1 March 2002
Optics Communications 203 (2002) 67–77 www.elsevier.com/locate/optcom
Parameterized multi-dimensional data encryption by digital optics Lingfeng Yu a,*, Xiang Peng b, Lilong Cai a a
Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong b National Laboratory of Precision Measurement Technology and Instrumentation, Tianjin University, 300072 Tianjin, China Received 17 December 2001; received in revised form 17 December 2001; accepted 18 December 2001
Abstract A new parameterized multi-dimensional data encryption method is presented in this paper. Several geometric and physical parameters derived out from a configuration of digital optics have been suggested as tools for designing multiple locks and keys for data encryption in hyperspace. Numerical experiments are performed to validate proposed method and parameters sensitivities are also quantitatively analyzed and illustrated with numerical simulations. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Information security; Encryption/decryption; Digital optics; Geometric parameters
1. Introduction Information security is very important in many application areas in the field of information technology, such as Communication Security, Internet Security, Multimedia Security, E-commerce/Mcommerce Security, Computer Network Security, Software and Hardware Security, ATM Security, Banknotes and other (e.g., ID Card or Credit Card or Passport, etc.) anti-counterfeits, IP and copyright protection, Personal Identification, Authentication, Verification, to name a few. The study of communications security includes not just encryption but also traffic security, whose essence lies in *
Corresponding author. Tel.: +852 2358 7209; fax: +852 2358 1543. E-mail address:
[email protected] (L. Yu).
hiding information. The general model of hiding data in other data can be described as: the embedded data are the message that one wishes to send secretly. A specific-designed key is used to control the hiding process so as to restrict detection and/or recovery of the embedded data. One of the major purposes of information hiding is that one is able to encode information to be sent in such a way that is difficult for unauthorized party to decode the information without having a proper key(s). Recently, optical information processing has been explored to be as a means of data encryption [1–5]. Particularly, all-optical [1] and hybrid [2–4] (e.g., digital holography) methods, nonlinear joint transform correlator (NJTC) [5], as well as computer-generated hologram (CGH) [6] are all suggested for data encryption and decryption. Other optical method such as data encryption based on
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orthogonal-phase-code multiplexing [7] has also been proposed for the application of information security. More recently, a combination version of digital holography and phase shifting technique is investigated for encrypting and decrypting threedimensional object [8,9]. These pioneer works have shown that optical information techniques are promising tools in the field of information security. Most of the aforementioned works utilized phase random mask(s) and holographic encoding to encrypt data, thus the hologram of such random mask(s) was used for a key(s) for decrypting data. However, there are other possibilities existed to encrypt data except for using a random mask(s). For instance, the geometric parameters of optical system can possibly be employed for data encryption/decryption. In this paper, we present a new parameterized multi-dimensional method to encrypt and decrypt information, which is based on geometric parameters derived from digital optics or virtual optics, in addition to utilize a random covering mask (RCM). We introduce new concepts and methodology for the design of multiple and multi-dimensional ‘‘lock(s)’’ and ‘‘keys’’ for virtual-opticsbased security system. The terminology ‘‘digital optics’’ or ‘‘virtual optics’’ used here means that we shall implement both encoding/encrypting and decoding/decrypting processes in all-digital fashion. The remaining parts of this paper are organized as follows: Section 2 gives a theoretical background of parameterized multi-dimensional encryption and decryption method. The parameters of virtual optical setup such as the spatial position of the host information, the spatial position of the reference wave point source, and the virtual wavelength, etc. are explored as tools for designing multiple and multi-dimensional ‘‘lock(s)’’ and ‘‘keys’’. Section 3 describes numerical experiments and dimension of possible keys is estimated. Section 4 gives a conclusion.
optics. Geometric parameters such as spatial position of the host information to be hidden, spatial position of the reference wave point source, and physical parameters such as the virtual wavelength will be explored for the purpose of enhancing information security level, respectively.
2. Theoretical analysis
jp 2 I~ðn; nÞ ¼ A exp ½n þ g2 kzo Z Z Iðxo ; yo Þ jz¼zo
In this section, we present some fundamental analysis on parameterized multi-dimensional encryption and decryption with the use of virtual
2.1. Spatial positions sensitivity A virtual optics setup for data encryption and decryption is schematically shown in Fig. 1, where ðxo ; yo ; zo Þ is an arbitrary point on the signal plane in which host information is located; ðxr ; yr ; zr Þ is the spatial position of virtual reference wave point; k1 is a virtual wavelength used for ‘‘recording’’ digital hologram, and k2 is assumed to be that of ‘‘reconstructing wave’’. The origin of z-axis is defined in the n–0–g plane as shown in Fig. 1. An RCM is numerically generated and used to convert host information, and the spatial position of RCM is denoted by zRCM . The whole information to be hidden is a set of linear combination of points ðxo ; yo ; zo Þ and is denoted by Iðxo ; yo Þ jz¼zo . The RCM is denoted by RCMðx; yÞ jz¼zRCM . The host information is embedded with the RCM in a cascade version. We can mathematically express such a cascade as Iðxo ; yo Þ jz¼zo þRCM ðx; yÞ jz¼zRCM . Thus, the output plane adjacent to the RCM becomes a virtual ‘‘scattered light field’’, and therefore the host data are converted. Furthermore, we assume that the propagations from one plane to another excluding that from the signal plane to the RCM plane shown in Fig. 1 are all within Fresnel diffraction region. Therefore, the complex amplitude of the converted data propagating to n–0–g plane can be expressed as: ~M ~C ~ ðn; gÞ; F ðn; gÞ ¼ I~ðn; gÞ þ R ð1Þ ~M ~C ~ ðn; gÞ will interfere with where I~ðn; gÞ and R each other at the n–0–g plane, and then together interfere with a reference wave. They are, respectively, defined as:
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Fig. 1. Data encryption and decryption with one layer random covering mask (RCM).
jp 2 2 exp ½x þ yo kzo o j2p exp ½xo n þ yo g dxo dyo ; ð2Þ kzo ~M ~C ~ ðn; gÞ ¼ B exp jp ½n2 þ g2 R kzRCM Z Z RCMðx; yÞ jz¼zRCM jp 2 2 exp ½x þ y kzRCM j2p exp ½xn þ yg dx dy: kzRCM ð3Þ The coefficients A¼
ejzo 2p=k jkzo
and
jp 2 2 Iðxo ; yo Þ jz¼zo exp ½x þ yo kzo o and RCMðx; yÞ jz¼zRCM exp
respectively. Now we assume that a virtual spherical wave with a point source is utilized as a reference to record digital hologram, and it can be approximated by quadratic phase expression as: jp 2 2 Rðn; gÞ ¼ Ar exp ½ðxr nÞ þ ðyr gÞ ; k1 z r ð4Þ where A¼
B¼
ejzRCM 2p=k jkzRCM
in front of the above integrals are complex constants and Eqs. (2) and (3) can be regarded as Fourier transforms of the complex functions,
jp 2 2 ½x þ y ; kzRCM
ejzr 2p=k1 jk1 zr
is a complex constant, ðxr ; yr ; zr Þ is the spatial position of the point source of spherical reference wave, and k1 is a virtual wavelength used for recording the digital hologram. Then the intensity distribution obtained by the interference between
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the converted signal wave and the reference wave at the hologram plane can be described as: He ðn; gÞ ¼ ½F ðn; gÞ þ Rðn; gÞ
½F ðn; gÞ þ Rðn; gÞ ¼ j F ðn; gÞ j2 þ j Rðn; gÞ j2 þF ðn; gÞ R ðn; gÞ þ F ðn; gÞ Rðn; gÞ:
ð5Þ
In order to decode the digital hologram, we need a suitable conjugate reference wave to reconstruct the real image (i.e., F ðn; gÞ), but leave a linear carrier wave for separating the real image from other terms in Eq. (5). To this end, we rewrite Eq. (4) as: jp 2 2 Rðn; gÞ ¼ Ar exp ½n þ g k1 z r j2p exp ½xr n þ yr g k1 z r jp 2 2 ¼ Ar exp ½n þ g k1 z r o n exp j2p½xr n0 þ yr g0 ; ð6Þ where we have applied an appropriate approximation, ððx2r þ yr2 Þ=k1 zr Þ 1, since Fresnel diffraction propagation has been assumed. The same assumption is also applied to Eq. (4). From Eqs. (2), (3) and (6), we find out that if the spatial positions of the signal plane, RCM, and the reference are secretly selected when we encode a digital hologram, then all these geometric parameters derived from virtual optics can be utilized to design ‘‘lock(s)’’ and ‘‘key(s)’’ for data encryption. Also, those parameters can be used in an either individual or combinable way so that multiple lock(s)/key(s) and multi-dimensional lock(s)/key(s) can be designed with such a concept and methodology in order to obtain a high-level information security system. Now, we select a spherical reference wave as a reconstruction wave which has the following form, so as to achieve the goal of decoding hologram with a linear carrier left: jp 2 2 ~ R ðn; gÞ ¼ Ar exp ½n þ g : ð7Þ k1 z r This digital reconstruction spherical wavefront converges to a point on the z-axis, i.e., ðxr ¼ 0;
yr ¼ 0Þ. If such a reconstruction wave is used to decode the digital hologram, then we have reconstructed complex amplitude distribution described as the following expression: Hd ðn; gÞ ¼ j F ðn; gÞ j2 R~ ðn; gÞþ j Rðn; gÞ j2
R~ ðn; gÞ þ F ðn; gÞ R ðn; gÞ R~ ðn; gÞ o n þ F ðn; gÞ exp j2p½xr n0 þ yr g0 ; ð8Þ where Hd ðn; gÞ denotes the ‘‘decoded’’ hologram with a specific conjugate spherical reference wave as defined in Eq. (7). The last term in Eq. (8) propagates in different directions with the other terms: the first two terms propagate along the z-axis ððxr ¼ 0; yr ¼ 0; zr ÞÞ, but the last term (i.e., real image) propagates along with a direction governed by spatial position ðxr ; yr ; zr Þ, where xr 6¼ 0; yr 6¼ 0, and the third term also propagates along a different direction. In order to reconstruct this ‘‘decoded hologram’’, a spectrum manipulation algorithm is performed similarly as in [10]. First, we can get the spatial frequency spectrum of the decoded hologram by taking the Fourier transform of Eq. (8), and if the spatial position ðxr ; yr ; zr Þ is properly selected, the non-overlapping spectrum of the fourth term in Eq. (8) can be separated from the other irrelevant terms. By following a filtering and a shifting operation, we can further obtain the spectrum of F ðn; gÞ (defined as Spctrum½F ). In addition, a propagation factor, ( 1=2 ) 1 2 2 exp j2pzi n0 g0 k2 is multiplied with the selected spectrum components, Spctrum½F , to localize its correct space position. Furthermore, we take an inverse FFT (IFFT) to get the real image of the object (composed with both the signal plane and the RCM) at any position along the z-direction, which is defined as real jz¼zi ðF Þ. For more details about this algorithm, reader can refer to [10]. In the process of decryption, the authorized parties should know the information of the RCM, and they should also follow the same procedure as above to procure a hologram of the RCM plane
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with all the other geometric or physical parameters unchanged. This hologram is expressed as: h i ~M ~C ~ ðn; gÞ þ Rðn; gÞ HRCM ðn; gÞ ¼ R h i ~M ~C ~ ðn; gÞ þ Rðn; gÞ :
R ð9Þ Similarly, the same sphere reference as in Eq. (7) is used to decode the new hologram of RCM. Then ~M ~C ~ (the spectrum of we can get Spctrum½R ~M ~C ~ ), which can be further used to localize the R real image of the RCM plane along the z-direction, ~M ~C ~ Þ. labeled as real jz¼zi ðR Thus the real image of the signal plane, defined as real jz¼zi ðI~ Þ, can be finally decoded as: ~M ~C ~ Þ: real jz¼zi ðI~ Þ ¼ real jz¼zi ðF Þ real jz¼zi ðR ð10Þ Generally speaking, for unauthorized third parties, not only the RCM serves as the lock for encryption, but also all the geometrical or physical parameters for hologram recording and reconstructing are unknown. For instance, geometric and virtual optical parameters xo ; yo ; zo ; ZRCM ; xr ; yr ; zr ; k1 ; k2 can all be secretly selected to serve as the locks and keys in the process of data encryption and decryption. The parameterized encryption process based on spatial position sensitivity can be briefly summarized as the following major steps: 1. Record the digital hologram of the object (both the signal plane and the RCM plane). The geometrical parameters, such as the spatial positions of the signal sheet and the RCM, the spatial position of the virtual reference wave, are all secretly selected. And the physical parameter as the virtual-wavelength can also be secretly selected from a huge numerical range instead of one coming from physically existed laser source. 2. Decode the above hologram with a linear carrier left by using a spherical reference wave as Eqs. (7) and (8) and the spectrum of the real object, Spctrum½F , can be easily obtained by some manipulations in spectrum domain. 3. In addition, a propagation factor, expfj2pzi 1=2 ðð1=k2 Þ n20 g20 Þ g, is multiplied with Spctrum½F to localize a correct space position
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of the real object. Furthermore, we take an IFFT to get the reconstructed image of the real object, which is labeled as real jz¼zi ðF Þ. With this algorithm, the real object can be reconstructed at any xy plane along the z-direction. 4. In practice, the position zi of the reconstructed object can also be secretly selected and serves as a key for encryption. The whole reconstructed information will serve as the encrypted data sheet (EDS) and can be sent from a transmitter as an ordinary electronic file through a communication network. On the side of receiver, the EDS will be received and decrypted with an authorized technology and suitable key(s). An authorized receiver will be informed of the corresponding decryption method and be delivered proper key(s). Three major steps for data decryption are as follows: 1. First, the information of RCM is properly delivered and serves as a key for decryption, and a hologram of the RCM is calculated with the informed parameters according to Eqs. (3), (6) and (9). This step involves some critical points: (1) correct spatial positions of the original RCM, (2) correct spatial positions of the virtual reference wave, and (3) correct virtual wavelength. Any limited disparity of geometric parameters or virtual optical parameters (e.g., the recording wavelength) would cause the disparity of the hologram of RCM, thus cause the failure of decryption. In the following section, we shall present the numerical experiments to test the sensitivities of the spatial positions of the signal plane and the reference wave, and the sensitivity of the virtual wavelength drifting. All these simulations will prove the prediction here. 2. After getting the hologram of the RCM, the same steps 2–3 as in encryption are performed to decode the spectrum of real RCM, ~M ~C ~ , and to reconstruct the real imSpctrum½R age of the RCM at the correct position along the z-axis. 3. Finally, the real image of the signal plane can be decrypted according to Eq. (10). In conclusion, in order to unlock the encrypted data, one key is made from the digital information of RCM, other key(s) arise from the secretly selected parameters that are derived out from virtual
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optics. No other than all keys are available, encrypted data can be unlocked. Oppositely, it will be extremely difficult for unauthorized third-parties who conspiratorially try to decrypt hidden information without the knowledge of encoding techniques or without proper key(s), since there are large degrees of freedom existed for designing a security lock(s)/key(s), including multiple lock(s)/ key(s) and multi-dimensional lock(s)/key(s). So, if the design concept and methodology based on geometric parameters of virtual optics is adopted for the purpose of data encryption, it will dramatically increase the level of information security.
3. Numerical experiments and results In this section we describe the implementations of numerical experiments designed to demonstrate our theoretical analysis about proposed method, namely, parameterized multi-dimensional data encryption. Numerical experiments are organized into three groups. The first experiment aims at testing the sensitivity of the spatial position disparity of the signal plane. The second one aims at testing the sensitivity of the spatial position disparity of the point-source-based virtual reference wave. The third experiment aims at testing the sensitivity of the virtual wavelength mismatch. At last, the possible dimension of these keys is roughly estimated. 3.1. Sensitivity of the spatial position disparity of the signal plane For simplicity, we degenerate some parameters to make simulation easier in this subsection. First, we place the reference source point at infinite distance, leading to a plane reference wave for digitally recording. That is, let zr ! 1. In addition, we assume that both recording and reconstructing waves have the same wavelengths, i.e., k1 ¼ k2 . All other geometric parameters are virtually designed for recording a digital hologram, in which zo ¼ 1:2 m; zRCM ¼ 1:0 m; k1 ¼ 0:623 lm. The size of digital hologram is limited to 6 mm 6 mm with 256 256 pixels. The RCM is numerically generated, and it is a two-dimensional white random
noise pattern possessing a probability distribution of Gaussian. Actually, the RCM is just a realization of the noise. The RCM is superimposed, in a cascade fashion, with the host data ‘‘THE FILE’’ as a converted digital object wavefront for data encryption. Both the signal plane and the RCM plane will be digitally transformed to the hologram with a discrete Fresnel diffraction (DFD) formula. As we have assumed, a digitally generated inclined-plane reference wave is employed to interfere with the wavefront of the object at n–0–g plane to form a digital-hologram. So the host information is converted and encoded with the secretly selected spatial position of the object aside from the detail of the RCM. Following manipulations on the digital hologram have been described in Section 2, and an EDS is obtained at zi ¼ 1:2 m. It is worthy to point out that the EDS is just a complicated digital image but nothing else hence it can be transmitted via a communication link as an ordinary electronic file. By the same procedure, we can also record another digital hologram of the RCM to prepare a ‘‘key-mould’’ for key fabrication. Figs. 2(a) and (b) show the host data sheet and the RCM, Fig. 2(c) is the digital hologram of the object with the predetermined spatial positions of both the signal plane and the RCM, while Fig. 2(d) shows the digital hologram of the RCM that will be used for the ‘‘key mould’’. Fig. 3(a) demonstrates the decrypted information with appropriate conditions imposed onto those parameters, for example, zi ¼ zo ; zc ¼ zr , and k1 ¼ k2 , and the correct delivered information of the RCM. Now let us look at what will happen if slight changes are introduced to spatial position of the signal plane, say, zo ¼ 1:201 m (i.e., Dzo ¼ 0:001 m) and zo ¼ 1:202 m (i.e., zo ¼ 0:002 m), respectively. Decrypted results with the correct key of the RCM, but with the disparities of spatial position of the signal plane, are shown in Figs. 3(b) and (c), from which one can see that the quality of data decryption is quite sensitive to the change of spatial position of signal sheet. When an amount of disparity is up to 0.002 m, the encrypted data become disappeared. This conclusion is equivalent to say that the tolerance of the positioning deviation is within two parts in one thousand. This
L. Yu et al. / Optics Communications 203 (2002) 67–77
(a)
(b)
(c)
(d)
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Fig. 2. (a) Data sheet to be encrypted; (b) the RCM; (c) digital hologram of the object; (d) digital hologram of the RCM.
phenomenon observed from numerical experiment shows that one is able to design new locks and keys for the applications of data encryption with the use of geometric and virtual optical parameters. 3.2. Sensitivity of the spatial position disparity of the reference wave Now we consider that the spatial position of a point-source-based reference wave is limited to a finite position in three-dimensional space and denoted by ðxr ; yr ; zr Þ. In this case, we utilize a spherical reference wave to record the digital hologram instead of using a plane wave. Except for applying a spherical wave for recording, other geometric parameters such as zo ; zRCM , and k1 , and the procedures for converting host data sheet,
encoding digital hologram, manipulation in spatial frequency domain, and decoding digital hologram are all the same as those steps already described in previous sections. Another numerical experiment is performed to test the sensitivity of spatial position disparity of the reference point source. First, we assume that reference wave is positioned at zr ¼ 1:1 m; xr ¼ 6:0 103 m, and yr ¼ 6:0 103 m. In order to examine the sensitivity of the position disparity of the reference wave, let us slightly adjust the value of xr and keep yr unchanged, leading to subsequent position disparities. Fig. 4(a) shows the EDS of the object, which is defined as real jz¼zi ðF Þ in Section 2. When we let xr ¼ 6:002 103 m ðDxr ¼ 2 106 mÞ, and yr ¼ 6:0 103 m ðDyr ¼ 0Þ, with all the other parameters unchanged, such as the precise image
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L. Yu et al. / Optics Communications 203 (2002) 67–77
(a)
(b)
(c) Fig. 3. Sensitivity of the spatial position of the signal plane: (a) decrypted data with correct RCM and correct position ðzo ¼ 1:200 mÞ; (b) decrypted data with correct RCM, but with position-disparity at 1.201 m; (c) decrypted data with correct RCM, but with positiondisparity at 1.202 m.
position as well as the precise reconstructed wavelength, then the reconstructed real image of ~M ~C ~ Þ, at the the RCM plane, the term real jz¼zi ðR position of zi ¼ zo ¼ 1:2 m, is shown as Fig. 4(b), and the reconstructed real image of the signal plane is calculated as Eq. (10) and shown as Fig. 4(c). Decryption with imprecise position of the reconstructed wave results in the deviation of the real image of the RCM, thus resulting in the effect of the decryption. The decrypted result with Dxr ¼ 4 106 m and Dyr ¼ 0, but with other parameters unchanged, is shown as Fig. 4(d), from which we can see that the host data have disappeared. Now we are able to estimate the sensitivity of the spatial position disparity of the reference. When an amount of the spatial position disparity
of the reference wave is up to about 4 106 m, the encrypted data become disappeared. 3.3. The sensitivity of virtual wavelength dispersion We have known that the spatial position of the reconstructed image is dependent on the encoding wavelength, k1 , and the decoding wavelength, k2 . Now we perform the third numerical experiment to test the sensitivity of the wavelength mismatch caused by wavelength drift. The virtual recording and reconstructing configuration of digital hologram is the same as that previously employed for testing spatial position sensitivity. The decoding procedure and the process of data encryption are also the same as described in the foregoing sec-
L. Yu et al. / Optics Communications 203 (2002) 67–77
(a)
(b)
(c)
(d)
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Fig. 4. Sensitivity of the spatial position disparity of the point-source-based reference wave: (a) EDS of the object; (b) reconstructed real image of the RCM with position disparity Dxr ¼ 2 106 m; (c) encrypted result with the disparity Dxr ¼ 2 106 m; (d) encrypted result with the position disparity Dxr ¼ 4 106 m.
tions. All the other parameters remain unchanged except for the decoding wavelength. Now we deliberately introduce some dispersion to the decoding wavelength, i.e., Dkc ¼ 0:01 nm; Dkc ¼ 0:02 nm and Dkc ¼ 0:03 nm. The numerical experiment results for data decryption are shown in Figs. 5(a)–(c), from which we can see that the sensitivity of the wavelength drifting is limited to amount at 0.03 nm. In other words, if the wavelength dispersion of the reconstructed wave exceeds 0.03 nm, then one is not able to decrypt host data even if one utilizes a correct key to unlock the RCM. This result proves our prediction that virtual wavelength can be used to design multiple and/or multi-dimensional lock(s) and key(s) for data encryption.
3.4. Estimated dimension of the keys As we have discussed in the previous sections, except the RCM, geometrical or physical parameters can also be selected as tools for designing multiple locks and keys for data encryption in hyperspace. And since the sensitivities of some geometrical and physical parameters have been studied in the above simulations, we could roughly estimate the total dimension of these keys. One important thing to be mentioned here is that whatever parameters selected for encryption and decryption, we should guarantee the Fresnel approximation to be satisfied since Fresnel diffraction propagation has been assumed in our algorithm.
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(a)
(b)
(c) Fig. 5. Sensitivity of the wavelength dispersion: (a) decrypted data with correct RCM but with wavelength drifting Dkc ¼ 0:01 nm; (b) decrypted data with correct RCM, but with wavelength drifting Dkc ¼ 0:02 nm; (c) decrypted data with correct RCM, but with wavelength drifting Dkc ¼ 0:03 nm.
It is obvious that the RCM will introduce a large number of key dimensions. For example, if the RCM is realized as 8-bit random noise in image, and the size of the RCM is 32 32 pixels, thus the total number of possible RCMs is about 25632 32 , which means that the key dimension of the RCM is a huge number. Now let us study the dimension of other possible keys derived from geometrical or physical parameters. This dimension may relate to some different factors such as the total number of geometrical or physical parameters selected, the scopes of all these possible parameters, and their sensitivities. As illustrated in the above simulations, if the spatial position of the signal plane is randomly selected from a scope between 1.2 and 3:2 m (could be larger), and since its sensitivity is
about Dzo ¼ 0:002 m as illustrated in Section 3.1, the possible key dimension resulted from this parameter could be 103 . Similarly, if the scopes of the x; y spatial positions of the reference wave are both selected from 4 103 to 8 103 m, thus the possible dimension of these two parameters could be around 106 since their sensitivity is about 4 106 m. The physical parameter of wavelength could also be selected from a huge numerical range instead of one coming from physically existed laser source. For example, if we randomly choose the decoding wavelength between 0.5 and 0:8 lm, and from simulations in Section 3.3 we get to know that the sensitivity of the decoding wavelength drifting is about 0.03 nm, thus the possible key dimension relative to decoding wavelength could be more than 104 . Thus the total dimension of the
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above four parameters is about 1013 . In practice, we could select even larger parameter scopes, or design much more possible parameters as keys for encryption, for example, the z position of the reference wave, the spatial position of the RCM plane, the spatial position of EDS, or the view angles of the reconstructed images, and the physical parameter of recoding wavelength could also be selected differently as the decoding wavelength, etc. The above analysis has roughly estimated the possible dimensions of the keys and shows a relatively high security strength that could be obtained by the proposed method, but it is not really proved that blind like deconvolution techniques or other attack techniques cannot be used to recover the input signal. This could be studied in the future researches. And some optical experiments could also be carried out to test the sensitivities of different parameters. For physical parameter such as wavelength, the experiments might be more complicated, depending on whether there is any easy way to fulfill wavelength drifting of 0.03 nm or even shorter drifting.
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tem. As we have proved that the advantages of the use of digital optics strategy attribute to the fact that large degrees of freedom have brought into information security technologies and therefore the security and imperceptibility level are dramatically increased. In addition, the proposed method is suitable to manage and control an information security system in a hierarchical manner. We have made quantitative evaluations about the sensitivities with respect to spatial position of the signal plane, the point-source-based reference wave, and the dispersion of the virtual wavelength, and the dimension of possible keys is roughly estimated. Numerical experiment results have proved the proposed concept and methodology for data encryption by making use of geometric and physical parameters derived from digital optics.
Acknowledgements The authors would like to thank financial support to this work by the Research Grants Council of the Hong Kong SAR, China (Project No. HKUST6175/00E and HKUST6015/02E).
4. Conclusion References In conclusion, we have introduced a new methodology of data encryption for digital optical security and demonstrated that parameters derived from digital optics or virtual optics can be utilized to design tools for multiple and/or multi-dimensional lock(s) and key(s) in the information security systems. During the encryption process, aside from the random mask, we can either use those virtual optical parameters in an individual manner or in a combinable fashion, so that it could be possible to develop new lock(s) and key(s) to enhance the reliability of information security sys-
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