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Some New Maximally Chaotic Discrete Maps.

Hyojeong Choi1, Gangsan Kim1, Hong-Yeop Song1

  • 1Department of Electrical and Electronic Engineering, Yonsei University, Seoul 03722, Republic of Korea.

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|January 28, 2026
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Summary
This summary is machine-generated.

This study proves a new discrete chaotic map exhibits optimal chaotic divergence. The map ensures inputs with the same output have identical parity, enhancing cryptographic applications.

Keywords:
chaotic mapdiscrete Lyapunov exponentdiscrete chaosfinite precisionrandom sequencesskew tent map

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Area of Science:

  • Number Theory
  • Discrete Mathematics
  • Cryptography

Background:

  • Discrete skew tent maps are foundational in chaos theory.
  • Understanding input-output relationships in chaotic maps is crucial for their application.
  • Parity properties of chaotic maps can reveal underlying structures.

Purpose of the Study:

  • To introduce a novel discrete chaotic map with proven bijective properties.
  • To demonstrate that the proposed map achieves maximal chaotic divergence among permutation maps.
  • To analyze the chaotic behavior of the new map through numerical experiments.

Main Methods:

  • Proving the parity property of symmetric discrete skew tent maps (Theorem 1).
  • Defining and proving the bijective nature of the new discrete chaotic map (Definition 1, Theorem 2).
  • Calculating and analyzing the discrete Lyapunov exponent (dLE) to assess chaotic properties (Theorem 3).
  • Conducting numerical experiments including approximation entropy, permutation entropy, NIST SP800-22 tests, and correlation analysis.

Main Results:

  • Established that inputs yielding the same output in symmetric discrete skew tent maps share the same parity.
  • Developed a new discrete chaotic map proven to be a bijection for all parameters.
  • Demonstrated that the proposed map's dLE approaches the maximum possible value for permutation maps, indicating high chaotic divergence.
  • Numerical experiments confirmed the map's chaotic behavior via entropy calculations and statistical tests.

Conclusions:

  • The proposed discrete chaotic map possesses desirable properties like bijection and maximal chaotic divergence.
  • The parity property offers a unique characteristic for potential cryptographic applications.
  • The map serves as a strong candidate for pseudorandom number generation and secure communication systems.