Keywords

1 Introduction

With substantial innovations in transmission and information processing technologies, it has become a challenge to convey the information with security. An optical 4f system of DRPE in Fourier domain proposed by Refregier and Javidi was a great accomplishment in this arena in 1995 [1]. This set off the path for many researchers for constructing other cryptosystem centered on DRPE with superior security and elevated noise diminution. These systems were still exposed to many attacks known as known plaintext attack (KPA) [2, 3], chosen plaintext or ciphertext attack (CPA, CCA), etc. [4] An idea has been taken in the direction to model an asymmetric system [5] that deliver solution to insecure transmission concern and make it resilient from numerous attacks by boosting the randomness and improving chaotic mask key parameters.

2 Key Generation: Chaotic Zone Plate Phase Mask

Chaotic random phase masks (CRPM) [6] are used to increase randomness and to add chaotic parameters which makes the system highly secure in terms  of private keys. Logistic map is one dimension (1D) non-linear chaotic maps and is used to generate the randomness in the CRPM. It can be expressed [7, 8].

$$x_{{n + 1}} = px_{n} \left( {1 - x_{n} } \right)$$
(1)

It is iterated \(n\) times. Where \(p\) is bifurcation parameter and lies between \(~0 < p < 4\), \(x_{0}\) is an initial value and \(x_{n} \in \left[ {0,1} \right]\) is an iterative value where \(n\) varies from 0 to M × N. \(M\) and \(N\) are size of a chaotic random mask in pixels. The 1D sequence \(X = x_{1} ,x_{2} ,~x_{3} ,x_{4} , \ldots ,~x_{{M \times N}}\) that is produced by Eq. (1) is rearranged into 2D matrix as \(Y = y_{{ij}}\), where \(i\) varies from \(\left\{ {1,2, \ldots ,M} \right\}\) and \(j\) varies from \(\left\{ {1,2, \ldots ,N} \right\}\) and \(y_{{ij}} \in \left( {0,1} \right)\). CRPM a 2D matrix is expressed as

$${\text{CRPM}}\left( {x,y} \right) = \exp \left( {i2\pi y_{{ij}} \left( {x,y} \right)} \right)$$
(2)

The Fresnel lens is built on quadratic phase change and the efficacy of zone plates. It is given [3] by

$$~L_{{\lambda ,f}} \left( r \right) = \exp ( - i\pi r^{2} /\lambda f)$$
(3)

where \(r\) is the lens radius, \(f\) is the focal length, and λ is the wavelength of incident light. Now, \({\text{CZPPM}}\left( {x,y} \right)\) is obtained by multiplying above two functions \(L_{{\lambda ,f}} \left( r \right)\) and \({\text{CRPM}}\left( {x,y} \right)\), i.e., Equation (1) and (2) and can be stated as

$$C\left( {x,y} \right) = \exp \left\{ {i\pi \left[ {2y_{{ij}} \left( {x,y} \right) + \frac{{r^{2} }}{{\lambda f}}} \right]} \right\}$$
(4)

3 Proposed Cryptosystem

Asymmetric encryption technique built on CZPPM and \({\text{AT}}^{\omega }\) [9, 10] is proposed in this paper. The \({\text{AT}}^{\omega }\) scrambles the image \(I\left( {x,y} \right)\) of size \(M~ \times ~M\) by shuffling the pixels arbitrarily as in Eq. (5). Then the obtained result is transformed in gyrator domain [10], and phase is truncated to produce \(G\left( {u,v} \right)\) as in Eq. (6). The encrypted image is generated by multiplying \(G\left( {u,v} \right)\) with arbitrary phase mask in gyrator domain and further truncating its phase as in Eq. (7). The CZPPM and \({\text{AT}}^{\omega }\) applied through encryption operates as supplementary key because it comprises number of parameters as well as expands the randomness which further enhances the security.

$$~I\left( {x^{\prime } ,~y^{\prime } } \right) = {\text{AT}}^{\omega } \left( {I\left( {x,y} \right)} \right) \times C_{1} \left( {x,y} \right)~$$
(5)
$$G\left( {u,v} \right) = {\text{PT}}\left[ {{\text{GT}}^{\alpha } \left( {I\left( {x^{\prime } ,~y^{\prime } } \right)} \right)} \right]$$
(6)
$$E\left( {x,y} \right) = {\text{PT}}\left[ {{\text{GT}}^{{ - \beta }} \left[ {G\left( {u,v} \right) \times C_{2} \left( {x,y} \right)} \right]} \right]~$$
(7)

Two decryption keys \({\text{DK}}_{1}\) and \({\text{DK}}_{2}\) were generated in the process where amplitude truncation is performed as shown in Eqs. (8) and (9)

$${\text{DK}}_{1} = {\text{AT}}\left[ {{\text{GT}}^{\alpha } \left( {I\left( {x^{\prime } ,~y^{\prime } } \right)} \right)} \right]~$$
(8)
$${\text{DK}}_{2} = {\text{AT}}\left[ {{\text{GT}}^{{ - \beta }} \left[ {G\left( {u,v} \right) \times C_{2} \left( {x,y} \right)} \right]} \right]$$
(9)

The current graph of the planned encryption stated is displayed in Fig. 1.

Fig. 1
figure 1

Flow graph for proposed encryption scheme

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The decipherment procedure is the reversal process of the encryption which retrieves the original image \(I\left( {x,y} \right)\). Following are the equations to decipher the encoded image \(E\left( {x,y} \right)\).

$$G\left( {u,v} \right) = \left[ {{\text{IGT}}^{{ - \beta }} \left[ {E\left( {x,y} \right) \times {\text{DK}}_{2} ~} \right] \times C_{2}^{*} \left( {x,y} \right)} \right]~$$
(10)
$$~I\left( {x^{\prime } ,~y^{\prime } } \right) = {\text{IGT}}^{\alpha } \left[ {G\left( {u,v} \right) \times {\text{DK}}_{1} } \right] \times C_{1}^{*} \left( {x,y} \right)$$
(11)
$$I\left( {x.y} \right) = ~{\text{AT}}^{\omega } \left( {~I\left( {x^{\prime } ,~y^{\prime } } \right)} \right)~~$$
(12)

4 Simulations Results

This paper demonstrates the statistical analysis and simulation results verifies the sensitivity of the indicated procedure and reveals that it accomplishes superior performance of recuperating a good superiority image. Findings also confirms the protection assessment based on correlation coefficient, noise attack, and key sensitivity. Contemplate initial image of baboon with size 256 × 256.

For simulation values use are p = 3.96, x0 = 0.35, λ = 6328 Å, f = 200 mm, ω = 5 and orders of GT are α = 0.4π and β = 0.7π. Figure 2 shows the CZPPM key generated from CRPM and Fresnel zone plate (FZP).

Fig. 2
figure 2

a Fresnel zone plate, b chaotic random phase masks, c chaotic zone plate phase mask

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4.1 Statistical Analysis

For statistical analysis, entropy [11] of the image has been analyzed, and the obtained value for baboon image is 6.319 which is close to standard value that guarantees failure of information is nearby zero. The information is homogenously scattered, and it does not deliver any valuable information to the attacker. It can be confirmed from the value obtained above that the projected system is extremely efficient.

Peak signal to noise ratio (PSNR) [12] achieved 23 dB after first level of phase retrieval and 28 dB attained after the second level of phase retrieval.

4.2 Performance Analysis

The mean square error (MSE) [13] determined for baboon image is 5.6346 × 10–22. Figure 3 presents the number of iterations versus MSE. Here, MSE  drops with the increase in number of iterations and after 50 iterations, MSE becomes constant therefore image randomization stabilizes.

Fig. 3
figure 3

MSE versus number of iteration graph

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4.3 Robustness Analysis

The proposed algorithm is verified against occlusion attack [14, 15] where the encrypted image has been occluded for different sizes, but the images are still visible from human eye till 75% which proves the system is robust. 45% and 70% occluded encrypted images and its retrieved images are presented in Fig. 4.

Fig. 4
figure 4

a 45% occluded encoded image, b 70% occluded encoded image, c recovered image after 45% of occlusion, d recovered image after 70% of occlusion

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4.4 Robustness Analysis

This system has been investigated against KPA, CPA, and special attack to verify its strength. Sets of initial and cipher images have been picked to uncover the authentic key, the erroneous value of parameter headed to incorrect generation of key. To verify the conjunction of the iteration procedure, MSE versus iterations are employed to recover the decryption keys. It can be viewed from MSE plot that decipherment key cannot be recovered even with large iteration number as shown in Fig. 5.

Fig. 5
figure 5

MSE plot versus number of iterations for decipherment key generation with proposed scheme for 1000 iterations

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5 Conclusion

The proposed asymmetric cryptosystem improves the security of the scheme as chaotic mask shifts the phase of the retrieved output and scrambling creates dispersion in the system. The introduced masking accomplishes superior execution of PSNR, MSE, occlusion, and noise as equated to existing systems. The CZPPM is further protected and cannot recapture initial image without the knowledge of all the parameters utilized for producing key. Experimental result reveals the viability and robustness of asymmetric cryptosystem.