Some properties of the Logistic map over the finite field and its application

The Logistic map is a classical chaotic system and has been used as a chaotic cipher in the real number field. This inevitably leads to the degradation of finite precision under the computer environment, and it also is very difficult to guarantee the security of the cryptosystem using the Logistic map over the real number field. To overcome these drawbacks, in this paper, we extend the Logistic map-3 to the finite field Z3n, and give theoretical analysis about some period properties of the Logistic map-3 over Z3n. Moreover, we discuss the control parameters which can change the behavior of the mapping, and also find that the maximum period can change with the control parameter. Furthermore, we analyze some characteristics including maximum period, generation time, NIST test, power spectrum, correlation property, phase diagram and Lyapunov exponent. Simulation results show that the period of Logistic map-3 over Z3n is longer than Logistic map-1 over Z3n, and much longer than Logistic map-1 over Z2n, and the same as Logistic map-2 over Z3n. Simulation experiments also show that the Logistic map-3 over the finite field has some good features including non-periodic, random, noise-like properties, continuous spectrum, good correlation, well uniform distribution, controllable length of generated sequence, positive Lyapunov exponent, and good sequence generation speed. Therefore, we attempt to design a nonlinear transformation based key stream generator by using the Logistic map-3 over Z3n and apply to image encryption. The performance analyses including statistical analysis, Shannon entropy analysis, differential attack analysis, and speed analysis also show that our proposed method may be suitable for practical cryptographic application.