Construction Of Non-degenerate Multi-dimensional Hyper Chaotic Maps And Applications In Cryptography

Chaotic maps have been widely used in cryptography due to its unpredictability,ergodicity and sensitivity to initial conditions.However,in many classical low-dimensional chaotic maps,there are some shortcomings,such as no ergodicity,intensive period window,short iteration period and small key space.If they are applied to cryptography,the encryption system will be lack of security.In order to solve the above problems,we constructed non-degenerate multi-dimensional hyper chaotic maps and analyzed their dynamic characteristics.The results show that the chaotic maps have ergodicity,larger Lyapunov exponents,better randomness and longer iteration period.This paper mainly focuses on the construction of non-degenerate multi-dimensional hyper chaotic maps and the design of high security image encryption schemes based on hyper chaotic maps.In order to eliminate the weakness of 1-dimensional chaotic maps,we have constructed non-degenerate multi-dimensional hyper chaotic maps to enhance dynamic performance.We have designed image encryption schemes based the excellent characteristics of non-degenerate hyper chaotic maps.The main research contents are as follows:(1)We have introduced the research background and significance of chaotic cryptography,and reviewed the development of chaos theory and the research progress of chaotic image encryption schemes.Then we have introduced the dynamic analysis tools of chaotic maps and the security analysis tools of image encryption scheme in detail.(2)Due to the characteristics of non-volatile memory,switching mechanism,nonlinearity and plasticity in memristor,adding memristor in chaotic system could produce complex nonlinear dynamics as well.In the past few years,the memristor has been applied in many fields and the study of memristor-based chaotic system has become a topical issue.In order to improve the security of chaotic maps,we constructed a memristor chaotic map(3D-MCM),and designed S-Boxes and a bit-level image encryption scheme based on 3D-MCM.The novelty is that each half-pixel is substituted with a half-pixel substitution box(HPS-Box)before S-Box substitution with random times.In the process of image encryption,we shuffled the pixels through bit-level permutation and diffusion to ensure that the pixels changed more thoroughly.(3)We designed a novel image encryption scheme based on a chaotic map and an S-Box.To solve the weaknesses in some 1-dimensional chaotic maps,we constructed a non-degenerate 3D hyper chaotic map(3D-HCM),and its dynamic analysis results demonstrate that,compared with 1-dimensional seed maps,the 3D-HCM has ergodicity,better randomness and a larger chaotic range.To counteract dynamic degradation,we adjusted the exponent in real time in each iteration.Based on 3D-HCM,we constructed keyed S-Boxes without fixed point,reverse fixed point or short period ring.To enhance the ability of the image encryption scheme for resisting common attacks,we blurred the plain image tinily before encryption and applied cross-plane permutation and diffusion to shuffle all the pixels.