Fresnel diffraction distance key is chosen as 100 mm. Input plane Fresnel diffraction sampling interval is 10 . For DPH1 shown in Fig. 1, in Eq. (5) is set to be across the hologram plane to generate a pure phase key. The off-axis angles for two reference waves are both.
During the reconstruction process, the Fourier transform spectrum filtering algorithm is pre-conducted to separate and isolate specific terms in cypher text; four terms on rhs of Eq. (8) can be isolated as shown in Fig. 4(a). Spectrum on four corners represents zero order term. After spatial filtering and inverse Fourier transform, we would like to reserve the two terms as and.
Afterwards, the right decryption keys known as reciprocal random amplitude key and conjugate random phase are utilized to recovery both the horizontal and vertical element of object wave on CCD surface plane as shown in Fig. 4(b) and Fig. 4(c) respectively. Then, an inverse Fresnel diffraction with correct distance 100 mm is performed. The results are shown in Fig. 4(d) and Fig. 4(e) in horizontal and vertical directions. Finally, an inverse polarization rotation matrix is utilized to conceal the random polarization scrambling. The decrypted image is shown in Fig. 4(f). This QR code can be recognized and scanned to recover original information as shown in Fig. 4(g). When the amplitude key and phase key for decryption are totally wrong, the intermediate decrypted images after inverse Fresnel transform are shown in Fig. 4(h) and Fig. 4(i) for horizontal and vertical components. The wrong decrypted image is shown in Fig. 4(j).
We can conclude from the simulation results that the proposed cryptosystem effectively randomized the plaintext and recovered the original information. Without correct decryption keys, the recovered structure of QR code as shown in Fig. 4(j) can hardly be recognized.
4. Key sensitivity and fault tolerance analyses
The feasibility for the proposed system has been basically proved above. However, whether this proposal is feasible in application or not may still be a question. Here, we present an analysis on the key sensitivity and fault tolerance property for the proposal.
4.1 Key space and key dimension discussion
We would like to emphasize that the key dimension (KD) of our proposal is four including amplitude, phase, polarization and diffraction distance. All parameters of light wave have been utilized simultaneously and effectively in the system.
Key space of this proposal can be approximately calculated as without considering the levels of Fresnel diffraction distance supposing that the level for amplitude key and phase key are both 4. The numbers of possible decryption keys are significantly larger than DRPE technique which is actually. is the number of pixels. What’s more important, the multiple freedoms of keys may make the cryptosystem more difficult for attackers to analyze and hack.
4.2 Key sensitivity and Fault tolerance analysis
We expect for a high key sensitivity for this cryptosystem because in essence, an insensitive key may reduce the key searching space for an attacker who conducts a brute-force attack. In the meanwhile, we also expect for a high fault tolerance because precision of implementing the amplitude key, phase key, polarization key even diffraction distance key may be limited by the core device such as LCD and the system noise for authorized users. So, there must be a compromise between the key sensitivity and fault tolerance for a specific cryptosystem.
We conduct some partial wrong analyses for each dimension of keys to clarify the key sensitivity and a whole wrong analysis to define the fault tolerance. The thresholds are found for every dimension of keys by the criterion that whether the reading program can recognize the decrypted QR code and recover the original information contained in QR code or not.
Amount of simulations are performed to increase the error percentage from the central area to the border area of QR code. We also make a comparison when the original information of Fig. 3(a) is represented with version 8 type H QR code shown in Fig. 5(a).
For random amplitude key (RAK) in reference arm, the decrypted QR code cannot be read out until the error percentage of RAK exceeds 15.3% for version 6 type M QR code shown in Fig. 5(b). For the same error percentage, the decrypted QR code in version 8 type H can be successfully read out as shown in Fig. 5(c) and Fig. 5(d). For the random phase key (RPK), decrypted QR code cannot be read out when the error percentage of RPK exceeds 6.3% for version 6 type M shown in Fig. 5(e). For version 8 type H QR code, a success recovery can be obtained as shown in Fig. 5(f) and Fig. 5(g). For the random SOP key (RSOPK) in object arm, this unrecognition threshold becomes 11.0% for version 6 type M QR code shown in Fig. 5(h). For version 8 type H QR code, a success recovery can also be obtained as shown in Fig. 5(i) and Fig. 5(j).
As to the Fresnel diffraction distance key, we make a successive decryption with an error step of 0.1 mm. When the decryption inverse Fresnel transform distance is bias away the right value in the range of + 0.1-5 mm. The decrypted QR code for version 6 type M is displayed in a format of movie (Media 1) at a speed of 10 frames per second. The top right corner in the movie is the value of error quantity for wrong distance. Three typical frames are shown in Fig. 6(a), Fig. 6(c) and Fig. 6(d) in which Fig. 6(a) can be read out with an error distance of 1 mm. For version 8 type H QR code (Media 2), the three frames are shown in Fig. 6(e), Fig. 6(g) and Fig. 6(h) in which Fig. 6(g) can be read out with an error distance of 2.5 mm after block processing and binarization . Figure 6(b) and Fig. 6(f) are direct scanning results of Fig. 6(a) and Fig. 6(e) respectively which indicate noise-free recovery on original information.
All the above simulation results with partial wrong decryption keys indicate that the key sensitivity for the proposal cryptosystem is relatively high. The key sensitivity can be ranked with phase>polarization>amplitude.
Then, a common case is that decryption RAK is not exactly the reciprocal of encryption RAK and decryption RPM is also not exactly the conjugate of encryption RPM, also the decryption RSOPK. We conduct an analysis on this issue. For the random number matrixes in RAK, RPK and RSOPK, when the values of 0~1 are incorrect with 0.13, 0.15 and 0.26 as shown in Fig. 7(a)-(c), the decrypted QR code can hardly be read out anymore. This indicates that the fault tolerance for the proposed system is in a satisfactory level.
Based on the above analysis, we found that the key sensitivity and fault tolerance for the proposed system are both in a relatively high level. This excellent characteristic makes the proposed security system more practical and useful in application environment.
In this work, we have put the multiple dimension property of optical security technique into effect. The key dimension has been designed as four in amplitude, phase, polarization and Fresnel diffraction distance. An all-optical encryption configuration is constructed based on the principle of single shot polarization digital holography with orthogonal reference waves. The decryption process is accomplished with inverse Fresnel diffraction reconstruction method digitally. To achieve a noise-free recovery, the QR code is introduced to represent the secret message. Under this condition, the QR code must be retrieved using decryption reference wave illumination with correct decryption key in amplitude and phase and then correct diffraction distance key together with inverse polarization rotation key. Then, after a successful scanning, original information can be read out. Numerical simulation results are presented to verify the feasibility of this proposal. Apart from this, the key sensitivity and fault tolerance property for this proposal is demonstrated. The partially wrong decryption results indicate that the sensitivity of this proposal on every dimension of keys is relatively high. A relatively high fault tolerance can also be observed from the whole wrong decryption results. The polarization digital holography technique enables manipulation on all parameters of optical lightwave simultaneously to design keys in security system which we expect to open up new perspective for the design of optical cryptosystem with multi-dimension keys, thus, enlarging the key space and key dimension of security systems. Beside, the robustness of this proposal against various kinds of attacks may be presented in our next work.
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