A cubic map chaos criterion theorem with applications in generalized synchronization based pseudorandom number...

Area of Science:

• Chaos theory
• Discrete dynamical systems
• Cryptography

Background:

• Chaos criterion theorems are essential for understanding discrete maps.
• Generalized synchronization (GS) is a key concept in chaotic systems.
• Pseudorandom number generators (PRNGs) are crucial for secure communication.

Purpose of the Study:

• To establish a chaos criterion theorem for cubic polynomial discrete maps.
• To construct an eight-dimensional chaotic GS system.
• To develop and evaluate a chaos-based pseudorandom number generator (CPRNG) for cryptographic applications.

Main Methods:

• Development of a novel chaos criterion theorem for cubic polynomial discrete maps.
• Construction of an eight-dimensional chaotic GS system using existing theorems.
• Implementation and testing of a CPRNG based on the chaotic GS system.
• Validation using FIPS 140-2 and Generalized FIPS 140-2 test suites.
• Application in image encryption with perturbed keys.

Main Results:

• The constructed eight-dimensional chaotic GS system passed theoretical verification.
• The CPRNG achieved high pass rates (99.9%/98.5%) on FIPS 140-2/Generalized FIPS 140-2 randomness tests.
• The CPRNG offers a large key space (> 2^1345).
• Image encryption experiments showed high security, with decrypted texts being independent of the original image, even with perturbed keys.

Conclusions:

• The proposed chaos criterion theorem and the resulting chaotic GS system are effective.
• The CPRNG is a viable tool for generating high-quality pseudorandom numbers for cryptography.
• The image encryption scheme demonstrates strong resistance against attacks, including those with perturbed keys.