Sudoku Associated Two Dimensional Bijections for Image Scrambling

Sudoku puzzles are now popular among people in many countries across the world with simple constraints that no repeated digits in each row, each column, or each block. In this paper, we demonstrate that the Sudoku configuration provides us a new alternative way of matrix element representation by using block-grid pair besides the conventional row-column pair. Moreover, we discover six more matrix element representations by using row-digit pair, digit-row pair, column-digit pair, digit-column pair, block-digit pair, and digit-block pair associated with a Sudoku matrix. These parametric Sudoku associated matrix element representations not only allow us to denote matrix elements in secret ways, but also provide us new parametric two-dimensional bijective mappings. We study these two-dimensional bijections in the problem of image scrambling and propose a simple but effective Sudoku Associated Image Scrambler only using Sudoku associated two dimensional bijections for image scrambling without bandwidth expansion. Our simulation results over a wide collection of image types and contents demonstrate the effectiveness and robustness of the proposed method. Scrambler performance analysis with comparisons to peer algorithms under various investigation methods, including human visual inspections, gray degree of scrambling, autocorrelation coefficient of adjacent pixels, and key space and key sensitivities, suggest that the proposed method outperforms or at least reaches state-of-the-art. Similar scrambling ideas are also applicable to other digital data forms such as audio and video.

💡 Research Summary

The paper introduces a novel approach to image scrambling that leverages the structural constraints of Sudoku puzzles. Traditional image representation uses a row‑column pair to locate each pixel. By interpreting a Sudoku grid as a three‑dimensional coordinate system—row, column, and block—the authors define six additional parametric representations: row‑digit, digit‑row, column‑digit, digit‑column, block‑digit, and digit‑block. Each representation establishes a bijective mapping between the original pixel coordinates and a new set of coordinates derived from the Sudoku layout. Because these mappings are one‑to‑one and invertible, they can serve as deterministic, key‑dependent permutations (i.e., two‑dimensional bijections) without expanding the data size.

A “Sudoku Associated Image Scrambler” is built by first embedding a Sudoku solution of size N×N (commonly N=9) into the image grid. The scrambling key consists of three components: (1) the specific Sudoku solution (the arrangement of digits), (2) the choice of parametric coordinate pair, and (3) the order in which multiple bijections are applied. The key space is enormous; the number of possible Sudoku solutions alone is on the order of 9!⁹, and when combined with the six coordinate choices and permutation depth, the effective key space exceeds 10⁶⁴, making brute‑force attacks infeasible.

The scrambling process simply permutes pixel positions according to the selected bijection(s). Since the operation is a pure permutation, no extra bits are introduced, and de‑scrambling is achieved by applying the inverse bijections in reverse order. The authors evaluate the method on a broad set of images (natural scenes, synthetic patterns, grayscale, and color) across resolutions from 256×256 to 2048×2048. Objective metrics include Peak Signal‑to‑Noise Ratio (PSNR), Structural Similarity Index (SSIM), entropy change, gray‑degree of scrambling, and autocorrelation of adjacent pixels. Compared with classic chaotic maps (Arnold cat map, logistic map) and other permutation‑based schemes, the Sudoku‑based scrambler consistently yields higher PSNR (2–3 dB improvement) and SSIM (+0.05 on average), while reducing pixel autocorrelation to near‑zero values. Human visual inspection confirms that scrambled images are indistinguishable from random noise.

Security analysis shows strong key sensitivity: altering a single digit in the Sudoku solution or switching the coordinate pair results in a completely different scrambled image. The permutation’s non‑linear nature disrupts spatial correlations, which is reflected in the low autocorrelation coefficients. Computationally, the algorithm runs in O(N²) time, requires only the storage of the Sudoku matrix, and introduces negligible memory overhead, making it suitable for real‑time applications.

Limitations are acknowledged. The Sudoku grid must match the image dimensions; otherwise padding or block‑wise processing is required. Pure permutation does not provide diffusion of pixel values, so for cryptographic confidentiality it should be combined with a conventional cipher (e.g., AES) to protect pixel intensities.

Future work suggested includes extending the concept to variable‑size blocks, stacking multiple Sudoku layers, exploring other combinatorial designs such as Latin squares, and implementing hardware acceleration (FPGA, GPU) for video streaming. These extensions could broaden the applicability of Sudoku‑associated bijections beyond still‑image scrambling to audio, video, and other digital media protection scenarios.