General Construction Method and Proof for a Class of Quadratic Chaotic Mappings

Wenxia Xu, Xiangkun Chen, Ziwei Zhou (), Guodong Li and Xiaoming Song ()
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Wenxia Xu: School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541002, China
Xiangkun Chen: School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541002, China
Ziwei Zhou: School of Mathematics and Data Science, Changji University, Changji 831100, China
Guodong Li: School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541002, China
Xiaoming Song: School of Computer Science and Information Security, Guilin University of Electronic Technology, Guilin 541002, China

Mathematics, 2025, vol. 13, issue 15, 1-10

Abstract: The importance of chaotic systems as the main pseudo-random cryptographic generator of encryption algorithms in the field of communication secrecy cannot be overstated, but in practical applications, researchers often choose to build upon traditional chaotic maps, such as the logistic map, for study and application. This approach provides attackers with more opportunities to compromise the encryption scheme. Therefore, based on previous results, this paper theoretically investigates discrete chaotic mappings in the real domain, constructs a general method for a class of quadratic chaotic mappings, and justifies its existence based on a robust chaos determination theorem for S single-peaked mappings. Based on the theorem, we construct two chaotic map examples and conduct detailed analysis of their Lyapunov exponent spectra and bifurcation diagrams. Subsequently, comparative analysis is performed between the proposed quadratic chaotic maps and the conventional logistic map using the 0–1 test for chaos and SE complexity metrics, validating their enhanced chaotic properties.

Keywords: robust chaos; quadratic mapping; S single-peaked mappings; chaotic map; 0–1 test; SE complexity metrics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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