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A Robust Cryptosystem Based Chaos for Secure Data Abir Awad, Safwan El Assad, Daniel Carragata IREENA, Ecole polytechnique de l’université de Nantes. Rue Christian Pauc BP 50609 Nantes Cedex 3, France.
[email protected],
[email protected] ABSTRACT It is well known that images are different from texts in many aspects, such as highly redundancy and correlation, the local structure and the characteristics of amplitude frequency. In the past few years, a number of image encryption algorithms based on chaotic maps have been proposed. In this paper, we present a new algorithm for image encryption. We use a perturbed PWLCM that has desirable dynamical statistical properties, a key size of 128 bits, substitution and permutation operations on multiple rounds to ensure confusion and diffusion properties. Theoretical and experimental results also show that our scheme is efficient and very secure.
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ALGORITHM
In this section, we present the developed Algorithm (see figure 1) for Image Encryption that we implemented in Matlab/Simulink. Let I be an MxN image with b-byte pixel values, where a pixel value is denoted by I(i), 0 ≤ i < MxNxb. A block cipher is an encryption scheme which breaks up the plaintext messages to be transmitted into blocks of fixed length and decrypts one block at a time.
INTRODUCTION
In recent years, a large amount of work on chaos-based cryptosystems has been published. Since 1990s, many researchers have noticed that there exists an interesting relationship between chaos and cryptography [1]. Many properties of chaotic systems have their corresponding counterparts in traditional cryptosystems. As a result of investigating the above relationship, a rich variety of chaos-based cryptosystems for end to-end communications have been put forward. Unfortunately, in many applications, conventional encryption algorithms are not suitable for real time data transmission. In order to overcome this problem, many fast encryption algorithms specifically designed for digital images have been proposed [2, 3]. The image encryption methods based on chaotic maps attract considerable attention recently due to their potential for digital multimedia encryption. The paper is organized as follows. Section II describes the proposed algorithm, Section III introduces the perturbed generator used. The simulation results and security analysis are given in section IV. The last section concludes this paper.
Figure1. Encryption Algorithm
The algorithm, characteristics and steps are [3, 4]: (1) The key size is 128-bits. (2) The piecewise linear chaotic map currently used is substituted by a perturbed one (perturbed PWLCM) to improve statistical properties. (3) A pseudo-random permutation generator is used in the encryption and decryption process, forming a permutation box (P-box) and adding
IEEE, ISIVC Conference On, Image/Video Communications over fixed and mobile networks, Bilbao Spain, July 2008, 4 pages.
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diffusion to the system. (4) A more complex substitution box (S-box) is applied. (5) Multiple rounds for encryption and decryption processes are used. The encryption algorithm transforms an image I using an SPnetwork generated by a one-dimensional chaotic map and a 128-bit secret key. The algorithm performs r rounds of an SP-network on each pixel. The decryption algorithm slightly differs from the encryption one. To decrypt an encrypted image, one has to perform the sequence of inverse transformations. 3. PERTURBED PWLCM MAP
The perturbing bit sequence can be generated every n clock as follows: Qk+−1 ( n ) = Qk ( n ) = g 0Q0 ( n ) ⊕ g1Q1 ( n ) ⊕ ... ⊕ g k −1Qk −1 ( n ) with n = 0,1, 2,...
Where ⊕ represents ‘exclusive or’, g = [ g 0 g1 ... g k −1 ] is the tap sequence of the primitive polynomial generator, and Q0 Q1...Qk −1 are the initial register values of which at least one is non zero. The perturbation begins at n= 0, and the next ones occur periodically every Δ iterations ( Δ is a positive integer), with n= l × Δ , l=1,2,…, The perturbed sequence is given by the equation (4):
A piecewise linear chaotic map (PWLCM) is a map composed of multiple linear segments. x( n) = F[x( n −1)] 1 ⎧ ⎪x( n −1) × p ⎪ ⎪ 1 = ⎨⎡⎣x( n −1) − p⎤⎦ × 0.5 −p ⎪ ⎪F[1− x( n −1)] ⎪ ⎩
if 0 ≤ x( n −1) < p if p ≤ x( n −1) < 0.5
(1)
⎧⎪ F [ xi ( n − 1)] xi ( n ) = ⎨ ⎪⎩ F [ xi ( n − 1)] ⊕ Q N − i ( n )
x i ( n ) ∈ {0,1} i = 1, 2,..., N
(4)
(
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T = σ ×Δ× 2k −1
“transient branch” and “cycle”. Accordingly, l and n+1 are respectively called “transient length” and “cycle period”, and l+n is called “orbit length”. To improve the dynamical degradation, a perturbation based algorithm is used [5]. The cycle length is expanded and so good statistical properties are reached. Here, for computing precision N, each x can be described as: x ( n ) = 0. x1 ( n ) x 2 ( n )... x i ( n )... x N ( n )
1≤ i ≤ N − k N − k +1≤ i ≤ N
Where F[ xi (n)] represents the ith bit of F [ x ( n)] . The perturbation is applied on the last k bits of F [ x( n)] . When n ≠ l × Δ , no perturbation occurs, so x ( n ) = F [ x ( n − 1)] . The system cycle length is given by the following relation
if 0.5 ≤ x( n −1)
