Four-pixel sum-preserving encryption using the chaotic map for high thumbnail-preserving image security

The utilization of cloud-based photo-sharing services has been observed to result in the exposure of thumbnails, thereby engendering a privacy risk. Thumbnail-preserving encryption (TPE) is a method of mitigating this issue by encrypting images while maintaining the usability of thumbnails. Existing sum-preserving encryption (SPE) typically operates on 3-pixel groups, which are not aligned with power-of-two block sizes and can leave residual plaintext when tiling. The proposed mapping is a four-pixel SPE (4px-SPE) mapping, which is a piecewise bijection on 8-bit pixel quadruples that exactly preserves the four-pixel sum and is perfectly invertible. The scheme is integrated into a permutation-substitution TPE framework. It uses a 256-bit master key with deterministic, hash-based key derivation to reproduce per-image and per-block schedules, and to derive per-block keystreams and permutations without relying on plaintext-derived key material. A mathematical proof and exhaustive verification of all 256⁴ quadruples are provided to validate invertibility under edge cases. Experiments on both grayscale and color images evaluate entropy, histogram behavior, correlation/robustness metrics, and runtime. The results demonstrate high entropy and lossless recovery while maintaining thumbnail usability.