Towards Analysable Chaos-based Cryptosystems: Constructing Difference Distribution Tables for Chaotic Maps

Chaos-based cryptography has yet to achieve practical, real-world applications despite extensive research. A major challenge is the difficulty in analysing the security of these cryptosystems, which often appear ad hoc in design. Unlike conventional cryptography, evaluating the security margins of chaos-based encryption against attacks such as differential cryptanalysis is complex. This paper introduces a straightforward approach of using chaotic maps in cryptographic algorithms in a way that facilitates cryptanalysis. We demonstrate how a chaos-based substitution function can be constructed using fixed-point representation, enabling the application of conventional cryptanalysis tools such as the difference distribution table. As a proof-of-concept, we apply our method to the logistic map, showing that differential properties vary based on the initial state and number of iterations. Our findings demonstrate the feasibility of designing analysable chaos-based cryptographic components with well-understood security margins.