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9 pages, 262 KB

Open AccessArticle

by Idriss J. Aberkane

Mathematics 2022, 10(11), 1835; https://doi.org/10.3390/math10111835 - 26 May 2022

Viewed by 16923

The algebraic topology of Collatz attractors (or “Collatz Feathers”) remains very poorly understood. In particular, when pushed to infinity, is their set of branches discrete or continuous? Here, we introduce a fundamental theorem proving that the latter is true. For any odd x [...] Read more.

A_{x}^{n}

as the set of all odd numbers with

S y r (x)

in their Collatz orbit and up to n more digits than x in base 2. We then prove

{lim}_{n \to \infty} \frac{| A_{x}^{n} |}{2^{n + c}} \geq 1

with

c > - 4

for all x and, in particular,

c = 0

for

x = 1

, which is a result strictly stronger than that of Tao 2019. Full article

(This article belongs to the Section C2: Dynamical Systems)

11 pages, 6442 KB

Open AccessArticle

A Novel S-Box Dynamic Design Based on Nonlinear-Transform of 1D Chaotic Maps

by Wenhao Yan and Qun Ding

Electronics 2021, 10(11), 1313; https://doi.org/10.3390/electronics10111313 - 30 May 2021

Cited by 36 | Viewed by 3200

Abstract

In this paper, a method to enhance the dynamic characteristics of one-dimension (1D) chaotic maps is first presented. Linear combinations and nonlinear transform based on existing chaotic systems (LNECS) are introduced. Then, a numerical chaotic map (LCLS), based on Logistic map and Sine map, is given. Through the analysis of a bifurcation diagram, Lyapunov exponent (LE), and Sample entropy (SE), we can see that CLS has overcome the shortcomings of a low-dimensional chaotic system and can be used in the field of cryptology. In addition, the construction of eight functions is designed to obtain an S-box. Finally, five security criteria of the S-box are shown, which indicate the S-box based on the proposed in this paper has strong encryption characteristics. The research of this paper is helpful for the development of cryptography study such as dynamic construction methods based on chaotic systems. Full article

(This article belongs to the Special Issue Optical Electronic Systems, Communications and Security)

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