A robust and efficient scheme for image encryption-decryption

Transcription

1 A robust and efficient scheme for image encryption-decryption Prerana Sharma 1, Dr. Shiv Kumar 1 M.Tech. Scholar, ACEIT, Kukas, Jaipur (Raj.), India Professor, ACEIT, Kukas, Jaipur (Raj.), India ABSTRACT Fractional Fourier transform and chaos functions play a key role in many of encryption-decryption algorithms. In this work performance of image encryption-decryption algorithms is quantified and compared using the resemblance of input image to the restored image, quantified by MSE. This work verifies robustness of the proposed image encryption-decryption algorithms. Further, simulations are performed using MATLAB7.7 to quantify performance of the proposed algorithms using MSE. The results obtained in these simulations prove that for the proposed algorithms MSE between restored and original images has a very high value, if decrypted at an incorrect thereby maintaining the robustness of the algorithms. These algorithms are found sensitive to a variation of 1x10-1 in the fractional orders used in encryption-decryption process. Keywords: Fractional Fourier Transform, Discrete Wavelet Transform, Tent map, Robust algorithm, Computation Complexity. 1. INTRODUCTION Optical systems process -dimensional complex data in parallel and at a high speed. Therefore, these systems are of growing interest for image encryption. Many optical methods have been proposed in past [1] [] [3] [4] [5] [6] [7]. Double random phase encoding as presented in [1], [4] has been widely used and remains highly successful. The fractional Fourier transform also has been potentially used in the field of optical security [] [3] [4] [5] [7].. FRACTIONAL FOURIER TRANSFORM The a th order fractional Fourier transform is a linear operation defined by the integral K a f u K u, u' f u' du' u, u' A expi cot u cscuu' cotu' 1 icot Volume, Issue 5, May 013 Page 436 A a When a j and K a (u, u ) Ξ (u u ) when a = 4j and K a (u,u )Ξ (u + u ) when a = 4j±, where j is an integer. The a th order transform is sometimes referred to as the th order transform [7]. 3. CHAOS FUNCTIONS Chaos functions have been mainly used to develop the mathematical models of non-linear systems. These functions generate completely random iterative values. But these values are limited between bounds. Convergence of the iterative values after any value of iterations is kept hidden. Chaos functions are sensitive to the initial conditions. Three chaos functions are discussed below: (a) The first chaos function is the logistic map [8] and is defined as: p. x.(1 x) This function is bounded for 0 < p < 4 and can be written in the iterative form as: x p. x.(1 x n 1 n n with x 0 as the initial value. This is also known as the seed value for the chaos function. (b) The second chaos function is the tent map [8] and is defined as: a. x 0 x0 0.5 a.(1 x) 0.5 x0 1 This function is bounded for 0< a and can be written in the iterative form as: )

2 a. xn 0 x0 0.5 xn 1 a.(1 xn ) 0.5 x0 1 with x 0 as the initial value. (c) The third chaos function is the Kaplan Yorke map [8] and is defined as: a. x.mod1 0 x0 0.5 f ( by cos(4x) 0.5 x0 1 This is bounded for 0 a and 0 b 1 and can be written in the iterative form as: x a. x mod1 y n 1 n by cos(4x n 1 n n with x 0 as the initial value. These chaos functions are used to generate random phase masks. Logistic map and tent map are one-dimensional chaos functions and the Kaplan Yorke map is a two-dimensional chaos function. For the two-dimensional chaos function, two seed values are required to generate the CRPM (Chaotic random phase mask). 4. PROPOSED SCHEME Here we are considering that an image I = f(x, of size MXN consists of three primary color channels viz. RED, GREEN and BLUE. For simplicity, we assume the image size to be N X N. As presented in [1], the image encryption and decryption algorithms in this work are based on the Discrete Wavelet Transform, fractional Fourier transform and use chaotic logistic map, Tent map and Kaplan-Yorke map. These algorithms shall be further referred as Algorithm 1, Algorithm and Algorithm 3 respectively throughout the paper. Each of these algorithms follows a similar sequence of operations but they differ in their implementation details. A general overview of the encryption and decryption process is as follows. For an input image 3 I f i1 The encryption process of I produces I 1 as below [1] I g( x, IDWT F F DWT,, i x, y, where1 x, y N ) f ( x, *expi S1( x, *expi S ( x, i 1 where α, β, γ, δ denote orders of the fractional Fourier transform. The decryption of I 1 is carried out as per the following formulae [1] ' ' I f x, IDWT F, F, DWT g ( x, * conjexpi S ( x, * conjexp i S ( x, ( 1 -α, -β, -γ and δ denote fractional orders used for decryption. 5. SIMULATION RESULTS In order to investigate the quality of encryption, decryption i.e. robustness of proposed algorithms, digital simulations were performed in an environment as under: Processor Intel Core i3 CPU, M380.53GHz. RAM 3.00 GB. Operating System 64-bit Windows 7. Simulation tool MATLAB R008b. Fig 1 Input image to algorithm 1, and 3(Lena.tif, 56X56, color). Volume, Issue 5, May 013 Page 437

3 The input image chosen for analysis is Lena (TIFF image; Size = 193 KB; pixel by pixel resolution = 56X56) as shown in Fig 1. Lena was encrypted using algorithm 1, and 3 for α = β = γ = δ = 0.5 fractional orders of FRT. The resulting encrypted images obtained after application of algorithm 1, and 3 are shown in Fig (a), (b) and (c) respectively. These images can be decrypted on any fractional order of FRT, but the restored image may differ from the input image depending on the order of FRT. The decrypted images obtained using incorrect fractional orders α = β = γ = δ = 0.49 and α = β = γ = δ = 0.51 of inverse FRT are shown in Fig 3(a) 3(c) and Fig 4(a) Fig 4(c). Fig. 5(a) - 5(c) show decrypted images obtained when correct fractional orders of inverse FRT are used i.e. α = β = γ = δ = Fig. Images encrypted at α = β = γ = δ = 0.5 using (a) Algorithm 1, (b) Algorithm and (c) Algorithm 3. Fig 3. Images decrypted at α = β = γ = δ = using (a) Algorithm 1, (b) Algorithm and (c) Algorithm 3. Fig 4. Images decrypted at α = β = γ = δ = using (a) Algorithm 1, (b) Algorithm and (c) Algorithm 3. Fig 5. Images decrypted at α = β = γ = δ = 0.5 using (a) Algorithm 1, (b) Algorithm and (c) Algorithm 3. Volume, Issue 5, May 013 Page 438

4 (a) (b) (c) Fig 6. Graph, showing MSE between input image and decrypted image using (a) Algorithm 1, (b) Algorithm and (c) Algorithm 3. The order of decryption ranging between Fig 6(a) 6(c) show the variation of MSE between decrypted (restored from the encrypted) image and the input image at different fractional orders. From the graphs it is observed that the MSE value at α = β = γ = δ = orders of decryption is much higher than MSE value at α = β = γ = δ = -0.5 orders of decryption. This ratio of these MSEs is of the order of 10 9, which can be observed from the graph. 6. CONCLUSION From the results shown in previous section, the algorithms presented in this work are found to be robust in nature. The algorithms are sensitive to a variation of fractional order which is of the order of Besides this, these algorithms are efficient also [1]. Due to their efficient and robust behavior, the algorithms may find various applications. References [1.] P. Sharma, D. Mishra, A. Agarwal, Efficient Image Encryption and Decryption Using Discrete Wavelet Transform and Fractional Fourier Transform, SIN 1, pp , October 01. [.] M. Joshi, Chandrashakhar, K. Singh, Color image encryption and decryption using fractional fourier transform, Optics communications, Vol. 79 Issue 1, pp 35-4, 1 November 007. Volume, Issue 5, May 013 Page 439

5 [3.] N. Singh, A. Sinha, Optical image encryption using fractional Fourier transform and chaos, Optics and Lasers in Engineering, Vol. 46 Issue, pp , February 008. [4.] R. Tao, Y. Xin, Y. Wang, Double image encryption based on random phase encoding in the fractional fourier domain, Optics Express, Vol. 15 Issue 4, 007, pp [5.] B. M. Hennelly, J. T. Sheridan, Image encryption and the fractional Fourier transform, Optik - International Journal for Light and Electron Optics, Volume 114 Issue 6, 003, pp [6.] C. H. Chuang, G. Lin, Data steganography for optical color image cryptosystems, International Journal of Image Processing (IJIP), Vol 3 Issue 6, January 010, pp [7.] H. M. Ozaktas, M. AlperKutay, Z. Zalevsky, The fractional Fourier transform: with applications in optics and signal processing, New York, John Wiley & Sons, 001. [8.] R. C. Alligood, T. D. Sauer, J. A. Yorke, Chaos; an introduction to dynamical systems, New York, Springer, 001 Volume, Issue 5, May 013 Page 440