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Period analysis of the Logistic map for the finite field

基于有限域N=3^n 的Logistic 映射周期分析

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 = 3n, 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.

摘要

创新点

  1. (1)

    Logistic 映射已被用作实数域的混沌密码系统。 在计算机环境下, 实数域的计算必然导致有限精度的退化。 因此, 我们将 Logistic 映射推广至有限域 N=3^n, 并分析了基于有限域 Z_N 的 Logistic 映射生成序列的周期特性, 讨论了映射的控制参数将改变映射的周期行为, 性能分析表明此推广更有利于实际应用。

  2. (2)

    基于前述的周期分析, 我们推导在基于不同控制参数的两个 Logistic 映射之间存在自同构映射, 设计了基于有限域 Z_N 的 Logistic 映射自同构序列生成算法。

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Correspondence to Xiaofeng Liao.

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Yang, B., Liao, X. Period analysis of the Logistic map for the finite field. Sci. China Inf. Sci. 60, 022302 (2017). https://doi.org/10.1007/s11432-015-0756-1

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Keywords

  • chaos
  • Logistic map
  • period analysis
  • sequence generated algorithm
  • Galois ring

关键词

  • 混沌
  • Logistic 映射
  • 周期分析
  • 序列生成算法
  • Galois 环 Z_(3^n)