Hybrid Color and Grayscale Images Encryption Scheme Based on Quaternion Hartley Transform and Logistic Map in Gyrator Domain
 Author: Li Jianzhong
 Publish: Current Optics and Photonics Volume 20, Issue1, p42~54, 25 Feb 2016

ABSTRACT
A hybrid color and grayscale images encryption scheme based on the quaternion Hartley transform (QHT), the twodimensional (2D) logistic map, the double random phase encoding (DRPE) in gyrator transform (GT) domain and the threestep phaseshifting interferometry (PSI) is presented. First, we propose a new color image processing tool termed as the quaternion Hartley transform, and we develop an efficient method to calculate the QHT of a quaternion matrix. In the presented encryption scheme, the original color and grayscale images are represented by quaternion algebra and processed holistically in a vector manner using QHT. To enhance the security level, a 2D logistic mapbased scrambling technique is designed to permute the complex amplitude, which is formed by the components of the QHTtransformed original images. Subsequently, the scrambled data is encoded by the GTbased DRPE system. For the convenience of storage and transmission, the resulting encrypted signal is recorded as the realvalued interferograms using threestep PSI. The parameters of the scrambling method, the GT orders and the two random phase masks form the keys for decryption of the secret images. Simulation results demonstrate that the proposed scheme has high security level and certain robustness against data loss, noise disturbance and some attacks such as chosen plaintext attack.

KEYWORD
Quaternion Hartley transform , Twodimensional logistic map , Gyrator transform , Image encryption

I. INTRODUCTION
Optical technology for security applications has received increasing interest in the last few decades [13]. In particular, the double random phase encoding (DRPE) [1] technique, in which the image is encoded to be a white noise pattern with two statistically independent random phase masks (RPM), has attracted much attention. The DRPE encryption technique is constructed by using two RPMs as keys: one is located at the input domain and the other is at the Fourier domain. To enlarge the key space, it has been further extended from the Fourier transform domain to the Fresnel transform domain [4, 5], the fractional Fourier transform domain [6], the Gyrator transform (GT) domain [7] and the Hartley transform (HT) domain [8], etc. In these encryption systems, the parameters (e.g. the propagation distance, the fractional order, etc) are introduced and served as extra keys. Since the HT is mathematically equivalent to the Fourier transform but is purely real, it has a computational advantage over the aforementioned transforms for not having to manage the real and imaginary parts [9]. Though most of the DRPEbased encryption techniques are quite robust and secure, the recovered image loses its color information, which is useful in image processing and practical applications, and in decryption since the original image is illuminated with monochromatic light [10].
Using the DRPE technique, Zhang and Karim reported a singlechannel encryption algorithm for color image in the Fourier domain [11]. Chen
et al . suggested a method using coherent diffractive imaging for optical colorimage encryption and synthesis in the Fresnel domain [3]. Joshiet al . [12] have proposed an approach for the encryption of color images via using a fractional Fourier transform. By use of the Arnold transform and an interference method, a color image encryption technique is proposed in [13]. A method for securing color image based on the Arnold transform and the gyrator transform is reported in [10]. Based on polarization encoding, Nishchalet al . proposed two asymmetric color image cryptosystem techniques [1415]. Li and Zhao [16] designed a color image encryption based on the fractional HT. However, the aforementioned encryption methods for color image ignore the potential interactions between different color channels [17, 18]. In these methods, a color image is separated into the three color channels, and encryption is carried out in each channel independently. Recently, quaternion theory [20] has been used in DRPEbased encryption. Based on the discrete quaternion Fourier transform (DQFT) and DRPE, a color image encryption method is reported [17]. With an iterative phase retrieval algorithm, an algorithm to encrypt double color images in quaternion gyrator domain is described [18]. Using quaternion representation, the color image is processed holistically as a vector field rather than as separated color components in both of the two methods. However, the method in [17] is vulnerable to some attacks (such as chosenplaintext attack, etc) caused by the linearity of the cryptosystem, while the algorithm in [18] has high computational cost due to the phase retrieval iteration process.In this paper, based on the quaternion Hartley transform (QHT), logisticbased scrambling technique, GTbased DRPE and threestep phaseshifting interferometry (PSI), a novel hybrid color and grayscale image encryption method is presented. In the proposed approach, the QHT is defined and its calculation for a quaternion matrix is developed first. Then a color image and a grayscale image are processed holistically in a vector manner using QHT. To resist an attack such as chosen plaintext attack, the components of the QHTtransformed images are permuted by use of the designed scrambling algorithm based on logistic maps. Since the flexible configuration of the GT based system, which means fixed distances between the generalized lenses and the manipulation of the transformation angle using lens rotation, makes the setup useful for image encryption [19], the GTbased DRPE technique is used to encrypt the images after QHT and permutation. The resulting encrypted signal from the cryptosystem is recorded as interferograms by the threestep PSI. Numerical simulations have been made for demonstrating the feasibility and performance of this encryption.
II. RELATED BACKGROUND
In this section, we come back to some related theories before extending the traditional HT to the quaternion domain, showing how to calculate the QHT of a quaternion matrix and developing the logistic mapsbased scrambling technique, etc.
2.1. Quaternion Number
Quaternions can be viewed as generalizations of complex numbers. A quaternion number can be represented as follows [18]:
where
a, b, c, andd are real numbers, andi, j, andk are three imaginary units obeying the following rulesThe conjugate and modulus of a quaternion are respectively defined by
When
a = 0,q is a pure quaternion.2.2. Hartley Transform
In mathematics, the HT, which transforms realvalued functions to realvalued functions, is an integral transform closely related to the Fourier transform. HT has two main properties: one is that it is a real transform; the other is that it and its inverse transform are identical [21]. Therefore it can have computational advantages over the Fourier transform [21]. The HT [8] of a real function
f (x ,y ) is defined asand the inverse Hartley transform (IHT) is defined as
where
cas =cos +sin ,HT [•] andIHT [•] denote the Hartley and inverse Hartley operators.2.3. Double Random Phase Encoding in the Gyrator Domain
In mathematics, the GT with respect to parameter
α , called as a fractional order [7], of a twodimensional functionf (x ,y ) can be expressed as [7, 9]where the
G_{α} (p ,s ) andf (x _{0},y _{0}) are the output and input of the transform, respectively.GTα [•] represents gyrator operator. The inverse GT (IGT) corresponds to the GT with respect to α and is given bywhere
IGT [•] denotes the inverse gyrator operator. The GT can be achieved either by using an optical system [9] or by using a fast algorithm [22].In Fig. 1, the optical setup of DRPE in the gyrator domain is shown. Three planes are defined as the input plane, the transform plane, and the output plane. The corresponding coordinates of the three planes are denoted by (
x _{0},y _{0}), (p ,s ) and (x _{1},y _{1}), respectively. The two random phase masksRPM 1 andRPM 2 are represented by exp[j2πφ (x _{0},y _{0})] and exp[j2πϕ ( )], wherep ,s φ (x _{0},y _{0}) andϕ ( ) are two random functions distributed uniformly in the interval [0,1]. Using a system which is composed of three generalized lenses L_{1}, L_{2} and L_{1} with fixed equal distances between them, the GT can be implemented optically. GT at orderp ,s α is reached by rotation of these lenses [9]. In Fig. 1, the dashed block GT1 which contains lenses L_{1}, L_{2} and L_{1} indicates the first optical GT with respect toα _{1}, and the other GT2 containing lenses L'_{1}, L'_{2} and L'_{1} indicates the second optical GT with respect toα _{2}. When the security system is illuminated perpendicularly by a plane wave, the encrypted datag (x _{1},y _{1}) is obtained at the output plane [9]:By applying IGT to the encrypted data with the conjugates of the
RPM 1 andRPM 2, the decrypted image can be obtained as follows [9]:2.4. Phaseshifting Interferometry
Since digital holography [2] provides a convenient form of recording the complex encrypted images after passing through the DRPE systems, phaseshifting interferometry is employed to record the complex resulting encrypted signal in the proposed scheme. A variety of PSI techniques have been developed, including threestep, fourstep, etc [23].
Let
A (x ,y )exp[jψ (x ,y )] andA_{r} exp(jδ_{k} ) be the complex amplitude distributions of the object wave in the recording plane and the reference wave in that plane at thek th exposure, respectively. Here,A_{r} is a constant,δ_{k} is the phase shift of the reference wave between two adjacent steps andk =1,2,…,N . Thek ^{th} interference patternI_{k} (x ,y ) can be represented as [24],For a known set {
δ_{k} }(k =1,2,...,N ), a digital hologramU (x ,y ) can be derived as a function {I_{k} } and {δ_{k} }[24]. In the threeframe case,N is 3. Whenδ _{1}=0,δ _{2}=π /2 andδ _{3}=π , a digital hologramU (x ,y ) from the three interferogramsI _{1},I _{2} andI _{3} can be expressed as [24, 25]2.5. Logistic Map
Chaos theory is an evolutionary theory, which describes that the nonlinear dynamical systems change from ordered state to disordered state [26]. The dynamical systems are established based on various chaos functions such as logistic maps, which are extremely sensitive to the initial conditions. These functions generate iterative values which are completely random in nature. In this paper, the twodimensional (2D) logistic map is used to make the change of sequence of image pixels. It is defined as [26]
The dynamic behavior of 2D logistic is controlled by the parameters
ε ,λ _{1},λ _{2} andρ shown in Eq. (12). Whenε =4,ρ =0.1, 0.65≤λ _{1}≤0.9 and 0.65≤λ _{2}≤0.9, the dynamical system is in a chaotic state and slight variations of the initial value will yield a drastically different result which is a nonperiodic and nonconverging sequence over time [26].III. QUATERNION HARTLEY TRANSFORM
3.1. Definition
Due to the noncommutative multiplication property of quaternions, there are different types of QHT that can be defined. In this work, the leftside QHT (QHT^{L}) and the rightside QHT (QHT^{R}) are defined:
• Leftside QHT:
• Rightside QHT:
where
f_{q} (x ,y ) is a twodimensional quaternion function andµ is a pure unit quaternion which meets the constraint thatµ ^{2} =1.QHT^{L} () andQHT^{R} () are the leftside QHT and rightside QHT operations, respectively.Corresponding to QHT, two forms of the inverse QHT (IQHT) are defined as follows
• Leftside IQHT (IQHTL):
• Rightside IQHT (IQHT^{R}):
Here,
IQHT^{L} () andIQHT^{R} () are the inverse leftside QHT and rightside QHT operations, respectively. QHT and IQHT are transformation pairs of each other. They ensure that a quaternion functionf_{q} (x ,y ) which is transformed into the QHT domain can be reconstructed completely by the inverse process without losing any information.3.2. QHT Calculation
In this subsection, the method which makes full use of the existing HT algorithm to calculate the QHT of a quaternion matrix is presented. Since the leftside QHT is used in this work, only the calculation method for it is described. By using the HT algorithm, the QHT can be implemented efficiently.
Since
f_{q} (x ,y ) is a quaternion function, it can be represented aswhere
f _{0}(x ,y ),f _{1}(x ,y ),f _{2}(x ,y ) andf _{3}(x ,y ) are real value functions.For leftside QHT, substituting (17) into (13), we have
Considering the general unit pure quaternion
µ =ξi + ηj +γk (ξ, η andγ are real numbers), substitutingµ into (18) and using the properties of the quaternion shown in Eq. (2), we havewhere
Similarly, applying leftside IQHT to Eq. (19), the reconstructed
f’_{q} (x ,y ) can be obtained.where
It can be observed from formulas (18)~(22), the leftside QHT and IQHT of a quaternion matrix can be calculated effectively by using the traditional HT and IHT algorithms. Notice that the rightside QHT can be processed in a similar way.
IV. THE 2D LOGISTIC MAPBASED IMAGE SCRAMBLING METHOD
By using a scrambling technique, He
et al . claimed that an optical security system can resist some potential attacks such as chosen plaintext attack [27]. Since the dynamic response of the chaotic system is highly sensitive to the initial values and parameters of chaotic variables, and the chaotic trajectory is unpredictable, a 2D logistic mapbased image scrambling technique is proposed to permute the position of image pixels.Let
I (m ,n ) be the tobepermuted image. The scrambling method is described as follows1) Calculate the height
M and the widthN ofI (m ,n ).2) Initialize
X (1) andY (1) randomly and choose an arbitrary natural numbert first, then iteratively generate the chaotic sequencesX (i ) andY (i ) whose lengths both areMN +t by using Eq. (12). Here,i =1,2, …,MN +t .3) Generate two integers
s _{1} ands _{2} which are between 1 andt randomly first. In other words, 1≤s _{1}≤t and 1≤s _{2} ≤t . Then truncateNM elements ofX (i ) from thes _{1}^{th} element to obtain a chaotic sequenceL 1={X (i ),i =s _{1},s _{1}+1,…,s _{1}+MN 1}. Similarly, another chaotic sequenceL 2={Y (i ),i =s _{2},s _{2}+1,…,s _{2}+MN 1} can be obtained by truncating fromY (i ).4) Compute the chaotic sequences
L 1 andL 2 using the following equationwhere round(
Z ) is the operation that rounds the elements ofZ to the nearest integers, andd is an integer. After calculation, the irregularity and distribution uniformity of the sequencesL 1 andL 2 can be enhanced, and all the elements ofL 1 andL 2 are larger than 0.5 and less than 0.5 [28]. That is 0.5<L 1<0.5 and 0.5<L 2<0.5.5) Sort the sequences
L _{1} andL _{2} in a certain (ascending or descending) order to obtain two new sequenceL ’1 andL ’2 and their corresponding permutation indicesIX 1 andIY 2. InIX 1 andIY 2, there areMN elements respectively. The relations betweenL 1 and L’1, and betweenL 2 andL ’2 areL ’1=L 1(IX 1) andL ’2=L 2(IY 2), respectively. For example, them ^{th} element inL ’1 corresponds to theIX 1(m )^{th} element inL 1.6) Map
I (m ,n ) into a onedimensional (1D) arrayI ’ using the zigzag algorithm [29]. So the length ofI ’ isMN 7) Use the permutation indices
IX1 andIY2 to permuteI ’ and the scrambled vectorI ''' can be obtained as follows8) Finally, the scrambled image
SI can be achieved by applying inverse zigzag scan [29] process toI '''.The inverse image scrambling process is similar to the image scrambling process. In decryption (inverse scrambling process), obtain the permutation indices
IX 1 andIY 2 as described in steps (1)~(5) with the same parameters first. Then map the scrambled imageSI into a 1D vectorSI ’ using the zigzag algorithm. Subsequently, permuteSI ’ back to their original position using the following equationsFinally, the decrypted image
DI can be retrieved by applying the inverse zigzag algorithm toSI '''. Since the proposed logisticbased scrambling method is highly sensitive to the parameters, including the initial valuesX (1) andY (1),λ _{1},λ _{2},s _{1}, ands _{2}, it offers a huge level of security.V. THE PROPOSED HYBRID COLOR AND GRAYSCALE IMAGE ENCRYPTION AND DECRYPTION
Let
f_{RGB} (x ,y ) be a color image with sizeM×N in the RGB color space. In principle, each image pixel off_{RGB} (x ,y ) can be treated as a pure quaternion number with real part equal to zero [17]:where
f_{R} (x ,y ),f_{G} (x ,y ), andf_{B} (x ,y ) are the red, green and blue channels off_{RGB} (x ,y ), respectively. By using this representation, each color triple can be treated as a whole unit and be applied in encryption and watermarking [17, 18, 30]. Hence, a color image which is represented by a pure quaternion can be processed holistically in a vector manner using QHT. Furthermore, it should be noticed that a quaternion contains four parts, which results that more than a simple color image can be processed in encryption. For example, the real part of the quaternion matrix can be used to represent a grayscale image while the three imaginary parts represent a color image.Therefore, based on the QHT, the scrambling method and the GTbased DRPE, a hybrid color and grayscale images encryption scheme is proposed. In the proposed method, the three step PSI is used to record the encrypted data. The optoelectronic setup of the proposed encryption process is shown in Fig. 2. Supposing
f_{gs} (x ,y ) denotes a grayscale image with sizeM×N , the proposed method is described as follows1) Normalize
f_{RGB} (x ,y ) andf_{gs} (x ,y ) first. To treat the color image and grayscale image in a holistic manner,f_{RGB} (x ,y ) andf_{gs} (x ,y ) are represented by quaternion: 1) Normalizef_{RGB} (x ,y ) andf_{gs} (x ,y ) first. To treat the color image and grayscale image in a holistic manner,f_{RGB} (x ,y ) andf_{gs} (x ,y ) are represented by quaternion:f_{q} (x ,y ) =f_{gs} (x ,y ) +if_{R} (x ,y ) +if_{G} (x ,y ) +kf_{B} (x ,y ) .2) Apply QHTL to fq(x,y):
HF (u ,v )=QHT^{L} [f_{q} (x ,y )]=A _{0}(u ,v ) +iA _{1}(u ,v ) +jA _{2}(u ,v ) +jA _{2}(u ,v ) +kA _{3}(u ,v ) .Here,
QHT^{L} () is the leftside QHT operation.A _{0},A _{1},A _{2} andA _{3} can be computed by using Eq. (20).3) With
A _{0},A _{1},A _{2} andA _{3}, form a new matrix using the following formula.Since the length and width of
f_{RGB} (x ,y ) andf _{gs} (x ,y ) are bothM×N , the size ofAI is2M×2N .4) Permute
AI to obtainAIS by the proposed chaosbased scrambling method mentioned in Section 4 with the parametersX (1),Y (1),λ 1,λ 2,s _{1} ands _{2}. That isAIS =scramble (AI, X (1),Y (1),λ _{1},λ _{2},s _{1},s _{2}). Here scramble() is the scrambling operation.5) Let
EA =AIS _{1}+jAIS _{2}, whereAIS _{1}=AIS [1:M , 1:2N ] is the first half part of matrixAIS andAIS _{2}=AIS [M +1:2M , 1:2N ] is the second half part ofAIS , respectively.EA can be regarded as the complex amplitude that will be encrypted by the DRPE technique in the gyrator domain. The size ofEA isM×2N .6) The complex amplitude
EA is multiplied byRPM 1, mathematically represented by functionU (x _{0},y _{0}) =EA (x _{0},y _{0}) exp[j2πφ (x _{0},y _{0})].7) As shown in Fig. 2, import
U (x _{0},y _{0}) into the first spatial light modulator (SLM) SLM1 which is located in the input plane under the control of the computer, and then optically transformed by first GT at orderα _{1}. The resulting complex distributionU _{1}( ) can be represented byp ,s 8) Subsequently, the complex function
U_{1} ( ) is multiplied byp ,s RPM 2 displayed on the SLM2 which is located in GT plane, and then optically transformed by second GT at orderα _{2}. So the object wave becomeswhere
A_{c} (x _{1},y _{1}) andψ (x _{1},y _{1}) are the amplitude and the phase ofU_{c} (x _{1},y _{1}), respectively.9) For convenience of storage and transmission, the object wave and reference wave overlap to produce realvalued interferograms. As shown in Fig. 2, the resulting encrypted data is recorded as three interference patterns
I_{1} ,I_{2} andI_{3} in threestep PSI case mentioned in Subsection 2.4.In the encryption process mentioned above, the steps (1)~(6) will be performed digitally. In Fig. 2, M1, M2 are mirrors, and BS1 and BS2 are beam splitters. GT1 and GT2 are corresponding to the gyrator transform systems GT1 and GT2 shown in Fig. 1, respectively. As shown in Fig. 2, the complex function
U (x _{0},y _{0}) obtained in step (6) and theRPM 2 are entered into the SLM1 and SLM2 respectively under the control of computer in order to be transformed, respectively. The phase shiftδ_{k} (k =1, 2, 3) is introduced by a mirror attached with a Piezoelectric Transducer (PZT) controlled by a computer at thek exposure [31]. Finally, the resulting encrypted signal is captured and recorded as three interference patterns by a charge coupled device (CCD) which is placed at the output plane of the encryption system. The flowchart of the encryption process is shown in Fig. 3.The parameters of the proposed encryption method, including
X (1),Y (1),λ _{1},λ _{2},s _{1},s _{2},RPM 1,RPM 2 and the orders of GTα _{1} andα _{2}, form a very large key space enhancing the security level of the encryption system. Only when all the right keys are simultaneously used for decryption can the encrypted data be reconstructed correctly.The decryption process, which is similar to the encryption process, but in the reversed order is depicted as follows:
1) Using the three interferograms
I _{1},I _{2} andI _{3}, an encrypted digital hologramUE (x _{1},y _{1}) can be achieved using Eq. (11).2) Apply a GT to
UE (x _{1},y _{1}) with order α2 and then multiply the obtained complex distribution by the conjugate of theRPM 2.3) Make another GT with order 
α _{1} and then multiply the achieved complex distribution by the conjugate of theRPM 1. The resulting distribution function is as followswhere
EA_{re} (x _{0},y _{0}) andEA_{im} (x _{0},y _{0}) are the real part and imaginary part ofEA’ (x _{0},y _{0}), respectively.4) With
EA_{re} andEA_{im} , form a new matrix using the following formula.5) With the parameters
X (1),Y (1),λ _{1},λ _{2},s _{1} ands _{2},AIS’ is permuted by the proposed inverse scrambling process mentioned in Section 4 to obtainAI’ .6) First, let
A ’_{0}=AI’[1:M , 1:N ],A ’_{1}=AI’ [1:M ,N +1:2N ],A ’_{2}=AI’ [M +1:2M , 1:N ] andA’ _{3}=AI’ [M +1:2M ,N +1:2N ]. Then letHF’ (u ,v )=A’ _{0}+iA’ _{1}(u ,v )+jA’ _{2}(u ,v )+kA’ _{3}(u,v) .7) Apply IQHT to
HF’ (u ,v ) to achievef’q (x ,y ).8) With the normalized
f’_{R} (x ,y ),f’_{G} (x ,y ) andf’_{B} (x ,y ), the decrypted color imagef’_{RGB} (x ,y ) can be obtained and the normalizedf’_{gs} (x ,y ) is the decrypted grayscale image.Figure 4 depicts the decryption process. In Fig. 4, conj(
RPM 2) and conj(RPM 1) are the conjugate ofRPM 2 and the conjugate ofRPM 1, respectively.VI. NUMERICAL SIMULATION RESULTS
To verify the feasibility of the proposed encryption technique, numerical simulations are performed on the color image “Peppers” and the grayscale image “Lena”, which sizes are both 512×512, shown in Fig. 5(a) and Fig. 4(b). We carried out tests on a notebook computer with Intel(R) Core(TM) i74700HQ CPU @ 2.40GHz and 8G DDRL3 and with the MATLAB R2013a. In the experiments, the system parameters are
µ =(i+j+k )/3^{1/2},α _{1}=0.92,α _{2}=0.75,ε =4,ρ =0.1,X (1) = 0.24,Y (1) = 0.67,t =50000,λ _{1}=0.716,λ _{2}=0.881,s _{1}= 12345,s _{2}= 36578 andd =4. To evaluate the performance of image reconstruction quantitatively, the normalized mean square error (NMSE) [18] was used to calculate the similarity between the decrypted image and the original image, which is expressed aswhere,
f_{o} (x ,y ) andf_{d} (x ,y ) are the normalized original and the decrypted images respectively, andM×N is the size of the images.6.1. Performance of the Encryption System
Using the proposed encryption scheme, the “Peppers” and “Lena” are encrypted and three encrypted interferograms are obtained, which are shown in Figs. 5(c)~5(e). Figs. 5(f)~5(g) display the retrieved color and grayscale images with the correct keys respectively, which are perfect, without any noise or distortion. As shown in Table 1, the NMSE between Fig. 5(a) and Fig. 5(f) and that between Fig. 5(b) and Fig. 5(g) are 1.7017×10^{30} and 1.173×10^{30}, respectively.
Now we investigate the sensitivity of retrieved image to small change of the parameters of 2D logistic mapbased scrambling technique on the decrypted results. Figure 6 showed the derivation of NMSE versus the parameters
X (1),Y (1),λ _{1},λ _{2},s _{1} ands _{2}, respectively. Figs. 7(a)~7(l) show the decrypted color and grayscale images with wrong keysX (1)=0.24+1.0×10^{15},Y (1)=0.671.0×10^{15},λ _{1}=0.716 +1.0×10^{16},λ _{2}=0.8811.0×10^{16},s _{1}= 12344 ands _{2}= 36579, respectively.The corresponding NMSEs are listed in the upper part of Table 1. Please note that in the above experiments, the other keys remain correct while a key is changed in decryption. As illustrated in Figs. 6(a)~6(d) and Figs. 7(a)~7(h), we cannot obtain any information from the decryption images visually when the absolute values of deviations of
X (1) andY (1) are up to 10^{15} and those ofλ _{1} andλ _{2} are up to 10^{16}. In addition, we know from Figs. 6(e)~6(f) and Figs. 7(i)~7(l) that if the parameterss _{1} ands _{2} are less 1 or more 1 than the correct value, the decrypted images cannot afford any valid information. From the NMSE values shown in Table 1, it can also be concluded that the decrypted images cannot be recognized even if the keys of the scrambling system are changed slightly. So, the parametersX (1),Y (1),λ _{1},λ _{2},s _{1} ands _{2} are highly sensitive to the proposed method.In the experiments, one of the two
RPM s was removed or just one of them was shifted transversely by one pixel. The NMSE values of the decrypted images are listed in the middle part of Table 1. Figs. 8(a)~8(b) display the decrypted color and grayscale images, which are totally unrecognizable, with other correct keys but withoutRPM 1. The recovered color and grayscale images, which were obtained fromRPM 1 shifted transversely by one pixel, are shown in Figs. 8(c)~8(d). These two images cannot be recognized too. The lack of theRPM 2 or small changes toRPM 2 in the decryption step also leads to similar results which are exhibited in Figs. 8(e)~8(h). Experimental results convincingly demonstrate that bothRPM s are necessary in decryption and that the illegal users cannot access the security system without theRPM s.To examine the sensitivity of small change of the orders of the GT, the decryption processes are performed by fixing one order and varying the other. The relationship curves between the NMSE and the deviation of the GT order are shown in Fig. 9, in which the deviation ranges from 0.1 to 0.1 and the step is 0.001. As can be seen from Fig. 9, the NMSE value approximates to zero when
α _{1} orα _{2} is correct while the value sharply increases whenα _{1} orα _{2} slightly departs from the correct value, which indicates that any tiny fluctuation will result in false decryption. The NMSE values shown in the lower part of Table 1 also indicate that if the orders of inverse GT are changed by 0.004 from their correct values, the decrypted images are noiselike images as displayed in Fig. 10 and cannot be recognized.Next we estimate the key space of the proposed encryption scheme. According to the description of the proposed method, we know that the key space of the cryptosystem consists of the
RPM s, the orders of GT and the parameters of 2D logistic map. In addition, the chaotic permutation module is independent from the GTbased DRPE system. In the logisticbased permutation process, the key spaces of the parametersX (1),Y (1),λ _{1},λ _{2},s _{1} ands _{2} should be analyzed, which are denoted byZ _{1},Z _{2},Z _{3},Z _{4},Z _{5} andZ _{6}, respectively. In the GTbased DRPE system, the key spaces ofRPM 1,RPM 2 and the ordersα _{1} andα ,_{2}, should be analyzed, which are denoted byZ _{7},Z _{8},Z _{9} andZ _{10} respectively. From Figs. 6(a)~6(d) and Figs. 7(a)~7(h), the parametersX (1),Y (1),λ _{1} andλ _{2} maintain 15, 15, 16 and 16 digits after decimal point respectively, soZ _{1}×Z _{2}×Z _{3}×Z _{4}=10^{62}. Since 1≤s _{1}≤t and 1≤s _{2}≤t ,Z _{5}×Z _{6}=t ^{2} . Therefore,Z _{1}×Z _{2}×Z _{3}×Z _{4}×Z _{5}×Z _{6}=t ^{2}×10^{62}. In our experiments, Z_{1}×Z_{2}×Z _{3}×Z _{4}×Z _{5}×Z _{6}=2.5×10^{71} sincet =50000. Suppose that 256 levels are chosen for a pixel in initial random functionsφ (x _{0},y _{0}) andϕ ( ) ofp ,s RPM 1 andRPM 2, the twoRPM s with size ofM ×2N need to be decoded with 2×256^{M×2N} attempts. In this case,M =N =512, soZ _{7}×Z_{8} =2×256^{512×1024}. Considering the sensitivity to one GT order of encryption system is 0.004, the fractional ordersα _{1} andα _{2} can be varied with steps of 0.001 in fractional orders range [0, 1] [7], which results inZ _{9}×Z _{10}=(1/0.001)^{2} =10^{6}. Therefore, the entire key space of the cryptosystem isZ _{1}×Z _{2}×Z _{3}×Z _{4}×Z _{5}×Z _{6}×Z _{7}×Z _{8}×Z _{9}×Z _{10}=5×256^{512×1024}×10^{77}≈ 256^{512×1024}×2^{258}. As stated in [32], to acquire a high level of security, the size of key space should at least be larger than 2^{100}. It is apparent that the key space of the proposed encryption system is far larger than 2^{100} and enormous enough to resist brute force attack.6.2. Robustness of the Method Against Attacks
Information loss or noise contamination may occur during data transmission. Now, the robustness of this scheme against occlusion attack which is regarded as data loss is tested. Figure 11(a) shows one of the interferograms occluded by 50%. The recovered images obtained with all correct keys are illustrated in Figs. 11(b)~11(c). It’s well known that Gaussian noise [33] and salt & pepper noise [33] are frequently appearing noises during the information transmission. The robustness test is further verified against noise attacks on the encrypted results. Figure 12(a) is one of the interferograms distorted by Gaussian noise with mean value 0 and standard deviation 20. The corresponding retrieved images are displayed in Figs. 12(b)~12(c). Fig. 12(d) exhibits one of the interferograms damaged by salt & pepper noise with density 0.02 added. Figs. 12(e)~12(f) depicts the corresponding retrieved images. Although all the results shown in Figs. 11(b)~11(c), Figs. 12(b)~12(c) and Figs. 12(e)~12(f) are interfered with seriously by noise and the corresponding NMSEs are big, the secret images among the noise fluctuation can still be distinguished.
Four attacks, including cipher only attack, known plaintext attack, chosen plaintext attack and chosen ciphertext attack [32], are often used to attack the DRPEbased optical security system to recover the ciphered images. Among these attacks, the chosenplaintext attack is the most powerful attack [32]. He
et al . claimed that permutation techniques can prevent the attacker from accessing the ciphertext obtained from the DRPE system in such attacks [27]. In addition, Zhang and Xiao stated that the security system should resist other attacks if it can resist chosen plaintext attack [32]. For the proposed method, we assume that the attacker has acquired the GT ordersα _{1} andα _{2} of the GTbased DRPE system but does not know the parameters of the logisticbased scrambling algorithm. Using two fake color and gray images as the plaintexts, the two random phase masks of the proposed method can be produced by utilizing the chosen plaintext attack described in [34]. By use of these achieved keys, the attacker can crack the original ciphertext. Two keysRPM 1’ andRPM 2’ are generated by choosing Figs. 13(a)13(b) as the fake plaintext images and then employed to decrypt the ciphertext of “Peppers” and “Lena”. As shown in Figs. 13(c)~13(d), the retrieved images cannot reveal any information visually. Therefore, the robustness against chosenciphertext attack is greatly enhanced because of the suggested logisticbased scrambling technique, which strengthens the nonlinearity in QHT domain. According to [32], the proposed encryption approach can also resist the other attacks mentioned above.6.3. Complexity Analysis
In the numerical simulation, the encrypted results were obtained by computer. So the computational complexity of the proposed encryption technique is calculated. The calculation depends on three main factors. First, to deal with a color image and a grayscale image holistically in a vector manner, the QHT is computed one time. According to the developed calculation method in subsection 3.2, the QHT can be implemented via calculating HT four times. For an image with size
M ×N , the cost to compute the HT isO (MIN log_{2}MN ) [35]. Hence, the complexity of QHT calculation isO (4MIN log_{2}MN ). Second, to scramble a matrix using the proposed chaosbased technique, the 2D logistic map operation, the round operation, the quicksort algorithm [36], the zigzag scan operation, the inverse zigzag scan operation and the sequence permutation operation are performed one time, two times, two times, one time, one time and one time, respectively. As shown in the steps (3) and (4) of the proposed encryption method, the size of the tobescrambled matrixAI is 2M ×2N though the sizes of the color image and grayscale image are bothM ×N . Therefore, the computational cost of the scrambling procedure isO (2(4MN +t ) + 2(4MN +t ) + 2(4MN +t )_{log2}(4MN +t )) + (4MN +t ) + (4MN +t ) + (4MN +t ). Here,t is the natural number shown in step (2) of the proposed scrambling method. Since 4MN +t ≈4MN and log_{2}(4MN +t )≈2+log_{2}MN asMN tends to infinity, the complexity of the scrambling procedure isO (8MN log_{2}MN + 44MN ). The last factor is the complexity of the virtual optical encryption process. In the proposed encryption method, an encrypted interferogram can be obtained after the complex amplitude being modulated two times by use of random phase masksRPM 1 andRPM 2, transformed two times by use of GT and interfered one time by use of the reference wave. For a matrix with sizeM×N , the complexities of GT operation, modulation operation and interference operation areO (2MN log_{2}MN ) [22],O (MN ) [37] andO (4MN ) [37], respectively. Because the size of complex amplitude isM×2N , the computational cost of the above process isO (3(8MN log_{2} 2MN + 4MN + 8MN )) =O (24MN log_{2}MN + 60MN ). Thus, the total computational complexity of the proposed encryption scheme isO (36MN log_{2}MN +104MN ). SinceO (MN log_{2}MN ) >O (MN ), compared with the quaternion gyrator transformbased image encryption method whose computational complexity isO (4000MN log_{2}MN ) [18], the proposed scheme is more efficient. In the experiments, the execution time for the encrypted results is 6.7728 seconds.VII. CONCLUSION
In the paper, a novel definition of quaternion Hartley transform and its implementation are presented first. Then a new method for hybrid color and grayscale images encryption is proposed, in which the QHT combined with the developed chaosbased pixel scrambling technique, the GTbased DRPE and the threestep PSI is used to encrypt the color and grayscale images. By use of QHT, a color image and a grayscale image are processed holistically in a vector manner without separating the color image into three channels and manipulating the grayscale image independently, so that the complexity of the security system can be reduced effectively without any reduction of its security. The components of the QHTtransformed original images are combined as the complex amplitude which will be strengthened the nonlinearity by the designed chaosbased scrambling technique and encrypted by the GTbased DRPE technique. In addition, the encrypted interferograms which are recorded by threestep PSI are realvalued, so they are convenient for storage and transmission. One needs to specify all the right keys to recover the original images correctly. Simulation results demonstrate that the proposed scheme has a quite sensitivity to the decryption keys, a enormous key space to resist brute force attack and a good robustness against Gaussian noise, salt & pepper noise and some attacks such as chosen plaintext attack.

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[FIG. 1.] Optical setup for the double random phase encoding in the gyrator domain.

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[FIG. 2.] Optoelectronic hybrid system implementing the proposed encryption method.

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[FIG. 3.] Flowchart of encryption process.

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[FIG. 4.] Flowchart of decryption process.

[FIG. 5.] Results of the proposed image encryption. (a) The original color image “Peppers”; (b) the original grayscale image “Lena”; (c), (d), (e) interferograms I1, I2 and I3, respectively; (f) correctly decrypted color image; (g) correctly decrypted grayscale image.

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[FIG. 6.] The NMSEs of the proposed method: (a), (b) the NMSEs for deviation of X(1) and Y(1); (c), (d) the NMSEs for deviation of λ1 and λ2; (e), (f) the NMSEs for deviation of s1 and s2.

[FIG. 7.] The decrypted color “Peppers” and grayscale “Lena” with: (a), (b) incorrect X(1)=0.24+1.0×1015; (c), (d) incorrect Y(1)=0.671.0×1015; (e), (f) incorrect λ 1=0.716+1.0×1016; (g), (h) incorrect λ2=0.8811.0×1016; (i), (j) incorrect s1= 12344; (k), (l) incorrect s2= 36579.

[TABLE 1.] Comparison between the NSMEs of the decrypted “Peppers” and “Lena” by use of correct and incorrect keys

[FIG. 8.] The decrypted color “Peppers” and grayscale “Lena” with: (a), (b) lack of RPM1; (c), (d) RPM1 is shifted transversely by one pixel; (e), (f) lack of RPM2; (g), (h) RPM2 is shifted transversely by one pixel.

[FIG. 9.] NMSEs versus (a) the deviation of order α1 and (b) the deviation of order α2.

[FIG. 10.] The decrypted color “Peppers” and grayscale “Lena” with: (a), (b) incorrect α1= 0.92+0.004; (c), (d) incorrect α2=0.750.004.

[FIG. 11.] Robustness against occlusion. (a) one of the interferograms with 50% occlusion; (b) the recovered “Peppers” (NMSE=0.498); (c) the recovered “Lena” (NMSE=0.4165).

[FIG. 12.] Robustness against noise. (a) one of the interferograms after Gaussian noise with mean value 0 and standard deviation 20; (b) the recovered “Peppers” from interferograms that have undergone Gaussian noise addition (NMSE=0.4471); (c) the recovered “Lena” from interferograms that have undergone Gaussian noise (NMSE=0.4133); (d) one of the interferograms after salt & pepper noise with density 0.02 added; (e) the recovered “Peppers” from interferograms that have undergone salt & pepper noise (NMSE =0.4247); (f) the recovered “Lena” from interferograms that have undergone salt & pepper noise (NMSE=0.4157).

[FIG. 13.] Robustness against chosen plaintext attack. (a), (b) Two fake plaintext images “Fruits” and “Elaine”; (c), (d) decrypted “Peppers” and “Lena”, the corresponding NMSEs are 0.5161 and 0.4924, respectively.