Optical cryptosystem based on phase-truncated Fresnel diffraction and transport of intensity equation

Chenggong Zhang; Wenqi He; Jiachen Wu; Xiang Peng

doi:10.1364/OE.23.008845

1. Introduction

Optical information security technique has drawn a lot of attention due to its inherent property of parallelism and ultrafast processing speed [1]. Since Refregier and Javidi proposed the double random phase encoding (DRPE) method, which encrypted a plaintext into a complex-valued distribution with a 4-f optical processor in 1995 [2], various optical setups have been introduced to this research realm. In 2000, Nomura and Javidi adopt a joint transform correlator (JTC) architecture to encrypt an image into the joint power spectrum, a real-valued distribution, which is convenient to record and transmit [3]. Moreover, the JTC method does not require accurate optical alignment and an extra complex conjugate of the key as the DRPE demands. However, with development of cryptanalysis, more and more security issues were exposed by researchers. Carnicer et al schemed a chosen-ciphertext attack (CCA) on the DRPE-based systems in 2005 [4]. Soon afterwards, the same system was attacked by known-plaintext attacks (KPAs) [5–7]. Besides, it was also proved that the JTC-based systems have some security risks. Chosen-plaintext attacks (CPAs) were first implemented on these cryptosystems [8, 9] and KPAs were also proved effective [10, 11]. Before long, this system was attacked by COA method based on a hybrid input-output algorithm [12]. To resist the mentioned attacks such as CPAs, KPAs and COA, Qin and Peng proposed an asymmetric cryptosystem based on phase-truncated Fourier transform (PTFT) which use different keys in encryption and decryption process [13–15]. However, it was reported that the PTFT is suffered from known public key attack (KPKA) by a modified amplitude-phase retrieval algorithm [16, 17]. In recent years, scholars proposed modified asymmetric methods [18, 19]; whereas some researchers argued that it is rather a one-time pad than an asymmetric system, since the decryption keys are not independent of the plaintext [20, 21].

In optical encryption methods, phase information plays an important role. For example, RPMs are used as encryption keys in cryptosystems based on DRPE and JTC [2, 3], and truncated phases are used as decryption keys in PTFT-based systems [14, 19]. Therefore, it is vital to record or retrieve the phase information. To our knowledge, there are two kinds of phase retrieval techniques which are widely used in optical cryptography and cryptanalysis: phase-shifting interferometry and iterative phase retrieval algorithm (e.g. Gerchberg-Saxton algorithm, GS). In this paper, we introduce the transport of equation (TIE) method [22, 23], a third type of method to obtain the phase information in our decryption process. As discussed in above paragraph, the PTFT-based systems have a security enhancement due to the nonlinear operator of the phase-truncation. However, it requires recording the truncated phase each time using complicated interferometry in encryption process, and then the truncated phase are regard as secret keys for decryption. Besides, the security risks would arise if the encryption keys are used as public keys. To avoid its deficiency, we use a phase-truncated Fresnel diffraction (PTFD) optical setup for encryption, but utilize a generalized amplitude-phase retrieval (GAPR) algorithm and the TIE method in decryption process, which have no need of the truncated phase. Therefore, rather than generate two truncated phase each time as decryption keys in the PTFT-based systems [14, 19], our method only use two fixed keys which is more convenient in practice. Above all, we proposed an optical cryptosystem based on PTFD and TIE, in which the decryption process is different from the encryption. And this scheme not only enhances the security of the system but also avoids recording truncated phases as the traditional PTFT-based systems do. The rest of this paper is organized as follows. In section 2, we give a theoretical description of the proposed cryptosystem. Section 3 provides simulation results and a discussion. At last, a conclusion is made in section 4.

2. Theoretical description

For encryption, a phase-encoded plaintext bonded with a random amplitude mask (RAM) Fresnel diffracts a short distance, then the phase is truncated and its amplitude is recorded by its intensity. Subsequently the resultant amplitude is attached by a random phase mask (RPM), and Fresnel diffracts another distance, then the phase is truncated again and its intensity is recorded as ciphertext. The flow chart of the encryption process is shown in Fig. 1. $R_{1} (x, y)$ and $R_{2} (x, y)$ are two random distributions, which form the RAM (Key 1) and the RPM (Key 2), respectively. The RAM and RPM are regarded as two secret keys.

Fig. 1 The flow chart of the encryption process. PTFD: phase-truncated Fresnel diffraction; E: ciphertext.

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To formulate the encryption process, suppose that an image (plaintext) $f (x, y)$ is phase-encoded as $\exp [i 2 π f (x, y)]$ , which multiply by the first key, RAM $R_{1} (x, y)$ , to form a complex amplitude distribution:

u_{0} (x, y) = R_{1} (x, y) \cdot \exp [i 2 π f (x, y)]

Then the complex amplitude Fresnel diffracts a distance $d z_{1}$ , which could be formulated by the angular spectrum theory [24]:

\begin{array}{l} u_{1} (x, y) = IFT {FT [u_{0} (x, y)] H_{1} (q_{x}, q_{y})} \\ = IFT {FT [u_{0} (x, y)] \exp [i k d z_{1} \sqrt{1 - {(λ q_{x})}^{2} - {(λ q_{y})}^{2}}]} \end{array}

Here $λ$ is wavelength, $k = 2 π / λ$ and $H_{1} (q_{x}, q_{y})$ is the transfer function. $(q_{x}, q_{y})$ are the variables conjugate to $(x, y)$ in the Frequency domain. FT and IFT denote Fourier transform and inverse Fourier transform, respectively. CCD records its intensity as:

I_{1} (x, y) = {| u_{1} (x, y) |}^{2}

Here $| \cdot |$ represents the modulus operator. In this record process the phase is truncated after Fresnel diffraction so that we name it as phase-truncated Fresnel diffraction (PTFD). Now we get the intermediate amplitude $| u_{1} (x, y) | = \sqrt{I_{1} (x, y)}$ as shown in Fig. 1. Then this amplitude is multiplied by the second key, RPM $\exp [i 2 π R_{2} (x, y)]$ . This process could be written as:

u_{2} (x, y) = | u_{1} (x, y) | \cdot \exp [i 2 π \cdot R_{2} (x, y)]

Then this complex amplitude Fresnel diffracts another distance $d z_{2}$ and is recorded by a CCD as ciphertext:

\begin{array}{l} E (x, y) = {| u_{3} (x, y) |}^{2} = {| IFT {FT [u_{2} (x, y)] H_{2} (q_{x}, q_{y})} |}^{2} \\ = {| IFT {FT [u_{2} (x, y)] \exp [i k d z_{2} \sqrt{1 - {(λ q_{x})}^{2} - {(λ q_{y})}^{2}}]} |}^{2} \end{array}

As described in the encryption process, the proposed method need not record the truncated phase as the PTFT-based cryptosystems do. We have also given a potential optical setup to realize our proposed encryption process. As shown in Fig. 2, SLM1 and SLM2 are amplitude-modulated and phase-modulated space-light modulators, respectively. The 4-f imaging system consisted by Lens1 and Lens2 enables the SLM1 and SLM2 to optically come into contact with each other. The Lens3 and Lens4 play identical role to transmit the complex amplitude on the SLM2 to the imaging plane. An intensity measurement device such as CCD is placed at a distance of dz from the imaging plane to measure the intensity of its corresponding complex amplitude. All the SLMs and CCD are controlled by a personal computer (PC). This optical setup could be used to implement the PTFD in the encryption process. As discussed in following paragraphs, the TIE method requires computing a finite difference which means that the Fresnel diffraction formulated in Eq. (2) only diffracts a very short distance. In other words, the CCD plane must be close to the image plane as shown in Fig. (2). It will be explained detailed in paragraphs following Eq. (14). Here we would like to emphasize that, the TIE method is widely used in the biomedical microscopy and X-ray imaging and its feasibility has been extensively verified by theories and experiments [23, 25, 26].

Fig. 2 Sketch of optical setup for PTFD. SLM: space-light modulator; PC: personal computer.

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To decrypt the ciphertext, as shown in Fig. 3, the ciphertext $E (x, y)$ and the RPM (Key 2) are taken as two constraints to retrieve the intermediate amplitude $\sqrt{I_{1} (x, y)}$ using the GAPR algorithm, and then the amplitude $\sqrt{I_{1} (x, y)}$ and the RAM (Key 1) are together to compute the phase $\exp [i 2 π f (x, y)]$ using the TIE method [22, 23]. The following paragraphs will formulate the GAPR and TIE methods in detail.

Fig. 3 The flow chart of the decryption process. GAPR: generalized amplitude-phase retrieval algorithm; TIE: transport of intensity equation.

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The GAPR method that we used to retrieve the intermediate amplitude from two constraints is similar to the GS algorithm [27, 28]. It mainly consists of five steps: (1) generate a random distribution as the initial phase $Φ_{0} (x, y)$ to initialize an estimation of the complex amplitude $G_{0} (x, y) = \sqrt{E (x, y)} \exp [i Φ_{0} (x, y)]$ in the output plane; (2) inverse Fresnel diffracts the estimation in the output plane; (3) replace the phase of the result in the step (2) with the RPM $\exp [i 2 π \cdot R_{2} (x, y)]$ (Key 2), which is regarded as the phase constraint, to form an estimation of the complex amplitude in the input plane; (4) Fresnel diffracts the result of step (3) leading to another complex amplitude in the output plane; (5) replace the modulus of the result in step (4) with $\sqrt{E (x, y)}$ , which is regarded as the amplitude constraint, to form an estimation of the complex amplitude in the output plane and it is then taken as the new input of the next iterative loop. The kth iteration process is illustrated in Fig. 4 and the aforementioned steps (2)-(5) are formulated as follows:

Fig. 4 The flow chart of the GAPR algorithm. FD and IFD denote Fresnel diffraction and inverse Fresnel diffraction, respectively.

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\begin{array}{l} G_{k} (x, y) = \sqrt{E (x, y)} \exp [i Φ_{k} (x, y)] \\ g_{k} (x, y) = IFD [G_{k} (x, y)] = \sqrt{{I^{'}}_{k} (x, y)} \exp [i φ_{k} (x, y)] \\ g_{k + 1} (x, y) = \sqrt{{I^{'}}_{k} (x, y)} \exp [i 2 π \cdot R_{2} (x, y)] \\ G_{k + 1} (x, y) = FD [g_{k + 1} (x, y)] = | G_{k + 1} (x, y) | \exp [i Φ_{k + 1} (x, y)] \end{array}

In Eq. (6), FD and IFD denote Fresnel diffraction and inverse Fresnel diffraction, respectively.

Once obtained the intermediate amplitude $\sqrt{I^{'} (x, y)}$ using the aforementioned GAPR method, we need to retrieve the plaintext using the amplitude $\sqrt{I^{'} (x, y)}$ and the RAM (Key 1). Generally speaking, iterative phase retrieval algorithm could not guarantee to retrieve the true solution, as the algorithm may converge to a local minimum. Therefore, as reported by J. R. Fienup, it is difficult to retrieve the true phase from two amplitude constraints using an iterative phase retrieval algorithms [26]. However, if the two amplitudes are in close proximity, its phase distribution could be recovered from the finite difference of the two amplitudes’ intensities using the TIE method [22, 23]. Here we give a brief derivation of the TIE formulation. A complex amplitude satisfies approximately the parabolic equation [22]:

(i \frac{\partial}{\partial z} + \frac{\nabla^{2}}{2 k} + k) u_{0} (x, y) = 0

Where $k = 2 π / λ$ , $\nabla = \vec{x} \frac{\partial}{\partial x} + \vec{y} \frac{\partial}{\partial y}$ , $\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}$ and $u_{0} (x, y)$ is such that,

I_{0} (x, y) = {| u_{0} (x, y) |}^{2}

$u_{0} (x, y)$ has been defined in Eq. (1) in above paragraphs. So that the complex amplitude may be expressed in terms of the intensity and the phase,

u_{0} (x, y) = {[I_{0} (x, y)]}^{1 / 2} \exp [i φ (x, y)] = R_{1} (x, y) \cdot \exp [i 2 π f (x, y)]

Let Eq. (7) be multiplied by $u_{0}^{*} (x, y)$ on the left-hand side and the complex conjugate of Eq. (7) be multiplied by $u_{0} (x, y)$ on the left-hand. To subtract the two resulting equations one gets:

k \frac{\partial I_{0} (x, y)}{\partial z} = - \nabla \cdot [I_{0} (x, y) \nabla φ (x, y)]

Now, we need further introduce an auxiliary function $ψ (x, y)$ , which defined as:

\nabla ψ (x, y) = I_{0} (x, y) \nabla φ (x, y)

Then Eq. (10) can be rewritten by:

\nabla^{2} ψ (x, y) = - k \frac{\partial I_{0} (x, y)}{\partial z}

According to the Fourier derivative theorem:

FT [\partial_{x}^{(n)} w (x, y)] = i^{n} q_{x}^{n} FT [w (x, y)]

Here $q_{x}$ is the variable conjugate to $x$ in the Fourier domain. So we get:

ψ (x, y) = IFT [{(q_{x}^{2} + q_{y}^{2})}^{- 1} FT (k \frac{\partial I_{0} (x, y)}{\partial z})]

The longitude intensity derivative $\partial I_{0} (x, y) / \partial z$ in Eq. (14) along the Fresnel diffraction direction could not be directly measured. Conventionally, it has to make an approximation by a finite difference between two adjacent intensities. In the encryption process, as shown in Fig. 2, we set a very short distance between the image plane and the CCD plane, which ensures that the derivative $\partial I_{0} (x, y) / \partial z$ could be approximated by its finite difference. In decryption process, the derivative is approximated as follows:

\frac{\partial I_{0} (x, y)}{\partial z} \approx \frac{I^{'} (x, y) - {| R_{1} (x, y) |}^{2}}{d z_{1}}

$I^{'} (x, y)$ is obtained by using GAPR algorithm, $R_{1} (x, y)$ is the Key 1 and $d z_{1}$ is the diffraction distance that we set in the encryption process. Substitute Eq. (14) into Eq. (11), we get:

φ (x, y) = - IFT {{(q_{x}^{2} + q_{y}^{2})}^{- 1} FT {\nabla \cdot [\nabla ψ (x, y) / I_{0} (x, y)]}}

We noticed that the Eq. (16) has to compute $\nabla ψ (x, y) / I_{0} (x, y)$ , where $I_{0} (x, y) = {| R_{1} (x, y) |}^{2}$ , so we set $R_{1} (x, y) \in [0.3, 1]$ in the encryption process to avoid the division by zero error. So we obtain the plaintext directly by:

f (x, y) = φ (x, y) / 2 π

As depicted in the decryption process, the plaintext could be decrypted from the ciphertext by only using the two secret keys. The truncated phases in encryption process are needless in our cryptosystem. It is an essential difference and advantage of our method compared with the traditional PTFT-based cryptosystems.

3. Simulation results and discussion

The computer simulations are performed under the environment of MATLAB R2013b. To illustrate the performance of our algorithm and the reasonability of the diffraction distances, we calculate the correlation coefficient (CC) values vary with the diffraction distances. As shown in Fig. 5, the horizontal axis is the diffraction distance with the unit of micron ( $μm$ ); the vertical axis is the CC values between the recovered results and the plaintext. The algorithm performance affected by diffraction distances of the TIE is shown in Fig. 5(a), which shows CC values varying with the diffraction distance $d z_{1}$ from 1 $μm$ to 300 $μm$ . Besides, if we fix the diffraction distance $d z_{1} = 10 μm$ , after encryption and decryption, the CC values between the decrypted results and the plaintext is shown in the Fig. 5(b). It is clear that there is a down trending for CC values with the increasing of the diffraction distance. The simulation results provide guidance for parameters setting. We would like to point out that in this simulation the secret key $R_{1} (x, y) \in [0.3, 0.8]$ , in which the minimum has been explained in the paragraph following the Eq. (16), and the maximum is set to constraint the intensity distribution within a reasonable scope after Fresnel diffraction.

Fig. 5 (a) The CC values between the original plaintext and the recovered results using the TIE method, it declines while the diffraction distance $d z_{1}$ increases, (b) fix the diffraction distance $d z_{1} = 10 μm$ , the CC values between the original plaintext and the decrypted results using the MAPR and TIE methods, it also declines while the diffraction distance $d z_{2}$ increases.

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In encryption process, we set parameters as $λ = 632.8 nm$ , $d z_{1} = 0.01 mm$ and $d z_{2} = 0.02 mm$ . The proposed method requires that the intermediate result and the ciphertext to be quantized and recorded to high-bit format (16-bit in our method). Otherwise the recovered plaintext will be unrecognizable. In all the simulation experiments, we assume that the pixel sizes of SLMs and the CCD in the proposed optical setup are both 8 $μm$ , and the CCD works in the 16-bit imaging mode. Two images are taken as plaintexts which are shown in Fig. 6(a) (Lena, 512 × 512 pixels) and Fig. 6(d) (Treasure map, 512 × 512 pixels). Based on the PTFD strategy mentioned above, the corresponding ciphertexts obtained according to the Eqs. (1)-(5) are shown in Fig. 6(b) and Fig. 6(e) respectively, which look like noise patterns. With the help of the two keys (RAM and RPM), the ciphertexts are decrypted by using the GAPR and TIE methods, which are shown in the Fig. 6 (c) and Fig. 6(f). The CC values between the decrypted images and the original plaintexts are 0.9501 and 0.9936, respectively.

Fig. 6 The simulation results. (a) Original plaintext ‘Lena’, (b) ciphertext corresponding to (a), (c) decrypted result using the GAPR and TIE methods. The CC value between the decrypted image (c) and the plaintext (a) is 0.9501. (d) Original plaintext ‘Treasure map’, (e) ciphertext corresponding to (d), (f) decrypted result using the GAPR and TIE methods. The CC value between the decrypted image (f) and the plaintext (d) is 0.9936.

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And the CC is defined as:

C C = \frac{\sum_{m} \sum_{n} (A_{m n} - \bar{A}) (B_{m n} - \bar{B})}{\sqrt{(\sum_{m} \sum_{n} {(A_{m n} - \bar{A})}^{2}) (\sum_{m} \sum_{n} {(B_{m n} - \bar{B})}^{2})}}

Where A and B denote two different two-dimensional distributions, m and n are the indexes of the rows and columns, respectively.

Here we would like to emphasize that thanks to introducing of the nonlinear phase-truncated operator, the security level of the proposed cryptosystem can thus get improved compared with other linear cryptosystems. For example, a plenty of optical cryptosystems based on the DRPE with a linear nature have already been proved to be vulnerable to the CCA and CPA [4, 7]. However, the proposed cryptosystem could be immune to these attacks. Meanwhile, referring to the more powerful KPA, the attackers usually adopt the phase retrieval technique to estimate the secret keys based on some prior knowledge, say a plaintext-ciphertext pair [5, 6]. But for our encryption scheme, the attackers use a plaintext-ciphertext pair cannot deduce the keys because they cannot aware of the RPM which is an essential constraint for a traditional phase retrieval algorithm [25]. As shown in Fig. 7, KPKA based on phase retrieval algorithm [16, 17] is also not feasible for our proposed method due to be lack of two secret keys.

Fig. 7 The flow chart of the phase retrieval process. GAPR: generalized amplitude-phase retrieval algorithm. The red parts are not available to the attackers.

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To further demonstrate the security of our cryptosystem, suppose an attacker who has intercepted the ciphertext attempts to retrieve the plaintext without the knowledge of the Keys, he might try to decode the ciphertext with arbitrarily selected random keys. Figure 8(a) shows the decrypted result with a fake RPM and a fake RAM. Moreover, our simulation shows that, even if one of the two secret keys is divulged, it is still impossible to deduce the plaintext. Figure 8(b) shows the decrypted result with the true RPM (Key 2) and a fake RAM; Fig. 8(c) shows the decrypted result with a fake RPM and the true RAM (Key 1). Only possessed the true pair of keys, the correct decrypted result could be obtained. The CC values between the plaintext and the decrypted results with fake keys shown in Fig. 8(a)-8(c) are 0.0716, 0.0699 and 0.0718, respectively.

Fig. 8 The decrypted results using fake keys. (a) Decrypted result using a fake RPM and a fake RAM; (b) decrypted result using the true RPM and a fake RAM; (c) decrypted result using a fake RPM and the true RAM.

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Moreover, different from the asymmetric cryptography based on PTFT which works with a one-time pad manner (each encryption process will generate a new set of keys used for decryption) [14, 18–21], our cryptosystem based on PTFD use only two fixed keys both in the encryption and decryption processes, this is more convenient in practice. Meanwhile, the PTFT-based methods have to record the truncated phase as decryption keys [14, 19]; by contrast, our decryption process based on the GAPR and TIE methods have no need of the truncated phase and the generated ciphertext is an intensity which could be recorded conveniently by a CCD.

4. Conclusion

In summary, we have proposed a novel optical cryptosystem based on the PTFD and TIE in which the decryption process is different from the encryption. A RAM and a RPM are employed as two secret keys to encrypt the input image into a real-valued noise-like intensity distribution. With the help of two secret keys, the ciphertext could be decrypt with the GAPR and TIE methods. Other than the published PTFT-based encryption methods which have to record the truncated phase as the decryption keys (different from the encryption keys), our method only requires the presence of the two encryption keys in the decryption process, and the decrypt plaintext is nearly lossless. Moreover, the generated ciphertext is an intensity which is convenient to record and transmit. Owing to the nonlinear operation of phase truncation as well as the decryption strategy, a higher security level could be achieved. A set of computer simulations are carried out to show the feasibility and security of the proposed method.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61171073 and 61307003), the Sino-German Center for Research Promotion (GZ 760) and China Postdoctoral Science Foundation (2013M540662, 2014T70823).

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Optical cryptosystem based on phase-truncated Fresnel diffraction and transport of intensity equation

Abstract

1. Introduction

2. Theoretical description

3. Simulation results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (18)

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