[PDF] Optical single-channel color image asymmetric cryptosystem based

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Optical single-channel color image asymmetric cryptosystem based on hyperchaotic system and random modulus decomposition in Gyrator domains Hang Chen,1,2* Li Zhu,2 Zhengjun Liu,3 Camel Tanougast,4 Feifei Liu,1 and Walter Blondel2

1School of Electrical Engineering and Automation, Jiangxi University of Science and Technology,

Ganzhou, 341000, China 2 Université de Lorraine, CNRS, CRAN UMR 7039, Nancy 54000 , France

3Department of Automation Measurement and Control, Harbin Institute of Technology, Harbin

150001, China

4Université de Lorraine, Laboratoire Conception Optimisation et Modélisation des Systèmes,

Metz 57070, France

*Corresponding author: hitchenhang@foxmail.com An enhanced optical single-channel asymmetric cryptosystem for color image is proposed b y u sing hyperch aotic system and random m odulus decomp osition (R MD) in gyrator domains. For increasing the security of the encryption scheme, a novel 4D four-winged hyperchaotic system is utilized to generate several hyperchaotic phases in the encryption approach. Subsequently, the RMD is implemented to obtain the effective trapdoor one-way f unction in the asymmetric process. Finally, a scrambling encoding algorithm is designed to combine and randomize the intermediate and hyperchaotic data, and the result can be regarded as the ciphertext and private key of the cryptosystem. In the decryption approach, the ciphertext and private key are imported into the input plane of gyrator transform. In fact, the proposed cryptosystem is also applicable in information authentication since the hyperchaotic data is prerequisite both in encryption and decryption approaches. Various numerical simulations are made to demonstrate the validity and capability of the proposed cryptosystem.

Keywords: Asymmetric cryptography, color image encryption, optical transform. 1. Introduction With the rapid development of the modern communication techniques, the optical information

security technology has drawn wide attention in recent decades due to its parallel processing and high spee d processing capabilities. Various opti cal cryptosystems ha ve been reporte d since Refrégiér and Javid i first pr oposed dou ble random phase encoding (D RPE) system in 1995, which can encrypt the image into stationary white noise in 4f 2 interference [19], randomized lens-phase functions [20] and compressive sensing [21] have also been introduced and utilized to break the linear relationship between the plaintext and ciphertext in optical encryption system. Besides, some random encoding algorithm has also been utilized in the image cryptosystem [22]. Most of the schemes [1-22] described above are symmetric encryption algorithms, in which the decryption key is identical to the encryption key. To overcome the linearity characteristics of DRPE, various asymmetric cryptosystems have been proposed based on phase-truncated Fourier transform (PTFT) [23-25]. However, the PTFT-based asymmetric cryptosystem was proved to be vulnerable to some specific attack using iterative amplitude-phase retrieval algorithm in the further research [26]. Moreover, the iterative phase retrieval algorithm, such as Gerchberg- Saxton algorithm and Yang-Gu algorithm [27-29] , have also been deeply researched and introduced in optical encryption system, which show high robustness against various potential

attacks. However, the iterative calculation makes the retrieval algorithm based cryptosystem

involved large amount of calculation. Besides the retrieval algorithm, several asymmetric encryption schemes based on equal modulus decomposition (EMD) have been presented in recent years [30-32]. Recently, Wang and his colleagues reported that the attacker is able to obtain precise result from EMD-based encryption system by performing a novel designed attack using phase iterative algorithm [33]. To improve the security of EMD, RMD is investigated, in which a random decomposition idea is utilized in creating the effective trapdoor one-way function. RMD can largely reduce the constraints compared with the EMD and it achieves high robustness while resisting the attack of iterative algorithm [33-34].

However, the RMD is

evolved from the original EMD, therefore the parameters of RMD may valuable to exhaustive attack. This attack process may waste much time, but it work in theory. Thus, one can introduce the hyperchaotic system into the cryptosystem, since the high randomness and extreme sensitivity to initial condition of hyperchaotic system can enhance the security of the cryptosystem dramatically. Moreover, some hyperchaotic systems have been employed in some cryptosystem to improve the security since the high sensitive characteristic of the initial value in chaotic system can break the linearity relationship [35-42]. In this paper, a single-channel optical color asymmetric cryptosystem using four-winged hyperchaotic system and random modulus decomposition in gyrator transform domain is proposed. Firstly, a 4D four-winged hyperchaotic system is introduced and utilized to generate the chaotic random phase mask (CRPM), which will be regarded as the public key of asymmetric encryption system. The original color image is encode into two-dimensional data and then combined with the random phase by propagating CRPM placed in the light path. Subsequently the image data is transformed as the light field in output plane of the gyrator transform. At the output plane of gyrator transform, the scrambled data is decomposed into two masks by performing RMD. Finally, a newly designed scrambling pixels position operation according to another chaotic random phase is employed to obtain the ciphertext and private key synchronously. In the encryption approach, the chaotic random phase serves as the public key in this cryptosystem. The parameter in gyrator transform and the initial conditions of the hyperchaotic system can be regarded as the extra keys to guarantee the security. Some corresponding numerical simulations are made to validate the performance of the proposed asymmetric cryptosystem.

The rest of the paper is organized as follow: In Section 2, the proposed cryptosystem is

introduced in detail. In Section 3, numerical simulation results are made to demonstrate the validity of the algorithm. Concluding remarks are summarized in the final section. 3

2. Optical color image asymmetric cryptosystem

Firstly, the 4D four-winged hyperchaotic system, random modulus decomposition and scrambling pixels position operation are introduced briefly, respectively. Then, the gyrator transform and its optical implementation are presented. Finally, the complete asymmetric encryption algorithm is addressed in detail.

2.1 Hyperchaotic system and scrambling operation

The highly unpredictable and random-like behavior of the hyperchaotic signals are the most attractive features in cryptography. For the hyperchaotic system, the extreme sensitivity to initial conditions and parameters can improve the security of the data against the brute force attack. It is believes that the performance of higher dimensional chaotic systems (with higher dimensional attractors) are much better in image encryption, therefore a 4D four-winged hyperchaotic system is considered and utilized in this paper and the system can be modeled as follows [43-44] x ax yz y byz cu xz z xy dz u y ku= - (1) where a, b, c, d and k are constant parameters of the hyperchaotic system. The symbol '×' represents the operation of derivation. When

3a=, -8b=, 5c d= = and 0.5k=, the Lyapunov

exponents of the hyperchaotic system are

10.32l=,21.23l=,30.81l= - and 411.04l= -.

Apparently, the first and second Lyapunov are larger than 0, which means the system is in hyperchaotic state. In calculation, the initial conditions are set as

0 0 0 010x y z u= = = = -, and

the four-winged ( )x y- hyperchaotic Lorenz attractors is illustrated in Fig.1. Fig.1 The hyperchaotic Lorenz attractor in x-y plane. 4 To obtain the chaotic data for the following encryption process, the hyperchaotic sequences can be generate by the following operation

4 410 ( 10 ), ( 1,2,3,4).i i is s round s i*= ´ - ´ = (2)

where is denotes the original chaotic sequence from the hyperchaotic system described in Eq.1, which can be also expressed as ix, iy, iz or iu. The symbol is* represents the improved sequences corresponding to is. Therefore, four improved sequences ix*, iy*, iz* and iu* can be obtained by implementing Eq.2. Finally, the obtained improved sequences are converted into two-dimensional format. To enhance the security of the final ciphertext and private key, the highly unpredictable and random chaotic sequence is considered for designing a scrambling pixels position operation.

Here, we suppose the image

( , )I x y to be scrambled and the chaotic mask ( , )M u v having the same size of 256×256. Firstly, the chaotic mask ( , )M u v is rearranged according to the pixel value, which can be expressed as follows '( , ) [ ( , )] , , , 1,2,3,...,256 '( , ) [ '( , )] , 1,2,3,...,256 t k i j t k t kM u v sort M u v i j t k

I x y array M u y t k

 (3) where the parameters ' i', 'j', 't' and 'k' represent the pixel position before and after the rearrangement. The symbol indicates the pixel values. In addition, 'sort' and 'array' denote the operation of rearrangement by size and by the specified position.

Therefore, the original

image ( , )I x y can be scrambled by the position of the random chaotic mask.

In the following step, two threshold pixel value

1V and 2V in the chaotic mask '( , )t kM u v are

selected randomly. According to the position corresponding to the threshold

1V and 2V, the

scrambled image '( , )I x y is divided into three sections. Finally, the pixels in these sections are synchronously arranged in parallel and vertical, as depicted in Fig.2. The mathematical definition of the scrambling pixels position operation can be written as follows 11 2 1 2 32
( , ) '( , ), ( , )t k t k t k t k t k t kI x y I x y M u v V

I x y I x y V M u v V

I x y I x y M u v V

(4)

where 1( , )I x y, 2( , )I x y and 3( , )I x y are the three sections of the scrambling image according to

the threshold

1V and 2V, respectively.

5 Fig.2 Schematic diagram of the scrambling operation.

2.2 Random modulus decomposition

Different from the equal modulus decomposition, random modulus decomposition can highly reduce the constraints since the modulus are not identical according the decomposition theory. The RMD is a kind of unequal modulus decomposition, in which the spectrum is randomly divided into two complex value masks [33]. Therefore, one two-dimensional image can be divided into two statistically independent masks randomly. To simplify the illustration, one complex number in two-dimensional Cartesian coordinate is considered as a vector I and it is divided into two vectors randomly as shown in Fig.3. Fig.3 Schematic (vector) representation of the decomposition process in RMD.

Suppose that ( , )I u vin Fig.3 is an image, the function ( , )u va,( , )u vb and( , )u vj can be

expressed as equation 6 ( , )=2 ( , ) ( , )=2 ( , ) ( , ) arg ( , ) u v rand u v u v rand u v u v I u v a p b p j´ = (5) where the function ( , )rand u v denotes a random matrix satisfying normal distribution in the interval [0, 1] and the symbol 'arg' represents the argument function. Besides, the amplitude of

( , )I u v is given by ( , )= ( , )A u v I u v. According to Fig.3, the result of the decomposition masks

1( , )P u v and 2( , )P u v can be obtained by geometrical deduction as follows

( , ) sin[ ( , )]1( , ) exp{[ ( , ) ( , )},sin[ ( , )+ ( , )]A u v u vP u v i u v u vu v u v bj aa b= - (6) and ( , ) sin[ ( , )]2( , ) exp{[ ( , ) ( , )]}.sin[ ( , ) ( , )]A u v u vP u v i u v u vu v u v aj ba b= ++ (7)

where the ( , )A u v and ( , )u vj are the amplitude and phase of ( , )I u v, respectively. In addition,

( , )u vq is a random function distributed uniformly in interval[0, 2 ]p. Besides, the phase function

( , )u va and ( , )u vb are given by ( , ) 2 ( , )u v rand u va p= ´ and ( , ) 2 ( , )u v rand u vb p= ´ as

illustrated in Eq.5, respectively.

2.3 Optical gyrator transform

In signal processing filed, the optical gyrator transform can be regarded as a kind of linear

canonical integral transform which produces the twisted rotation in position-spatial frequency planes of phase space [45]. In optical implementation, a coherent optical system consisting of six thin cylinder lenses can be considered to realize the optical gyrator propagation [46-47]. In fact, the gyrator transform has only two-dimensional format and the mathematical definition of gyrator transform for the function ( , )I x y can be expressed as + cos 1 ( , )exp[ 2 ]d d ,sin sinF u v G I x y u v xy uv xv yuI x y i x y a apa a (8) where ( , )I x y and ( , )F u v represent the input and output of gyrator transform, respectively. x and y are the input coordinates at spatial domain, while u and v denote the output coordinates in gyrator domain. Besides, the parameter 'a' is the rotation angle of gyrator transform, which usually serves as an additional key in encryption system . When / 2a p=, the gyrator transform reduces Fourier transform with the rotation of coordinates at / 2p. Finally, the inverse transform of gyrator transform with rotation angle a is transform with rotation angle a-or 2p a-. Also, 7 the gyrator transform has the property of index additive and periodic with the parameter a. The optical gyrator transform is utilized to complete the proposed asymmetric cryptosystem.

2.4 Asymmetric cryptosystem

The flowchart of the intact proposed cryptosystem is depicted in Fig.4. To implement the

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