Research On Characteristics For Logistic Map Over The Finite Field And Its Application In Image Encryption

During the information age,we can hardly find the industry without being changed by the Internet.So people regard the Internet as a sign of entering the information age from the industrial age.The rapid development and extensive application of science and technology,especially network technology,bring great convenience to people's daily life,work,research and so on.While enjoying the convenient transmission of information,people are also suffering from information leakage.Image is a visual expression of infor-mation and is widely used in network.Compared with text information,image is more intuitive,and image information becomes the best way that one of the most important transmission mediums in the former society.However,because of the large amount of data,the strong correlation between adjacent pixels,and the uneven distribution of en-ergy,the space of the image information is large and the information bandwidth of the transmission image is high.In practical engineering applications,reducing the bandwidth of image information transmission,efficient real-time transmission of image information is a topic of great concern for people.It is also a hot issue for scientific and technological workers to continue to study.At the same time,due to the insecurity of communication channels,and the existence of additional thieves and attackers,it is also an important topic to protect image information from leakage.Chaotic systems have many excellent characteristics such as sensitivity to initial values and control parameters,random-like,noise-like and ergodicity,which are expected by a good cryptographic system.The tradi-tional chaotic map works in the real domain.In this case,the mapping has a shortcoming,that is,when the computer implements this mapping,the computational complexity of the floating point number will be multiplied,which will seriously affect the practical en-gineering application of chaotic map.In order to overcome this problem,it is an urgent problem to extend the chaotic map to the finite field.In recent years,combining cryp-tography and chaos theory,researchers proposed a variety of chaos-based cryptography technologies to implement a variety of chaotic image encryption.However,none of these encryption schemes can resist all kinds of attacks.Therefore,it is of great practical sig-nificance to research and implement image encryption technology based on chaotic map over the finite field.In this thesis,we study the characteristics of discrete chaotic Logistic map over the finite field and its application in image encryption.The main work of this thesis is divided into the following aspects.?1?This thesis introduces the characteristics of Logistic map over the finite field and its application in image encryption,as well as its research background,research signifi-cance and research contents in this thesis.?2?Three traditional discrete chaotic Logistic maps are extended from the real num-ber domain to the finite field,and three Logistic map forms over the finite field are ob-tained.?3?Periodic characteristics of three Logistic maps over the finite field are analyzed theoretically,including the periodic characteristics of Logistic map-1 over the Finite Field Z₂n?Z₃n,the periodic characteristics of Logistic map-2 over the Finite Field Z₃nand the periodic characteristics of Logistic map-3 over the Finite Field Z₃n.The results of the analysis show that the behavior of the generated sequence by the Logistic map over the finite field will be affected by the mapping control parameters,the periodic characteristics of Logistic map-3 over the finite field Z₃nis the same as that of Logistic map-2 over the finite field Z₃n,and the period of the generated sequence by Logistic map-3 over the finite field Z₃nis longer than that of Logistic map-1 over the finite field Z₃n,and also the period of the generated sequence by Logistic map-2 over the finite field Z₃nis longer than that of Logistic map-1 over the finite field Z₃n.?4?From the aspects of maximum period,pseudorandom property,power spectrum,correlation,phase diagram,Lyapunov exponent,and generation time,the other charac-teristics of three Logistic maps over the finite field Z₃nare analyzed.The results of the analysis show that the three sequences generated by three Logistic maps over the finite field Z₃nhave good characteristics,such as random-like,noise-like properties,continuous spectrum,good correlation,well uniform distribution,controllable length of generated se-quence,positive Lyapunov exponent,long period and fast sequence generation speed.At the same time,it also shows that in terms of the generation time of a sequence,over the finite field Z₃n,the time of generating for Logistic map-3 is faster than that of Logistic map-2,and the time of generating for Logistic map-2 is faster than that of Logistic map-1.?5?The existence of automorphism mapping of Logistic map-1 over the finite field Z₃nis proved theoretically and its automorphism mapping form is derived.Using the Logistic map-1 and its automorphism mapping over the finite field Z₃n,we design an automorphism sequence generator and apply it to image encryption and decryption.The results of the analysis show that the image encryption and decryption methods using the Logistic map-1 and its automorphism mapping over the finite field Z₃nhave good perfor-mance.?6?The key sequence generator is designed by the Logistic map-3 over the finite field Z₃nand used in image encryption and decryption.The results of the analysis show that the image encryption and decryption methods using the Logistic map-3 over the finite field Z₃nalso have good performance.