Security enhancement of the phase-shifting interferometry-based cryptosystem by independent random phase modulation in each exposure

Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Security enhancement of the phase-shifting interferometry-based cryptosystem by independent random phase modulation in each exposure Meihua Liao, Wenqi He n, Dajiang Lu, Jiachen Wu, Xiang Peng n College of Optoelectronic Engineering, Key Laboratory of Optoelectronics Devices and Systems, Education Ministry of China, Shenzhen University, Shenzhen 518060, China

ar t ic l e i nf o

a b s t r a c t

Article history: Received 25 January 2016 Received in revised form 24 February 2016 Accepted 11 March 2016

The traditional phase-shifting interferometry (PSI)-based cryptosystem is one of the most classical optical cryptosystems. It employs the Mach–Zahnder interferometer to record the intensity distributions to partly overcome the inconvenience while storing the complex-valued ciphertext in some other optical cryptosystems (e.g., double random phase encoding technique). However, it has been proven to be vulnerable to chosen-plaintext attack and known-plaintext attack. In this manuscript, we propose an alternative method to enhance the security strength of the traditional PSI-based cryptosystem. By substituting the fixed random phase mask (RPM) and the phase retarder in the reference arm with four independent and different RPMs (served as secret keys) in four exposures, we can correspondingly capture four intensity-only patterns (regarded as ciphertexts). Theoretical analysis, especially with respect to security characteristics, as well as the numerical simulations are presented to verify the feasibility and reliability of the proposed cryptosystem. & 2016 Published by Elsevier Ltd.

Keywords: Fourier optics and signal processing Image encryption Phase-shifting interferometry

1. Introduction Since a novel image encryption method based on double random phase encoding (DRPE) was firstly invented in 1995 [1], optical image encryption has attracted more and more researchers' attention because of its advantages such as high-speed operation capability and the possibility of encoding data in multiple dimensions [2]. For the past two decades, besides extending DRPE into different transform domains, e.g., fractional Fourier-transform domain [3–5], Fresnel transform domain [6], gyrator domain [7,8], fractional Mellin transform domain [9], researchers have also reported various methods based on different optical principles and structures, such as joint transform correlator architecture [10], digital holography [11,12], phase-shifting interferometry (PSI) [13–20], two-beam interference [21–26], diffractive imaging [27,28], ghost imaging [29], integral imaging [30,31], ptychography [32,33], compressive sensing [34] and phase space theory [35,36]. Among these cryptosystems, the PSI-based cryptosystem is one of the most classical and interesting optical cryptosystems [13–20] because of the following advantages: (1) the complex-valued ciphertext is recorded by a group of intensity-only interferograms, which facilitates its storage n

Corresponding authors. E-mail addresses: [email protected] (W. He), [email protected] (X. Peng).

and transmission; (2) digital PSI can make more efficient use of the limited CCD resolution than off-axis digital holography. However, almost all the existing PSI-based cryptosystems, including four-step PSI, three-step PSI and two-step PSI, have shown their weakness against chose-plaintext attack (CPA) [37] and known-plaintext attack (KPA) [38]. Frankly to say, some kinds of security leaks to specific attacks could not totally deny the validation of the PSI-based cryptosystems, but rather the efforts on eliminating the weakness should be encouraged and it is exactly our motivation. In this manuscript, we present a novel method to enhance the security strength of the traditional PSIbased cryptosystem. In our proposed cryptosystem, the reference wave will be modulated by different RPMs in each exposure, other than by the same RPM in the traditional four-step PSIbased cryptosystems. This feature could bring in the abilities to resist against the aforementioned CPA and KPA, and also maintain its nature advantages. The rest of this contribution is organized as follows: Section 2 analyzes the security issue of the traditional four-step PSI-based cryptosystem. Section 3 describes our proposed method and discusses its security level. Section 4 performs a series of numerical simulations to verify the validity and robustness of the proposed scheme. Section 5 summarizes the major points of this approach.

http://dx.doi.org/10.1016/j.optlaseng.2016.03.015 0143-8166/& 2016 Published by Elsevier Ltd.

Please cite this article as: Liao M, et al. Security enhancement of the phase-shifting interferometry-based cryptosystem by independent random phase modulation in each exposure. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.03.015i

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2

2. Security analysis of the traditional four-step PSI-based cryptosystem In this section, we would like first to briefly review the encryption and decryption procedures of the traditional four-step PSI-based cryptosystem [13], and emphatically analyze why and how it is vulnerable to CPA [37] and AKPA [38]. For simplicity, the coordinates of the involved parameters in the following description are omitted. The encryption procedure is carried out on a Mach–Zehnder interferometer, as depicted in Fig. 1. The laser beam after spatial filtering and collimation splits in two by a beam splitter, which constitutes the reference and object arms of the interferometer. The plaintext P bonded with a RPM (exp½iϕ, the first secret key) is fixed at input plane in the object arm. Through a Fourier transform lens, the complex object field in the output plane is given by U ¼ FTfP U exp½iϕg;

ð1Þ

where FTf U g denotes the Fourier transform operation. Simultaneously, another RPM (exp½iφ, the second secret key) is placed at the reference plane with a distance z from the output plane. There are also two phase retarders, one quarter- and one half-wave plate, inserted in the path of the reference beam to provide the specific phase difference (π =2). Similarly, four complex reference fields in the output plane are given by Rn ¼ exp½iðn  1Þπ =2 U FrT λ;z fexp½iφg;

n ¼ 1; 2; 3; 4;

ð2Þ

where FrT λ;z f U g denotes the Fresnel diffraction with the illumination wavelength of λ and the distance of z. In this way, four corresponding phase-shifting interferograms are recorded by a CCD camera in the output plane and they can be expressed as I n ¼ jU þ Rn j2 ;

n ¼ 1; 2; 3; 4

ð3Þ

Let U ¼ AU exp½iΦU  and Rn ¼ AR expfi½ΦR þ ðn  1Þπ =2g, respectively. Then, Eq. (3) can be rewritten as I n ¼ AU 2 þ AR 2 þ 2AU AR cos ½ΦU  ΦR  ðn  1Þπ =2;

n ¼ 1; 2; 3; 4 ð4Þ

Thus, this was the end of the encryption and four noise-like interferograms I n serve as the ciphertexts. The decryption procedure is implemented by digital means. Let ΦE ¼ ΦU  ΦR and

L1

SF

M1

LASER

Retarder plates M3

BS1

2 P

b2

4

Reference plane

b1

z CCD

M2

Input plane

L2

BS2

AE ¼ AU U AR , and they can be respectively calculated by this PSI technique from the follow equations:
 I I ð5Þ ΦE ¼ arctan 4 2 ; I1  I3 AE ¼

(

1 I1  I3 ; 4 cos ðΦE Þ

ð6Þ

So, AO and ΦO can be calculated by two simple linear relations: AU ¼ AE =AR

ΦU ¼ ΦE þ ΦR

;

ð7Þ

Then, the plaintext P can be acquired immediately by P ¼ jFT  1 fAU U expðiΦU Þgj;

ð8Þ

1

where FT f Ug denotes the inverse Fourier transform operation. The whole encryption procedure looks like a nonlinear encoding process because we get four intensity distributions but not a complex one as some other cryptosystems. Unfortunately, according to Eqs. (5) and (6), ϕE and AE can be directly deduced by the recorded four interferograms without any of the decryption keys, which indicates ϕE and AE can be regarded as the real ciphertext of cryptosystem and the so-called PSI technique just a way to further encode the real ciphertext into intensity distributions but without introducing any further security measures. Meanwhile, Eq. (7) indicates there is a simple linear relation between the plaintext and real complex ciphertext. Therefore, in essence, the traditional PSI-based cryptosystem is a linear system from the view of cryptanalysis and various traditional attack methods could be easily applied in this cryptosystem [37,38]. We would like to give a brief description on two kinds of recently reported attack schemes, CPA and KPA. For CPA, it is assumed that attackers have the capability to choose arbitrary plaintexts to be encrypted and obtain the corresponding ciphertexts. In Ref. [37], the attackers choose a Dirac delta function δðx  x0 ; y  y0 Þ as the known plaintext of the traditional four-step PSI-based cryptosystem. According to Eq. (1), Uðx; yÞ can be written as Uðx; yÞ ¼ FTfδðx  x0 ; y y0 Þ U exp½iϕðx; yÞg ¼ exp½iϕðx0 ; y0 Þ;

ð9Þ

Thus, its amplitude AU ¼ 1 and its phase ΦU ¼ ϕðx0 ; y0 Þ. Then substituting them into Eq. (7), that is ( AR ¼ AE =AU ¼ AE ð10Þ ΦR ¼ ΦU  ΦE ¼ ϕðx0 ; y0 Þ  ΦE ; where ϕðx0 ; y0 Þ is a constant phase angle, corresponding to one pixel value of ϕðx; yÞ at the coordinate ðx ¼ x0 ; y ¼ y0 Þ. Hence, the decryption keys AR and ϕR could be obtained easily by Eq. (10) as AE and ϕE are the known resources. For KPA, it is assumed that attackers know several plaintext– ciphertext pairs [38]. Let fðP 1 ; C 1 Þ; ðP 2 ; C 2 Þg as the known resources in the traditional four-step PSI-based cryptosystem. Note that both C 1 and C 2 are the complex ciphertext (which are easy to acquire with the four corresponding encrypted interferograms by the PSI technique) and can be described as ( C 1 ¼ FTfP 1 U r 0 g þ AR expðiΦR Þ ; ð11Þ C 2 ¼ FTfP 2 U r 0 g þ AR expðiΦR Þ

Output plane

Fig. 1. The optical setup for encryption of the traditional PSI-based cryptosystem. SF and L1 , spatial filter lens assembly; L2 , Fourier transform lens; M 1 , M 2 and M 3 , mirrors; BS1 and BS2 , beam splitters; λ=2 and λ=4 are half- and quarter-wave plates;P, plaintext; b1 and b2 , RPMs; z, diffraction distance.

According to Eq. (11), the distribution of r 0 can be deduced by r0 ¼

FT  1 fC 2  C 1 g ; P2  P1

ð12Þ

Please cite this article as: Liao M, et al. Security enhancement of the phase-shifting interferometry-based cryptosystem by independent random phase modulation in each exposure. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.03.015i

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Then, the decryption keys AR and ϕR can be obtained as follows: ( AR ¼ jC 1  FTfP 1 Ur 0 gj ; ð13Þ ΦR ¼ argfC 1  FTfP 1 U r 0 gg

3. Description of our proposed scheme In this section, we will describe our proposed scheme which is a security enhanced version of the PSI-based cryptosystem. Our method is inspired by the single-exposure wave-splitting phaseshifting digital holography [39]. The encryption procedure is also implemented on the basis of Mach–Zehnder interferometer, as depicted in Fig. 2. The object arm is the same as the traditional four-step PSI-based cryptosystem's, and U ¼ FTfP U exp½iϕg is the complex object field in the output plane. However, in the reference arm, instead of introducing four fixed phase differences, we place four different RPMs time by time to modulate the reference wave in four exposures respectively. This can be done by a dynamic refreshed SLM together with a predetermined program in the connected PC. These RPMs mentioned above are independent with each other and can be given by r n ¼ exp½iφn ;

n ¼ 1; 2; 3; 4

ð14Þ

Similarly, through the Fresnel diffraction with the illumination wavelength of λ and the distance of z, four complex reference fields in the output plane are written as Rn ¼ FrT λ;z fexp½iφg;

n ¼ 1; 2; 3; 4

ð15Þ

In this way, four corresponding phase-shifting interferograms can be expressed as I n ¼ jUj2 þ jRn j2 þ URn þ U  Rn ;

n ¼ 1; 2; 3; 4

ð16Þ

where asterisk * means complex conjugate. Up to now, the whole encryption process of our method has been already completed. In decryption, the authorized receiver can analytically retrieve the exact original plaintext image. Firstly, we would like to introduce four intermediate variables as follows: H 12 ¼ I 1  I 2 ðjR1 j2  jR2 j2 Þ;

ð17Þ

L1

3

H 34 ¼ I 3  I 4  ðjR3 j2  jR4 j2 Þ;

ð18Þ

R12 ¼ R1  R2 ;

ð19Þ

R34 ¼ R3  R4 ;

ð20Þ

Eqs. (17)–(20) show that all the intermediate variables are formed by the recorded interferograms (ciphertext) and the complex amplitude distribution of the reference wave in the CCD plane (keys). That's to say, if the information of all correct keys is known, the Fourier distribution of the plaintext, U could be calculated out by a set of simple deductions, U¼

H 12 R34  H 34 R12 ; R12 R34  R12 R34

ð21Þ

It should be noted that R12 R34 must not equal to R12 R34 for the value of the denominator on the right side of upper equation should not be zero. Fig. 3 shows a certain condition of the reference wave on a complex plane. The complex amplitude quantities R1 , R2 , R3 and R4 are shown as vectors. In other words, the composed vectors R12 and R34 must not be parallel on a complex plane. Then, the plaintext P can be acquired immediately by     ð22Þ P ¼ FT  1 fUg; So far, the original plaintext image has been correctly recovered by digital means. Obviously, without the decryption keys, the complex object field U can not be deduced only from four interferograms. In this sense, we can also claim that there is no direct linear relationship between the plaintext and ciphertext. This is the nature difference of our scheme with the traditional PSI-based cryptosystems and some other related developments. Next, we will investigate the security strength of our proposed cryptosystem against CPA and KPA. (1) Assume that attackers choose a Dirac delta function δðx  x0 ; y y0 Þ as the known plaintext according to what they do in Ref. [37]. The complex object field in the CCD plane is Uðx; yÞ ¼ exp ½iϕðx0 ; y0 Þ by Eq. (9). Then, the corresponding ciphertexts (four interferograms) can be expressed as I n ¼ 1 þ jRn j2 þ exp½iϕðx0 ; y0 ÞRn þ exp½  iϕðx0 ; y0 ÞRn ; n ¼ 1; 2; 3; 4

ð23Þ

SF

Im

M1

LASER

R12 M3

BS1 r1 , r2 , r3 , r4

P

PC

R2

SLM

Re

r0

z

R4 CCD

M2

Input plane

R1

L2

BS2

Output plane

Fig. 2. The optical setup for encryption of our proposed cryptosystem. SF and L1 , spatial filter lens assembly; L2 , Fourier transform lens; M 1 , M 2 and M 3 , mirrors; BS1 and BS2 , beam splitters; P, plaintext; r 0 , r 1 , r 2 , r 3 and r 4 , RPMs; SLM, spatial light modulator.

R3 R34 Fig. 3. A specific case of the reference wave on a complex plane.

Please cite this article as: Liao M, et al. Security enhancement of the phase-shifting interferometry-based cryptosystem by independent random phase modulation in each exposure. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.03.015i

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M. Liao et al. / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎

where exp½iϕðx0 ; y0 Þ and exp½  iϕðx0 ; y0 Þ can be regarded as two constants. Even though four interferogram I n contains the information of four corresponding decryption key Rn , the attacks can not deduce Rn by using the same strategy in Ref. [38] because there is no any relationship among Rn . So, we can claim CPA is ineffective to our proposed cryptosystem. (2) Assume that attackers know one or more plaintext–ciphertext pairs, that is, the plaintext P and the corresponding four interferograms I n are the known resources for the situation of our proposed cryptosystem. Firstly, attackers try to recover r 0 in order to know the complex object field U. In the traditional four-step PSI-based cryptosystem, according to Eq. (12), r 0 can be retrieved easily by using two plaintext–ciphertext pairs as AR and ΦR is unchanged in four different exposures. However, in our proposed cryptosystem, AR and ΦR are changed simultaneously with RPMs r n at each exposure, which results in that

Fig. 4. The original plaintext image “Lena”.

r 0 can not be retrieved even if the complex ciphertext is known. Therefore, it is much harder to deduce out the keys Rn than CPA because the complex object field U is unknown. So, KPA is also failure to crack our proposed cryptosystem. From what have been discussed above, it is indicated that our proposed cryptosystem has higher security to resist CPA and KPA than the traditional four-step PSI-based cryptosystem. The crucial reason is that the phase difference of all pixels in the reference wave plane are the same for each exposure in the traditional fourstep PSI-based cryptosystem while they are totally different in our proposed cryptosystem.

4. Numerical simulations A series of numerical simulations are provided to verify the reliability and availability of our proposed cryptosystem. In all the following simulations, a standard gray-scale image “Lena” (256  256 pixels) with 0.2 mm pixel size and 256 Gy levels is chosen as the plaintext, as shown in Fig. 4. The RPMs are generated by computer which have the same size and gray levels with the plaintext image. Fig. 5(a)–(d) shows the distributions of four RPMs (r 1 , r 2 , r 3 and r 4 ) respectively. By performing the encryption procedure of our proposed cryptosystem, we obtain four interferograms (Fig. 5(e)–(h)) as the corresponding ciphertexts. It is obvious that the ciphertexts appear to be stationary white noise, implying the plaintext has been fully confused. Besides, the ciphertexts are represented as intensity-only images for easy storage and transmission. By performing the decryption procedure with all correct keys, we can retrieve the original plaintext image “Lena”, as shown in Fig. 6 (a), perfect and lossless. By contrast, Fig. 6(b) shows the decrypted result when one of the decryption keys is arbitrarily selected. To objectively evaluate the effectiveness of our proposed method, we calculate the correlation coefficient (CC) value between the

Fig. 5. (a–d) The distributions of four different RPMs r 1 , r 2 , r 3 and r 4 ; (e–h) the distributions of four corresponding interferograms I 1 , I 2 , I 3 and I 4 .

Please cite this article as: Liao M, et al. Security enhancement of the phase-shifting interferometry-based cryptosystem by independent random phase modulation in each exposure. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.03.015i

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original plaintext image and the decrypted image. The CC is defined as follows: P P m n ðAmn AÞðBmn  BÞ CC ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð24Þ P P P P ½ m n ðAmn  AÞ2 ½ m n ðBmn  BÞ2 ; where A and B denote the mean value of images A and B. Amn and Bmn are the pixel values at the coordinate ðm; nÞ of images A and B, respectively. Obviously, the CC value ranges from 0 to 1, and the higher CC value implies the more similar between two images. For Fig. 6, the calculation shows that the CC value between the original plaintext image and the decrypted image when all keys are correct is 1 while it is only 0.0081 when one of the keys is arbitrarily selected.

5

Then, the robustness of our proposed cryptosystem is tested as the ciphertexts may be contaminated by noise or occlusion in a practical case. For one thing, the robustness against noise is tested. A random noise is added to one of the interferograms I 1 in the following way I 01 ¼ I 1 ð1 þ kQ Þ;

ð25Þ

I 01

denotes noise-affected ciphertext I 1 , k is a weighting where factor, Q is a random noise with variable randomly distributed in the range of ½0; 1. Fig. 7(a)–(c) shows the decrypted results for k¼ 0.01, 0.05 and 0.1 respectively, and the corresponding CC values are 0.9647, 0.8331 and 0.7542. For another, the robustness against occlusion is tested. Fig. 7(d)–(f) shows the decrypted results for

Fig. 6. The decrypted result (a) when all keys are correct; (b) when one of the keys is arbitrarily selected.

Fig. 7. Robustness of our proposed method against noise and occlusion: (a–c) decryption results when ciphertext I 1 contains random noise (k ¼ 0.01, 0.05 and 0.1 respectively), (d–f) decryption results when the ciphertext I2 the occluded by 5%, 10% and 20% respectively.

Please cite this article as: Liao M, et al. Security enhancement of the phase-shifting interferometry-based cryptosystem by independent random phase modulation in each exposure. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.03.015i

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6

another interferogram I 2 occluded by 5%, 10% and 20%, and the corresponding CC values are 0.8021, 0.6395 and 0.5618 respectively. These results indicate that our proposed cryptosystem has a good robustness against noise and occlusion.

5. Conclusions In summary, we have proposed a method to enhance the security level of the PSI-based cryptosystem by independent random phase modulation in each exposure. After analyzing why and how the traditional PSI-based cryptosystem is vulnerable to CPA and KPA, we modify its encryption structure by placing four independent and different RPMs in the reference arm of the Mach–Zahnder interferometer at the time of four exposures respectively. The security analysis indicates our scheme has higher security strength than the traditional PSI-based cryptosystems. Numerical simulations have also verified the reliability and availability of our proposed cryptosystem.

Acknowledgment This work is supported by the National Natural Science Foundation of China (61377017, 61307003, and 61171073), and SinoGerman Center for Research Promotion (GZ 760).

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Please cite this article as: Liao M, et al. Security enhancement of the phase-shifting interferometry-based cryptosystem by independent random phase modulation in each exposure. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.03.015i