A two-level coprime-inverse framework for secure grayscale image encryption

This paper presents a grayscale image encryption scheme built on a two-level coprime-inverse construction. Key-dependent byte streams are obtained by inverting coprime residues modulo 1024 and then inverting the odd values of the result modulo 512, which yields bytes in the range $$\:\{\text{0,1},\dots\:,255\}.$$ Three such streams are generated: one supplies the ranking values that define the permutation, and the other two control forward and backward diffusion through modular addition, XOR and circular bit rotation. No substitution box, chaotic iteration or DNA coding rule is used, and every stage of the cipher has an explicit inverse. We also prove that each complete 256-byte block produced by the generator is a permutation of the whole byte alphabet, which accounts for the first-order balance seen in the ciphertext. The scheme was tested on seven benchmark images drawn from aerial, planetary, photographic and radiographic sources, and compared with seven recently published implementations representing substitution-box, quantum-operator, deep-learning-assisted, multi-chaotic and chaos-FPGA designs. All eight implementations were run at 256 × 256 resolution under a single protocol with one set of metric definitions, and exact recovery was confirmed for all 56 image-and-scheme combinations before any attack was applied. Entropy of the encrypted images lies between 7.9970 and 7.9977 bits per pixel, adjacent-pixel correlations stay below 0.01 in absolute value in all three directions, and plaintext-to-ciphertext NPCR ranges from 99.5911% to 99.6307%. Key sensitivity was measured over all 256 single-bit changes of the 256-bit master key: mean NPCR was 99.6077–99.6117% and mean UACI 33.4484–33.4943%, and no modified key recovered the plain image. Under ciphertext corruption the scheme gives the highest average reconstruction quality of the eight methods for Gaussian noise and remains competitive under cropping and salt-and-pepper noise, while reaching the highest core throughput, 8.167 MPixels/s for encryption and 21.679 MPixels/s for decryption. The proposed two-level coprime-inverse scheme therefore offers a clear, S-box-free, and robust algebraic route for secure grayscale image encryption.