Asymmetric multiple image encryption using a wavelet transform and gyrator transform

Muhammad Rafiq Abuturab

doi:10.1364/OSAC.1.001111

1. Introduction

With more and more threats to information communication systems, optical information security has attracted widespread interest due to their inherent properties of high-speed and multidimensional signal processing capabilities [1,2]. Since Réfrégier and Javidi proposed double random phase encoding (DRPE) [3], various methods for optical image encryption based on DRPE using different generalizations of the Fourier transform have been developed [4–13]. However, because of the inherent linearity property, DRPE-based encryption schemes are vulnerable to different types of attacks [14]. In order to remove the linearity, an asymmetric cryptosystem based on phase-truncated Fourier transforms has been proposed [15]. But it has been found susceptible to the specific attack based on an iterative amplitude-phase retrieval algorithm [16]. Several asymmetric encryption methods have been proposed to resist specific attack [17–19].

In recent years, multiple-image encryption (MIE) techniques particularly draw more and more attention due to its high application value in multiuser authentication, content distribution, enhancing the encryption capacity and the efficiency of secret information transmission. Many variations of MIE have been developed mainly to suppress the cross-talk noise effect and improve the data security [20–31].

Watermarking technique is an effective data security technique that protects the data by means of a watermark embedded within it for copyright protection purpose. Generally, the encryption and watermarking techniques are applied separately. If the watermark is encrypted prior to embedding into the host image, an unauthorized user cannot restitute the original watermark without using correct keys. Consequently, the joint encryption-watermarking technique can achieve a higher security level. A number of joint encryption-watermarking systems have been reported [32–37].

The newly introduced theory of compressive sensing (CS) [38] has attracted the interest of researchers. It is constructed on the inherent sparsity of images and can recover the compressed images with desirable quality from much fewer compressed data. Subsequently, various image encryption techniques based on compressive sensing have been studied [39–41].

Recently, an efficient double-image encryption system combining CS with the discrete fractional random transform (DFRT) has been designed. The measurement matrix in CS and the DFRT are constructed with the random circular matrix controlled by the 2D sine logistic modulation map. The images to be encrypted are represented in the discrete wavelet transform domain [42]. Further, a simultaneous image compression, fusion and encryption approach based on CS and chaos has been proposed to ensure the efficiency and security of image transmission [43]. A binary-tree encryption strategy has been put forwarded. In this approach, encryption units are regarded as nodes and plain images are input only into leaf nodes. This scheme is used to realize a secure authority management among the users sharing a cipher image [44]. Lately, a color image watermarking scheme based on CS in the gyrator transform domain has been presented. With the use of human visual characteristics, the significant blocks of the grayscale host image are chosen to formulate the appropriate reference image and the compressed color watermark is embedded into it to achieve camouflage property to some extent [45].

In this paper, for the first time to author’s knowledge, a new asymmetric multiple image security scheme using wavelet transform and gyrator transform is proposed. In this method, a set of four color images of each user are individually a single-level 2-D discrete wavelet transformed to split into their four sub-bands, corresponding $L L$ sub-bands are fused to get a single fused image as an input image and then divided into $R,$ $G,$ and $B$ channels. Each channel is compressed by compressive sensing with measurement matrices (which are constructed by exploiting the circulant matrices and controlling the original row vectors of the circulant matrices with logistic map) and modulated by random phase mask generated by using measurement matrices. The modulated $R,$ $G,$ and $B$ channels of $n$ sets are separately multiplexed and then gyrator transformed. The phase of encrypted $R,$ $G,$ and $B$ channels are, respectively, embedded into $R,$ $G,$ and $B$ channels of host image to get corresponding watermarked channels. The amplitudes of encrypted $R,$ $G,$ and $B$ channels supply common decryption keys. The decrypted images are free from cross-talk noise effects. The security system can be implemented by using optoelectronic setup. Numerical simulation results show the validity and reliability of the proposed scheme.

Optical encryption systems evolved into multiple image encryption because, for the majority of data communications that take place today, several users must simultaneously share a common channel resource in a controlled and effective way. The usual multiplexed package is synthesized by superimposing individual encrypted images together. Digitally speaking, all the images are superimposed in one composite CCD frame, and each one of them can be independently reconstructed through a digital spatial filtering. Since no direct combination of the information of multiple hidden images is employed in the encryption process, each image can be perfectly retrieved without cross-talk caused by the existence of the other.

The proposed technique has three advantages compared with reported MIE systems. First, each set has initial value and bifurcation parameter as remarkably sensitive decryption keys as well as individual decryption key. Second, the security system has a common decryption key, which provides extra security layer. Third, the phase of encrypted image is embedded into host image to get watermarked image in order to achieve imperceptibility and robustness.

2. Theory

2.1 Compressive sensing

Compressive sensing (CS) is a mathematical paradigm which allows reconstructing the data from a substantially smaller number of measurements than those imposed by the Shannon–Nyquist theorem [38].

Suppose one-dimensional compressible signal (or image) $f \in R^{N}$ with $N$ samples (or pixels) has a sparse representation under an arbitrary orthogonal basis matrix (or sparsifying operator) $Ψ \in R^{N \times N} .$

That is, f = Ψ θ with {‖ θ ‖}_{0} = S < < N

where

θ

denotes the transform coefficient vector. It is an

S

-sparse representation of signal (or image)

f

projected on

Ψ^{T},

meaning, that

θ

has only

S

nonzero entries while the remaining

N - S

has zero entries.

The sensing (or measurement) vector $g \in R^{M}$ can be written as

g = Φ f = Φ Ψ θ with M < N

where

Φ \in R^{M \times N}

is a measurement matrix (or optical sensing operator). If

Φ

satisfies the restricted isometry property (RIP) of order

S

with isometry constant

δ_{S} \in (0, 1)

,

That is, (1 - δ_{S}) {‖ f ‖}_{2}^{2} \leq {‖ Φ f ‖}_{2}^{2} \leq (1 + δ_{S}) {‖ f ‖}_{2}^{2}

for all

S

-sparse signals

f

. Then original signal (or image) is reconstructed by solving

l_{0}

norm minimization problem [46]. The estimated coefficients vector

\hat{θ}

is the solution of the non-convex optimization program:

\hat{θ} = \arg \min_{θ} {‖ θ ‖}_{0} subject to g = Φ Ψ θ

In brief, a signal or image is transformed into its sparse form. The measurement matrix $Φ$ is exploited to compress the data. The smoothed $l_{0} -$ norm $(S L_{0})$ algorithm is adopted to recover the signal.

2.2 Logistic map

The one-dimensional (1D) non-linear chaos function is a logistic map and its iterative form is expressed as

x_{n + 1} = p x_{n} (1 - x_{n})

where

p

is called bifurcation parameter:

0 < p < 4.

x_{n} \in [0,

1]

and

x_{0}

are iterative and initial values.

The measurement matrix $Φ$ is constructed by circulant matrix [42]. First, the random sequence with length $2 N$ is produced by logistic map with initial value $x_{0}$ . The preceding $N$ elements are discarded to get new sequence. Second, the new sequence is used as the first row vector of the circulant matrix $Φ$ . The first row vector is circulated to construct its other row vectors as

\begin{array}{l} Φ (i, 1) = λ Φ (i - 1, N) \\ Φ (i, 2 : N) = Φ (i - 1, 1 : N - 1) \end{array}

where

2 \leq i \leq M

and

| λ | > 1

. In order to reduce the relevance among the column vectors, the first element of vector

Φ (i, 1)

is set as

λ Φ (i - 1, N)

.

2.3 Gyrator transform

The gyrator transform (GT), at transformation angle $α$ , of a two-dimensional function $f_{i} (x_{i}, y_{i})$ is given by [6]

\begin{array}{l} f_{o} (x_{o}, y_{o}) = G^{α} [f_{i} (x_{i}, y_{i})] (x_{o}, y_{o}) \\ = \frac{1}{| \sin α |} \iint f_{i} (x_{i}, y_{i}) \exp (i 2 π \frac{(x_{o} y_{o} + x_{i} y_{i}) \cos α - (x_{i} y_{o} + x_{o} y_{i})}{\sin α}) d x_{i} d y_{i} \end{array}

where

G^{α} []

represents GT operator.

(x_{i}, y_{i})

and

(x_{o}, y_{o})

are the input and output coordinates, respectively. GT setup contains three generalized lenses with fixed distance z between them. Each generalized lens is an assembled set of two cylindrical lenses. The variation of the transformation parameter

α

is achieved by the rotation of the cylindrical lenses. The optical GT system does not require axial movements.

3. Proposed cryptosystem

The block diagram of proposed encryption method is shown in Fig. 1

Fig. 1 Block diagram of proposed encryption system, part a.

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. It consists of following steps:

Step 1: A color image $f (x_{i}, y_{i})$ is a single-level 2-D discrete wavelet transformed to decompose into $W_{L L}, W_{H L}, W_{L H},$ and $W_{H H}$ sub-bands where subscripts $L$ and $H$ represent low and high frequency parts, respectively.

[W_{L L}, W_{H L}, W_{L H}, W_{H H}] = DWT [f (x_{i}, y_{i})]

Each set contains four color images. Let $n {(= 1, 2,, N)}$ sets of four color images be $f_{n 1} (x_{i}, y_{i})$ , $f_{n 2} (x_{i}, y_{i})$ , $f_{n 3} (x_{i}, y_{i})$ , and $f_{n 4} (x_{i}, y_{i}) .$ The corresponding sub-bands are represented as

[W_{L L_{n 1}}, W_{H L_{n 1}}, W_{L H_{n 1}}, W_{H H_{n 1}}] = D W T [f_{n 1} (x_{i}, y_{i})]

[W_{L L_{n 2}}, W_{H L_{n 2}}, W_{L H_{n 2}}, W_{H H_{n 2}}] = D W T [f_{n 2} (x_{i}, y_{i})]

[W_{L L_{n 3}}, W_{H L_{n 3}}, W_{L H_{n 3}}, W_{H H_{n 3}}] = D W T [f_{n 3} (x_{i}, y_{i})]

[W_{L L_{n 4}}, W_{H L_{n 4}}, W_{L H_{n 4}}, W_{H H_{n 4}}] = D W T [f_{n 4} (x_{i}, y_{i})]

Step 2: The $n$ sets of four $L L$ sub-bands $W_{L L_{n 1}}$ $W_{L L_{n 2}}$ $W_{L L_{n 3}}$ and $W_{L L_{n 4}}$ are fused as

F_{n} (x, y) = [W_{L L_{n 1}}, W_{L L_{n 2}}; W_{L L_{n 3}} W_{L L_{n 4}}]

Step 3: The fused image is split into $R,$ $G,$ and $B$ channels denoted as $F_{R}^{n} (x, y)$ , $F_{G}^{n} (x, y)$ and $F_{B}^{n} (x, y)$ , respectively.

Step 4: The measurement matrices $Φ_{R}^{n},$ $Φ_{G}^{n},$ and $Φ_{B}^{n}$ with size of $(M / 2) \times N$ are constructed with corresponding parameters $(x_{0 R}^{n}, p_{R}^{n}, λ),$ $(x_{0 G}^{n}, p_{G}^{n}, λ),$ and $(x_{0 B}^{n}, p_{B}^{n}, λ)$ by using Eq. (6).

Step 5: $F_{R}^{n} (x, y)$ , $F_{G}^{n} (x, y)$ and $F_{B}^{n} (x, y)$ are, respectively, multiplied by $Φ_{R}^{n},$ $Φ_{G}^{n},$ and $Φ_{B}^{n}$ , and then modulated by corresponding phase masks $M_{R}^{n} (x, y) = \exp {[i 2 π Φ_{R}^{n} (x, y)]}^{}$ , $M_{G}^{n} (x, y) = \exp {[i 2 π Φ_{G}^{n} (x, y)]}^{}$ , and $M_{B}^{n} (x, y) = \exp {[i 2 π Φ_{B}^{n} (x, y)]}^{}$ .

\begin{array}{l} C_{R}^{n} (x, y) = Φ_{R}^{n} F_{R}^{n} (x, y) M_{R}^{n} (x, y) \\ C_{G}^{n} (x, y) = Φ_{G}^{n} F_{G}^{n} (x, y) M_{G}^{n} (x, y) \\ C_{B}^{n} (x, y) = Φ_{B}^{n} F_{B}^{n} (x, y) M_{B}^{n} (x, y) \end{array}}

Step 6: The $n$ sets of compressed $R,$ $G,$ and $B$ channels are separately combined together as

\begin{array}{l} C_{R} (x, y) = \prod_{n = 1}^{N} C_{R}^{n} (x, y) \\ C_{G} (x, y) = \prod_{n = 1}^{N} C_{G}^{n} (x, y) \\ C_{B} (x, y) = \prod_{n = 1}^{N} C_{B}^{n} (x, y) \end{array}}

Step 7: The combined $R,$ $G,$ and $B$ channels are, respectively, gyrator transformed at transformation angle $α_{R},$ $α_{G},$ and $α_{B}$ to obtain corresponding resultant images.

\begin{array}{l} E_{R} (x_{o}, y_{o}) = G^{α_{R}} [C_{R} (x, y)] \\ E_{G} (x_{o}, y_{o}) = G^{α_{G}} [C_{G} (x, y)] \\ E_{B} (x_{o}, y_{o}) = G^{α_{B}} [C_{B} (x, y)] \end{array}}

The phase and amplitude of the resultant images are given by

\begin{array}{l} P_{R} (x_{o}, y_{o}) = \arg [E_{R} (x_{o}, y_{o})] \\ P_{G} (x_{o}, y_{o}) = \arg [E_{G} (x_{o}, y_{o})] \\ P_{B} (x_{o}, y_{o}) = \arg [E_{B} (x_{o}, y_{o})] \end{array}}

\begin{array}{l} A_{R} (x_{o}, y_{o}) = | E_{R} (x_{o}, y_{o}) | \\ A_{G} (x_{o}, y_{o}) = | E_{G} (x_{o}, y_{o}) | \\ A_{B} (x_{o}, y_{o}) = | E_{B} (x_{o}, y_{o}) | \end{array}}

Equation (17) is used as the common asymmetric/decryption key.

Step 8: The host image $H (x_{o}, y_{o})$ of size $204 \times 512 \times 3$ pixels is decomposed into $R,$ $G,$ and $B$ channels denoted as $H_{R} (x_{o}, y_{o})$ , $H_{G} (x_{o}, y_{o})$ and $H_{B} (x_{o}, y_{o})$ , respectively. $P_{R} (x_{o}, y_{o})$ , $P_{G} (x_{o}, y_{o})$ and $P_{B} (x_{o}, y_{o})$ (employed $R,$ $G,$ and $B$ channels of watermark image) are embedded into corresponding $H_{R} (x_{o}, y_{o})$ , $H_{G} (x_{o}, y_{o})$ and $H_{B} (x_{o}, y_{o})$ to obtain watermarked image as

\begin{array}{l} W_{R} (x_{o}, y_{o}) = H_{R} (x_{o}, y_{o}) + γ . [P_{R} (x_{o}, y_{o})] \\ W_{G} (x_{o}, y_{o}) = H_{G} (x_{o}, y_{o}) + γ . [P_{G} (x_{o}, y_{o})] \\ W_{B} (x_{o}, y_{o}) = H_{B} (x_{o}, y_{o}) + γ . [P_{B} (x_{o}, y_{o})] \end{array}}

where

γ

is a real weight factor.

W (x_{o}, y_{o}) = [W_{R} (x_{o}, y_{o}), W_{G} (x_{o}, y_{o}), W_{B} (x_{o}, y_{o})]

The individual asymmetric/decryption keys of

R,

G,

and

B

channels are generated as

K_{R}^{m} (x, y) = \frac{1}{\prod_{\begin{array}{l} n = 1 \\ n \neq m \end{array}}^{N} C_{R}^{n} (x, y)}

K_{G}^{m} (x, y) = \frac{1}{\prod_{\begin{array}{l} n = 1 \\ n \neq m \end{array}}^{N} C_{G}^{n} (x, y)}

K_{B}^{m} (x, y) = \frac{1}{\prod_{\begin{array}{l} n = 1 \\ n \neq m \end{array}}^{N} C_{B}^{n} (x, y)}

The block diagram of proposed decryption method is shown in Fig. 2

Fig. 2 Block diagram of proposed decryption system, part b.

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. It consists of following steps:

Step 1: The $R,$ $G,$ and $B$ channels of watermark image are retrieved as

\begin{array}{l} P_{R} (x_{o}, y_{o}) = [W_{R} (x_{o}, y_{o}) - H_{R} (x_{o}, y_{o})] / γ \\ P_{G} (x_{o}, y_{o}) = [W_{G} (x_{o}, y_{o}) - H_{G} (x_{o}, y_{o})] / γ \\ P_{B} (x_{o}, y_{o}) = [W_{B} (x_{o}, y_{o}) - H_{B} (x_{o}, y_{o})] / γ \end{array}}

Step 2: The $R,$ $G,$ and $B$ channels of watermark image are multiplied with corresponding second set of decryption keys and then gyrator transformed at transformation angle $- α_{R},$ $- α_{G},$ and $- α_{B}$ _.

\begin{array}{l} D_{R} (x, y) = G^{- α_{R}} [\exp [i P_{R} (x_{o}, y_{o})] A_{R} (x_{o}, y_{o})] \\ D_{G} (x, y) = G^{- α_{G}} [\exp [i P_{G} (x_{o}, y_{o})] A_{G} (x_{o}, y_{o})] \\ D_{B} (x, y) = G^{- α_{B}} [\exp [i P_{B} (x_{o}, y_{o})] A_{B} (x_{o}, y_{o})] \end{array}}

Step 3: The obtained $R,$ $G,$ and $B$ channels are modulated by corresponding conjugate of phase masks and then multiplied by corresponding decryption keys.

\begin{array}{l} D_{R}^{m} (x_{i}, y_{i}) = D_{R} (x, y) {\exp [i 2 π Φ_{R}^{n} (x, y)]}^{*} K_{R}^{m} (x, y) \\ D_{G}^{m} (x_{i}, y_{i}) = D_{G} (x, y) {\exp [i 2 π Φ_{G}^{n} (x, y)]}^{*} K_{G}^{m} (x, y) \\ D_{B}^{m} (x_{i}, y_{i}) = D_{B} (x, y) {\exp [i 2 π Φ_{B}^{n} (x, y)]}^{*} K_{B}^{m} (x, y) \end{array}}

Step 4: The $R,$ $G,$ and $B$ channels are decrypted by using $S L_{0}$ algorithm with corresponding measurement matrices $Φ_{R}^{n},$ $Φ_{G}^{n},$ and $Φ_{B}^{n}$ .

\begin{array}{l} F_{R}^{m} (x_{i}, y_{i}) = S L_{0} [D_{R}^{m} (x, y)] \\ F_{G}^{m} (x_{i}, y_{i}) = S L_{0} [D_{G}^{m} (x, y)] \\ F_{B}^{m} (x_{i}, y_{i}) = S L_{0} [D_{B}^{m} (x, y)] \end{array}}

F_{m} (x_{i}, y_{i}) = [F_{R}^{m} (x_{i}, y_{i}), F_{G}^{m} (x_{i}, y_{i}), F_{B}^{m} (x_{i}, y_{i})]

Step 5: Finally, the fused image $F_{m} (x_{i}, y_{i})$ is decomposed into $W_{L L_{m 1}},$ $W_{L L_{m 2}},$ $W_{L L_{m 3}},$ and $W_{L L_{m 4}},$ which represent decrypted images as $f_{m 1} (x_{i}, y_{i})$ , $f_{m 2} (x_{i}, y_{i})$ , $f_{m 3} (x_{i}, y_{i})$ , and $f_{m 4} (x_{i}, y_{i})$ , respectively.

A single-level 2-D discrete wavelet is exploited to decompose each original image. The low-frequency part occupies most of energy. So low-frequency parts of four images are fused into a new image (as a set of four images). As the number of the original images increases, the level of discrete wavelet increases, which means that the number of pixels in the low frequency part decreases and corresponding energy occupies less. In other words, if the number of images increases (more than four), the qualities of decrypted images deteriorate.

Asymmetric cryptography, also known as public key cryptography, uses public (/encryption) and private (/decryption) keys to encrypt and decrypt data, respectively. The key distribution scheme is as follows:

(1) If the sender (say Ali) wants to send information to multi-user, he first needs to register common decryption keys (as authentication keys $A$ ) in information database of the multi-user.
(2) Ali encrypts $n$ sets of images $F_{n}$ using the individual public keys of multi-user and common public keys and then generates common decryption keys. The encrypted image $E$ embeds into a host image $H$ and then sends the resulting watermarked image $W$ to multi-user.
(3) User 1 exploits Ali’s common public and decryption keys to recover the ciphertext and then decrypts the set of images $F_{1}$ using the individual public and decryption keys.
(4) User 2 exploits Ali’s common public and decryption keys to recover the ciphertext and then decrypts the set of images $F_{2}$ using the individual public and decryption keys, and so on.

The proposed encryption and decryption processes can be performed with the optoelectronic setup as shown in Figs. 3(a) and 3(b)

Fig. 3 (a) Optoelectronic setup for proposed encryption system; (b) optoelectronic setup for proposed decryption system.

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, respectively. In encryption system, the pre-processing operations are achieved digitally by means of a computer system to obtain the final compressed red channel

C_{R} (x, y)

, which is imported into the spatial light modulator

(S L M)

and converted by the optical system of gyrator transform. The reference beam

P_{R} (x_{o}, y_{o})

, similar to inline holography, is used to record the phase distribution of output image by a charge-coupled device

(C C D)

camera. The post-processing operations are carried out digitally by using the computer system to get the watermarked red-channel

W_{R} (x_{o}, y_{o})

. In the same way,

W_{G} (x_{o}, y_{o})

and

W_{B} (x_{o}, y_{o})

are obtained.

In decryption system, the pre-processing operations are accomplished digitally by means of a computer system to get the red channel function $\exp [i P_{R} (x_{o}, y_{o})] A_{R} (x_{o}, y_{o})$ , which is modulated on $S L M$ and converted by the optical system of inverse gyrator transform. The reference beam is used to record the output image $D_{R} (x, y)$ by a $C C D$ camera and stored in the computer system to obtain decrypted image digitally. The output image is multiplied by corresponding conjugate of phase mask and decryption key. Subsequently, fused red channel is recovered by using $S L_{0}$ algorithm with measurement matrix. Similarly, fused blue and green channels are recovered using the same processes. Finally recovered fused red, blue, and green channels are combined to produce fused color image, which is decomposed into four color images of the selected set.

4. Numerical results

Numerical simulations have been performed on a Matlab 9.0 (R2016a) platform to test the validly and security of the proposed technique. The four images of set I, set II, and set III are, respectively, shown in Figs. 4(a)–4(d), 4(e)–4(h), and 4(i)–4(l)

Fig. 4 Set I [(a) Babu, (b) Ali, (c) Mahdi, (d) Barbara], Set II [(e) Olive fruits, (f) Honey, (g) Date tree, (h) Pomegranate tree], Set III [(i) Butterfly, (j) Goat, (k) Horse, (l) Camel], (m) Fused color images of Set I, (n) fused color images of Set II, and (o) fused color images of Set III [(a)-(o) are of size $512 \times 512 \times 3$ pixels].

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. The size of each image is

512 \times 512 \times 3

pixels. The fused color image of set I, set II, and set III are, respectively, shown in Figs. 4(m)–4(o). The parameters of set I, set II, and set III are, respectively,

(x_{0 R}^{1} = x_{0 G}^{1} = x_{0 B}^{1} = 0.32,

p_{R}^{1} = p_{G}^{1} = p_{B}^{1} = 3.90,

λ = 20),

(x_{0 R}^{2} = x_{0 G}^{2} = x_{0 B}^{2} = 0.36,

p_{R}^{1} = p_{G}^{1} = p_{B}^{1} = 3.94,

λ = 20),

and

(x_{0 R}^{3} = x_{0 G}^{3} = x_{0 B}^{3} = 0.40,

p_{R}^{1} = p_{G}^{1} = p_{B}^{1} = 3.98,

λ = 20) .

The compression ratio of column is taken as

C = 0.4,

so the measurement matrix becomes

204 \times 512 \times 3

pixels. The transformation angles of the GT for

R,

G,

and

B

channels are, respectively,

α_{R} = {0.70}^{\circ},

α_{G} = {0.80}^{\circ},

and

α_{B} = {0.90}^{\circ}

. The phase masks of set I, set II, and set III are, respectively, shown in Figs. 5(a)–5(c)

Fig. 5 (a) Phase mask for set I, (b) phase mask for set II, (c) phase mask for set III, (d) individual decryption phase key for set I, (e) individual decryption phase key for set II, (f) individual decryption phase key for set III, (g) amplitude of encrypted image, (h) phase of encrypted image, and (i) watermarked host image [Figures (a)-(i) are of size $204 \times 512 \times 3$ pixels].

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. The individual decryption phase keys of set I, set II, and set III are, respectively, demonstrated in Figs. 5(d)–5(f). The amplitude and phase of final encrypted images are depicted in Fig. 5(g) and Fig. 5(h), respectively. As the images are encoded into noise-like signals, no useful information can be detected. The watermarked host image is illustrated in Fig. 5(i). The size of each image displayed in Figs. 5(a)–5(i) is

204 \times 512 \times 3

pixels.

To weight the difference between the input image and output image, the correlation coefficient (CC) is given by

ρ = \frac{E {[I_{i} - E [I_{i}]]} {[I_{o} - E [I_{o}]]}}{\sqrt{E {{[I_{i} - E [I_{i}]]}^{2}}} \sqrt{E {{[I_{o} - E [I_{o}]]}^{2}}}}

where

I_{o}

and

I_{i}

are output and input images, respectively.

E [\cdot]

is the expected value operator. The CC has the value

ρ = 1

if the two images are fully correlated,

ρ = 0

if they are completely uncorrelated, and

ρ = - 1

if they are totally anti-correlated. For simplicity, only the fused color image of set I has been studied.

First, the security analysis of the proposed system has been investigated. The reconstructed images of individual images of set I without individual decryption keys (/asymmetric keys) are displayed in Figs. 6(a)–6(d)

Fig. 6 Decrypted individual images of set I: (a)-(d) without individual decryption keys, (e)-(h) without common decryption keys, (i)-(l) without conjugate phase masks.

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. The CC values of

R,

G,

and

B

channels are, respectively, (

- 0.0396,

- 0.0846,

- 0.0903

), (

- 0.1214,

0.0872,

0.1216

), (

- 0.0428,

- 0.0847,

- 0.1110

), and (

0.0035,

0.0508,

0.0823

). The CC values of

R,

G,

and

B

channels are very low. The reconstructed images of individual images of set I without common decryption keys (/asymmetric keys) are displayed in Figs. 6(e)–6(h). The CC values of

R,

G,

and

B

channels are, respectively, (

- 0.0015,

- 0.0554,

- 0.0294

), (

0.0133,

0.2268,

0.1889

), (

0.1074,

0.0393,

0.0221

), and (

0.0434,

- 0.0040,

0.0233

). The CC values of

R,

G,

and

B

channels are very low. The retrieved images of individual images of set I without conjugate phase keys are presented in Figs. 6(i)–6(l). The CC values of

R,

G,

and

B

channels are, respectively, (

- 0.0482,

- 0.0738,

- 0.0594

), (

0.0002,

0.1312,

0.1348

), (

- 0.0174,

- 0.0145,

- 0.0094

), and (

0.0319,

0.0306,

0.0352

). The CC values of

R,

G,

and

B

channels are very low. It can be inferred that the decrypted images will not visually render any information about the input image without using decryption keys or conjugate phase keys.

The initial values $x_{0 R}^{1},$ $x_{0 G}^{1},$ and $x_{0 B}^{1}$ of corresponding $R,$ $G,$ and $B$ channels fused color image of set I are changed by $1 \times 10^{- 16} .$ The CC values of $R,$ $G,$ and $B$ channels for individual images of set I as shown in Figs. 7(a)–7(d)

Fig. 7 Retrieved individual images of set I: (a)-(d) with $x_{0 R}^{1} = x_{0 G}^{1} = x_{0 B}^{1}$ changed by $1 \times 10^{- 16},$ and (e)-(h) with $p_{R}^{1} = p_{G}^{1} = p_{B}^{1}$ changed by $1 \times 10^{- 15} .$

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are, respectively, (

0.0247,

0.0708,

0.0804

), (

0.1790,

0.1326,

0.1456

), (

0.1299,

0.1282,

0.1627

), and (

0.0501,

0.0224,

0.0222

). The CC values of

R,

G,

and

B

channels are low. Thus, no information about the input image can be extracted. The bifurcation parameters

p_{R}^{1},

p_{G}^{1},

and

p_{B}^{1}

of corresponding

R,

G,

and

B

channels fused color image of set I are varied by

1 \times 10^{- 15} .

The CC values of

R,

G,

and

B

channels for individual images of set I as demonstrated in Figs. 7(e)–7(h) are, respectively, (

0.0391,

0.0451,

0.0575

), (

0.0955,

0.0401,

0.0888

), (

0.1343,

0.1794,

0.1862

), and (

0.0577,

0.0303,

0.0528

). The CC values of

R,

G,

and

B

channels are very low. Hence, no information about the input image can be observed.

The transformation angles $α_{R},$ $α_{G},$ and $α_{B}$ of multiplexed image are independently changed by $1 \times 10^{- 14} .$ The recovered individual images of set I are, respectively, depicted in Figs. 8(a)–8(d), 8(e)–8(h), and 8(i)–8(l)

Fig. 8 Recovered individual images of set I: (a)-(d) with $α_{R}$ changed by $1 \times 10^{- 14},$ (e)-(h) with $α_{G}$ changed by $1 \times 10^{- 14},$ (i)-(l) with $α_{B}$ changed by $1 \times 10^{- 14},$ and (m)-(p) with $α_{R} = α_{G} = α_{B}$ changed by $1 \times 10^{- 14} .$

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. It can be observed that only noise-like images are obtained. The transformation angles

α_{R},

α_{G},

and

α_{B}

of multiplexed image are together varied by

1 \times 10^{- 14} .

The extracted individual images of set I are, respectively, depicted in Figs. 8(m)–8(p). The corresponding CC values of

R,

G,

and

B

channels are (

- 0.0406,

- 0.1672,

- 0.0317

), (

0.0190,

0.1637,

0.0816

), (

0.0475,

- 0.0551,

- 0.0330

), and (

- 0.0069,

- 0.0643,

- 0.0208

). The CC values of

R,

G,

and

B

channels indicate that decrypted images will imperceptible. Figures 9(a)-9(c)

Fig. 9 Decrypted images with all correct keys: (a) Fused color images of Set I, (b) fused color images of Set II, and (c) fused color images of Set III, (d)-(g) individual images of set I, (h)-(k) individual images of set II, and (l)-(o) individual images of set III. [Figures (a)-(c) are of size $512 \times 512 \times 3$ pixels, and Figs. (d)-(o) are of size $256 \times 256 \times 3$ pixels].

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show the decrypted fused color images of Set I, Set II, and Set III with all correct keys, respectively. The individual images Set I, Set II, and Set III are depicted in Figs. 9(d)-9(g), 9(h)-9(k), and 9(l)-9(o), respectively. The corresponding CC values are greater than 0.988, which is close to one. So, input images are decoded correctly with slight distortion.

Second, the sensitivity analysis of security keys of fused color image of Set I of the proposed scheme has been tested. The CC values between the decrypted images and original images of $R,$ $G,$ and $B$ channels of fused color image of Set I are computed against the variation of initial value $x_{0}$ , bifurcation parameter $p,$ and transformation angle $α$ and plotted in Fig. 10(a)-10(c)

Fig. 10 (a) Correlation Coefficient versus variation in decryption keys (a) $x_{0}$ of set I, (b) $p$ of set I, (c) $α$ of set I.

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, respectively. The curves of

R,

G,

and

B

channels clearly show that the CC values of corresponding channels attain one when the security keys are correct during decryption process whereas the curves of

R,

G,

and

B

channels decrease rapidly for slight change in CC values of corresponding channels. The parameters

x_{0}

,

p,

and

α

provide sensitive keys and thus enhance the security the proposed method.

Finally, the attack analysis of the proposed scheme has been examined. In the proposed security system, different ciphertext of fused image has different decryption keys (/asymmetric keys) which mean that the chosen ciphertext attack [14], and chosen plaintext attack [15] will not work here. Moreover, the combined compressed channel is treated as a part of the input information; unauthorized users do not have enough constraints to set up the iterative amplitude-phase retrieval [15] even if they have obtained the correct transformation angle of corresponding channel as shown in Fig. 11

Fig. 11 Fused color images of Set I by using specific attack. (a) Decrypted image, and (b) the relation between MSE values and iteration number.

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.

5. Conclusion

A new nonlinear multiple image security method based on wavelet transform and compressive gyrator transform is put forwarded. The proposed technique has benefits of individual decryption keys, chaotic parameters as very sensitive decryption keys and common decryption keys, which enhance the nonlinear characteristics of the scheme. The phase of encrypted image is embedded into corresponding host image to obtain watermarked image so that imperceptibility and robustness can be maintained. The proposed algorithm not only reduces data volume but also simplify keys, which improve the efficiency of transmitting data and distributing keys. Thus, the proposed method is of great practical importance as the image fusion, compression, encryption, and watermarking are accomplished simultaneously. The security and viability are proved by the numerical simulation results.

Acknowledgments

The author is indebted to Abdul Aziz RA and Muhammad Sulayman RA for their inspiring supports.

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Asymmetric multiple image encryption using a wavelet transform and gyrator transform

Abstract

1. Introduction

2. Theory

2.1 Compressive sensing

2.2 Logistic map

2.3 Gyrator transform

3. Proposed cryptosystem

4. Numerical results

5. Conclusion

Acknowledgments

References

Cited By

Figures (11)

Equations (29)

OSA Continuum