Image Encryption Research Based On Reality-Preserving Fractional DCT And Compressive Sensing

The security problem of digital image becomes the focus of people’s attention. Image encryption is adopted to solve this problem. As encryption methods based on fractional Fourier transforms have been thoroughly studieded, new fractional transforms are obtained by the generalization of fractional order. They are applied to image encryption to construct new image encryption systems. Yet, the output of most transform-based encryption algorithm is complex, which increases the burden of data storage and transmission. Venturini proposed a method of reality-preserving treatment for a fractional transform, in which the coefficients of transform are real. In order to obtain reality-preserving character, a reality-preserving transform matrix must be constructed, but it does not inherit all properties of non-reality-preserving fractional transform. At present, concerning the fractional reality-preserving transform and its application in image encryption there still exist the following problems:(1) the sensitivity of fractional orders as cipher key is not high, which is due to the fact that the transform function is a continous function of the fractioal orders; (2) Althouth the optical realization technology of Fourier transform is fairly mature, the optical realization of reality-preserving transform, such as discrete cosine transform (DCT), has not yet been reported in literature. Hence, the reality-preserving fractional discrete cosine transform (FrDCT) and its application in image encryption are proposed in this paper. On the basis of G. Cariolaro’s method, FrDCT and the derived reality-preserving transform are obtained and applied in image encryption. In this thesis, the main work and research results are as follows:(1) An alternative fractional cosine transforms are proposed. According to the relationship of discrete Fourier transform (DFT), DCT and FrDCT, the optical implementation schemes of DCT and FrDCT are proposed. Since DCT has mathematically a certain relationship with DFT and FrDCT is achieved by integer multiple of DCT, the scheme of optical implementation of DCT and FrDCT with the aid of DFT’s is designed.(2) Since FrDCT has the character of real output and decorrelation, a series of reality-preserving image encryption schemes based on FrDCT are proposed. The introduction of generating sequence of integer type solves the nonuniqueness of fractional operator of FrDCT. The real output of transform is beneficial to storage and transmission of images. In FrDCT with random fractional orders and generating sequences (RFrDCT), fractional order and GS are all vectorized. RFrDCT applied in image encryption enlarges the key space of the fractional order itself and the key space of encryption scheme. Meanwhile, it also enhances the sensitivity of the quality of decrypted image with respect to cipher keys.(3) Since the security of the encryption algorithm based on FrDCT is not so high, image encryption scheme combining multiple generating sequences controlled FrDCT (MGSFrDCT) with dependent scrambling and diffusion (DSD) is proposed. In MGSFrDCT, the multiple-generating sequences greatly enlarge the key space of the encryption system. To enhance the security of the encryption system, the dependent scrambling and diffusion operation after quantizing transform coefficients is adopted. The cipher keys are highly sensitive to plaintext due to the application of DSD. Meanwhile, the scrambling and diffusion are realized simultaneously. In order to reduce the quantization error and computational complexity, an image encryption algorithm based on FrDCT via polynomial interpolation (PI-FrDCT) and nonlinear function is proposed. The encrypted image has the ability of resisting the statistical attacks, known-plaintext and chosen-plaintext attacks.(4) For three components of an RGB color image with a strong correlaton, a color image encryption based on FrDCT in HSI space is proposed. The spatial transform between RGB and HSI is nonlinear and the HSI space can decouple the three components in RGB space. The nonlinear spatial conversion and the independent three components in HSI make the propose encryption algorithm more secure than linear ones. In order to enhance the sensitivity of the encryption system, integer GS is produced by a two-dimensional chaotic mapping to ensure the uniqueness of the transform matrix and enlarge the cipher key space of the encryption system. Experimental results demonstrate that the proposed encryption algorithm is sensitive with respect to cipher keys and, to some extent, robust to noise and occlusion attacks.(5) An image encryption algorithm based on compressive sensing (CS) and chaotic mapping is proposed. The compression and the encryption of an image are achieved simultaneously in the encryption scheme. The burden of storage and transmission of encrypted images is reduced and the security of the encryption system is enhanced. The measurement matrix is obtained by adopting variable-parameter chaotic map, which enlarges the key space of encryption scheme based on CS. In this view, an image algorithm combining CS and FrDCT is proposed, which makes the encryption system more secure.