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Hyojeong Choi1, Gangsan Kim1, Hong-Yeop Song1

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

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Resumen

Este estudio demuestra que un nuevo mapa caótico discreto exhibe una divergencia caótica óptima. El mapa asegura que las entradas con la misma salida tengan la misma paridad, mejorando las aplicaciones criptográficas.

Palabras clave:
mapa caóticoexponente de Lyapunov discretocaos discretoprecisión finitasecuencias aleatoriasmapa de carpa sesgado

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Área de la Ciencia:

  • Teoría de números
  • Matemáticas discretas
  • Criptografía

Sus antecedentes:

  • Los mapas de carpa sesgados discretos son fundamentales en la teoría del caos.
  • La comprensión de las relaciones entrada-salida en mapas caóticos es crucial para su aplicación.
  • Las propiedades de paridad de los mapas caóticos pueden revelar estructuras subyacentes.

Objetivo del estudio:

  • Introducir un nuevo mapa caótico discreto con propiedades biyectivas probadas.
  • Demostrar que el mapa propuesto logra una divergencia caótica máxima entre los mapas de permutación.
  • Analizar el comportamiento caótico del nuevo mapa a través de experimentos numéricos.

Principales métodos:

  • Demostración de la propiedad de paridad de los mapas de carpa sesgados discretos simétricos (Teorema 1).
  • Definición y demostración de la naturaleza biyectiva del nuevo mapa caótico discreto (Definición 1, Teorema 2).
  • Cálculo y análisis del exponente de Lyapunov discreto (dLE) para evaluar las propiedades caóticas (Teorema 3).
  • Realización de experimentos numéricos que incluyen entropía de aproximación, entropía de permutación, pruebas NIST SP800-22 y análisis de correlación.

Principales resultados:

  • Se estableció que las entradas que producen la misma salida en mapas de carpa sesgados discretos simétricos comparten la misma paridad.
  • Se desarrolló un nuevo mapa caótico discreto que se demuestra que es una biyección para todos los parámetros.
  • Se demostró que el dLE del mapa propuesto se acerca al valor máximo posible para mapas de permutación, lo que indica una alta divergencia caótica.
  • Los experimentos numéricos confirmaron el comportamiento caótico del mapa a través de cálculos de entropía y pruebas estadísticas.

Conclusiones:

  • El mapa caótico discreto propuesto posee propiedades deseables como biyección y divergencia caótica máxima.
  • La propiedad de paridad ofrece una característica única para posibles aplicaciones criptográficas.
  • El mapa sirve como un fuerte candidato para la generación de números pseudoaleatorios y sistemas de comunicación segura.