The keys to holographic data security: Encrypted optical memory systems based on multidimensional keys for secure data storage and communication
There is a growing demand to store large sizes of data due to the development of digital computer technologies and digital television for the next generation. Up to now, only digital versatile disk (DVD) and compact disk (CD), which are bit-oriented storage methods, have been developed. Optical technology can provide a number of ways to solve the problem of large storage and fast transmission of data. The holographic technique discovered in 1948 records and reconstructs a complex-valued optical wave by using the interference between two coherent light beams. Using the holography technique, two- or three-dimensional images can be stored and reconstructed. Thus, development is expected of a three-dimensional television or a two- or three-dimensional image storage system operating at high speed that cannot be implemented in a magnetic-based storage system.
Holographic memories that use photorefractive materials are attractive due to their high storage capacity, high-speed access to data, and rewritability [1-6]. Unlike bit-oriented optical memories such as DVD and CD, two-dimensional data is stored as a hologram by illuminating the interference pattern formed by an object beam and a reference beam. In a photorefractive crystal, the intensity distribution of the interference pattern is stored as a refractive-index distribution [7, 8]. Using angle, wavelength, and phase multiplexing techniques, one can store multiple images at the same position. Maximum storage capacity using angular multiplexing techniques is estimated to be V/l3 (1500 CD-ROMs per cm3), where V is the volume of the photorefractive material and l is the wavelength of light. The readout rate can be more than 1 Gb/s, operating at 1 kHz readout rate with a 1 Mb image.
In practical systems, data security is an important issue. Optical encryption techniques provide a high level of security [9-15] because there are many degrees of freedom with which to encode the information, such as amplitude, phase, wavelength, and polarization. This article discusses three encrypted optical memory systems based on multidimensional keys.
Encryption Overview
Numerous methods have been described to use optical encoding schemes [10, 16-18] in order to secure data in holographic memory systems. One way to protect the stored information is to encrypt the data. Here the encryption means that the original data is converted into stationary white-noise data by key codes, and unauthorized users cannot obtain the original data without knowledge of the key code. Original data may be encoded optically by using encryption techniques such as double random phase encryption [10] or exclusive OR (XOR) operation [16, 17]. Another method is based on protecting the access to memory from unauthorized users by encoding the reference beam [18]. These techniques can be implemented by using a set of uncorrelated reference beams generated by orthogonal phase codes, such as random phase masks, or generated by speckle patterns from an optical fiber.
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1. (a) Block and (b) schematic diagrams of the secure communication system using secure holographic memory. ST and TS denote the space-to-time and the time-to-space converters, respectively. |
Encrypted memory using double random phase encryption can be used in a secure communication network using ultrashort pulses, as shown in Fig. 1 [19]. Figure 1(a) and Fig. 1(b) show block and schematic diagrams of the secure communication system using the encrypted memory and spatial-temporal converters. In this system, the original data is stored in an encrypted memory system. The encrypted data read-out from the memory is converted into a one-dimensional temporal pulse using the space-to-time converter [20] and then is transmitted to users via optical fibers. At the receivers, the temporal signal is converted again into the spatial signal by the time-to-space converter. The authorized users can decrypt the data using the correct key. This system can be expected to communicate at an ultrahigh speed of more than 1 Tb/s.
In the following sections we will discuss three encrypted optical memory systems using multidimensional keys [21-23]. Original data is encrypted by the multidimensional keys, which consist of two random phase masks, their three-dimensional positions, and the wavelength of the recording beams. It makes it difficult to decrypt the data without the key information because the total number of the multidimensional keys becomes extremely huge.
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2. Encrypted holographic memory system using multidimensional keys. |
Encrypted Memory Using Double Random Phase Encryption
Figure 2 shows an illustration of the encrypted optical memory systems used in this article. The encoding methods used in the proposed memory systems are based on the double random phase encryption technique [10]. We briefly review encryption and decryption of encrypted holographic memory using double random phase encryption [21]. Let gi(x,y) denote the ith positive real-valued image to be encrypted. Here, x and y denote the spatial-domain coordinates. The original data is converted into a white-noise-like image by using two random phase masks, exp{-jni(x,y)} and exp{-jhi(n,h)}, located at the input and Fourier planes. Here, ni(x,y) and hi(n,h) are two independent white sequences that are uniformly distributed on the interval [0,2p]. Note that n and h denote Fourier domain coordinates. The original data is illuminated by a collimated light beam and multiplied by a random phase function exp{-jni(x,y)}. The Fourier transform of the input data is multiplied by another random phase function H(n,h)=exp{-jhi (n,h)} and is given by
(1)
where
(2)
In Eq. (2), F[•] denotes the Fourier transform operation, l is the wavelength of the light, and f is the focal length of the Fourier-transform lens. Each encrypted data frame is obtained by taking another Fourier transform:
(3)
where ƒ denotes convolution. Equation (3) shows that the two phase functions, ni(x,y) and hi(n,h), convert the original data into a stationary-white-noise-like data [10].
The Fourier-transformed pattern of the encrypted data that is described in Eq. (1) is stored holographically together with a reference beam in a photorefractive material. To store many frames of data, angular multiplexing is employed. The interference pattern f(n,h) to be stored in a photorefractive material is written as
(4)
where M is the total number of stored images and Ri(n,h) is a reference beam at a specific angle used to record the ith encrypted data. Here we briefly describe the mechanism of the photorefractive effect when the carriers are electrons. In a photorefractive crystal, illumination with a sufficient wavelength content excites the electrons in the conduction band from the donor level between the valence and the conduction bands. The donor level is created by impurity ions or defects. The photoexcited electrons can move in the crystal by the diffusion, the drift, and the photovoltaic effect and then get trapped in the ionized donors. At the steady state, the space charge density is proportional to the interference pattern in the diffusion-dominant region. This space-charge density creates the space-charge field that can cause the refractive-index change via the electro-optic effect. The created refractive-index distribution is proportional to the interference pattern and can be stored for a long time (more than two months) in the dark in LiNbO3 crystal. Since a volume hologram is created in a photorefractive material, an appropriate angular separation between adjacent stored data can reduce the cross talk among the stored frames of data. Thus, we can store many frames by using angular multiplexing.
In the decryption process, the readout beam is the conjugate of the reference beam. The readout using the conjugate of the reference beam offers advantages. It is able to use the same random phase mask in the encryption and the decryption process, and it eliminates the aberration of the optical system. The data of the ith-stored image can be reconstructed when the readout beam is incident at the correct angle. The reconstructed data in the Fourier plane (FP), Di(n,h), is written as
(5)
where
(6)
The asterisk in Eq. (5) denotes the complex conjugate, and ki(n,h) is a phase key used in the decryption process. We can reconstruct the image by Fourier transforming Eq. (5). The reconstructed ith image, di(n,h), is written as
(7)
where
(8)
In Eq. 8, ƒ denotes correlation. When phase key ki(n,h) = hi(n,h), the conjugation of the original data is successfully recovered because Eq. (8) becomes a delta function. The random phase function in the input plane, exp{-jni(n,h)}, may be removed by an intensity-sensitive device, such as a charge-coupled device (CCD) camera. In a practical system, operating at high-speed detection of the reconstructed data, the parallel detection at each pixel of two- dimensional data is desirable. When one uses an incorrect phase key, ki(n,h) þ hi(n,h), the original data cannot be recovered.
We describe an encrypted memory system based on double random phase encryption [21]. Figure 3 shows the experimental setup. A 10 X 10 X 10 mm3 LiNbO3 crystal doped with 0.03 mol.% Fe is used as the recording medium. The c axis is on the paper and is at 45° with respect to the crystal faces. The crystal is mounted on a rotary stage and a three-dimensionally movable stage. An Ar+ laser beam of wavelength 514.5 nm is used as a coherent light source. The light beam is divided into an object and a reference beam by a beamsplitter (BS1) for holographic recording. The reference beam is again divided into two reference beams. One of the beams is used for the conjugate readout by another beamsplitter (BS2). An input image is displayed on a liquid-crystal display that is controlled by a computer. The input image is multiplied by an input random phase mask (RPM1) and is then Fourier-transformed by lens L1. The Fourier-transformed input image is multiplied by another random phase mask (RPM2) at the Fourier plane. The Fourier-transformed image is imaged at a reduced scale in the LiNbO3 crystal by lens L2. The encrypted image is observed by a CCD camera (CCD1) after the Fourier transform is produced by lens L3. The focal lengths of L1, L2, and L3 are 400 mm, 58 mm, and 50 mm, respectively. For holographic recording, the object and reference beams interfere at an angle of 90¡ in the LiNbO3 crystal. All of the beams are ordinarily polarized due to the creation of an interference fringe pattern. Shutters SH1 and SH2 are open, while SH3 is closed.
In the decryption process, the readout beam is the conjugate of the reference beam used for recording. Shutters SH1 and SH2 are closed, while SH3 is open. If the same mask is located at the same place as the one used to write the hologram, the original image is reconstructed at CCD2. This is because the ideal reconstructed beam read out by using the conjugate of the reference beam eliminates the phase modulation caused by the random phase mask. Otherwise, the original data may not be recovered. In the experiments, we use a pair of counterpropagating plane waves as the reference and the conjugate beams.
Angularly multiplexed recording of four digital images is demonstrated. One of the original digital images is shown in Fig. 4(a). This image consists of 32 X 32 randomly generated pixels. The size of the liquid crystal display that shows the input image is 28.5 mm X 20 mm. Two diffusers are used as the random phase masks, RPM1 and RPM2. Figure 4(b) shows the intensity distribution of the encrypted image. Random-noise-like images were observed. In the recording process, the optical intensities of the object and the reference beams were 78 mW/cm2 and 1.4 W/cm2, respectively. The exposure time of each image was 60 s. Angular multiplexing was achieved by rotating the LiNbO3 crystal in the plane of Fig. 2. The angular separation between adjacent stored images was 0.2¡. This angular separation is enough to avoid the cross talk between reconstructed images. Figure 5(a) shows the reconstructed images obtained using the correct key. The resolution of the reconstructed image is determined by the crystal size and the space-bandwidth product of the optical system. This key is the same as the phase mask in the Fourier plane used to record the hologram. This result shows that the stored images were reconstructed successfully. No noise due to the cross talk between the reconstructed images was observed. After the binarization of the reconstructed images, we confirmed that there is no bit error in the four-output digital data. Figure 5(b) shows the reconstructed images when incorrect keys were used. No part of the original image can be seen. The average bit-error rate obtained using incorrect keys was 0.384.
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3. Experimental setup. RPM denotes random phase mask; BS denotes beamsplitter; L denotes lens; M denotes mirror; BE denotes beam expander; SH denotes shutter; CCD denotes CCD camera. |
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4. (a) Original image and (b) encrypted image. |
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5. (a) Reconstructed image using correct phase key and (b) reconstructed image using incorrect phase key. |
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6. Experimental results. (a) Original image, (b) encrypted image. (c) and (d) are reconstructed images when positions of the phase masks are correct and incorrect, respectively. |
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7. (a) Reconstructed image using correct phase key and (b) reconstructed image using incorrect phase key. |
Encrypted Memory Using Three-Dimensional Keys in the Fresnel Domain
We can make the memory system more secure by using random phase masks in the Fresnel domain. In addition to the phase information, the positions of two phase masks are used as encryption keys. Even if the phase masks are stolen, the unknown positions of the masks can protect the data. The positions of the masks have as many as three degrees of freedom. We have demonstrated encryption and decryption of three binary images by angular multiplexing [22]. The experimental setup is the same as that shown in Fig. 3. Figure 6(a) shows one of the three original images. RPM1 and RPM2 were located at a distance of 100 mm from L1 and at the center of L1 and FP, respectively, as shown in Fig. 3. Figure 6(b) shows an encrypted image of Fig. 6(a). Random-noise-like images were observed. In the recording process, the optical powers of the object and the reference beams were 4 mW/cm2 and 500 mW/cm2, respectively. The exposure time was 110 s. Figure 6(c) shows one of the reconstructed images obtained by using the same masks located at the same positions used in the recording. This result shows that the original image was successfully reconstructed. Figure 6(d) shows the reconstructed image when the two phase masks were incorrectly located. We can see that the reconstructed image is still a white-noise-like image.
We estimate the available number of three-dimensional positions of two random phase masks. Let the dimensions of random phase masks be Lx X Ly, and Dx and Dy be correlation lengths of the random phase mask along the x- and y-axes, respectively. The x-, y-, and z-axes are defined as shown in Fig. 3. When a number of Nz resolvable positions along the optical axis can be used for the encryption key, the total number of three-dimensional positions to be examined in a three-dimensional key, P, is written by,
(9)
(10)
where f is the focal length of L1 in Fig. 3 and Dz is computed according to the sensitivity of the decryption to the shifts of the keys along the z axis. Since two three-dimensional keys are used in the system, the total number of three-dimensional positions to be examined is given by
(11)
In the memory system shown in Fig. 3, N = 3 X 1018 when Lx = Ly = 25 mm, L = 400 mm, Dx = Dy = 6 mm, and Dz = 4 mm. Note that Dx and Dy were calculated from the measurement of an autocorrelation function of the phase mask used in the experiments. When one searches 106 positions per second, it takes 95 years to finish the whole search. It is practically impossible to decrypt without the knowing the positions of two three-dimensional keys.
Encrypted Memory Using Wavelength-Code and Random Phase Masks
The wavelength of recording beams can be used as a key for security in a holographic memory system [23]. The wavelength code increases the key space by one dimension. Since an optical storage medium, such as a photorefractive material doped with impurities, has broad spectral sensitivity, we can use many wavelengths of light emitted from tunable laser sources, such as a dye laser. In this memory system, shown in Fig. 3, one original data frame is stored by using a set of two random phase masks at the input and Fourier planes as well as a wavelength key. The wavelength key can protect the decrypted data even if the phase masks have been illegally obtained. When the wavelength of the readout beam is different from that of the recording beam, the wavelength mismatch modifies the scale of coordinates at the Fourier plane in the readout process. Due to the incorrect wavelength, the phase modulation at the Fourier plane is not completely canceled because of a scale mismatch. If a substantial part of the phase modulation at the Fourier plane is not canceled, the original image cannot be recovered. We note that the wavelength mismatch results in decreased diffraction efficiency due to the Bragg condition, because the volume grating structure of the hologram is complex.
Figure 7(a) and Fig. 7(b) show an original and an encrypted image. The encrypted image is stored holographically using recording beams at a wavelength of 514.5 nm. In the decryption process, we use two readout beams at wavelengths of 514.5 nm and 632.8 nm. Note that in both cases the Bragg conditions are satisfied. Figure 7(c) and Fig. 7(d) show reconstructed images when wavelengths of 514.5 nm and 632.8 nm are used, respectively. When the wavelength of the readout beam is the same as that of the recording beam, and when the same mask is located at the same place as that used to record the hologram, we can obtain the reconstructed original image. The wavelength selectivity depends on the pixel size of the random phase mask at the Fourier plane. We can use many wavelengths by utilizing the small pixel size of the random phase mask.
Conclusions
We have presented three encrypted holographic memory systems. These systems are secure because the total number of mathematical possibilities of the multidimensional keys, which consist of two-dimensional phase masks, their three-dimensional positions, and wavelengths of light, is extremely large. The experimental results are very encouraging. We expect the encrypted memory system to play an important role in ultrafast secure communication systems using the spatial-temporal converters with ultrashort pulsse that enable communication at ultrahigh speed of more than Tb/s [19, 24].
Osamu Matoba is a research associate at the Institute of Industrial Science of the University of Tokyo, Japan. E-mail: matoba@ iis.u-tokyo.ac.jp. Bahram Javidi is a professor with the Department of Electrical and Computer Engineering of the University of Connecticut in Storrs, Connecticut, USA. E-mail: bahram@ engr.uconn.edu.
References