Modification of the Logistic Map Using Fuzzy Numbers with Application to Pseudorandom Number Generation and Image...

Area of Science:

• Chaos theory
• Fuzzy mathematics
• Cryptography

Background:

• The classic logistic map is a fundamental model in chaos theory, but its complexity can be limited.
• Fuzzy numbers offer a way to introduce uncertainty and imprecision into mathematical models.
• Pseudo-random bit generation is crucial for secure communication and data encryption.

Purpose of the Study:

• To propose a modified logistic map incorporating fuzzy triangular numbers.
• To analyze the complexity and chaotic behavior of the new map.
• To evaluate its application in pseudo-random bit generation and image encryption.

Main Methods:

• Modification of the classic logistic map using fuzzy triangular numbers.
• Analysis of the modified map using Lyapunov exponent (LE) and bifurcation diagrams.
• Generation of pseudo-random bit sequences from the modified map.
• Statistical testing of the generated sequences using the National Institute of Standards and Technology (NIST) suite.
• Application of the random bit generator (RBG) to image encryption.

Main Results:

• The modified logistic map exhibits higher complexity than the classic version.
• The map demonstrates phenomena such as antimonotonicity and crisis.
• The developed random bit generator (RBG) successfully passes all NIST statistical tests.
• The RBG is effectively applied to encrypt images, demonstrating its practical utility.

Conclusions:

• The fuzzy triangular number-based logistic map offers enhanced complexity and chaotic behavior.
• This modified map provides a robust foundation for developing high-quality pseudo-random bit generators.
• The proposed RBG is suitable for cryptographic applications, including image encryption, due to its statistical soundness.