New mathematical model makes image encryption more resilient to real-world uncertainty
image: An artistic padlock built from the mathematics of the study. Its glowing shackle is not drawn by hand: it is made of millions of points generated by a "fuzzy skew map", a chaotic equation whose key setting is deliberately uncertain. Each thread of light corresponds to a different possible value of the key; brighter, taller threads are the more likely values. The body of the lock is a mosaic of encrypted pixels produced by the same chaotic system, dissolving on the right into scattered fragments, as an image does when it is encrypted. The researchers showed that this kind of chaos stays strong and unpredictable even when the key is uncertain, making it a robust ingredient for protecting digital images.
Credit: Illych Alvarez et al., ESPOL / UCSG. Artistic visualization created for outreach from the equations in Alvarez et al., Mathematics 2026, 14, 1010 (MDPI), CC BY 4.0.
In today's digital world, everything from medical images and financial records to personal photos and secure communications depends on encryption systems that can protect sensitive information. But real-world technologies rarely operate under perfect conditions. Measurements can be imprecise, hardware introduces small errors, and communication channels are often affected by noise.
A new mathematical study proposes a way to make image encryption more resilient to these unavoidable uncertainties by combining robust chaos with fuzzy logic, two mathematical concepts that together help maintain security even when system parameters are not perfectly known.
The researchers developed a new framework called fuzzy skew maps, which extends robust chaotic systems to environments where uncertainty is part of the problem rather than something to ignore. Their work demonstrates that chaotic encryption can remain stable even when key parameters vary within realistic ranges of uncertainty.
"Real-world cryptographic systems cannot always rely on perfectly precise parameters," the researchers explain. "Our model shows that security can be preserved even when uncertainty is explicitly incorporated into the mathematical design."
Chaos has been widely used in cryptography because of its extreme sensitivity to initial conditions. A tiny change in the starting point of a chaotic process produces a completely different outcome, making chaotic systems highly effective for generating the unpredictable sequences needed to encrypt information.
However, traditional chaotic models have an important limitation. Many lose their chaotic behavior when control parameters change slightly, producing predictable periodic patterns that may reduce encryption strength.
The new study addresses this problem by using robust chaos, a form of chaos that remains stable across a continuous range of parameter values instead of only at carefully selected points. The researchers then incorporated fuzzy logic, a mathematical framework that represents uncertainty through degrees of confidence rather than exact numerical values.
Instead of assuming that encryption parameters are perfectly known, the model treats them as fuzzy numbers, allowing uncertainty caused by measurement errors, hardware limitations, numerical approximations, or imperfect key generation to become part of the mathematical analysis.
The researchers demonstrated that the proposed fuzzy skew maps preserve the essential properties required for secure encryption. Even under uncertainty, the systems maintain strong chaotic behavior, including positive Lyapunov exponents—an indicator that small changes continue to produce dramatically different outcomes—and avoid the periodic windows that can weaken conventional chaotic systems.
To evaluate the practical implications of their approach, the team applied the model to chaos-based image encryption.
In this scheme, the fuzzy parameter becomes part of the secret encryption key, while the chaotic system generates pseudorandom sequences that scramble and mask image pixels. The resulting encrypted images reveal virtually no visual or statistical information about the originals.
Performance tests showed strong cryptographic results. The encryption method achieved NPCR values close to 99.6%, UACI around 33.5%, entropy approaching the theoretical maximum of 8 bits, and nearly zero pixel correlation—all widely accepted indicators of secure image encryption.
Importantly, these performance levels remained stable across multiple fuzzy parameter configurations, including triangular, trapezoidal, and Gaussian fuzzy models. This suggests that the encryption system continues to perform reliably even when uncertainty affects its operating conditions.
The researchers also compared their approach with fuzzy versions of several classical chaotic maps, including logistic, tent, and Chebyshev maps. While all methods performed well, the fuzzy skew maps consistently produced slightly stronger security metrics, indicating that skew transformations provide an additional level of robustness that is preserved after fuzzification.
Beyond security, the proposed method is also computationally efficient. Its computational cost grows linearly with image size and the number of uncertainty levels analyzed, making it suitable for practical applications without introducing excessive processing overhead.
According to the researchers, the framework creates a bridge between abstract mathematical theory and practical cybersecurity by showing that uncertainty does not necessarily weaken encryption if it is incorporated into the design from the beginning.
The work could contribute to the development of more resilient encryption systems for applications including secure multimedia transmission, medical imaging, Internet of Things (IoT) devices, embedded systems, and real-time communications, where imperfect operating conditions are unavoidable.
The authors also identify several directions for future research, including hardware implementations of fuzzy skew maps, new mathematical measures of complexity for fuzzy chaotic systems, and large-scale evaluation of fuzzy-chaotic encryption protocols.
As digital technologies continue to expand into increasingly complex environments, the study highlights a simple but powerful idea: the strongest security systems should remain secure not only under ideal mathematical assumptions, but also under the uncertainty of the real world.