Fuzzy Skew Maps: Preserving Robust Chaos Under Uncertainty with Applications to Cryptography

We introduce fuzzy skew maps as a levelwise ($\alpha$-cut) extension of robustly chaotic skew transformations of S-unimodal maps to epistemically uncertain environments. Our central hypothesis is that the robust-chaos mechanism of the underlying skew family transfers to fuzzy parameter uncertainty in a set-based (not probabilistic) sense is as follows: for every $\alpha \in [0,1]$, the induced crisp family $\{F(·,q):q\in {\left[\tilde{q}\right]}_{\alpha }\}$ preserves the absence of periodic windows and maintains strictly positive Lyapunov exponents. This yields a precise notion of fuzzy robustness that is distinct from interval enclosures (pure bounds) and stochastic robustness (average-case guarantees). We also formalize fuzzy topological entropy via the extension principle and discuss its basic structural properties under mild continuity assumptions. For chaos-based image encryption, fuzzification provides an uncertainty-aware key representation and stabilizes cryptographic indicators across $\alpha$-cuts as follows: in our experiments, NPCR remains within $99.58$–$99.64\%$, UACI within $33.41$–$33.52\%$, and the cipher entropy is near 8 bits, while pixel correlation stays close to zero. These results support fuzzy skew maps as a robust primitive for secure information systems operating under parametric uncertainty.