Research On Robustness Of Chaotic Systems With Application In Image Encryption

The booming of science technology and information process has brought the rapid development opportunities of the global information technology. People are enjoying the huge benefits brought by information network, at the same time, also faces the great challenge of information security. Image as the important carrier and transmission medium of the information, the security has been attracted more and more attention.The core tool of information security is encryption. With the increasingly complex of communication environment and the improved decoding ability, some traditional cipher algorithm is breached or the security is threatened, the more advanced design cipher theory and technology innovation are needed. The discovery of chaos is the third revolution in the 20th century physics. Some attributes of chaotic systems are in conformity with the requirements of cryptography, the chaotic cryptography becomes a hot research topic in the field of cryptography.In 1975, Li and Yorke have defined the term chaos by using mathematical language, and established the famous theorem, which is that’period three implies chaos’. In 2009, using the Li-Yorke theorem, Zhou and Song have proposed and proved the robust chaos theorem of the quadratic polynomial. In 2015, Yang et al have proposed and proved the robust chaos theorem of the cubic polynomial. In 1999, Banerjee and Grebogi have proposed the corresponding robust chaos theorems on the normal form of two-dimensional piecewise smooth maps. In 2001, Andrecut and Ali have proposed the robust chaos theorem of S-unimodal maps. In 2013, Min and Chen have proposed an image encryption scheme with avalanche effect (SESAE) based on d-bit keystream. The scheme realizes the key avalanche effect, and increases the difficulty of interpreters.Based on the predecessors’ work, this thesis studies on the robustness of discrete chaotic systems, chaotic pseudorandom number generators, image encryption scheme with avalanche effect. The main contents and achievements are listed as follows.(1) Research on the robustness of a few kinds of chaotic systemsBased on the Li-Yorke theorem and the robustness chaos theorem of the S-unimodal maps, this thesis sets up the robustness chaos constructing theorem on one-dimensional piecewise nonlinear maps and the robustness chaos constructing theorem on cubic polynomial discrete maps. This thesis proposes the robustness chaos equivalence theorem on the normal form of two-dimensional piecewise smooth maps, by improving the theorem on the normal form of two-dimensional piecewise smooth maps, and presents the requirement for constructing the four dimensional discrete chaotic maps, provides the theoretical proof for constructing chaotic systems, provides the new tools for the chaotic application.(2) Pseudorandom number generators and pseudorandomness testsBased on the new proposed robustness chaos theorem, the triangle bounded function and the chaotic generalized synchronization theorem, this thesis constructs six new chaotic generalized synchronization systems. This thesis optimization designes six pseudorandom number generators (CPRNGs) using the six chaotic generalized synchronization systems.The improved FIPS 140-2 test and SP 800-22 test issued by the American National Institute of Standards and Technology (NIST) are used for testing the randomness of the pseudorandom numbers generated by CPRNGs, the RC4 algorithm PRNG and the ZUC algorithm. The testing results show that the proposed CPRNGs, the RC4 algorithm, and the ZUC algorithm do not have significant differences, and the 14th to 15th testing results of SP 800-22 test of the three CPRNGs are better than the RC4 algorithm and the ZUC algorithm.(3) Research on the image encryption scheme with avalanche effectThis thesis proposes a stream encryption scheme with both key avalanche effect and plaintext avalanche effect (SESKPAE) and a block encryption scheme with avalanche effect (BESAE), promotes the research work about the image encryption scheme with avalanche effect (SESAE).