Novel asymmetric cryptosystem based on distorted wavefront beam illumination and double-random phase encoding

Honghao Yu; Jun Chang; Xin Liu; Chuhan Wu; Yifan He; Yongjian Zhang

doi:10.1364/OE.25.008860

1. Introduction

With the rapid development of global internet and social network activity, information security technology is becoming increasingly important. Optical security—a type of information-security technology that utilizes optical theory and methods, has attracted the interest of researchers around world in recent decades [1, 2]. Compared with other information-security technologies based on electronics, optical technologies have unique characteristics. They exhibit multidimensional properties, are highly parallelizable, have a high capacity, and exhibit multiple degrees of freedom [1, 2].

The field of optical encryption was pioneered by Refregier and Javidi, who used double-random phase encoding (DRPE) in the standard 4f optical information processor in 1995 [1]. To expand the key space and add degrees of freedom for improving the information security, many scholars, including the pioneering team, have performed more research on the conventional DRPE optical cryptosystem in the past two decades. Among these studies, the representative results are the extension of the transform domain [3–6] and the improvement of conventional optical encryption schemes with new technology [7–13]. The aim of these methods is to reinforce the decoding resistance and simplify the implementation of the operation.

In practice, the conventional DRPE encryption scheme has weakness against specific types of attacks, such as chosen-plaintext attacks and known-plaintext attacks [14–16]. To improve the deficiencies of conventional DRPE systems, asymmetric cryptosystems have been developed [17–19]. Although they are easy to implement algorithmically on a computer, they are difficult to implement in optical experiments. Furthermore, they are generally vulnerable to the collision algorithm [20, 21]. This is because these asymmetric cryptosystems are derived from mathematical formulas and most of them lack a reasonable explanation from an optical viewpoint. Therefore, an asymmetric cryptosystem based on optical principles that is practical to implement and exhibits a high resistance against specific attacks is desirable.

Inspired by the aforementioned research, we propose a novel asymmetric cryptosystem based on distorted wavefront beam illumination and DRPE. The encryption and decryption can be performed entirely optically. The distorted spectrum of the encrypted image can be corrected according to the known wavefront-aberration keys and the double-random phase keys to acquire the spectrum of the decrypted image. Then, the decrypted image can be obtained by the Fourier transform. This approach has several benefits. First, a new type of vector key—the wavefront-aberration key, is added to the DRPE scheme. The magnitude of wavefront aberration is replaced by fringe Zernike coefficients, and each order of the Zernike coefficients and each direction coefficient is a key. The entire Zernike polynomial is a set of keys, which tremendously increases the key space. Second, asymmetric public-key cryptography can be implemented by wavefront aberration using wavefront sensing. The distorted wavefront and the ideal wavefront are used as public and private keys, respectively, and the direction-informations of wavefront aberration are included in them. For decrypting wavefront-aberration keys, both of these wavefronts must be known, and the wavefront aberrations must be solved. Third, this novel cryptosystem combines a symmetric key with an asymmetric key to enhance the resistance and exhibits a higher robustness against known-plaintext attacks and chosen-plaintext attacks than symmetric cryptosystems, as well as resistance to specific attacks in principle and simulation.

This paper is organized as follows. In Section 2, we discuss the aberration diffraction theory and use this theory to analyze the novel asymmetric cryptosystem based on distorted wavefront beam illumination and DRPE. We also evaluate the security of this encryption against known-plaintext attacks, chosen-plaintext attacks, and specific attacks in principle. The simulation results and analysis are presented in Section 3. Finally, our conclusions are presented in Section 4.

2. Theoretical analysis

2.1 Diffraction theory of optical aberrations in the presence of wavefront aberrations

When an optical system contains wavefront aberrations, the representation of a wavefront aberration is shown in Fig. 1, where $P_{0}$ denotes the object point; $P_{1}^{*}$ and $P_{1}$ denotes the Gaussian ideal image point and real image point, respectively; and $\bar{Q}$ and $Q$ are the points of intersection between the real ray with the Gaussian spherical wave and the real wavefront surface along a certain direction [22, 23]. We obtain the optical path lengths as follows.

Fig. 1 Wavefront aberration and the ray aberration.

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When $i = K$ ,

Φ_{K} = [\bar{Q} Q] = [P_{0} Q] - [P_{0} \bar{Q}],

\sum_{ξ} \sum_{η} W (ξ, η) = \sum_{i} Φ_{i},

where [ ] represents the optical path length,

Φ_{K}

denotes the optical path difference (OPD) between the ideal wavefront and the distorted wavefront in a certain direction K, and

W (ξ, η)

is the distribution of the OPD in the pupil.

Assuming that the ideal wavefront phase and the distorted wavefront phase in the pupil are $ϕ_{0} (ξ, η)$ and $ϕ (ξ, η)$ , respectively, the phase difference $k W (ξ, η)$ is given as follows:

k W (ξ, η) = Δ ϕ = ϕ (ξ, η) - ϕ_{0} (ξ, η) .

According to the diffraction theory for optical aberrations in the presence of small aberrations, we define a generalized pupil function $P (ξ, η)$ according to the wavefront aberration, which is as follows:

P (ξ, η) = {\begin{cases} \exp [j k W (ξ, η)] \sqrt{ξ^{2} + η^{2}} \leq \frac{D_{p u p i l}}{2} \\ 0 \sqrt{ξ^{2} + η^{2}} > \frac{D_{p u p i l}}{2} \end{cases} .

where

k = 2 π / λ

is the wave number, i.e., the scalar quantity of the wave vector k;

λ

is the input wavelength; and

D_{p u p i l}

is the diameter of pupil. The two blank parentheses are the physical black-box model of an optical system with wavefront aberration.

2.2 Double-random phase encoding with distorted wavefront beam illumination

The conventional DRPE scheme was first proposed by Refregier and Javidi and has since been widely studied. The basic principle can be described as follows. The incident beam is a plane wave, and two statistically independent random phase-only masks $n (x, y)$ and $B (μ, ν)$ are placed in the input image plane and the frequency spectrum plane domain. The input image can be encrypted as stationary white noise by double-random phase. The complex-valued wavefront of ciphertext can be described by Eq. (5) [1, 2]:

o (x', y') = I F T {F T [f (x, y) \cdot n (x, y)] \cdot B (μ, ν)},

where

f (x, y)

denotes the input signal;

o (x', y')

denotes the complex-valued wavefront obtained using DRPE in the detector plane. FT and IFT denote Fourier transform and inverse Fourier transform, respectively.

When the incident beam is a distorted complex wave, the effect of the wavefront aberration should be considered when the DRPE scheme is implemented:

o' (x', y') = o (x', y') \otimes h (x', y'),

where

\begin{array}{l} h (x', y') = c \iint P (ξ, η) \exp [j \frac{k}{d} (x' ξ + y' η)] d ξ d η . \\ = c' F T {P (λ d f_{ξ}, λ d f_{η})} \end{array}

Here, $f_{ξ} = \frac{ξ}{λ d}$ , $f_{η} = \frac{η}{λ d}$ , $d$ denotes the diffraction distance from the pupil to the image plane and $o' (x', y')$ is the output of the DRPE system with distorted wavefront beam illumination.

And:

H_{1} (f_{ξ}, f_{η}) = F T [h (x', y')] = c' P (- λ d f_{ξ}, - λ d f_{η}) = \exp [j k W (- λ d f_{ξ}, - λ d f_{η})],

where

H_{1} (0, 0) = 1, P (0, 0) = 1 \Rightarrow c' = 1

.

The complex-valued wavefront of the ciphertext in the proposed system is described by Eq. (9):

o' (x', y') = I F T {F T [f (x, y) \cdot n (x, y)] \cdot B (μ, ν) \cdot \exp [j k W (- λ d μ, - λ d ν)]},

where

(μ, ν)

is the frequency spectrum plane.

At this point, the encryption effects are the superposition of random phase masks and wavefront aberrations. The wavefront aberration in this cryptosystem can be regarded as a special public key, which is asymmetric. It is related to the ideal wavefront and the distorted wavefront in the pupil. The wavefront aberrations can be reconstructed by the Zernike modes. The two point arrays for wavefront-aberration reconstruction are obtained via wavefront sensing using a Shack–Hartmann wavefront sensor [24].

According to the relation between the fringe Zernike coefficients and the Seidel aberration, the wavefront aberration magnitude of an optical system is determined with fringe Zernike coefficients. According to the relative position difference between the ideal wavefront point array and the distorted wavefront point array, the wavefront aberration direction of an optical system is determined by slope matrix G.

The distorted wavefront as a public key to encryption information and the ideal wavefront as a private key are reserved by the CCD1. The wavefront-aberration keys can only be obtained when the distorted wavefront and the ideal wavefront are known. Even if an opponent knows the distorted wavefront, it is difficult for him to infer the ideal wavefront, let alone recover the wavefront-aberration key $\exp [j k W (ξ, η)]$ . In this novel asymmetrical optical cryptosystem, apart from double-random phase keys in spatial and frequency domains, there is another asymmetric wavefront-aberration key. Both the distorted wavefront and the ideal wavefront are required for determining the wavefront aberrations for correcting the distortion spectrum. The entire setup of the cryptosystem is shown in Fig. 2. It is necessary to point out that the f-number and optical path length of the two wavefront sensing optical paths should be same. In addition, the CCD1 needs to be meshed and the wavefront sensor should be calibrated before wavefront sensing.

Fig. 2 Schematic cryptosystem based on distorted wavefront beam illumination and DRPE.

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2.3 Analysis of the security

Regarding known-plaintext attacks [14], suppose that attackers know a pair of plaintext-ciphertext ${f (x, y), ψ (μ, ν)}$ , where:

ψ (μ, ν) = F T {f (x, y) \cdot \exp [j ϕ (x, y)]} \cdot \exp [j φ (μ, ν)] \cdot \exp [j k W (- λ d μ, - λ d ν)] .

The random phase function $\exp [j ϕ (x, y)]$ in the spatial domain can be calculated using the phase retrieval algorithm. However, the random phase function $\exp [j φ (μ, ν)]$ in the frequency domain is unavailable under the premise of not knowing the wavefront-aberration keys, because the wavefront-aberration keys are directional. As shown in Eq. (11):

\exp [j φ (μ, ν)] = \frac{ψ (μ, ν)}{F T {f (x, y) \cdot \exp [j ϕ (x, y)]} \cdot \exp [j k W (- λ d μ, - λ d ν)]} .

At this time, the wavefront-aberration key is an asymmetric key. For attackers to obtain it, they must know both the public key and the private key. Because the private key is retained by the recipient, it is difficult for the attacker to obtain.

Similarly, the analysis for chosen-plaintext attacks is the same as that for known-plaintext attacks. The plaintext is an impulse function $δ (x - i, y - i)$ [15], and the encryption equation is as follows.

When $i = 0, j = 0$ ,

\begin{array}{l} ψ (μ, ν) = F T [f (x, y) \cdot n (x, y)] \cdot B (μ, ν) \cdot \exp [j k W (- λ d μ, - λ d ν)] \\ = F T [δ (x, y) \cdot n (x, y)] \cdot B (μ, ν) \cdot \exp [j k W (- λ d μ, - λ d ν)] \\ = n (0, 0) \cdot B (μ, ν) \cdot \exp [j k W (- λ d μ, - λ d ν)], \end{array}

When $i \neq 0, j \neq 0$ ,

ψ' (μ, ν) = n (i, j) \cdot B (μ, ν) \cdot \exp [j k W (- λ d μ, - λ d ν)] .

Attackers can recover the random phase function

n' (x, y)

using the following relationship:

n' (x, y) = \frac{ψ' (μ, ν)}{ψ (μ, ν)},

n' (x, y) = \frac{n (x, y)}{n (0, 0)},

Then, another plaintext $f_{2} (x, y)$ is encrypted, using the encryption equation following:

ψ_{2} (μ, ν) = F T [f_{2} (x, y) \cdot n (x, y)] \cdot B (μ, ν) \cdot \exp [j k W (- λ d μ, - λ d ν)] .

Inserting $n' (x, y)$ into the encryption equation,

ψ_{2} (μ, ν) = F T [f_{2} (x, y) \cdot n' (x, y)] \cdot B' (μ, ν) \cdot \exp [j k W (- λ d μ, - λ d ν)],

B' (μ, ν) = \frac{ψ_{2} (μ, ν)}{F T [f_{2} (x, y) \cdot n' (x, y)] \cdot \exp [j k W (- λ d μ, - λ d ν)]} .

It can be seen from these equations indicate that the random phase keys in the frequency domain are unavailable to attackers for chosen-plaintext attacks, unless the attackers know the wavefront-aberration keys in advance.

The foregoing analysis indicates that the asymmetric wavefront-aberration key has a protective effect on the symmetric double random key.

For the asymmetric cryptosystem, strictly speaking, the public and private keys should be independent of the input plaintext and symmetric keys. In this study, the public and private keys are obtained via the wavefront sensing of the distorted wavefront and the ideal wavefront, respectively; thus, our optical encryption scheme satisfies the requirements of asymmetric encryption.

Most existing specific attacks against asymmetric cryptosystem are based on the amplitude–phase retrieval algorithm, for example, the two-step iterative amplitude–phase-retrieval algorithm [20]. As mentioned in the above point, only the magnitude of wavefront aberration could be obtained by amplitude–phase retrieval algorithm, but it is infeasible to obtain the direction of wavefront aberrations without knowing the relative position difference between distorted wavefront and ideal wavefront by these methods. In summary, the wavefront-aberration key can be obtained neither by input–output phase-retrieval algorithm attack nor by public key and chosen-plaintext attacks. Because of the protective effect of the asymmetric wavefront-aberration keys, the symmetric double random keys cannot be obtained accurately.

In principle, the novel asymmetric cryptosystem shows high robustness against known-plaintext attacks, chosen-plaintext attacks, and specific attacks based on the amplitude–phase retrieval input–output algorithm.

3. Simulation results and analysis

For the system discussed in Section 2, computer simulations are performed to verify the proposed asymmetric cryptosystem based on wavefront beam distortion and DRPE. The distorted wavefront and the ideal wavefront, i.e., the public and private keys, respectively, can be obtained and constructed via wavefront sensing. The measurement of the optical path is shown in Figs. 3(a) and 3(b).

Fig. 3 (a) Ideal wavefront for measuring the optical path; (b) Distorted wavefront for measuring the optical path.

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By exploiting the measurement of the optical path and Shack–Hartmann wavefront sensors, we can obtain two point array maps of the distorted wavefront and ideal wavefront formed by the micro-lens array. The point array maps and wavefront maps under these two circumstances are presented in Fig. 4. We chose λ = 537.8 nm and f = 100 mm as a reference.

Fig. 4 When λ = 537.8nm, f = 100mm: (a) Ideal wavefront point array; (b) Distorted wavefront point array; (c) Ideal wavefront map; (d) Distorted wavefront map. See Data File 1.

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The wavefront slope matrix G can be acquired according to the ideal wavefront point array and the distorted wavefront point array. By differentiating M-order Zernike polynomials in the X and Y directions and then computing the average of the sub aperture area, we can obtain a transform matrix D. The wavefront-aberration keys can be fitted by Zernike modes using the wavefront slope matrix G and the transform matrix D. According to the relation between the fringe Zernike coefficients and the Seidel aberration, the wavefront aberration of optical systems is determined with the fringe Zernike coefficient [25].

We generate a 256 × 256 matrix of the OPD $W (- λ d f_{ξ}, - λ d f_{η})$ in MATLAB, and the unit of $W (- λ d f_{ξ}, - λ d f_{η})$ is λ. The wavefront-aberration key is $\exp [j k W (- λ d f_{ξ}, - λ d f_{η})]$ , the magnitude of wavefront aberration decryption key is the complex conjugate of the wavefront-aberration key function and the direction of wavefront-aberration decryption key are determined by wavefront slope matrix G. We chose f = 100 mm and λ = 537.8 nm as a reference, and Fig. 5 shows these two generated surfaces.

Fig. 5 For λ = 537.8nm and f = 100mm: (a) Wavefront aberration map; (b) Wavefront aberration decryption map; (c) Wavefront aberration surface shadow in XY plane; (d) Wavefront aberration decryption surface shadow in XY plane.

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We employed Townlet (256 × 256 pixels, the copyright of the photograph belongs to the corresponding author of this paper) as the input image and performed computer simulations using the ZEMAX and the MATLAB R2014a platform to demonstrate the correctness and security of the cryptosystem, as shown in Figs. 6, 7, and 8.

Fig. 6 (a) Input plaintext (Townlet, 256 × 256 pixels); Only wavefront-aberration keys are used; (b) f = 100mm, λ = 441.6nm, the ciphertext; (c) f = 100mm, λ = 537.8nm, the ciphertext; (d) f = 100mm, λ = 632.8nm, the ciphertext; (e) f = 200mm, λ = 537.8nm, the ciphertext; (f) f = 300mm, λ = 537.8nm, the ciphertext.

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Fig. 7 (a) Input plaintext (Townlet, 256 × 256 pixels); (b) Encryption with two encryption keys; (c) Two decryption keys are both wrong; (d) False double-random phase decryption keys and true wavefront aberration decryption keys; (e)True double-random phase decryption keys and false wavefront aberration decryption keys; (f) Two decryption keys are both true.

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Fig. 8 (a) SSEs of phase-key retrieval versus the number of iterations for the WA-DRPE and DRPE; (b) Retrieved result of the proposed scheme; (c) Retrieved result of the DRPE scheme.

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Figures 6(a)–6(f) show the encryption effects at different wavelengths and different focal lengths when only the wavefront-aberration keys are imposed, and Fig. 7 shows the encryption effects of two hybrid keys and the decryption results under different circumstances.

The normalized mean squared error (NMSE) values between the original image and the different images shown in Figs. 6(a)-6(f) and Figs. 7(a)-7(f) are calculated using the following equation:

N M S E = \frac{\sum_{i = 1}^{M} \sum_{j = 1}^{N} {[f (i, j) - f^{'} (i, j)]}^{2}}{\sum_{i = 1}^{M} \sum_{j = 1}^{N} {[f (i, j)]}^{2}},

where M and N represent the number of pixels in the image length and width, respectively;

f (i, j)

is a sampled version of the original image; and

f^{'} (i, j)

is a sampled version of a processed image. The NMSE values corresponding to Figs. 6(b)–6(f) are 0.0915, 0.0730, 0.0658, 0.0893 and 0.0933, respectively. The NMSE values corresponding to Figs. 7(b)–7(f) are 0.5151, 0.5177, 0.5171, 0.2474 and 2.7335e-31, respectively.

The normalized signal-noise ratio (NSNR) values between the input plaintext and the different images shown in Figs. 6(a)–6(f) and Figs. 7(a)–7(f) are calculated using the following equation:

N S N R = - 10 \log_{10} {\frac{\sum_{i = 1}^{M} \sum_{j = 1}^{N} {[f (i, j) - f^{'} (i, j)]}^{2}}{\sum_{i = 1}^{M} \sum_{j = 1}^{N} {[f (i, j)]}^{2}}} = - 10 \log_{10} (N M S E) .

The NSNR values corresponding to Figs. 6(b)–6(f) are 10.3840, 11.3697, 11.8149, 10.4896 and 10.3031, respectively. The NSNR values corresponding to Figs. 7(b)–7(f) are 2.8814, 2.8589, 2.8643, 6.0657 and 305.6328, respectively. These results demonstrate that the cryptosystem has very high validity and robustness against different attacks.

As shown in Fig. 6 and 7, the aberration key alone is insufficient for the image to be fully encrypted, but when the aberration key and the double random key are combined, as illustrated in Section 2.3, the wavefront-aberration keys have a protective effect on the double-random keys, and this special system can resist the input–output phase retrieval algorithm for double-random keys. When the phase retrieval algorithm is used for DRPE and the wavefront-aberration keys are unknown, the correct double-random phase keys cannot be obtained. The situation shown in Fig. 7(e) is almost impossible, unless attackers obtain the double-random keys directly. Thus, for this system, to recover the double-random phase keys using the phase retrieval algorithm, the wavefront-aberration keys must be decrypted.

Finally, we employed the input–output algorithm (HIO) proposed by Fienup [26] to test the effectiveness of the proposed scheme for generating the keys. We used the sum square error (SSE) as the evaluation index:

S S E = 10 \log_{10} {\frac{\sum {[f (i, j) - f^{N} (i, j)]}^{2}}{\sum f {(i, j)}^{2}}},

where

f (i, j)

denotes the known amplitude distribution on the object plane, and

f^{N} (i, j)

denotes the computed amplitude distribution after N iterations.

Figure 8 shows the effectiveness of the system when the phase retrieval algorithm is implemented.

Figure 8(a) shows the SSEs of phase-key retrieval with respect to the number of iterations for the wavefront-aberration DRPE (WA-DRPE) and DRPE. Figures 8(b) and 8(c) show the results retrieved using the proposed scheme and the DRPE scheme, respectively. The results of the proposed scheme show higher robustness against phase retrieval algorithms than the DRPE scheme, demonstrating the validity of our security-enhancing method.

In addition, we have made related tolerance analysis of the sensitivity for the optical system with the wavefont aberration. We use RMS wavefront to evaluate the tolerance, the worst RMS wavefront induced by Default Tolerance (Maximum Tolerance) is only 0.28 waves, while the Nominal Criterion is 2.26637962. Thus, the proposed system is insensitive to vibrations from the results and it is feasible to implement optically.

4. Conclusion

We proposed a novel asymmetric cryptosystem based on distorted wavefront beam illumination and DPRE. Double random phase masks and incident wavefront distortion are employed as encryption keys to form a hybrid cryptosystem containing both symmetric and asymmetric keys. The wavefront-aberration keys are vector keys and they have a protective effect on the double-random keys, therefor this novel cryptosystem can resist the input–output phase retrieval algorithm for double-random keys. From the perspective of optical encryption, this cryptosystem is a bold attempt to improve the information processing and recovery of the general diffraction-limited system. If wavefront aberrations are considered, a more general lens imaging system may replace 4f Fourier lens systems in optical information processing.

Funding

National Natural Science Foundation of China (NSFC) (Grant No. 61471039).

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Novel asymmetric cryptosystem based on distorted wavefront beam illumination and double-random phase encoding

Abstract

1. Introduction

2. Theoretical analysis

2.1 Diffraction theory of optical aberrations in the presence of wavefront aberrations

2.2 Double-random phase encoding with distorted wavefront beam illumination

2.3 Analysis of the security

3. Simulation results and analysis

4. Conclusion

Funding

References and links

Supplementary Material (1)

Cited By

Figures (8)

Equations (21)

Optics Express