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Quick response code and interference-based optical asymmetric cryptosystem
Journal of Information Security and Applications 45 (2019) 35–43
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Quick response code and Interference-based optical asymmetric cryptosystem Avishek Kumar, Naveen K. Nishchal∗ Department of Physics, Indian Institute of Technology Patna, Bihta 801 106, Patna, Bihar, India
a r t i c l e
i n f o
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Keywords: Image encryption Arnold cat map Quick response codes Fresnel transform Interference
a b s t r a c t In this paper, a novel image encryption scheme based on quick response code and interference in Fresnel domain is proposed. The framework is asymmetric in nature due to amplitude- and phase-truncation operation of encoded spectrum. The input image is scrambled using Arnold cat map and divided into pixel blocks. Each block is encoded into a quick response code, which are then multiplexed to generate a binary matrix. The phase-truncated binary matrix is encoded into analytically generated two phaseonly masks. During decryption the phase-only masks are recombined to yield the binary matrix, which is grouped into quick response codes and decoded into pixel blocks. The pixel blocks are recombined and descrambled to retrieve the original image. Numerical simulations have been carried out to demonstrate the effectiveness of the proposed method. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Securing data/information and secure dissemination of information are of prime concern in today’s digital era. Digital computational techniques to secure data have been flourishing through decades. In last two decades, optical image encryption techniques have also emerged and are attracting increased attention because of their high speed and parallel processing capability. An optically processed image may be represented as an amplitude image, phase-only image or a combination of both. A groundbreaking research was reported by Refregier and Javidi, who proposed a scheme to encrypt an image into a stationary white noise employing double random phase encoding (DRPE) architecture [1]. The method was further explored and expanded into other optical domains, such as, fractional Fourier, Fresnel, gyrator transform domains, etc. [2–4]. Despite widespread acclamation of DRPE, the scheme was proved to be prone to security attacks. The decryption key was easily obtained if Dirac delta function was used as the test image in the encryption process [5]. Vulnerabilities of DRPE to various other applicable attacks were also reported [6–8]. A method of scrambling and then applying DRPE to the test image also proved to be vulnerable to brute force attack [9]. To address the vulnerabilities of the basic DRPE, security enhanced versions of the DRPE scheme were reported [10,11]. The major drawback of symmetric
∗
Corresponding author. E-mail address: [email protected] (N.K. Nishchal).
https://doi.org/10.1016/j.jisa.2019.01.004 2214-2126/© 2019 Elsevier Ltd. All rights reserved.
cryptosystems is the key management. Since the same key is used during encryption and decryption process, the keys needs to be sent to the authorized receiver through a dedicated channel and are thus vulnerable to attacks. Asymmetric cryptographic schemes use different encryption and decryption keys, in which encryption key is made publicly available. Even if the attacker gets any of the pairs of plaintext, cipher text, and encryption key, it would be very difficult to derive the decryption key and hence decrypt the information. Amplitude- and phase-truncation based optical asymmetric cryptosystems are very popular among the research community and various contributions exist in literature [12]. However, a security leak has been discovered in the basic scheme. The scheme was prone to known-plaintext attack, in which the decryption keys are obtained using a pair of known-plaintext and ciphertext using the modified Gerchberg–Saxton phase retrieval algorithm [13]. The basic scheme was also proved vulnerable to a specific attack [14,15]. Chaos, a non-linear phenomenon has gained lot of ground and continues to be an active element of research owing to its highly unpredictable nature and sensitivity to the initial conditions. A slight difference between the initial states produces entirely different trajectories, which grows exponentially with time and do not converge. Singh and Sinha proposed a chaos-based DRPE scheme in which the random phase masks (RPM) are generated by chaos functions namely Logistic map, Tent map, and Kalpan–Yorke map [16]. A proposal to enhance the security of basic DRPE scheme by randomizing the input image pixels using a chaotic Baker map have also been reported [20]. It can therefore be inferred that optical technologies linked with chaos theory may be a good
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candidate to strengthen the basic optical asymmetric schemes [16–22]. Off late, interference based optical image encryption has been reported, in which the image to be encrypted is encoded into analytically generated two phase-only masks (POMs). Zhang and Wang proposed that any input image can be directly recovered as an intensity pattern recorded at the output plane by illuminating the POMs [23]. It is a simple scheme that involves non-iterative process. However, a loophole was found in this scheme. If the attacker has the information of any one of the generated POMs, then different combinations of POMs can be tried to obtain the silhouette of the original image. To overcome this issue, Zhang et al. [24] proposed a scheme in which a subpart of the POMs are randomly exchanged. The scheme had a drawback with respect to computational time. Another approach was proposed by Wang and Zhao to divide the POMs into three POMs [25]. However, it resulted into a complicated decryption setup. Also, method to encrypt binary image employing the interference based optical encryption concept has been reported [26]. Rajput and Nishchal reported a method employing asymmetric keys in the Fresnel domain [27]. It may be noted that interference based encryption scheme is quite popular and could be exploited and linked with other optical technologies to yield reliable results [28–31]. A general outcome of any optical cryptosystem using a coherent light source is speckle noise. Other types of noises also exist, which corrupt the encrypted image and hence results into poor quality of decrypted image. Barrera et al. reported an encryption scheme employing QR codes as an information container [32]. In this scheme, the input image is transformed into a QR code and is fed as an input to the DRPE scheme. Although the decrypted image of the QR code was noisy but it was successfully decoded. QR code is a two-dimensional (2D) bar code which is represented as a matrix [33]. It is an arrangement of black modules in a square grid on a white background. The data is encoded as bits and any mode of encoding ensures that the shortest possible sequence of bits is generated. While encoding the input data, the inherently embedded error correction code is also employed. This remarkable feature of a QR code makes it resistant to noise. Off late, QR codes have gained popularity and are being used in the field of optical image encryption [34–48]. An idea to encrypt QR code and a grey scale image as two separate entities in an interference based scheme using position multiplexing was reported by Qin et al. [38]. The scheme was not asymmetric and also the potential of a QR code was not properly exploited. A scheme to encode a grey scale image into multiple QR codes in Fresnel domain using DRPE has also been reported [41]. The scheme relied on QR code generators and readers and hence cannot be considered as standalone. Also, the scheme was not asymmetric in nature. The basic schemes related to amplitude- and phase- truncation were vulnerable to various types of attacks [13–15]. The nonlinearity of the asymmetric based cryptosystems may be enhanced by unifying it with the chaotic systems to make it more resistant to attacks [16–22]. In this paper, the shortcomings of both the amplitude- and phase- truncated cryptosystem and the interference-based cryptosystem are addressed by unifying them with the features of chaotic scrambling and QR code. The pixel positions of the input image are scrambled by a chaotic Arnold Cat map and is further divided into a number of blocks. Each block is encoded to the corresponding QR code and finally each QR coded matrix is multiplexed to obtain a binary matrix, which is processed by an asymmetric cryptosystem in the Fresnel domain appended by the interference based encryption method. To the best of our knowledge, this is the first report in which QR code has been unified with asymmetric scheme of image encryption and decryption.
2. Arnold cat map Arnold cat map is a 2D invertible chaotic map, which when applied to an image, randomizes the original organization of pixels. The period of the Arnold Cat map is determined by the number of iterations. The 2D Cat map is given as [49],
x x 1 = A mod (256 ) = y y q
p pq + 1
x mod (256 ) y
(1)
Here, p and q are positive integers. The Cat map is an area preserving map since det(A) = 1. (x , y ) and (x, y) denote the coordinates and transformed coordinates of pixels of an (N × N) image, respectively. The Arnold transformation of the input image may be denoted as,
T (x, y ) = AR[I (x, y )]
(2)
Here, I(x, y) is the input image and AR is the Arnold transform operator. The Arnold transformed image, T(x, y) is divided into N equal size matrices. Each matrix is encoded to a QR code. The obtained N number of QR codes are multiplexed to obtain a binary matrix denoted by f(x, y), which is an input to the amplitude- and phase-truncation based asymmetric encryption process. 3. Amplitude- and phase-truncation based Fresnel domain encryption The function f(x, y) is multiplied by a random phase mask (RPM) and its Fresnel transform (FrT) is calculated.
F (u, v ) =
exp(ikz ) iλz
× exp
{ f (x, y ) × exp[i2π r1 (x, y )]}
ik [(u − x )2 + (v − y )2 ]dxdy 2z
(3)
Here, (x, y) and (u, v) are the coordinates of the input and output planes, respectively. λis the wavelength of the incident light, k = 2λπ , and z denotes the free space propagation distance. r1 (x, y) is a random phase distributed in the interval [0,1]. The obtained spectrum F(u, v), is phase-truncated (PT) and multiplied by another RPM and Fresnel transformed which may be mathematically expressed as,
G(ξ , η ) = F rT [P T {F (u, v )} × r2 (u, v )]
(4)
The function r2 (u,v) is another RPM. The obtained spectrum given by Eq. (4) is amplitude-, and phase-truncated.
A(ξ , η ) = AT [G(ξ , η )]
(5)
H ( ξ , η ) = P T [ G ( ξ , η )]
(6)
The term (AT) denote the amplitude-truncation operation. The function, A(ξ , η) serves as a decryption key. The second decryption key is obtained as,
k=
AT [F rT { f (x, y ) × exp[i2π r1 (x, y )]}] P T [F rT { f (x, y ) × exp[i2π r1 (x, y )]}]
(7)
The function, H(ξ , η) is encoded into two POMs as described in the interference-based optical encryption method [12,15]. 4. Generating phase-only masks using the principle of interference The phase-truncated output H(ξ , η) is used to analytically generate two POMs. A new RPM, ϕ (ξ , η) is bonded with the function H(ξ , η).
H˙ (ξ , η ) = H (ξ , η ) exp[i2π φ (ξ , η )]
(8)
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The complex function, H˙ (ξ , η ) may be expressed as the interference of two POMs, P1 and P2 [23,27].
H˙ (ξ , η ) = exp[iP1 (x, y )] × h(x, y, l ) + exp[iP2 (x, y )] × h(x, y, l ) (9) where,
h(x, y, l ) =
exp( i2λπ l ) iπ 2 exp ( x + y2 ) il λ lλ
(10)
is the point pulse function of the Fresnel transform, l is the distance between the phase mask and the output plane. The Eq. (9) can be rewritten as,
exp[iP1 (x, y )] × h(x, y, l ) + exp[iP2 (x, y )] × h(x, y, l ) (H˙ (ξ , η )) = −1 (h(x, y, l ))
(11)
where, {} denotes the Fourier transform and − 1 {} is the inverse Fourier transform operation. Let us take
D=I
−1
I H˙ (ξ , η )
(12)
I(h(x, y, l ) )
The Eq. (11) is rewritten as
exp[iP2 (x, y )] = D − exp[iP1 (x, y )]
Since, phase-only information is present in the two POMs, we may express it as
|D − exp(iP1 (x, y ))|2 = [D − exp(iP1 (x, y ))] × [D − exp(iP2 (x, y ))] = 1
(14)
Finally, we can arrive to an expression defining the POMs as,
P1 (x, y ) = arg(D ) − arccos
abs(D ) 2
Fig. 1. Flow chart for the image encryption process.
(13)
P2 (x, y ) = arg(D − exp(iP1 (x, y ))
(15)
Table 1 Results of CC values between original image (Fig. 2(a)) and its scrambled versions (Fig. 2(b–f)) with respective iterations. No. of iterations used for Arnold scrambling
CC value between original image and its scrambled version
1 5 10 100 10 0 0
0.028 0.0023 0.0050 0.0047 0.0021
(16)
where, arg(D ) returns the phase of a complex function and abs(D) returns its magnitude. 5. Simulation results and discussion The encryption process is portrayed using a flowchart shown in Fig. 1. The computer simulation has been carried out on the MATLAB platform (Version: 8.5, R2015 a). To prove the effectiveness of the proposed scheme, a grey scale image of M10 NUT of size 128 × 128 pixels as shown in Fig. 2(a) has been used. The input image is first subjected to the Arnold transform. The scrambled images of M10 NUT obtained after 1, 5, 10, 10 0, 30 0, and 10 0 0 iterations of the Arnold Cat map have been shown in Fig. 2(b–f), respectively. The correlation coefficient (CC) has been computed between the original image and the respective scrambled images. The values have been shown in Table 1. A visual inspection of the scrambled images along with the corresponding CC values suggests that the Arnold Cat map is periodic and increasing the number of iterations do not help in increasing the security of system. Hence, in the proposed scheme, keeping in mind the computational cost, scrambling of the input image is done for 14 iterations. The scrambled image is divided into 64 blocks of row vectors and is subsequently encoded into 64 QR codes, which are then multiplexed. The values for Arnold parameters, p and q have been taken as 1. The multiplexed function is multiplied with an RPM and is Fresnel transformed. The output is amplitude-, and phasetruncated and retained amplitude part is again bonded with another RPM and the product is Fresnel transformed. For obtaining
the Fresnel transform, propagation distances d1 = 8 cm, d2 = 10 cm, have been used. The output is phase-truncated and used to construct the POMs. The final step of encryption consists in generating the POMs. For this, initial wavelength, λ = 633nm and separation between the POMs and output screen, l = 12 cm is used for implementing the steps explained in Eqs. (8–16). During decryption, for experimental realization, the interference of the POMs may be realized by half mirror arrangement illuminated by a laser light of correct wavelength [23]. Use of correct RPMs, generated decryption keys, Arnold parameters, Fresnel propagation distances, and used wavelength retrieves the binary matrix. The separation distance between POMs and the output plane is the same as used in encryption process. The retrieved binary matrix is grouped into N QR codes and decoded to corresponding pixel blocks. The pixel blocks are recombined and descrambled to obtain the original image successfully. It is not possible to encode or decode a large image in a single QR code with a software tool or a smartphone because of the size and data handling capability restriction of QR code. Error detection and correction capability is affected negatively on increasing the size of the QR code. In this study, the encoding and decoding of the QR codes have been done in MATLAB platform using ZXing library, which is a standalone feature of the cryptosystem. ZXing ("zebra crossing") is an open-source, multi-format 1D/2D barcode image processing library implemented in Java, which has been ported to MATLAB environment. Version 24 QR code (113 × 113) with error correction level L (bit limit-2812 bits) has been used. The results of
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Fig. 2. (a) Input image of M10 NUT, (b) resultant image after one iteration of Arnold transform, (c) resultant image after five iterations of Arnold transform, and (d) resultant image after ten iterations of Arnold transform, (e) resultant image after hundred iterations of Arnold transform, and (f) resultant image after thousand iterations of Arnold transform.
Fig. 3. Phase-only masks obtained as a result of encrypting the image of M10 NUT: (a) phase-only mask P1 , (b) phase-only mask P2 , and (c) decrypted M10 NUT obtained after using correct decryption parameters.
encrypting the image of M10 NUT [Fig. 2(a)] into two POMs, P1 and P2 . The subsequent decryption with correct decryption parameters are shown in Fig. 3(a–c), respectively. Since encoding and decoding of QR codes are performed on the same software platform automatically, there are no issues regarding the size of the data block as well as the size of the QR code. The results of decryption with one of the POMs being incorrect have also been shown in Fig. 4. Fig. 4(a) depicts the image of a wrong POM P1 and Fig. 4(b) shows the decrypted image of M10 NUT obtained after using wrong POM P1 . In order to verify the features of QR code, we carried out an experiment and tried retrieval of original image with different number of QR codes scanned through a smartphone. For experimental realization, out of the 64 QR codes obtained during decryption some QR codes are manually decoded with the help of a standard QR decoder app downloaded and installed on a smartphone (Samsung Galaxy J5 Prime). Each decoded QR code is a matrix containing 256 pixels ranging from 0 to 255. These QR codes are scanned by a smartphone one by one and the corresponding decoded matrices, are saved in an excel sheet in the smartphone. Finally, we
Fig. 4. (a) Incorrect phase-only mask P1 and (b) decrypted M10 NUT obtained after using wrong POM P1.
get an excel sheet comprising of entries pertaining to each QR code. The excel sheet is imported to the computer, and processed using MATLAB. The entire entry of the excel sheet is transformed to a matrix and descrambled using Arnold cat map. This retrieves a
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Fig. 5. Partially decrypted image of M10 Nut using (a) 32 QR codes, (b) 16 QR codes, and (c) 8 QR codes. (d) 25th QR code obtained in the decryption process and (e) a screenshot of the decoded QR code data using smartphone.
section of the image of M10 NUT. Three cases have been discussed below considering 32, 16, and 8 QR codes out of total 64 QR codes.
be wirelessly transferred to the computer which upon further processing would result in the decrypted image.
Case 1. 32 QR codes numbered from 10 to 41 are taken. These QR codes are scanned by a smartphone, which gives an excel sheet comprising of 32 entries. A section of the image of M10 NUT of size 128 × 64 pixels is retrieved as shown in Fig. 5(a).
5.1. Attack analysis
Case 2. 16 QR codes numbered from 15 to 30 are taken. These QR codes are scanned by a smartphone, which gives an excel sheet comprising of 16 entries. A section of the image of M10 NUT of size 128 × 32 pixels is retrieved as shown in Fig. 5(b). Case 3. 8 QR codes numbered from 25 to 32 are taken. These QR codes are scanned by a smartphone, which gives an excel sheet comprising of 8 entries. A cross-section of the image of M10 NUT of size 128 × 16 pixels is retrieved as shown in Fig. 5(c). For the purpose of representation, we picked the QR code number 25 obtained during decryption, depicted in Fig. 5(d) and took a screenshot of the decoded data by the smartphone, which is shown in Fig. 5(e). Visually we can affirm that the app was able to retrieve all the 256 pixels encoded in the QR code. Similarly, if all the 64 QR codes are provided, the smartphone would decode all of them and save the decoded data in an excel sheet. The excel sheet may
Let us consider a situation in which an attacker tries to decipher the encrypted image. Assuming that the attacker has knowledge of all the decryption keys, decryption parameters (free space propagation distances, wavelength, and Arnold parameters) and one of the POMs, say P2 . Also, assuming that the attacker has the knowledge of certain percentage of pixels of P1 . In this case, the rest information of the POM P1 can be considered as random. Based on this fact the attacker tries to decrypt the encrypted image [41]. We computed the CC (correlation coefficient) values between decrypted image and the original image with respect to the pixels of P1 known to the attacker. Fig. 6 represents a plot between percentage of pixels of P1 known to the attacker and CC values. Fig. 7(a) and (b) show the decrypted images of M10 NUT obtained with 99.0234% and 99.9999% of pixels of P1 known to the attacker, respectively. Fig. 7(b) refers to the situation when attacker does not have the correct information of a single pixel of one of the POMs, P1 . We find that even silhouette is not seen in the output. Thus, it
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Fig. 6. A plot between percentage of pixels of phase-only mask P1 known to the attacker and the CC values (CC between original image of M10 NUT and the image deciphered by the attacker).
Fig. 7. Decrypted images of M10 NUT by the attacker for a percentage of pixels of phase-only mask P1 known. (a) P1 = 99.0234% and (b) P1 = 99.9999%.
can be inferred that for successful retrieval of information correct knowledge of POMs is must. 5.2. Noise analysis In an optical setup, it is evident that noise may deteriorate the quality of the decrypted image. In this study, the generation of POMs has been done analytically. We assume it to be noise free. However, it is possible that during decryption process noise may contaminate the binary matrix which had been obtained as a result of interference of the two POMs. The effect on the decrypted image as a result of polluted binary matrix has been studied. A random binary noise matrix of same size as the binary matrix contaminates some random pixel locations of the binary matrix by swapping of bits. Fig. 8(a) represents the decrypted image of M10 NUT as a result of 50 0 0 contaminated pixels out of 1024 × 1024 pixels. The CC value between the original M10 NUT and decrypted image is 0.5478. A noise analysis of the proposed cryptosystem has been done without using Arnold Cat Map. The decrypted image of M10 NUT as a result of 50 0 0 contaminated pixels out of 1024 × 1024 pixels without using Arnold Cat Map is shown in Fig. 8(b). The CC value between the original M10 NUT and decrypted image stands out to be 0. 5452. It is interesting to note that however the CC values are similar, the visual quality of Fig. 8(a) is better than Fig. 8(b). Similarly, Fig. 8(c) and (d) represent decrypted images with 19,0 0 0 contaminated pixels with and without using Arnold Cat map and CC values of 0.1231 and 0.1276, respectively have been obtained. Hence, Arnold Cat map solves the purpose of its use in the proposed cryptosystem. Fig. 9 presents a plot between the CC value and number of contaminated pixels. Beyond 19,0 0 0 contaminated
Fig. 8. Decrypted image of M10 NUT with contaminated pixels of binary matrix (the binary matrix is obtained by multiplexing the QR codes). (a) using Arnold Cat map and 50 0 0 contaminated pixels, (b) without using Arnold Cat map and 50 0 0 contaminated pixels, (c) using Arnold Cat map and 19,0 0 0 contaminated pixels, and (d) without using Arnold Cat map and 19,0 0 0 contaminated pixels.
pixels, the CC value falls to a very low value making the decrypted image unperceivable. The efficient use of QR code as a data container solves the problem of inherent speckle noise present in an optical cryptosystem using coherent light sources. In can be inferred that the combined effect of data scrambling and QR codes provide good quality decrypted images. 5.3. Occlusion attack It may happen that some amount of data in the binary matrix obtained in the decryption phase gets lost. A special case of occlusion is considered where random region(s) in the binary matrix may get replaced by bit zero. Occlusion tests on the binary matrix have been performed and the decryption results are presented. Fig. 10(a–d) show occluded binary matrix situations with 1.6093%,
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Fig. 9. A plot between the number of contaminated pixels of the binary matrix (matrix obtained as a result of multiplexing of QR codes) and the correlation coefficient values (correlation coefficient between original image of M10 NUT and decrypted images of M10 NUT resulting from respective contaminated binary matrices).
Fig. 11. Decrypted images of M10 NUT obtained with (a) 1.6093% occlusion, (b) 2.15% occlusion, (c) 2.174% occlusion, and (d) 3.647% occlusion. Fig. 10. Occlusion of binary matrix obtained after multiplexing QR codes. (a) 1.6093% occlusion, (b) 2.15% occlusion, (c) 2.174% occlusion, and (d) 3.647% occlusion.
2.15%, 2.1745%, and 3.6478% occlusion, respectively. Fig. 11(a–d) show the corresponding decrypted images. With these results, we infer that QR code can tolerate some amount of data loss. 5.4. Statistical analysis A histogram of an image plots the frequency of occurrence of all the pixels in an image. Ideally the frequency distribution of the pixel values should be uniform in the encrypted image [50–51]. The histograms of the original image of M10 NUT and the POMs, P1 and P2 have been evaluated. Fig. 12(a) shows the histogram of the original image and Fig. 12(b) shows the histogram of POM P1 . A uniform distribution of pixels in the range [0,1] for P1 suggests that the CC value between the adjacent pixels have been brought down
to a very low value, which is a desirable feature of an encrypted image. 5.5. Mean absolute error analysis The proposed cryptosystem takes an 8-bit digital image as an input and produces two POMs. These POMs are a matrix containing phase information of a complex quantity which lies in the range [−2π , 2π ]. The mean absolute error between two 2-D images is given by the formula [52],
M N MAE =
j=1
i=1
|C (i, j ) − D(i, j )| M×N
(17)
Here, M and N denote the number of rows and columns respectively of the images C and D. The original plain image of M10 NUT was taken to obtain the POMs. MAE was employed to determine the sensitivity of a slight change in the input plain image to the
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Fig. 12.
Fig. 13.
Histogram of (a) original image of M10 NUT used for encryption and (b) generated phase-only mask P1 .
Mean absolute error plot for a number of iterations between the POM P1 of original plain image of M10 NUT and its one-bit changed version.
one of the POMs say P1 . One hundred iterations were carried out by varying a pixel of the plain image at random locations by onebit to obtain POM P1 respectively for each iteration. The POM P1 of the original plain image of M10 NUT and its one-bit changed version was taken and MAE was calculated. A plot of MAE value Vs iteration number is shown in Fig. 13. The plot indicates that each of the POM P1 obtained after every iteration is distinct and different.
cause of the use of phase-truncated version of the binary matrix. Results of computer simulation demonstrate the practicability and efficacy of the method. Acknowledgments The authors acknowledge the funding from the Council of Scientific and Industrial Research (CSIR), Government of India, under Grant No. 03/ (1351)/16/EMR-II.
5.6. Time analysis Supplementary material The simulation study has been carried out on MATLAB platform (Version: 8.5, R2015 a) in a Windows 10 environment on a computer with core i5 processor and 4GB RAM. The cryptosystem took an encryption time of 14.214 s and decryption time of 15.022 s for a test image of size 128 × 128 pixels. 6. Conclusion To conclude, we present a novel technique to encrypt an image employing asymmetric framework in the Fresnel domain and the principle of interference. The original image has been encoded into 64 QR codes, which are tolerant to external noise and partial data loss because of its inherent error detecting and error correcting capabilities. The analytically generated POMs are real and hence ease the problem of storage and transmission. An attack condition was created and the results reflect the immunity of the proposed cryptosystem. Also, the problem of silhouette has been resolved be-
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Journal of Information Security and Applications journal homepage: www.elsevier.com/locate/jisa
Quick response code and Interference-based optical asymmetric cryptosystem Avishek Kumar, Naveen K. Nishchal∗ Department of Physics, Indian Institute of Technology Patna, Bihta 801 106, Patna, Bihar, India
a r t i c l e
i n f o
Article history:
Keywords: Image encryption Arnold cat map Quick response codes Fresnel transform Interference
a b s t r a c t In this paper, a novel image encryption scheme based on quick response code and interference in Fresnel domain is proposed. The framework is asymmetric in nature due to amplitude- and phase-truncation operation of encoded spectrum. The input image is scrambled using Arnold cat map and divided into pixel blocks. Each block is encoded into a quick response code, which are then multiplexed to generate a binary matrix. The phase-truncated binary matrix is encoded into analytically generated two phaseonly masks. During decryption the phase-only masks are recombined to yield the binary matrix, which is grouped into quick response codes and decoded into pixel blocks. The pixel blocks are recombined and descrambled to retrieve the original image. Numerical simulations have been carried out to demonstrate the effectiveness of the proposed method. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Securing data/information and secure dissemination of information are of prime concern in today’s digital era. Digital computational techniques to secure data have been flourishing through decades. In last two decades, optical image encryption techniques have also emerged and are attracting increased attention because of their high speed and parallel processing capability. An optically processed image may be represented as an amplitude image, phase-only image or a combination of both. A groundbreaking research was reported by Refregier and Javidi, who proposed a scheme to encrypt an image into a stationary white noise employing double random phase encoding (DRPE) architecture [1]. The method was further explored and expanded into other optical domains, such as, fractional Fourier, Fresnel, gyrator transform domains, etc. [2–4]. Despite widespread acclamation of DRPE, the scheme was proved to be prone to security attacks. The decryption key was easily obtained if Dirac delta function was used as the test image in the encryption process [5]. Vulnerabilities of DRPE to various other applicable attacks were also reported [6–8]. A method of scrambling and then applying DRPE to the test image also proved to be vulnerable to brute force attack [9]. To address the vulnerabilities of the basic DRPE, security enhanced versions of the DRPE scheme were reported [10,11]. The major drawback of symmetric
∗
Corresponding author. E-mail address: [email protected] (N.K. Nishchal).
https://doi.org/10.1016/j.jisa.2019.01.004 2214-2126/© 2019 Elsevier Ltd. All rights reserved.
cryptosystems is the key management. Since the same key is used during encryption and decryption process, the keys needs to be sent to the authorized receiver through a dedicated channel and are thus vulnerable to attacks. Asymmetric cryptographic schemes use different encryption and decryption keys, in which encryption key is made publicly available. Even if the attacker gets any of the pairs of plaintext, cipher text, and encryption key, it would be very difficult to derive the decryption key and hence decrypt the information. Amplitude- and phase-truncation based optical asymmetric cryptosystems are very popular among the research community and various contributions exist in literature [12]. However, a security leak has been discovered in the basic scheme. The scheme was prone to known-plaintext attack, in which the decryption keys are obtained using a pair of known-plaintext and ciphertext using the modified Gerchberg–Saxton phase retrieval algorithm [13]. The basic scheme was also proved vulnerable to a specific attack [14,15]. Chaos, a non-linear phenomenon has gained lot of ground and continues to be an active element of research owing to its highly unpredictable nature and sensitivity to the initial conditions. A slight difference between the initial states produces entirely different trajectories, which grows exponentially with time and do not converge. Singh and Sinha proposed a chaos-based DRPE scheme in which the random phase masks (RPM) are generated by chaos functions namely Logistic map, Tent map, and Kalpan–Yorke map [16]. A proposal to enhance the security of basic DRPE scheme by randomizing the input image pixels using a chaotic Baker map have also been reported [20]. It can therefore be inferred that optical technologies linked with chaos theory may be a good
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candidate to strengthen the basic optical asymmetric schemes [16–22]. Off late, interference based optical image encryption has been reported, in which the image to be encrypted is encoded into analytically generated two phase-only masks (POMs). Zhang and Wang proposed that any input image can be directly recovered as an intensity pattern recorded at the output plane by illuminating the POMs [23]. It is a simple scheme that involves non-iterative process. However, a loophole was found in this scheme. If the attacker has the information of any one of the generated POMs, then different combinations of POMs can be tried to obtain the silhouette of the original image. To overcome this issue, Zhang et al. [24] proposed a scheme in which a subpart of the POMs are randomly exchanged. The scheme had a drawback with respect to computational time. Another approach was proposed by Wang and Zhao to divide the POMs into three POMs [25]. However, it resulted into a complicated decryption setup. Also, method to encrypt binary image employing the interference based optical encryption concept has been reported [26]. Rajput and Nishchal reported a method employing asymmetric keys in the Fresnel domain [27]. It may be noted that interference based encryption scheme is quite popular and could be exploited and linked with other optical technologies to yield reliable results [28–31]. A general outcome of any optical cryptosystem using a coherent light source is speckle noise. Other types of noises also exist, which corrupt the encrypted image and hence results into poor quality of decrypted image. Barrera et al. reported an encryption scheme employing QR codes as an information container [32]. In this scheme, the input image is transformed into a QR code and is fed as an input to the DRPE scheme. Although the decrypted image of the QR code was noisy but it was successfully decoded. QR code is a two-dimensional (2D) bar code which is represented as a matrix [33]. It is an arrangement of black modules in a square grid on a white background. The data is encoded as bits and any mode of encoding ensures that the shortest possible sequence of bits is generated. While encoding the input data, the inherently embedded error correction code is also employed. This remarkable feature of a QR code makes it resistant to noise. Off late, QR codes have gained popularity and are being used in the field of optical image encryption [34–48]. An idea to encrypt QR code and a grey scale image as two separate entities in an interference based scheme using position multiplexing was reported by Qin et al. [38]. The scheme was not asymmetric and also the potential of a QR code was not properly exploited. A scheme to encode a grey scale image into multiple QR codes in Fresnel domain using DRPE has also been reported [41]. The scheme relied on QR code generators and readers and hence cannot be considered as standalone. Also, the scheme was not asymmetric in nature. The basic schemes related to amplitude- and phase- truncation were vulnerable to various types of attacks [13–15]. The nonlinearity of the asymmetric based cryptosystems may be enhanced by unifying it with the chaotic systems to make it more resistant to attacks [16–22]. In this paper, the shortcomings of both the amplitude- and phase- truncated cryptosystem and the interference-based cryptosystem are addressed by unifying them with the features of chaotic scrambling and QR code. The pixel positions of the input image are scrambled by a chaotic Arnold Cat map and is further divided into a number of blocks. Each block is encoded to the corresponding QR code and finally each QR coded matrix is multiplexed to obtain a binary matrix, which is processed by an asymmetric cryptosystem in the Fresnel domain appended by the interference based encryption method. To the best of our knowledge, this is the first report in which QR code has been unified with asymmetric scheme of image encryption and decryption.
2. Arnold cat map Arnold cat map is a 2D invertible chaotic map, which when applied to an image, randomizes the original organization of pixels. The period of the Arnold Cat map is determined by the number of iterations. The 2D Cat map is given as [49],
x x 1 = A mod (256 ) = y y q
p pq + 1
x mod (256 ) y
(1)
Here, p and q are positive integers. The Cat map is an area preserving map since det(A) = 1. (x , y ) and (x, y) denote the coordinates and transformed coordinates of pixels of an (N × N) image, respectively. The Arnold transformation of the input image may be denoted as,
T (x, y ) = AR[I (x, y )]
(2)
Here, I(x, y) is the input image and AR is the Arnold transform operator. The Arnold transformed image, T(x, y) is divided into N equal size matrices. Each matrix is encoded to a QR code. The obtained N number of QR codes are multiplexed to obtain a binary matrix denoted by f(x, y), which is an input to the amplitude- and phase-truncation based asymmetric encryption process. 3. Amplitude- and phase-truncation based Fresnel domain encryption The function f(x, y) is multiplied by a random phase mask (RPM) and its Fresnel transform (FrT) is calculated.
F (u, v ) =
exp(ikz ) iλz
× exp
{ f (x, y ) × exp[i2π r1 (x, y )]}
ik [(u − x )2 + (v − y )2 ]dxdy 2z
(3)
Here, (x, y) and (u, v) are the coordinates of the input and output planes, respectively. λis the wavelength of the incident light, k = 2λπ , and z denotes the free space propagation distance. r1 (x, y) is a random phase distributed in the interval [0,1]. The obtained spectrum F(u, v), is phase-truncated (PT) and multiplied by another RPM and Fresnel transformed which may be mathematically expressed as,
G(ξ , η ) = F rT [P T {F (u, v )} × r2 (u, v )]
(4)
The function r2 (u,v) is another RPM. The obtained spectrum given by Eq. (4) is amplitude-, and phase-truncated.
A(ξ , η ) = AT [G(ξ , η )]
(5)
H ( ξ , η ) = P T [ G ( ξ , η )]
(6)
The term (AT) denote the amplitude-truncation operation. The function, A(ξ , η) serves as a decryption key. The second decryption key is obtained as,
k=
AT [F rT { f (x, y ) × exp[i2π r1 (x, y )]}] P T [F rT { f (x, y ) × exp[i2π r1 (x, y )]}]
(7)
The function, H(ξ , η) is encoded into two POMs as described in the interference-based optical encryption method [12,15]. 4. Generating phase-only masks using the principle of interference The phase-truncated output H(ξ , η) is used to analytically generate two POMs. A new RPM, ϕ (ξ , η) is bonded with the function H(ξ , η).
H˙ (ξ , η ) = H (ξ , η ) exp[i2π φ (ξ , η )]
(8)
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The complex function, H˙ (ξ , η ) may be expressed as the interference of two POMs, P1 and P2 [23,27].
H˙ (ξ , η ) = exp[iP1 (x, y )] × h(x, y, l ) + exp[iP2 (x, y )] × h(x, y, l ) (9) where,
h(x, y, l ) =
exp( i2λπ l ) iπ 2 exp ( x + y2 ) il λ lλ
(10)
is the point pulse function of the Fresnel transform, l is the distance between the phase mask and the output plane. The Eq. (9) can be rewritten as,
exp[iP1 (x, y )] × h(x, y, l ) + exp[iP2 (x, y )] × h(x, y, l ) (H˙ (ξ , η )) = −1 (h(x, y, l ))
(11)
where, {} denotes the Fourier transform and − 1 {} is the inverse Fourier transform operation. Let us take
D=I
−1
I H˙ (ξ , η )
(12)
I(h(x, y, l ) )
The Eq. (11) is rewritten as
exp[iP2 (x, y )] = D − exp[iP1 (x, y )]
Since, phase-only information is present in the two POMs, we may express it as
|D − exp(iP1 (x, y ))|2 = [D − exp(iP1 (x, y ))] × [D − exp(iP2 (x, y ))] = 1
(14)
Finally, we can arrive to an expression defining the POMs as,
P1 (x, y ) = arg(D ) − arccos
abs(D ) 2
Fig. 1. Flow chart for the image encryption process.
(13)
P2 (x, y ) = arg(D − exp(iP1 (x, y ))
(15)
Table 1 Results of CC values between original image (Fig. 2(a)) and its scrambled versions (Fig. 2(b–f)) with respective iterations. No. of iterations used for Arnold scrambling
CC value between original image and its scrambled version
1 5 10 100 10 0 0
0.028 0.0023 0.0050 0.0047 0.0021
(16)
where, arg(D ) returns the phase of a complex function and abs(D) returns its magnitude. 5. Simulation results and discussion The encryption process is portrayed using a flowchart shown in Fig. 1. The computer simulation has been carried out on the MATLAB platform (Version: 8.5, R2015 a). To prove the effectiveness of the proposed scheme, a grey scale image of M10 NUT of size 128 × 128 pixels as shown in Fig. 2(a) has been used. The input image is first subjected to the Arnold transform. The scrambled images of M10 NUT obtained after 1, 5, 10, 10 0, 30 0, and 10 0 0 iterations of the Arnold Cat map have been shown in Fig. 2(b–f), respectively. The correlation coefficient (CC) has been computed between the original image and the respective scrambled images. The values have been shown in Table 1. A visual inspection of the scrambled images along with the corresponding CC values suggests that the Arnold Cat map is periodic and increasing the number of iterations do not help in increasing the security of system. Hence, in the proposed scheme, keeping in mind the computational cost, scrambling of the input image is done for 14 iterations. The scrambled image is divided into 64 blocks of row vectors and is subsequently encoded into 64 QR codes, which are then multiplexed. The values for Arnold parameters, p and q have been taken as 1. The multiplexed function is multiplied with an RPM and is Fresnel transformed. The output is amplitude-, and phasetruncated and retained amplitude part is again bonded with another RPM and the product is Fresnel transformed. For obtaining
the Fresnel transform, propagation distances d1 = 8 cm, d2 = 10 cm, have been used. The output is phase-truncated and used to construct the POMs. The final step of encryption consists in generating the POMs. For this, initial wavelength, λ = 633nm and separation between the POMs and output screen, l = 12 cm is used for implementing the steps explained in Eqs. (8–16). During decryption, for experimental realization, the interference of the POMs may be realized by half mirror arrangement illuminated by a laser light of correct wavelength [23]. Use of correct RPMs, generated decryption keys, Arnold parameters, Fresnel propagation distances, and used wavelength retrieves the binary matrix. The separation distance between POMs and the output plane is the same as used in encryption process. The retrieved binary matrix is grouped into N QR codes and decoded to corresponding pixel blocks. The pixel blocks are recombined and descrambled to obtain the original image successfully. It is not possible to encode or decode a large image in a single QR code with a software tool or a smartphone because of the size and data handling capability restriction of QR code. Error detection and correction capability is affected negatively on increasing the size of the QR code. In this study, the encoding and decoding of the QR codes have been done in MATLAB platform using ZXing library, which is a standalone feature of the cryptosystem. ZXing ("zebra crossing") is an open-source, multi-format 1D/2D barcode image processing library implemented in Java, which has been ported to MATLAB environment. Version 24 QR code (113 × 113) with error correction level L (bit limit-2812 bits) has been used. The results of
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Fig. 2. (a) Input image of M10 NUT, (b) resultant image after one iteration of Arnold transform, (c) resultant image after five iterations of Arnold transform, and (d) resultant image after ten iterations of Arnold transform, (e) resultant image after hundred iterations of Arnold transform, and (f) resultant image after thousand iterations of Arnold transform.
Fig. 3. Phase-only masks obtained as a result of encrypting the image of M10 NUT: (a) phase-only mask P1 , (b) phase-only mask P2 , and (c) decrypted M10 NUT obtained after using correct decryption parameters.
encrypting the image of M10 NUT [Fig. 2(a)] into two POMs, P1 and P2 . The subsequent decryption with correct decryption parameters are shown in Fig. 3(a–c), respectively. Since encoding and decoding of QR codes are performed on the same software platform automatically, there are no issues regarding the size of the data block as well as the size of the QR code. The results of decryption with one of the POMs being incorrect have also been shown in Fig. 4. Fig. 4(a) depicts the image of a wrong POM P1 and Fig. 4(b) shows the decrypted image of M10 NUT obtained after using wrong POM P1 . In order to verify the features of QR code, we carried out an experiment and tried retrieval of original image with different number of QR codes scanned through a smartphone. For experimental realization, out of the 64 QR codes obtained during decryption some QR codes are manually decoded with the help of a standard QR decoder app downloaded and installed on a smartphone (Samsung Galaxy J5 Prime). Each decoded QR code is a matrix containing 256 pixels ranging from 0 to 255. These QR codes are scanned by a smartphone one by one and the corresponding decoded matrices, are saved in an excel sheet in the smartphone. Finally, we
Fig. 4. (a) Incorrect phase-only mask P1 and (b) decrypted M10 NUT obtained after using wrong POM P1.
get an excel sheet comprising of entries pertaining to each QR code. The excel sheet is imported to the computer, and processed using MATLAB. The entire entry of the excel sheet is transformed to a matrix and descrambled using Arnold cat map. This retrieves a
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Fig. 5. Partially decrypted image of M10 Nut using (a) 32 QR codes, (b) 16 QR codes, and (c) 8 QR codes. (d) 25th QR code obtained in the decryption process and (e) a screenshot of the decoded QR code data using smartphone.
section of the image of M10 NUT. Three cases have been discussed below considering 32, 16, and 8 QR codes out of total 64 QR codes.
be wirelessly transferred to the computer which upon further processing would result in the decrypted image.
Case 1. 32 QR codes numbered from 10 to 41 are taken. These QR codes are scanned by a smartphone, which gives an excel sheet comprising of 32 entries. A section of the image of M10 NUT of size 128 × 64 pixels is retrieved as shown in Fig. 5(a).
5.1. Attack analysis
Case 2. 16 QR codes numbered from 15 to 30 are taken. These QR codes are scanned by a smartphone, which gives an excel sheet comprising of 16 entries. A section of the image of M10 NUT of size 128 × 32 pixels is retrieved as shown in Fig. 5(b). Case 3. 8 QR codes numbered from 25 to 32 are taken. These QR codes are scanned by a smartphone, which gives an excel sheet comprising of 8 entries. A cross-section of the image of M10 NUT of size 128 × 16 pixels is retrieved as shown in Fig. 5(c). For the purpose of representation, we picked the QR code number 25 obtained during decryption, depicted in Fig. 5(d) and took a screenshot of the decoded data by the smartphone, which is shown in Fig. 5(e). Visually we can affirm that the app was able to retrieve all the 256 pixels encoded in the QR code. Similarly, if all the 64 QR codes are provided, the smartphone would decode all of them and save the decoded data in an excel sheet. The excel sheet may
Let us consider a situation in which an attacker tries to decipher the encrypted image. Assuming that the attacker has knowledge of all the decryption keys, decryption parameters (free space propagation distances, wavelength, and Arnold parameters) and one of the POMs, say P2 . Also, assuming that the attacker has the knowledge of certain percentage of pixels of P1 . In this case, the rest information of the POM P1 can be considered as random. Based on this fact the attacker tries to decrypt the encrypted image [41]. We computed the CC (correlation coefficient) values between decrypted image and the original image with respect to the pixels of P1 known to the attacker. Fig. 6 represents a plot between percentage of pixels of P1 known to the attacker and CC values. Fig. 7(a) and (b) show the decrypted images of M10 NUT obtained with 99.0234% and 99.9999% of pixels of P1 known to the attacker, respectively. Fig. 7(b) refers to the situation when attacker does not have the correct information of a single pixel of one of the POMs, P1 . We find that even silhouette is not seen in the output. Thus, it
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Fig. 6. A plot between percentage of pixels of phase-only mask P1 known to the attacker and the CC values (CC between original image of M10 NUT and the image deciphered by the attacker).
Fig. 7. Decrypted images of M10 NUT by the attacker for a percentage of pixels of phase-only mask P1 known. (a) P1 = 99.0234% and (b) P1 = 99.9999%.
can be inferred that for successful retrieval of information correct knowledge of POMs is must. 5.2. Noise analysis In an optical setup, it is evident that noise may deteriorate the quality of the decrypted image. In this study, the generation of POMs has been done analytically. We assume it to be noise free. However, it is possible that during decryption process noise may contaminate the binary matrix which had been obtained as a result of interference of the two POMs. The effect on the decrypted image as a result of polluted binary matrix has been studied. A random binary noise matrix of same size as the binary matrix contaminates some random pixel locations of the binary matrix by swapping of bits. Fig. 8(a) represents the decrypted image of M10 NUT as a result of 50 0 0 contaminated pixels out of 1024 × 1024 pixels. The CC value between the original M10 NUT and decrypted image is 0.5478. A noise analysis of the proposed cryptosystem has been done without using Arnold Cat Map. The decrypted image of M10 NUT as a result of 50 0 0 contaminated pixels out of 1024 × 1024 pixels without using Arnold Cat Map is shown in Fig. 8(b). The CC value between the original M10 NUT and decrypted image stands out to be 0. 5452. It is interesting to note that however the CC values are similar, the visual quality of Fig. 8(a) is better than Fig. 8(b). Similarly, Fig. 8(c) and (d) represent decrypted images with 19,0 0 0 contaminated pixels with and without using Arnold Cat map and CC values of 0.1231 and 0.1276, respectively have been obtained. Hence, Arnold Cat map solves the purpose of its use in the proposed cryptosystem. Fig. 9 presents a plot between the CC value and number of contaminated pixels. Beyond 19,0 0 0 contaminated
Fig. 8. Decrypted image of M10 NUT with contaminated pixels of binary matrix (the binary matrix is obtained by multiplexing the QR codes). (a) using Arnold Cat map and 50 0 0 contaminated pixels, (b) without using Arnold Cat map and 50 0 0 contaminated pixels, (c) using Arnold Cat map and 19,0 0 0 contaminated pixels, and (d) without using Arnold Cat map and 19,0 0 0 contaminated pixels.
pixels, the CC value falls to a very low value making the decrypted image unperceivable. The efficient use of QR code as a data container solves the problem of inherent speckle noise present in an optical cryptosystem using coherent light sources. In can be inferred that the combined effect of data scrambling and QR codes provide good quality decrypted images. 5.3. Occlusion attack It may happen that some amount of data in the binary matrix obtained in the decryption phase gets lost. A special case of occlusion is considered where random region(s) in the binary matrix may get replaced by bit zero. Occlusion tests on the binary matrix have been performed and the decryption results are presented. Fig. 10(a–d) show occluded binary matrix situations with 1.6093%,
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Fig. 9. A plot between the number of contaminated pixels of the binary matrix (matrix obtained as a result of multiplexing of QR codes) and the correlation coefficient values (correlation coefficient between original image of M10 NUT and decrypted images of M10 NUT resulting from respective contaminated binary matrices).
Fig. 11. Decrypted images of M10 NUT obtained with (a) 1.6093% occlusion, (b) 2.15% occlusion, (c) 2.174% occlusion, and (d) 3.647% occlusion. Fig. 10. Occlusion of binary matrix obtained after multiplexing QR codes. (a) 1.6093% occlusion, (b) 2.15% occlusion, (c) 2.174% occlusion, and (d) 3.647% occlusion.
2.15%, 2.1745%, and 3.6478% occlusion, respectively. Fig. 11(a–d) show the corresponding decrypted images. With these results, we infer that QR code can tolerate some amount of data loss. 5.4. Statistical analysis A histogram of an image plots the frequency of occurrence of all the pixels in an image. Ideally the frequency distribution of the pixel values should be uniform in the encrypted image [50–51]. The histograms of the original image of M10 NUT and the POMs, P1 and P2 have been evaluated. Fig. 12(a) shows the histogram of the original image and Fig. 12(b) shows the histogram of POM P1 . A uniform distribution of pixels in the range [0,1] for P1 suggests that the CC value between the adjacent pixels have been brought down
to a very low value, which is a desirable feature of an encrypted image. 5.5. Mean absolute error analysis The proposed cryptosystem takes an 8-bit digital image as an input and produces two POMs. These POMs are a matrix containing phase information of a complex quantity which lies in the range [−2π , 2π ]. The mean absolute error between two 2-D images is given by the formula [52],
M N MAE =
j=1
i=1
|C (i, j ) − D(i, j )| M×N
(17)
Here, M and N denote the number of rows and columns respectively of the images C and D. The original plain image of M10 NUT was taken to obtain the POMs. MAE was employed to determine the sensitivity of a slight change in the input plain image to the
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Fig. 12.
Fig. 13.
Histogram of (a) original image of M10 NUT used for encryption and (b) generated phase-only mask P1 .
Mean absolute error plot for a number of iterations between the POM P1 of original plain image of M10 NUT and its one-bit changed version.
one of the POMs say P1 . One hundred iterations were carried out by varying a pixel of the plain image at random locations by onebit to obtain POM P1 respectively for each iteration. The POM P1 of the original plain image of M10 NUT and its one-bit changed version was taken and MAE was calculated. A plot of MAE value Vs iteration number is shown in Fig. 13. The plot indicates that each of the POM P1 obtained after every iteration is distinct and different.
cause of the use of phase-truncated version of the binary matrix. Results of computer simulation demonstrate the practicability and efficacy of the method. Acknowledgments The authors acknowledge the funding from the Council of Scientific and Industrial Research (CSIR), Government of India, under Grant No. 03/ (1351)/16/EMR-II.
5.6. Time analysis Supplementary material The simulation study has been carried out on MATLAB platform (Version: 8.5, R2015 a) in a Windows 10 environment on a computer with core i5 processor and 4GB RAM. The cryptosystem took an encryption time of 14.214 s and decryption time of 15.022 s for a test image of size 128 × 128 pixels. 6. Conclusion To conclude, we present a novel technique to encrypt an image employing asymmetric framework in the Fresnel domain and the principle of interference. The original image has been encoded into 64 QR codes, which are tolerant to external noise and partial data loss because of its inherent error detecting and error correcting capabilities. The analytically generated POMs are real and hence ease the problem of storage and transmission. An attack condition was created and the results reflect the immunity of the proposed cryptosystem. Also, the problem of silhouette has been resolved be-
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