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June 15, 2000 / Vol. 25, No. 12 / OPTICS LETTERS
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Optical encryption by double-random phase encoding in the fractional Fourier domain G. Unnikrishnan, J. Joseph, and K. Singh Phontonics Group, Indian Institute of Technology, New Delhi 110016, India Received February 22, 2000 We propose an optical architecture that encodes a primary image to stationary white noise by using two statistically independent random phase codes. The encoding is done in the fractional Fourier domain. The optical distribution in any two planes of a quadratic phase system (QPS) are related by fractional Fourier transform of the appropriately scaled distribution in the two input planes. Thus a QPS offers a continuum of planes in which encoding can be done. The six parameters that characterize the QPS in addition to the random phase codes form the key to the encrypted image. The proposed method has an enhanced security value compared with earlier methods. Experimental results in support of the proposed idea are presented. 2000 Optical Society of America OCIS codes: 070.2580, 070.2590, 070.5040, 200.4740, 210.2860.
Optical processing techniques are increasingly being used for data security applications. R´efr´egier and Javidi1 proposed a method, referred to in the literature as double-random phase encoding, to encrypt a primary image to stationary white noise. The encoding is done by use of two statistically independent random phase codes (key) in the input and Fourier planes with their phases uniformly distributed in the interval 关0, 2p兴. Matoba and Javidi2 added a third dimension to the random phase codes by shifting the random phase codes away from the input and Fourier planes. As an optical wave front propagates through a quadratic phase system (QPS) the distributions in any two planes are related, in general, by a fractional Fourier transform of the appropriately scaled distribution in the two planes. Thus a QPS offers a continuum of planes in which encoding can be done. We use this f lexibility of a QPS to propose an optical architecture that encrypts a primary image to stationary white noise by using two statistically independent random phase codes. We consider three planes of this encryption system, referred to as input, encryption, and output planes. The input plane contains the image to be encrypted, and the encrypted image is obtained in the output plane. One does the encoding by multiplying the distributions in the input plane and the encryption plane by random phase codes. In general, the input, encryption, and output planes are in fractional Fourier domains. The distributions in these planes are related by a fractional Fourier transform (FRT) with three parameters, input and output scale factors and order of FRT.3,4 The random phase codes as well as the parameters of the FRT that relate the input, encryption, and output planes constitute the key to the encrypted image. The proposed system is a QPS with two lenses separated by a stretch of free space. The input and the encryption planes are, in general, two planes located asymmetrically with respect to the first lens. Similarly, the encryption plane and the output plane are two planes located asymmetrically with respect to the second lens. The encryption system is shown in Fig. 1. The image to be encrypted is in the input plane 0146-9592/00/120887-03$15.00/0
and is denoted f 共x兲, which is assumed to be positive. Although the image is two dimensional, we follow a one-dimensional representation for convenience and clarity. x represents the input domain coordinate. The first random phase code, R1, is represented by exp关if1 共x兲兴, where f1 共x兲 is a random white sequence uniformly distributed in the interval 关0, 2p兴. Coherent light, amplitude modulated by f 共x兲 and phase modulated by R1, propagates through a distance d1 , passes through lens L1, and further propagates a distance d2 to reach the encryption plane. The distribution in the encryption plane is denoted g共u兲, where u is the coordinate in the encryption plane. g共u兲 is related to f 共x兲exp关if1 共x兲兴 by a FRT with three parameters and is referred to as the extended FRT.4 g共u兲 is given by the expression Z g共u兲 苷 K f 共x兲exp关if1 共x兲兴 µ ∂ a 2 x 2 1 b 2 u2 abux 3 exp ip 2 i2p dx , (1) tan a sin a where a, b, and a are the three parameters of the FRT and K is a complex constant, a, b, and a are in general complex quantities. Performing an extended FRT on a function is equivalent to expanding the function a times, performing a FRT of order a, and contracting the resultant distribution b times. The parameters a, b, and a are related to the distances d1 and d2 and the focal length f1 of the lens through the expressions p 1 f1 2 d2 1 2 , p (2) a 苷 l f1 2 d1 关f1 2 2 共f1 2 d1 兲 共f1 2 d2 兲兴1/2
Fig. 1. Optical setup for encryption. 2000 Optical Society of America
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OPTICS LETTERS / Vol. 25, No. 12 / June 15, 2000
"p
a 苷 arccos b2 苷
# p 共 f1 2 d1 兲 共f1 2 d2 兲 , f1
p 1 f1 2 d1 1 . p l f1 2 d2 关f1 2 2 共f1 2 d1 兲 共f1 2 d2 兲兴1/2
(3)
(4)
The distribution g共u兲 is encoded by random phase code R2, represented by exp关if2 共u兲兴, in fractional domain a. f2 共 兲 is a random white sequence uniformly distributed in the interval 关0, 2p兴 and is statistically independent from f1 共 兲. The distribution g共u兲exp关if2 共u兲兴 propagates a distance d3 , passes through lens L2, and further propagates a distance d4 to give the encrypted image C共 y兲 in the output plane. y is the output-plane coordinate: Z C共y兲 苷 K 0 g共u兲exp关if2 共u兲兴 ∂ µ cduy c 2 u2 1 d 2 y 2 2 i2p du , (5) 3 exp ip tan b sin b where C共 y兲 is related to g共u兲exp关if2 共u兲兴 through an extended FRT with scale parameters c and d and FRT order b. One can obtain the expressions that relate the parameters c, d, and b to distances d3 and d4 and focal length f2 by replacing a, b, a, d1 , d2 , and f1 with c, d, b, d3 , d4 , and f2 , respectively, in Eqs. (2)–(4). Substituting Eq. (1) for g共u兲 in Eq. (5), we can get an expression for the encrypted function in terms of input function f 共x兲. It can be shown that the encrypted function is a wide-sense stationary noise. A schematic of the encryption system is shown in Fig. 2. The schematic for decoding is as shown in Fig. 3. The conjugate of the encrypted image is expanded d times, a FRT of order b is performed, and the result is contracted c times. The resultant distribution gd 共u0 兲 is given by Z gd 共u0 兲 苷 K C ⴱ 共 y兲
performed by a CCD gives information about the primary image f 共x兲. The optical setup (Fig. 4) for decryption is the same as for encryption but in the reverse direction (the input port of the decryption system is the output port of the encryption system). The conjugate of the encrypted function is generated in the input plane of the decryption setup through phase conjugation in a photorefractive crystal5 – 7 and propagates through the optical system. Random phase code R2 is used in the encryption plane for decoding. The phase distortions undergone by the optical wave front during encryption get canceled during decryption, and the original function is obtained in the output plane of the decryption system by an intensity-sensing device such as a CCD. This cancellation does not take place if R2 is used in some plane other than the encryption plane. Hence to decrypt the data correctly it is essential to know the FRT parameters used for encryption. The use of phase conjugation during decryption ensures that the decryption QPS parameters (corresponding to any given set of encryption QPS parameters) will translate into realizable physical parameters, i.e., focal lengths of the lenses and distances. The proposed method has been experimentally demonstrated with the setup shown in Fig. 5. A diode-pumped Nd:YAG laser (Coherent Model DPSS Mini-YAG; l 苷 532 nm) was used as the coherent-light source. A transparency carried the image to be encrypted [Fig. 6(a)]. A photorefractive Fe:LiNbO3 crystal was used as the holographic medium to record the encrypted image. The angle between the reference and the signal beams was 60±. The signal beam’s arm contains the encryption setup, whose parameters are f1 苷 13.5 cm, f2 苷 9 cm, d1 苷 24 cm, d2 苷 12 cm, d3 苷 27 cm, and d4 苷 18 cm. The parameters of the quadratic phase system to encrypt the image as calculated from Eqs. (2)–(4) are a 苷 15.9 2 15.9i, b 苷 42 1 42i, a 苷 1.57 2 0.29i, c 苷 27.17 2 27.17i,
∂ d2 y 2 1 c2 u02 cdu0 y dy . (6) 3 exp ip 2 i2p tan b sin b µ
The superscript d indicates that the signal is obtained during decryption. The distribution gd 共u0 兲 is multiplied by R2, which cancels the random phase accrued during encryption. The resultant distribution is expanded b times, a FRT of order a is performed, and the result is contracted a times: Z f d 共x0 兲 苷 K gd 共u0 兲exp关if2 共u0 兲兴
Fig. 2. Schematic of the encryption system.
Fig. 3. Schematic of the decryption system.
µ
∂ b2 u02 1 a2 x02 abu0 x0 3 exp ip 2 i2p du0 . tan a sin a (7) By substituting gd 共u0 兲 from Eq. (6) into Eq. (7) one can show that the decrypted signal f d is equal to f 共x兲exp关2if1 共x兲兴. Inasmuch as f 共x兲 is a positive function, a square modulus of the decrypted signal as
Fig. 4. Optical setup for decryption.
June 15, 2000 / Vol. 25, No. 12 / OPTICS LETTERS
Fig. 5. Experimental setup: BS’s, beam splitters, BE, beam expander; M’s, mirrors; R1, R2, random phase masks; L1 – L3, lenses; PRC, photorefractive crystal; O, object transparency.
Fig. 6. (a) Primary image used for the study, ( b) encrypted image, (c) decrypted image with the correct key, (d ) decrypted image with the correct random phase codes but the wrong QPS parameters.
d 苷 38.43 2 38.43i, and b 苷 3.14 2 0.88i. For decryption, the encrypted images are read out by use of the conjugate of the reference beam used for recording. The conjugate of the stored encrypted images gets reconstructed and retraces the path followed by the wave front during encryption. The distortions that occurred to the wave front during encryption get canceled, and the original image is retrieved in CCD2 (Fig. 5). The decoded image is shown in Fig. 6(c). The image, when it is decoded with a set of QPS parameters 共a 苷 16.1 2 16.1i, b 苷 41.3 1 41.3i, a 苷 1.57 2 0.29i, c 苷 27 2 27i, d 苷 38.43 2 38.43i, and b 苷 3.14 2 0.88i, corresponding to f1 苷 13.5 cm, f2 苷 9 cm, d1 苷 24 cm, d2 苷 11.9 cm, d3 苷 27.1 cm, and d4 苷 18 cm, respectively), which are different from those used for recording (but with the correct random codes), is as shown in Fig. 6(d). It can be seen that, even if the random codes used are correct, if the QPS with the correct parameters is not used for decoding, the original image cannot be retrieved. This experiment demonstrates the use of fractional order as a key
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for encryption and the robustness of this technique to blind decryption (i.e., an attempt to decrypt without knowledge of the fractional order used for encryption). The security of an encryption technique depends on how secure the key is. In conventional double-random phase encoding1 the security of the key is related to the space –bandwidth product of the random phase codes used. The space–bandwidth product of the random phase codes can be increased only to the extent that it does not exceed that offered by the optical system. Hence an alternative and meaningful approach would be to include the additional degrees of freedom offered by a QPS (in terms of the FRT parameters). We consider the diff iculty in decrypting data if only the random phase codes are known but the fractional order parameters are not known. In the QPS considered in Fig. 1, if distances d1 , d2 , d3 , and d4 and focal lengths f1 and f2 can take N different values, then the QPS parameters can have N 6 different values. For the experiment demonstrated in this paper, N is of the order of 100. Therefore the number of different values that the QPS parameters can take is of the order of 1012 . The probability of guessing the correct QPS parameters is quite low. To summarize, we propose an optical architecture that encrypts a primary image to stationary white noise by using two statistically independent random phase codes. The encoding is done in the fractional Fourier domain. The input, encryption, and output planes are, in general, three planes of a QPS. The distributions in these planes are related by a FRT with three parameters. The encrypted image is a function of six parameters in addition to the random codes. The six parameters in addition to the random codes serve as the key to the encrypted image. Knowledge of the key in its entirety is essential for successful decryption, which enhances the security value of the proposed method. The proposed method could find application in securing data in holographic storage.6,7 Experimental results in support of the proposed idea have been presented. K. Singh’s e-mail physics.iitd.ernet.in.
address
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References 1. P. R´efr´egier and B. Javidi, Opt. Lett. 20, 767 (1995). 2. O. Matoba and B. Javidi, Opt. Lett. 24, 762 (1999). 3. H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, Chap. 4, pp. 239 –291. 4. J. Hua, L. Liu, and G. Li, J. Opt. Soc. Am. A 14, 3316 (1997). 5. G. Unnikrishnan, J. Joseph, and K. Singh, Appl. Opt. 37, 8181 (1998). 6. O. Matoba and B. Javidi, Appl. Opt. 38, 6785 (1999). 7. O. Matoba and B. Javidi, Appl. Opt. 38, 7288 (1999).
