Affine cryptosystem of double-random-phase encryptionbased on the fractional Fourier transform

Zhou Xin, Yuan Sheng, Wang Sheng-wei, and Xie Jian

An affine mapping mathematical expression of the double-random-phase encryption technique has beendeduced utilizing the matrix form of discrete fractional Fourier transforms. This expression clearlydescribes the encryption laws of the double-random-phase encoding techniques based on both the frac-tional Fourier transform and the ordinary Fourier transform. The encryption process may be regarded asa substantial optical realization of the affine cryptosystem. It has been illustrated that the encryptionprocess converts the original image into a white Gaussian noise with a zero-mean value. Also, thedecryption process converts the data deviations of the encrypted image into white Gaussian noises,regardless of the type of data deviations. These noises superimpose on the decrypted image and degradethe signal-to-noise ratio. Numerical simulations have been implemented for the different types of noisesintroduced into the encrypted image, such as the white noise with uniform distribution probability, thewhite noise with Gaussian distribution probability, colored noise, and the partial occlusion of the en-crypted image. © 2006 Optical Society of America

OCIS codes: 070.0070, 070.4560, 070.6020.

1. Introduction

Many optical information processing techniques havebeen applied in the field of data security. Among theseimage encryption techniques, the double-random-phase encryption technique, which was proposed in1995 by Refregier and Javidi,1 has attracted muchattention. In this method, the original image is disar-ranged in the space domain and the Fourier frequencydomain by two independent random-phase masks, andthen the encrypted image obtained on the output planeis a stationary white noise in which the statisticalproperty does not change along with the translation oftime.1,2 The double-random-phase encryption tech-nique cannot only be implemented in an optical setup,but it can also be used in digital image processing.There are many correlated application reports in manyfields, such as optical security, image embedding, dig-ital watermarks, and so on.2–4 In addition, there isnumerous and thorough research on the quality ofthe decrypted image and the signal-to-noise ratio ofthe transmitted image in the process of decoding an

image.5,6 The results of this research indicate that thedouble-random-phase encryption technique can pro-vide an extremely high security effect and better qual-ity of the decrypted image under suitable conditions. Inrecent years, many new methods have been proposedthat popularized the double-random-phase encryptedprinciple from the Fourier domain to the Fresnel do-main and the fractional Fourier domain.7–10 These de-velopments expand the extent of research, analysis,and application of the random-phase encryption tech-nique. However, to our knowledge, the cryptographyprinciples of the optical encryption technique based ondouble-random-phase encryption have not yet beenreported. In this paper, based on the matrix form ofdiscrete fractional Fourier transform, we propose anaffine cryptosystem explanation of the double-random-phase encryption technique. This explanation includesthe double-random-phase encryption based on boththe fractional Fourier transform and the ordinary Fou-rier transform, which can be implemented on the 4fsystem.

2. Affine Cryptosystem Explanation

In the double-random-phase encryption based on theordinary Fourier transform, we know that the de-cryption keys are the random-phase masks. In thecase of fractional domain, encoding a fractional-orderparameter is additional information required for suc-cessful retrieval of the data.9 It seems to be much

The authors are with the Department of Opto-electronics Scienceand Technology, Sichuan University, Chengdu 610065, China. Thee-mail address for Z. Xin is zhoxn@21cn.com.

Received 10 May 2006; revised 31 July 2006; accepted 5 August2006; posted 9 August 2006 (Doc. ID 70662).

0003-6935/06/338434-06$15.00/0© 2006 Optical Society of America

8434 APPLIED OPTICS � Vol. 45, No. 33 � 20 November 2006

safer for the system based on the fractional Fouriertransform.

We consider two fractional-Fourier-transform unitsin cascade, and two independent random-phase masksare put on the input plane of the cascade system andthe media plane of the two units, respectively. Thediagram of the implementation is shown in Fig. 1.

For simplicity, we use a one-dimensional notation.Let f�x� and g�x� denote the original image and theencrypted image, respectively. �0�x� and �0�u� aretwo independent white sequences, uniformly distrib-uted in �0, 1�. ��x� and ��u� represent two randomfunctions with values ��x� � exp�i2��0�x�� and ��u�� exp�i2��0�u��, respectively. If the sample pixelnumber in the one-dimensional direction is N, thefunctions f�x�, g�x�, ��x�, and ��u� can be expressed,respectively, as

f � �f�0�, f�1�, . . . , f�N � 1��T,

g � �g�0�, g�1�, . . . , g�N � 1��T,

� � ���0�, ��1�, . . . , ��N � 1��T,

� � ���0�, ��1�, . . . , ��N � 1��T.

We introduce the matrix symbol to the discrete frac-tional Fourier transformation using the standardweighted form defined by Shih10–12:

F� � �l�0

3

Al���F l, (1)

where � is the fractional-order parameter, andAl����l � 0, 1, 2, 3� are continuous functions of �:

Al��� �14

1 � exp��i2��� � l��

1 � exp��i2��� � l�

4 �� cos��� � l��

4 �cos�2�� � l��

4 �exp��3i�� � l��

4 �.

(2)

F is the discrete Fourier-transform matrix:

F �1

NW0�0 W0�1 · · · W0��N�1�

W1�0 W1�1 · · · W1��N�1�

É É Ì É

W�N�1��0 W�N�1��1 · · · W�N�1���N�1��, (3)

where W � exp��i�2��N��, and the exact value in themth row and lth column element of the matrix F isFml � 1�N exp��i�2�ml�N��.

Let F��m, l� denote the mth row and lth columnelement of the matrix F�; the double-random-phaseencryption process in Fig. 1 can then be obtained bythe following steps.

(1) Making the �1th power discrete Fourier trans-formation to ��x�f�x�, we obtain

X�m� � �l�0

N�1

F�1�m, l���l�f�l�, �m � 0, 1, . . . , N � 1�.

(4)

The result is equivalent to first, making a set of ele-mentary column transforms to F�1, i.e., every elementin the zeroth, first, second, . . . , (N � 1)th column ofthe matrix F�1 is multiplied, respectively, by ��0�,��1�, . . . , ��N � 1�, and obtaining a matrix denoted asF�

�1; second, we multiply the matrix F��1 by the vec-

tor f .(2) Multiplying the random-phase function ��u�,

we obtain

X�m���m� � ��m� �l�0

N�1

F�1�m, l���l�f�l�,

�m � 0, 1, . . . , N � 1�. (5)

This is equivalent to first making a set of elementaryrow transformations to F�

�1, i.e., every element in thezeroth, first, second, . . . , (N � 1)th row of the matrixF�

�1 is multiplied, respectively, by ��0�, ��1�, . . . ,��N � 1�, and obtaining a matrix denoted as F��

�1;second, we multiply the matrix F��

�1 by the vector f .(3) After making the a2th power discrete frac-

tional Fourier transformation, we obtain the vector gof the encoded image:

g�m� � �k�0

N�1

�l�0

N�1

��k�F�2�m, k�F�1�k, l���l�f�l�,

�m � 0, 1, . . . , N � 1�. (6)

This result is equivalent to first obtaining the matrixH by the multiplication of F�2 and F��

�1 and second bymultiplying the matrix H by the vector f . Finally, theprocess of random-phase encryption can be denotedby the multiplication of the matrix and the vector as

g � Hf , (7)

where H is a N � N matrix. Since the matrix F isnonsingular and the elementary matrix transforma-tion does not change the matrix order, the matrix Hshould be nonsingular also, i.e., the inverse matrixH�1 exists, making f � H�1g.

Fig. 1. Diagram of the implementation.

20 November 2006 � Vol. 45, No. 33 � APPLIED OPTICS 8435

Usually, the original image f�x� is a real function,and the encrypted image g�x� is a complex function.Hence, Eq. (7) actually describes an affine mappingfrom the domain of the real number to the domain ofthe complex number. If the original image is regardedas the plaintext and the encrypted image is regardedas the ciphertext, considering matrix H is nonsingu-lar, and the process of encryption described in Eq. (7)is actually an affine cryptosystem belonging to cryp-tography. The affine cipher is one of the classicalcryptosystems, it is a symmetrical key system and itssecurity depends on the complication of the mappingmatrix.13

From Eq. (6) it is obvious that all the pixels of theplaintext f make a contribution to every pixel of theciphertext g; in other words, all the pixels of the ci-phertext g are influenced by every pixel of the plain-text f . So the corresponding relations between theplaintext and the ciphertext are complicated by theaffine mapping. This process is called diffusion andconfusion in cryptography, and it can prevent some-one from decoding the information related to theplaintext through statistical analysis of the cipher-text. It is also the basic principle of keeping the in-formation secret by an affine cryptosystem. Then wecould learn that the property and security of an opticalencoding system, based on the double-random-phaseencryption technique, may be analyzed according tothat of an affine cryptosystem.

Moreover, when �1 � 1, �2 � �1, the fractional-Fourier-transform matrix F�1 and F�2 degenerate intothe discrete Fourier-transform matrix F and its in-verse matrix F�1. Under this situation, the process inFig. 1 is the double-random-phase encoding systemproposed by Refregier and Javidi,1 based on the or-dinary Fourier transform, and it can be realized inthe 4f system.

3. Characteristics of Encrypted Images

The encrypted image will present the random distrib-uted characteristics of the amplitude and the phase.If it is regarded as a stochastic process, and the dig-ital characteristic value of the function g�x� can becomputed.

The mathematical expectation of the mean func-tion is14,15

E� 1N �

m�0

N�1

g�m���1N �

m�0

N�1

E� �k�0

N�1

�l�0

N�1

��k�F�2�m, k�

� F�1�k, l���l�f�l��, (8)

where E denotes the mathematical expectation. Since��x�, ��u�, and f�x� are the independent statistical,functions, we could obtain

E� 1N �

m�0

N�1

g�m��� 0. (9)

The mathematical expectation of the autocorrela-tion function is

E� 1N �

m�0

N�s�1

g�m�g*�m � s���

1N �

m�0

N�s�1

E�� �k1�0

N�1

�l1�0

N�1

��k1�F�2�m, k1�F�1�k1, l1�

���l1�f�l1��� �k2�0

N�1

�l2�0

N�1

�*�k2��F�2�*�m � s, k2�

��F�1�*�k2, l2��*�l2�f�l2�� , (10)

where the asterisk denotes the complex conjugate ands � 0, 1, . . . , N � 1. Let R�s� � E�g�m�g*�m � s��;then

E� 1N �

m�0

N�s�1

g�m�g*�m � s��� R�s� �sN R�s�. (11)

Obviously, the autocorrelation function of the en-crypted image is related to s and is not related to theinitial location; hence the encrypted image can beregarded as a wide-sense stationary stochastic pro-cess. When N is very large, we can obtain14,15

E� 1N �

m�0

N�1

g�m�g*�m � s���� 1N �

l�0

N�1

�f�l��2, s � 0

0, s 0.

(12)

Equation (12) is identical with the conclusion in Ref.1. Furthermore, the double-random-phase encryptionsystem in Ref. 1, based on the ordinary Fourier trans-form, is an exception to the affine cryptosystem thatis established in this paper. Equation (12) means thatthe pixels of the encrypted image are unrelated toeach other. Usually, we call a stationary stochasticprocess a white-noise process when its mean value is0 and its autocorrelation function is A0���, where A0is a nonzero constant. For a white-noise process withGaussian distribution probability, its variance is�2 � A0.

Since ��x�, ��u�, and f�x� are statistically indepen-dent variables, the probability distribution of the en-crypted image tends to be a Gaussian distributionbecause of the central limit theorem. From Eqs. (9)and (12), we know that the encrypted image is azero-mean white Gaussian noise with the variancegiven by

�2 �1N �

l�0

N�1

�f�l��2. (13)

Obviously, the phase distribution functions ��x�and ��u� in the two random-phase masks are thedecisive factors that make the encrypted image theGaussian white noise. In the double-random-phaseencryption system based on the ordinary Fouriertransform, assuming that we remove one of the tworandom-phase masks [for instance, let ��u� � 1], andthe mapping relation decided by Eq. (6) becomes

8436 APPLIED OPTICS � Vol. 45, No. 33 � 20 November 2006

simple (i.e., the diffusion and confusion of the affinecryptosystem is not complicated enough), the en-crypted image can be recovered easily. On the otherhand, assuming ��x� � 1, as long as we dispose of theencrypted image in the Fourier frequency domain,Eq. (6) is still a simple mapping relation. Thus, bothof the two random masks are necessary in the double-random-phase encryption system based on the or-dinary Fourier transform. However, in the affinecryptosystem of the double-random-phase encryptionbased on the fractional Fourier transform, even ifboth random masks are removed, the fractional-Fourier-transform matrix can still contribute to thecomplicated mapping relation; the power of the frac-tional Fourier transform is the key of the decryptedimage, so the image information is still safe, but theencrypted image is no longer a white Gaussian noise.7

4. Error Diffusion and the Noise of the DecryptedImage

The diffusion and confusion make the plaintext’s in-formation diffuse in the whole of the ciphertext; onthe other hand, if some deviation occurs in the cipher-text’s data, this kind of deviation may also diffuse inthe decrypted plaintext, and it exhibits as noise in-formation. This is called the error diffusion. In fact,this kind of data deviation is inevitable; for example,a scrape on the antifraud marks of a credit card; thenoise raised in the channel where the coded signal istransmitted, as well as the abridgement; the jugglemade illegally in the encrypted image; and so on.

If we regard the inverse process of F�1 and F�2 asF��1 and F��2, the decryption process of Eq. (6) can beexpressed as

f�m� � �k�0

N�1

�l�0

N�1

�*�k�F��1�m, k�F��2�k, l��*�l�g�l�,

�m � 0, 1, . . . , N � 1�. (14)

If the encrypted image has the deviation g� � g �g, �g � ��g0, �g1, . . . , �gN�1�T, the decrypted im-age also has the deviation f� � f �f, �f ���f0, �f1, . . . , �fN�1�T. So

�fm � �k�0

N�1

�l�0

N�1

�*�k�F��1�m, k�F��2�k, l��*�l��gl,

�m � 0, 1, . . . , N � 1�. (15)

Similarly, computing the digital characteristic valueof the noise �f, the mathematical expectation of themean function is

E� 1N �

m�0

N�1

�fm�� 0, (16)

and the mathematical expectation of the autocorre-lation function is

E� 1N �

m�0

N�1

�fm�fm�s*��� 1N �

l�0

N�1

�gl�gl*, s � 0

0, s 0. (17)

Here we can obtain a significant conclusion, re-gardless of the type deviations �g. Ultimately theresult will be reflected on the decrypted image as azero-mean white Gaussian noise with the variancegiven by

�2 �1N �

l�0

N�1

�gl�gl*. (18)

5. Numerical Simulations

In this section we simulate the double-random-phaseencryption of a 256-level gray-scale original imagewith 256 � 256 pixels, based on the fractional Fou-rier transform. The power of two fractional-Fourier-transform units in the x, y directions are �0.9, 0.9�and �1.0, 1.0� respectively.

A. Statistical Characteristics of the Encrypted Image

The original image and the encrypted image (realpart) are shown in Figs. 2(a) and 2(b), respectively.Figure 3(a) shows the probability distribution of thepixel value of the encrypted image (real part), wherethe probability of the pixel value is calculated by theratio of the pixel number with same one-pixel value tothe total pixel number. The dotted curve in Fig. 3(a)is the Gaussian distribution having a zero mean anda variance equal to �2�2, where �2 is given by Eq. (13).The variance of the encrypted image is the sum ofthe variances of its real and imaginary parts be-cause the encrypted image could be regarded as acomplex stochastic process with equal variance ofreal and imaginary parts. The autocorrelation func-tion of the encrypted image is shown in Fig. 3(b). Itis shown that the encrypted image could be treatedas a zero-mean white Gaussian noise.

B. Statistical Characteristics of the Noise Added on theDecrypted Image

First we introduce several kinds of data deviationinto the encrypted image such as white noise with auniform distribution probability, white noise with aGaussian distribution probability, colored noise witha similar frequency spectrum to the original image,and occlusion of the encrypted image. Then, carryingon the decoding operation correspondingly, we canobtain the decrypted image with noise.

The white-noise and colored-noise data deviationsare multiplied by a constant range from 0.1 to 0.3 and

Fig. 2. (a) Original image, (b) encrypted image (real part).

20 November 2006 � Vol. 45, No. 33 � APPLIED OPTICS 8437

added on the encrypted image in our numerical sim-ulation. Figure 4 shows two of the data deviations,where Fig. 4(a) is the colored noise added on theencrypted image and Fig. 4(b) is one-fourth occlusionin area of the encrypted image. Corresponding tothese two types of data deviations, the probabilitydistributions of the pixel value of the noise (real part)added on the decrypted image are shown in Fig. 5.The dotted curves in Fig. 5 are the Gaussian distri-bution having a zero mean and a variance equal to�2�2, where �2 is given by Eq. (18). We could obtainsimilar results if the white noises with uniform andnormal distribution probability were introduced. Also,the autocorrelation functions of these noises added onthe decrypted image are similar to Fig. 3(b). Hence thenoises added on the decrypted image could be regardedas zero-mean white Gaussian noise.

6. Conclusion

In this paper, an affine mapping expression of thedouble-random-phase encryption technique has beenproposed by using the discrete fractional-Fourier-transform matrix. This expression clearly describesthe diffusion and confusion processes between the orig-inal and the encrypted images. They conceal enor-mously the connection between the plaintext and theciphertext; thus the double-random-phase encryptiontechnique provides high security. The encryption pro-cess of this technique could be regarded as an opticalrealization of an affine cryptosystem. Analysis showsthat the encrypted image could be treated as zero-mean white Gaussian noise, as well as the noise addedon the decrypted image, which is converted from thedata deviation of the encrypted image.

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Fig. 3. (a) Probability distribution of the pixel value of the en-crypted image (real part), (b) autocorrelation function of the en-crypted image.

Fig. 4. (a) Colored noise, (b) one-fourth occlusion of the encryptedimage (real part).

Fig. 5. Probability distribution of the pixel value of the noisecorresponding to data deviation: (a) colored noise, (b) one-fourthocclusion.

8438 APPLIED OPTICS � Vol. 45, No. 33 � 20 November 2006

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