Periodicity Analysis of the Logistic Map over Ring ℤ3n

Xiaoxiong Lu, Eric Yong Xie, Chengqing Li
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引用次数: 8

Abstract

Periodicity analysis of sequences generated by a deterministic system is a long-standing challenge in both theoretical research and engineering applications. To overcome the inevitable degradation of the logistic map on a finite-precision circuit, its numerical domain is commonly converted from a real number field to a ring or a finite field. This paper studies the period of sequences generated by iterating the logistic map over ring [Formula: see text] from the perspective of its associated functional network, where every number in the ring is considered as a node, and the existing mapping relation between any two nodes is regarded as a directed edge. The complete explicit form of the period of the sequences starting from any initial value is given theoretically and verified experimentally. Moreover, conditions on the control parameter and initial value are derived, ensuring the corresponding sequences to achieve the maximum period over the ring. The results can be used as ground truth for dynamical analysis and cryptographical applications of the logistic map over various domains.
环上Logistic映射的周期性分析
确定性系统产生的序列的周期性分析是理论研究和工程应用中一个长期存在的挑战。为了克服有限精度电路中逻辑映射不可避免的退化问题,通常将其数值域由实数域转换为环域或有限域。本文从环上逻辑映射的关联功能网络的角度研究环上逻辑映射迭代生成序列的周期[公式:见文],将环上的每一个数视为一个节点,将存在的任意两个节点之间的映射关系视为一条有向边。从理论上给出了从任意初值出发的序列周期的完备显式形式,并进行了实验验证。推导了控制参数和初始值的条件,保证了相应的序列在环上达到最大周期。该结果可作为逻辑映射在各种域上的动态分析和密码学应用的基础真理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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