Vol. 20, No. 3, June 2016, pp. 341-357
- 341 -
Asymmetric Multiple-Image Encryption Based on Octonion Fresnel Transform and Sine Logistic Modulation Map
Jianzhong Li*
Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, Guangdong 521041, China
(Received February 16, 2016 : revised April 25, 2016 : accepted May 19, 2016)
A novel asymmetric multiple-image encryption method using an octonion Fresnel transform (OFST) and a two-dimensional Sine Logistic modulation map (2D-SLMM) is presented. First, a new multiple-image information processing tool termed the octonion Fresneltransform is proposed, and then an efficient method to calculate the OFST of an octonion matrix is developed. Subsequently this tool is applied to process multiple plaintext images, which are represented by octonion algebra, holistically in a vector manner. The complex amplitude, formed from the components of the OFST-transformed original images and modulated by a random phase mask (RPM), is used to derive the ciphertext image by employing an amplitude- and phase-truncation approach in the Fresnel domain. To avoid sending whole RPMs to the receiver side for decryption, a random phase mask generation method based on SLMM, in which only the initial parameters of the chaotic function are needed to generate the RPMs, is designed. To enhance security, the ciphertext and two decryption keys produced in the encryption procedure are permuted by the proposed SLMM-based scrambling method. Numerical simulations have been carried out to demonstrate the proposed scheme's validity, high security, and high resistance to various attacks.
Keywords: Octonion Fresnel transform, Sine logistic modulation map, Asymmetric cryptosystem, Multiple- image encryption
OCIS codes : (060.4785) Optical security and encryption; (070.2590) ABCD transforms; (100.2000) Digital image processing
*Corresponding author: [email protected]
*
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/
licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
*Copyright 2016 Optical Society of Korea I. INTRODUCTION
Over the past two decades, optical image encryption techniques have played an important role in the information security field. Since development of the double random phase encryption (DRPE) technique [1], in which the image is encoded to be a white noise pattern with two statistically independent random phase masks (RPM), a number of improved DPRE-based image encryption methods have been proposed, to achieve higher security [2-7]. Most of the optical encryption methods can be viewed as symmetric cryptosystems, in which the keys for encryption and decryption are identical [8]. However, earlier studies indicated that these cryptosystems are lacking in security strength, because of the inherently linear property of mathematical or optical transformation, and are vulnerable to various
attacks such as chosen plaintext attack [8-11]. In addition, most schemes mainly focused on single-image encryption, which leads to deficiency in multiple-image encryption and transmission [8].
To avoid the problems that arise from the linearity and symmetry of DRPE, an asymmetric encryption scheme based on phase truncated Fourier transforms has also been proposed [12], in which the encryption keys (public keys) are different from the decryption keys (private keys).
However, when RPMs are used as public keys to encrypt different plaintexts, this method is still vulnerable to specific attacks such as known public key attack [13, 14]. To enhance security, it has been further extended from the Fourier transform (FT) domain to the Fresnel transform (FST) domain [15], the fractional Fourier transform (FrFT) domain [8], the Gyrator transform (GT) domain [11, 16],
DOI: http://dx.doi.org/10.3807/JOSK.2016.20.3.341
TABLE 1. Octonion multiplication table
i j k l m n o
i -1 k -j m -l -o n
j -k -1 i n o -l -m
k j -i -1 o -n m -l
l -m -n -o -1 i j k
m l -o n -i -1 -k j
n o l -m -j k -1 -i
o -n m l -k -j i -1
etc. To improve the efficiency of image transmission and communication over a network, more and more multiple- image encryption methods have been proposed [17-21].
Situ and Zhang reported the wavelength multiplexing-based multiple-image encryption scheme [16]. Using DRPE and an orthogonal encoding technique, Lee and Cho presented a multiple-image encryption method [18]. Based on the nonlinear amplitude-truncation (AT) and phase-truncation (PT) operations in the FT domain, Wang and Zhao developed an asymmetric multiple-image encryption scheme [19]. Based on coupled logistic maps and an iterative phase-retrieval process, Sui et al. introduced an asymmetric multiple-image encryption technique in the FrFT domain [20], but it has a high computational cost due to its iterative phase-retrieval process. Abuturab designed a multiple-color-image security system based on a generalized Arnold map in the GT domain [21]. However, the plaintexts in most multiple-image encryption methods are manipulated independently, resulting in complicated systems. Additionally, the increase in keys, such as the whole RPMs that have to be sent to the receiver side for decryption, in the aforementioned DRPE-based encryption schemes make it difficult to save and distribute the keys expediently and safely [22-23].
In this paper, based on the octonion Fresnel transform (OFST) and two-dimensional Sine Logistic modulation map (2D-SLMM), a novel asymmetric multiple-image encryption method is presented. In the proposed approach, first the OFST is defined and its calculation for an octonion matrix is developed. Then multiple images are processed holistically in a vectorial manner using OFST. Subsequently, the components of the OFST-transformed images are combined to construct the input complex amplitude. To improve robustness against attacks, the input information is modulated by a phase mask, which is generated based on 2D-SLMM.
The initial values of the SLMM, which is computed using the components of the OFST-transformed images, are used as decryption keys. With AT, PT, and RPMs, the complex amplitude is encrypted in the Fresnel domain, and two decryption keys are produced in the encryption process. To enhance security, the encrypted result and two decryption keys are permuted by the proposed SLMM-based scrambling method. During decryption, the eight original grayscale images can be recovered with the correct encryption and decryption keys. Numerical simulations are presented to demonstrate the feasibility and performance of this encryption.
II. RELATED BACKGROUND
In this section, we discuss some related theories before extending the traditional FST to the octonion domain, showing how to calculate the OFST of an octonion matrix, and designing the 2D-SLMM-based scrambling technique, etc.
2.1. Octonion Number
Octonions can be viewed as octets (or 8-tuples) of real
numbers [24] and have been applied in image processing, such as image saliency detection [25]. An octonion can be represented as follows [24, 25]:
0 1 2 3 4 5 6 7 ,
ON i j k l m n o (1) where ζ0, ζ1, ζ2, ζ3, ζ4, ζ5, ζ6, and ζ7 are real numbers, and i, j, k, l, m, n and o are seven imaginary units which satisfy the following rules [24]
2 2 2 2 2 2 2 1.
i j k l m n o (2) The addition and subtraction of octonions are performed by adding and subtracting corresponding terms [25]. However, the multiplication of octonions is neither commutative nor associative [24, 25]. The multiplication rules, which describe the result of multiplying the element in the pth (p= 1, 2,…, 7) row by the element in the qth (q= 1, 2,…, 7) column, are given in Table 1 [25].
The conjugate and modulus of an octonion are respectively defined by [25]
*
0 1 2 3 4 5 6 7
* 2 2 2 2 2 2 2 2
0 1 2 3 4 5 6 7
| |
ON i j k l m n o
ON ON ON
(3) When ζ0 = 0, ON is a pure octonion.
2.2. The Fresnel Transform
The Fresnel transform of a function f(x,y) is given by [26]
1 exp( 2 / )
( , ) D[ ( , )] ( , ) j D
F u v FST f x y f x y
D j
2 2
( ) ( )
exp[ u x v y ]
j dxdy
D
2 2 2
1 ( , ) exp[ 2 ( ) ( ) ]
f x y j j D u x v y dxdy
D D
(4)
FIG. 1. A schematic optical experimental arrangement for asymmetric cryptography in the Fresnel domain [15].
where FSTD[•] denotes the Fresnel transform with respect to D. In Eq. (4), D is the diffraction distance, λ is the wavelength, and (x,y) and (u,v) are the coordinates of the input and the transform planes respectively.
Since the FST of f(x,y) is a complex function, F(u,v) can also be expressed as follows
[ ( , )] Re{ [ ( , )]} Im{ [ ( , )]}
D D D
FST f x y FST f x y j FST f x y (5) where Re(x) and Im(x) represent respectively the real and imaginary parts of the complex number x.
For simplicity, let P=π[2D2+(u-x)2+(v-y)2]/(λD). According to Euler’s formula exp(jx)=cosx+jsinx, Eq. (4) can be rewritten as
[ ( , )] 1 ( , ) (cos sin )
1 ( , )sin ( , )cos ,
FST f x yD f x y j P j P dxdy D
f x y Pdxdy j f x y Pdxdy
D D
-
= -
(6) Comparing Eqs. (5) and (6), the real and imaginary parts of f(x,y) after FST are as follows
Re{ [ ( , )]} 1 ( , )sin ,
Im{ [ ( , )]} 1 ( , )cos .
D
D
FST f x y f x y Pdxdy D
FST f x y f x y Pdxdy D
-
(7)
The inverse Fresnel transform (IFST) means an FST of a Fresnel diffraction pattern with the distance parameter – D [27].
2 2 2
( , ) [ ( , )] [ ( , )]
1 ( , ) exp[ 2 ( ) ( ) ] .
( ) ( )
D D
f x y IFST F u v FST F u v
D u x v y
F u v j j dudv
D D
(8) 2.3. The Asymmetric Encryption Technique in The
Fresnel Domain
Based on the amplitude-truncated and phase-truncated strategy, an asymmetric cryptosystem in the Fresnel domain was reported in [15]. Figure 1 shows the schematic optical encryption setup for this asymmetric cryptography [15]. In Fig. 1, three planes are defined as the input plane, the transform plane, and the output plane. The corresponding coordinates of the three planes are denoted respectively by (x0,y0), (s,t) and (x1,y1). The distances between adjacent planes are D1 and D2. Two random phase masks RPM1 and RPM2 are located in the input plane and the transform plane, respectively.
In the asymmetric encryption method in the Fresnel domain, when the system is illuminated by a collimated plane wave of wavelength , the input image f(x0,y0) modulated by
RPM1 is transformed by FST with respect to D1, and a complex distribution can be obtained. The real amplitude and phase maps of the complex distribution just before RPM2 can be extracted by employing the angular spectrum algorithm, to describe the free-space propagation process [15]. Then the achieved real amplitude multiplied by RPM2 is Fresnel transformed with respect to D2. Similarly, the real amplitude and phase maps of the Fresnel spectrum in the charge-coupled device (CCD) plane can be extracted.
This last amplitude is the final ciphertext and the two phase maps are use as decryption keys. Since amplitude and phase maps can be displayed by a spatial light modulator (SLM), the asymmetric cryptosystem in the Fresnel domain can be implemented by an optical or digital approach [15].
2.4. 2D Sine Logistic Modulation Map
Due to their excellent properties, such as sensitivity to initial conditions and control parameters, chaotic maps are employed to encrypt the image, which can strengthen the nonlinearity of the image [8]. Since 2D-SLMM offers a wider chaotic range and better ergodicity and hyperchaotic properties than existing chaotic maps [28], it has been chosen to design the scrambling method and generate the RPMs in this study. 2D-SLMM is defined as [28]
1
1 1
[sin( ) ] (1 )
[sin( ) ] (1 ),
n n n n
n n n n
X Y X X
Y X Y Y
(9)
where 0 ≤ α ≤ 1 and 0 ≤ β ≤3 are control parameters.
When β is close to 3, 2D-SLMM exhibits good chaotic performance [28].
III. OCTONION FRESNEL TRANSFORM 3.1. Definition
Similar to Eq. (4), the right-side OFST with respect to D and is defined by Eq. (10), due to the noncommutative and nonassociative multiplication properties of octonions:
( , ) [ ( , )]
oc D oc
F u v OFST f x y
2 2 2
1 2 ( ) ( )
( , ) exp[ ] ,
oc
D u x v y
f x y dxdy
D D
μ μ(10)
where foc(x,y) is a two-dimensional octonion function and μ is a pure unit octonion meeting the constraint that μ2 = -1 [29, 30]. OFST() denotes the OFST operation.
Corresponding to OFST, the inverse right-side OFST (IOFST) is defined as
2 2 2
( , ) [ ( , )] [ ( , )]
1 ( , ) exp[ 2 ( ) ( ) ] .
( ) ( )
oc D oc D oc
oc
f x y IOFST F u v OFST F u v
D u x v y
F u v dudv
D D
μ μ
(11) Here, IOFST() is the IOFST operation. OFST and IOFST are transformation pairs of each other. They ensure that an octonion function foc(x,y) that is transformed into the OFST domain can be reconstructed completely by the inverse process, without losing any information.
3.2. OFST Calculation
The method that utilizes the existing FST algorithm to calculate the OFST of an octonion matrix is developed in this subsection. Using the FST algorithm, the OFST can be implemented efficiently.
Let P = π[2D2+(u-x)2+(v-y)2]/(λD). Using Euler’s Formula for Octonions, exp(μx)=cosx+μsinx [29], Eq. (10) can be rewritten as
( , ) [ ( , )]
oc D oc
F u v OFST f x y
1 foc( , ) (cosx y P sin )P dxdy,
D
μ μ (12)Since foc(x,y) is an octonion function, it can be represented as
0 1 2 3
( , ) ( , ) ( , ) ( , ) ( , ) f x yoc f x y f x y i f x y j f x y k
f x y l f x y m f x y n f x y o4( , ) 5( , ) 6( , ) 7( , ) , (13) where f0(x,y), f1(x,y), f2(x,y), f3(x,y), f4(x,y), f5(x,y), f6(x,y), and f7(x,y) are real-value functions. For simplicity, let foc, f0, f1, f2, f3, f4, f5, f6, and f7 denote foc(x,y), f0(x,y), f1(x,y), f2(x,y), f3(x,y), f4(x,y), f5(x,y), f6(x,y), and f7(x,y) respectively in the following equations.
Considering the general unit pure octonion μ= γ1i + γ2j + γ3k + γ4l + γ5m + γ6n +γ7o (where γ1, γ2, γ3, γ4, γ5, γ6, and γ7 are real numbers), substituting foc and μinto (12), we have
0 1 2 3 4 5 6 7
1 1
( , ) sin cos
1 ( )sin
oc oc oc
F u v f Pdxdy f Pdxdy
D D
f f i f j f k f l f m f n f o Pdxdy D
μ
0 1 2 3 4 5 6 7
1 (f f i f j f k f l f m f n f o)
D
(1i2j3k4l5m6n7o)cosPdxdy
(14)
Using Eq. (7), Eq. (15) can be obtained after calculating Eq. (14):
0 1 2 3
4 5 6 7
1 0 2 0 3 0 4 0
( , ) Re[ ( )] Re[ ( )] Re[ ( )] Re[ ( )]
Re[ ( )] Re[ ( )] Re[ ( )] Re[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
oc D D D D
D D D D
D D D D
F u v FST f FST f i FST f j FST f k
FST f l FST f m FST f n FST f o
FST f i FST f j FST f k FST f l
5 0 6 0 7 0 1 1 2
2 1 3 1 4 1 5 1
6 1 7 1 0 1 2 2
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )]
D D D D
D D D D
D D
FST f m FST f n FST f o FST f i
FST f ij FST f ik FST f il FST f im FST f in FST f io FST f ji
2 2
2 2 4 2 5 2 6 2
2
7 2 1 3 2 3 3 3
4 3 5 3
Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )]
D
D D D D
D D D D
D D
FST f j
FST f jk FST f jl FST f jm FST f jn
FST f jo FST f ki FST f kj FST f k FST f kl FST f
6 3 7 3
2
1 4 2 4 3 4 4 4
5 4 6 4 7 4 1 5
2
Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[
D D
D D D D
D D D D
D
km FST f kn FST f ko
FST f li FST f lj FST f lk FST f l FST f lm FST f ln FST f lo FST f mi FST
5 3 5 4 5 5 5 2
6 5 7 5 1 6 2 6
3 6 4 6 5 6 6
( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ (
D D D
D D D D
D D D D
f mj FST f mk FST f ml FST f m
FST f mn FST f mo FST f ni FST f nj
FST f nk FST f nl FST f nm FST
6 2
7 6 1 7 2 7 3 7
2
4 7 5 7 6 7 7 7
)]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
D D D D
D D D D
f n FST f no FST f oi FST f oj FST f ok
FST f ol FST f om FST f on FST f o
(15) From Table 1, we have
, n ,
, ,
, ,
i jk lm on kj ml no j ki l mo ik nl om k ij lo nm ji ol mn l mi nj ok im jn ko m il oj kn li jo nk n jl io mk lj oi km o ni jm kl in mj lk
(16) Substituting Eqs. (2) and (16) into Eq. (15), we have
0 1 2 3
( , ) ( , ) ( , ) ( , ) ( , ) F u voc F u v F u v i F u v j F u v k
F u v l F u v m F u v n F u v o4( , ) 5( , ) 6( , ) 7( , ) , (17) where
0 0 1 1 2 2 3 3
4 4 5 5 6 6 7 7
1 1 1 0 2 3 3
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ (
D D D D
D D D D
D D D D
F u v FST f FST f FST f FST f
FST f FST f FST f FST f
F u v FST f FST f FST f FST
2
4 5 5 4 6 7 7 6
2 2 1 3 2 0 3 1
4 6 5 7 6 4
)]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ (
D D D D
D D D D
D D D
f
FST f FST f FST f FST f
F u v FST f FST f FST f FST f
FST f FST f FST f
7 5
3 3 1 2 2 1 3 0
4 7 5 6 6 5 7 4
4 4 1 5 2
)] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[
D
D D D D
D D D D
D D
FST f
F u v FST f FST f FST f FST f
FST f FST f FST f FST f
F u v FST f FST f FST
6 3 7
4 0 5 1 6 2 7 3
5 5 1 4 2 7 3 6
4 1 5
( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ (
D D
D D D D
D D D D
D D
f FST f
FST f FST f FST f FST f
F u v FST f FST f FST f FST f
FST f FST
0 6 3 7 2
6 6 1 7 2 4 3 5
4 2 5 3 6 0 7 1
7 7 1
)] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[
D D
D D D D
D D D D
D
f FST f FST f
F u v FST f FST f FST f FST f
FST f FST f FST f FST f
F u v FST f F
6 2 5 3 4
4 3 5 2 6 1 7 0
( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
D D D
D D D D
ST f FST f FST f
FST f FST f FST f FST f
(18) It can be seen from Eq. (18) that each coefficient in F0, F1, F2, F3, F4, F5, F6, and F7 contains the information of f0, f1, f2, f3, f4, f5, f6, and f7. Similarly, applying the right-side IOFST to Eq. (17), the reconstructed f’oc(x, y) can be obtained:
'
0 1 2 3
( , ) [ ( , )]
[ ( , ) ( , ) ( , ) ( , )
oc D oc
D
f x y IOFST F u v
IOFST F u v F u v i F u v j F u v k
F u v l F u v m F u v n F u v o4( , ) 5( , ) 6( , ) 7( , ) ]
' ' ' ' '
0( , ) 1( , ) 2( , ) 3( , ) 4( , ) f x y f x y i f x y j f x y k f x y l
f x y m f x y n f x y o5'( , ) 6'( , ) 7'( , ) ,
(19) where
'
0 0 1 1 2 2 3 3
4 4 5 5 6 6 7 7
'
1 1 1 0 2 3
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ (
D D D D
D D D D
D D D
f x y IFST F IFST F IFST F IFST F
IFST F IFST F IFST F IFST F
f x y IFST F IFST F IFST F
3 2
4 5 5 4 6 7 7 6
'
2 2 1 3 2 0 3 1
4 6 5
)] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] I
D
D D D D
D D D D
D
IFST F
IFST F IFST F IFST F IFST F
f x y IFST F IFST F IFST F IFST F
IFST F
7 6 4 7 5
'
3 3 1 2 2 1 3 0
4 7 5 6 6 5 7 4
' 4
m[ ( )] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
( , )
D D D
D D D D
D D D D
IFST F IFST F IFST F
f x y IFST F IFST F IFST F IFST F
IFST F IFST F IFST F IFST F
f x y
4 1 5 2 6 3 7
4 0 5 1 6 2 7 3
'
5 5 1 4 2 3
Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[
D D D D
D D D D
D D D
IFST F IFST F IFST F IFST F
IFST F IFST F IFST F IFST F
f x y IFST F IFST F IFST F I
6
4 1 5 0 6 3 7 2
'
6 6 1 7 2 4 3 5
4 2 5
( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ (
D
D D D D
D D D D
D D
FST F
IFST F IFST F IFST F IFST F
f x y IFST F IFST F IFST F IFST F
IFST F IFST F
3 6 0 7 1
'
7 7 1 6 2 5 3 4
4 3 5 2 6 1 7 0
)] Im[ ( )] Im[ ( )]
( , ) Re[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
Im[ ( )] Im[ ( )] Im[ ( )] Im[ ( )]
D D
D D D D
D D D D
IFST F IFST F
f x y IFST F IFST F IFST F IFST F
IFST F IFST F IFST F IFST F
(20) In Eq. (20), F0, F1, F2, F3, F4, F5, F6, and F7 denote F0(u,v), F1(u,v), F2(u,v), F3(u,v), F4(u,v), F5(u,v), F6(u,v) and F7(u,v) respectively.
As can be seen from formulas (14)-(20), the right-side OFST and IOFST of an octonion matrix can be calculated effectively using the traditional FST and IFST algorithms.
IV. THE 2D SLMM-BASED RANDOM-PHASE-MASK-GENERATION
METHOD AND 2D SLMM-BASED SCRAMBLING METHOD
Singh et al. [22] and Liu et al. [31] stated that the security and the amount of RPM data of an encryption technique, in which the entire RPMs have to be sent to the receiver side to decrypt the original image, can be improved and reduced by employing a chaotic function to generate the chaotic random distributions that are use to construct the RPMs. He et al. [32] claimed that an optical security system can resist some potential attacks, such as chosen plaintext attack, by use of a scrambling technique.
Since the chaotic sequences generated by 2D SLMM have the characteristics of a good random distribution, and are extremely sensitive to the initial value of the chaotic function, an SLMM-based RPM generation (SBRPMG) method and an SLMM-based scrambling (SBS) method are presented to generate the RPM and permute the positions of image
pixels. Supposing the size of the input data image I(x,y) is M×N, the two methods are described as follows.
4.1. The SLMM-based Random-Phase-Mask-Generation Method
The procedure for generating the RPM can be described as follows:
1) Initialize X(1) and Y(1) randomly, then iteratively generate two one-dimensional (1D) chaotic sequences X = {X(p)} and Y = {Y(p)} using Eq. (9). Here the lengths of X and Y both are MN, and p = 1, 2, …, MN.
2) Compute the chaotic sequences X and Y using the following equation:
X 105X fix(105X), Y 105Y fix(10 ).5Y (21) Here, fix(ρ) rounds each element of ρ to the nearest integer, down toward zero. All elements of X and Y belong to [0, 1].
3) Construct a new 1D sequence R with length MN using the following rule:
( ) , if is an odd number;
( ) ( ) , otherwise.
X p p
R p Y p
(22)
4) Convert the sequence R into a 2D matrix Φ with size M×N, which can be considered a random distribution.
5) With the random distribution Φ, the RPM, represented by exp[j2πΦ(x,y)] can be obtained.
4.2. The SLMM-based Scrambling Method The scrambling method is described as follows:
1) Initialize S(1) and T(1) randomly, then iteratively generate two 1D chaotic sequences S = {S(q)} and T
= {T(q)} using Eq. (9). Here the lengths of S and T both are MN, and q = 1, 2, …, MN.
2) With S and T, produce a new 1D sequence W with length MN according to the following rule:
( ) , if is an odd number;
( ) ( ) , otherwise.
S q q
W q T q
(23)
3) Using the quicksort algorithm [33], sort W in a certain (ascending or descending) order to obtain a new sequence W’ and its corresponding permutation indices IW. There are MN elements in IW. The relation between W and W’ is W’ = W(IW). For instance, the qth element in W’ corresponds to the IW(q)th element in W.
4) With IW, use the following loop procedure (described
