A New Class Of Chaotic Maps And Their Image Encryption Applications

Chaos is a complex phenomenon in nat,ure that appears t.o be irregular and ran-dom but actually has inherent regularity.It is one of the great scientific discoveries of the 20th century.Chaotic systems are widely used in the field of pseudorandom sequence generation and cryptographic design because of their extreme sensitivity to initial conditions and control parameters,random-like behavior,and long-term unpredictability,which naturally correspond to the characterist,ics of diffusion,dis-order,and pse.udo-randomness required by cryptographic systems.When chaotic systems are applied to the field of digital cryptography,they often face Problems such as the degradation of dynamic characteristics caused by finite-precision ealeu-lation,whether a sequence can pass statistical randomness tests,and the efficiency of floating-point operat,ions.Therefore,it is particularly important to study and improve the properties of digital chaotic pseudorandom sequences to meet the requirements of measurement metrics and test standards in the application fields.In view of the fact t,hat some chaotic systems commonly used do not simul-taneously have excellent characteristics such as uniform distribut,ion of iterative sequences,global chaos in the whole parameter range,nonlinearity,sufficient.ly large Lyapunov exponent,sufficiently large se.t of parameters,and fast calculation speed,this thesis constructs a new class of one-dimensional nonlinear chaotic maps,i.e.,the chaotic reciprocal difference maps,analyzes their properties and compares them with other maps.A universal method for constructing similar one-dimensional chaotic maps is also proposed.The chaotic reciprocal difference maps provide good candidate chaotic systems for designing secure and efficient chaotic cryptographic algorithms.The universal construction method also provides a new idea for design-ing cryptographic algorithms.The main results of this t,hesis include the following three aspects:1.Construction of the reciprocal difference chaotic maps and analysis of their properties.The chaotic reciprocal difference maps based on the reciprocal differ-ence function have all the above-mentioned excellent characteristics.Compara-tively,many known one-dimensional chaotic maps,such as the logistic map,the tent map and the Chebyshev map,have only part of these characteristics.The analysis of their properties shows that the chaotic reciprocal difference maps have good cryptographic properties and broad application prospects.Furthermore,the Lyapunov exponent values of the constructed piecewise maps based on this class of maps can be as large as possible.Therefore,this class of maps would be good candidates for the design or construction of secure cryptosystems,hash functions,pseudo-random number generators,and so an.This the.sis also proposes a universal method for constructing one-dimensional chaotic maps with uniform distribution,and gives an expression for accurately solving their Lyapunov exponent.2.Extension of chaotic maps on Z(pn)and analysis of their properties.The digraphs and period properties of the logistic map on residue class rings Z(pn)are analyzed.Some theorems and conjectures are proven or given.Some numerical experiments are carried out to validate the properties.Based on the chaotic recip-rocal difference maps,"extended chaotic reciprocal difference maps,are proposed,which are realized on residue class rings Z(pn).The properties of the generating sequences by the new maps are analyzed,including periods and long chains and so on.The judgement problem of full-period maps on Z(pn)are studied,and a new congruential map with double modulus on Z(pn)is proposed.The full-period prop-erties of the sequences generated by this new map are studied completely.Some theorems including full-period judgement,theorem on Z(pn)are proven and validat-ed by some numerical experiments.The analysis and experiments show that these full-period maps have good randomness and can be applied in the pseudo-random generators,cryptgraphy,spread spectrum communications and so on.3.Design and security analysis of symmetric encryption algorithm,for RGB color images.A symmetric encryption algorithm is designed with t.he chaotic reciprocal difference maps.It permutates and diffuses the images with the new maps,and enerypts and decrypts RGB color images correctly.The security and performance of the algorithm are analyzed,tested and compared including key space,statistical histograms,key sensitivity,adjacent pixel corre-lation,anti-cut attack and anti-noise attack,encryption speed and so on.The theoretical analysis and experimental results show that the algorithm is secure and fast to be applied in the image encryption and embedded digital watermarking.