Period analysis of the Logistic map for the finite field

Abstract

Usually, the security of traditional cryptography which works on integer numbers and chaotic cryptosystem which works on real numbers is worthy of study. But the classical chaotic map over the real domain has a disadvantage that the calculation accuracy of the floating point number can be doubled when the map is implemented by computer. This is a serious drawback for practical application. The Logistic map is a classical chaotic system and it has been used as a chaotic cipher in the real number field. This inevitably leads to the degradation of finite precision under computer environment, and it is also very difficult to guarantee security. To solve these drawbacks, we extend the Logistic map to the finite field. In this paper, we consider the Logistic map for the finite field N = 3 n , and analyze the period property of sequences generated by the Logistic map over Z N . Moreover, we discuss the control parameters which may influence the behavior of the mapping, and show that the Logistic map over Z N may be suitable for application by performance analysis. Ultimately, we find that there exists an automorphic map between two Logistic maps with the different control parameters, which makes them suitable for sequence generator in cryptosystem. The automorphic sequence generated algorithm based on the Logistic map over Z N is designed and analyzed in detail. These sequences can be used in the pseudorandom number generator, the chaotic stream cipher, and the chaotic block cipher, etc.