2816 JOURNAL OF COMPUTERS, VOL. 8, NO. 11, NOVEMBER 2013 Single-Channel Color Image Encryption Using the Reality-Preserving Fractional Discrete Cosine Transform in YCbCr Space Jianhua Wu, Fangfang Guo Nanchang University/Department of Electronic Information Engineering, Nanchang 330031, China Email: [email protected] (JH Wu), [email protected] (FF Guo) Nanrun Zhou* Nanchang University/Department of Electronic Information Engineering, Nanchang 330031, China Beijing University of Posts and Telecommunications/Information Security Center, Beijing 100876, China Email: [email protected] Abstract—A novel single-channel color image encryption issues about illegal data access on Internet are becoming algorithm is proposed, which utilizes the reality-preserving more and more serious. fractional discrete cosine transform in YCbCr space. The Past two decades we have witnessed the appearance of color image to be encrypted is decomposed into Y, Cb, and various encryption methods for gray images. Amongst, Cr components, which are then separately transformed by the most famous and widely used one is the double Discrete Cosine Transform (DCT). The resulting three random phase encoding (DRPE) given by Refregier and spectra sequences, obtained by zig-zag scanning the spectra [1] matrices, are truncated and the lower frequency coefficients Javidi , which applies two random phase masks of the three components are scrambled up into a single arranged separately in the input and the Fourier planes to matrix of the same size with the original color image. Then encrypt the image into a stationary white noise. The two the obtained single matrix is encrypted by the fractional random phase masks are uniformly distributed in the discrete cosine transform, which is a kind of encryption with interval [0,2π] and the second one is taken as the main secrecy of pixel value and pixel position simultaneously. The cipher key. Except the Fourier domain, other different encrypted image is convenient for display, transmission and storage, thanks to the reality-preserving property of the domains such as fractional Fourier transform (FrFT) domain[2-5], Fresnel transform domain[6,7], Hartley fractional discrete cosine transform. Additionally, the [8,9] [10-12] proposed algorithm enlarges the key space by employing the transform domain and Gyrator transform domain generating sequence as an extra key in addition to the are explored for more new encryption methods. However, fractional orders. Simulation results and security analysis the decrypted images resulting from optical demonstrate the proposed algorithm is feasible, effective cryptosystems would lose their color information, which and secure. The robustness to noise attack is also makes these encryption algorithms inappropriate to guaranteed to some extent. encrypt color images. In response to this demand, many image encryption schemes especially for color images Index Terms—single-channel, color image encryption, reality-preserving fractional discrete cosine transform, have been designed, where three components of color image are encrypted using the traditional gray image generating sequence, spectrum truncation [13] encryption methods separately . In that case, it renders the cryptosystems sophisticated, since three channels I. INTRODUCTION must be involved. To address this problem, various single-channel color image encryption techniques have The reason of the use of color in image processing not been put forward successively[14-17]. Zhou et al.[14] only is that color is a powerful descriptor to provide proposed a single-channel color image encryption beauty in vision, but also is that humans can discern algorithm based on chaotic scrambling and the FrFT in thousands of color shades and intensities compared with HSI space, the output of the encryption system is not a about only two dozen shades of gray. Thus, color images color image but a gray and a phase matrix. Wu et al.[16] contain more information than gray images do and are made full use of the complex number mode to realize a widely used in real life. Color image encryption has single-channel color image encryption in fractional become a major task for information security since the Fourier domain. Although the above discussed encryption methods Manuscript received June 28, 2013; revised August 2, 2013; belong to single-channel, the encrypted images are accepted September 24, 2013. complex-valued possessing amplitude information as well *Corresponding author: [email protected]. as phase information, which makes them inconvenient to © 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.11.2816-2822 JOURNAL OF COMPUTERS, VOL. 8, NO. 11, NOVEMBER 2013 2817 display, transmit and store. In this paper, a novel single- 1 ⎛⎞()21nk+ channel color image algorithm based on the fractional C=cos2επ (1) [18] k ⎜⎟ discrete cosine transform (FrDCT) , which inherits the N ⎝⎠4N reality of the discrete cosine transform (DCT) matrix, is proposed. The original color image is converted into the where nk,0,1,,1= … N− and ε =1 , ε = 2 YCbCr space, where the Y component denotes the 0 k brightness, and the Cb and the Cr components for nonzero k. respectively denote the color differences of red and blue The eigen-decomposition of an NN× DCI-II matrix [16]. Since human eyes are more attuned to brightness and C is: less to color differences, hence the YCbCr color model * jϕn (2) allows more attention to be paid to the Y component, and C=UΛUU= ∑ ne less to the others. It is well known that the discrete cosine n transform (DCT)[19] has the property of energy where U is a unitary matrix, composed of columns concentration, namely, the energy of an image after DCT (eigenvectors) u , uu* = δ , and Λ is the diagonal concentrates towards the top left corner — the low n mn mn frequency, which human vision is more sensitive to. With matrix with diagonal entries, i.e. eignvalues λn , the help of spectrum truncation, the low frequency jϕn with . spectra truncated from the corresponding cosine spectra λn = e 0 < ϕn < π in accordance with the ratio of 2:1:1 are scrambled up The FrDCT matrix Cα can be written in a compact into one single matrix and sequentially encrypted by the jϕn FrDCT, which has a character of altering the pixel value form by substituting λn = e with their αth powers and the pixel position simultaneously. The resulting α α λ , i.e., the matrix Λ by its αth power Λ : cipher-text is a real gray-scale image, which is convenient n * for display, transmission and storage, and has camouflage CUα = ΛU (3) property to some extent. Furthermore, generating The matrix given by (3) can be rewritten in an sequence (GS), which results from the multiplicity of Cα FrDCT matrices’ roots, is introduced as an extra cipher alternative form according to the eigenstructure of C : [20] key. Spatiotemporal chaotic map is utilized to generate K ⎡⎤α α the random GS. Thus the high sensitiveness to initial α (4) CUV+Vα = 211ℜ+⎢⎥∑ nnλ 11()− ( − ) values and system parameters inherent in any chaotic ⎣⎦n=1 system provides high security naturally. Since the where KN=−−μμ 2 , μ and μ represents fractional orders are not so sensitive compared with the ( 11− ) 1 −1 chaotic maps, they can be abandoned or be used merely the multiplicities of the eigenvalues 1 and −1, as auxiliary keys. Simulation results and security analysis respectively. V1 collects the μ1 matrices Un verify the effectiveness and feasibility of the algorithm. Robustness to noise attack is also validated. corresponding to the eigenvalue 1 and similarly for V−1 . The rest of this paper is organized as follows. Section For NN= 4 0 and real α , the FrDCT matrix Cα II describes the theoretical background about the reality- preserving fractional discrete cosine transform. Section becomes a real-valued matrix because of the absence of III gives the details of the proposed algorithm including the eigenvalues ±1 and can be written as: NN22 the color model, spectrum truncation and the ⎡ α ⎤⎡jq()ϕπαnn+2 ⎤ spatiotemporal chaotic map. The procedures of the CUα =ℜ22⎢∑∑nnλ ⎥⎢ =ℜ U ne ⎥ proposed algorithm are also described in Section III. ⎣ nn==11⎦⎣ ⎦ (5) Simulations and discussions are given in Section IV. N 2 =+ABcosωα sin ωα Finally, conclusion is drawn in final section followed. ∑()nnnn n=1 ω = ϕπ+=21,2,,2qn… N II. THEORETICAL BACKGROUND nn n (6) 0 <<ϕπ The fractional discrete cosine transform (FrDCT) is a n ∗ generalization of the DCT. In current literatures, even where U=uu, AU=ℜ2 , BU=−2 ℑ and nnnnn[ ] nn[ ] though several versions of fractional cosine transform [18] , introduced due to the multiplicity have been derived, the FrDCT different from those q = (qq12,,,… qN 2) defined in [21,22] possesses the mathematical properties of the α th power of λ and called as generating of reality in addition to linearity, unitarily and additivity. n And the reality is of importance for image encryption, sequence (GS) of the FrDCT, is an arbitrary sequence of which ensures the outputs are real for real inputs. integers depending on the nature of the fraction and has The FrDCT is derived based on the eigen- strong effect on the results of the FrDCT since different decomposition and eigenvalue substitution of the DCT-II q leads to different Cα , so by taking the GS q as kernel denoted as: secret key can provide a huge key space. Readers can refer to the [18] for more information about q . The © 2013 ACADEMY PUBLISHER 2818 JOURNAL OF COMPUTERS, VOL. 8, NO. 11, NOVEMBER 2013 expansion of the FrDCT for a two-dimensional signal is truncated at an appropriate position to realize the low straightforward and simple through two FrDCTs frequency extraction. As described before, the Y successively by rows and by columns. component contains more image information, thus, the three corresponding spectra, of Y, Cb, and Cr, are III. DESCRIPTION OF THE METHOD truncated by a ratio of 1/2, 1/4, and 1/4, respectively, and scrambled up into a single combined spectrum for A.