A Content-Aware Pure Permutation with Intrinsic Avalanche Effect: Breaking the Diffusion-Permutation Dichotomy

This paper introduces the Triangular Content-Aware Permutation (TCA) algorithm, which challenges the long-standing assumption in the technical literature that permutation alone cannot create differential sensitivity; that is, changing a single pixel in the input, after applying permutation, merely relocates that same pixel in the output, producing no avalanche effect.

The proposed method creates a unique image partitioning by extracting edge points using the Canny operator and applying Delaunay Triangulation to a set comprising edges and corners, then uses this partitioning to construct the permutation pattern. Since "the triangulation structure is highly sensitive to image geometry and topology, changing even a single pixel alters the edge map, resulting in a fundamentally different triangulation and an entirely new global permutation pattern."

The paper's main conceptual contribution is demonstrating that a pure permutation can exhibit significant differential sensitivity. Unlike "classical dimension-based permutations—which are still currently used in the permutation stage of data hiding systems—and advanced content-aware modern permutations (2025–2026) employed in the permutation stage of image encryption systems, all of which lack differential sensitivity at this stage—TCA dramatically increases NPCR from near-zero values (in static-formula permutations) to 97.10% solely through pixel relocation."

The TCA algorithm operates in three main stages:

1. Content Feature Extraction: Edge points are extracted using the Canny algorithm

2. Geometric Partitioning: The extracted edge points, along with the four image corners, are used as seeds for Delaunay Triangulation, which inherently partitions the convex hull of its input points; that is, every pixel belongs to exactly one triangle, the triangles have no overlaps, and their union covers the entire region

3. Pixel Permutation: Pixels are permuted based on the geometric properties of the generated triangles, with the permutation map created by traversing the list of triangles and then performing an internal shuffling of the pixels within each triangle region

The experiments were conducted on 50 grayscale images of size 512×512 (48 from the UGR database, plus two additional images from USC-SIPI and MATLAB's default image). Key quantitative results include:

• Differential Sensitivity: TCA achieves NPCR = 97.10% and UACI = 20.06% with an average of 14.81 iterations, compared to conventional methods that maintain near-zero NPCR under identical conditions (the Arnold transform yields NPCR ≈ 0.0004% and PSNR > 100 dB even after 100 iterations)

• Statistical Performance: Low PSNR values (11.93 dB) and near-zero correlation coefficients (∼10−3) confirm the algorithm's superior statistical performance between two images initially differing by only one pixel

• Chaos Threshold: The number of iterations required to reach the chaos threshold varies from 6.4 to 30.7 depending on content complexity, rooted in deep geometric dependency that renders the algorithm's behavior unpredictable

• Computational Cost: The average execution time for a 512×512 image is 4.536 seconds for approximately 15 iterations. However, only 5 iterations (1.49 seconds) are sufficient to break spatial correlation, and even a single iteration (0.298 seconds) suffices to guarantee the uniqueness of the permutation pattern for each image

The paper compares TCA against three classical scrambling algorithms (Arnold, Zigzag, and Spiral transforms) across multiple metrics. While all four algorithms achieve similar visual and statistical disruption (PSNR ≈ 11.93 dB, SSIM ≈ 0.027), the key difference lies in differential sensitivity. The paper notes that content-independent permutations (such as the Arnold transform) inherently lack the avalanche effect, because changing one pixel in the original image only results in the relocation of two pixels in the output image.

The paper also reviews advanced encryption schemes from 2024–2026 (including MIE-SPD, block permutation with weighted bit-plane chain diffusion, and various hyperchaotic systems) and demonstrates that none have evaluated or claimed an avalanche effect in the permutation stage alone.

The paper provides both quantitative and analytical arguments for TCA's resistance to Known-Plaintext Attacks (KPA):

1. Quantitative argument: For a 512×512 grayscale image with roughly uniform histogram, the number of possible permutations yielding the same histogram is approximately 10631000, making the probability of an attacker guessing the exact pixel arrangement effectively zero (P ≈ 10−631000)

2. Analytical argument: "The permutation map generated by TCA is non-transferable. Consequently, even if an attacker can extract the permutation pattern for a specific image pair (P1, C1), this pattern is inherently tied to the unique topology of P1 and provides no usable information about another image P2, even if P2 differs from P1 by only one pixel."

The paper acknowledges several limitations:

• Sensitivity to noise and alterations: Any noise or manipulation in the transmission channel can challenge the accurate reconstruction of this map

• Focus on grayscale images: The current study focuses on grayscale images, overlooking the complexities arising from inter-channel color correlations

• Computational complexity: Execution time compared to classical methods is higher, though this cost is fully justified by achieving unparalleled security features such as KPA resistance and over 97% improvement in permutation differential sensitivity

Given "the non-analytic and content-dependent nature of the permutation pattern—which makes reconstructing the mapping impossible without access to the reference image—the algorithm is well-suited for applications in reference-based encryption, fragile watermarking, and non-blind steganography."

The paper concludes that by leveraging the topological features of an image and employing Delaunay triangulation, the permutation layer can be transformed from a passive component into an active contributor to achieving the avalanche effect. The authors state that TCA demonstrates that content dependency is the key to designing a new generation of permutation functions for image security applications, elevating permutation from a simple shuffling operation to a form of 'geometric diffusion.'

Improvement 1: Content-Aware Cryptographic Permutation Module

The AI system can integrate TCA’s geometric partitioning (Canny edge detection + Delaunay triangulation) as a plug-in layer for any image encryption pipeline. Instead of static permutations (e.g., Arnold), the system dynamically generates permutation maps based on each image’s unique topology. This enables the AI to achieve NPCR > 97% in a single permutation stage, turning permutation from a passive shuffle into an active diffusion mechanism. The system can now provide avalanche-effect-level sensitivity in encryption, making it resistant to differential attacks even before any diffusion stage.

Improvement 2: Non-Transferable Key Generation for Image Security

The AI can generate encryption keys or permutation seeds that are inherently bound to the specific image content. Because TCA’s permutation map is non-transferable (even a 1-pixel change yields a completely different map), the system can produce one-time, image-specific encryption patterns. This makes the AI ideal for reference-based encryption, fragile watermarking, and non-blind steganography where the receiver must possess the original image to reconstruct the mapping. The system can now guarantee that stolen keys or patterns from one image are useless for decrypting any other image, even near-identical ones.

Improvement 3: Chaos-Threshold Prediction for Adaptive Security

The AI can predict the number of TCA iterations needed to reach the chaos threshold (NPCR ≈ 100%) based on image content complexity (e.g., edge density, texture entropy). By analyzing the triangulation structure, the system can automatically adjust iteration counts (from 6 to 31) to balance security and computational cost. This enables adaptive encryption where simple images are processed faster (e.g., 1–5 iterations) while complex images receive more iterations, optimizing throughput without sacrificing security.

Improvement 4: Noise-Resilient Geometric Hashing for Authentication

The AI can use TCA’s triangulation structure as a geometric hash for image authentication. By extracting edge points and triangulating them, the system creates a content-fingerprint that is highly sensitive to pixel-level tampering. This can be used for fragile watermarking or tamper localization: any modification to a pixel alters the edge map and triangulation, allowing the AI to pinpoint the exact region of alteration. The system can now detect and localize even single-pixel forgeries in images, which is critical for forensic analysis and integrity verification.

Improvement 5: Geometric Diffusion for Neural Network Preprocessing

The AI can apply TCA as a preprocessing step for neural networks that require input invariance to geometric transformations. By permuting pixels based on Delaunay triangulation, the system creates a “geometric diffusion” that spreads local information globally. This can improve the robustness of image classifiers or generative models against adversarial perturbations, since a single-pixel attack would trigger a completely different permutation pattern, effectively destroying the adversarial noise’s structure. The system can now use TCA as a defense layer that forces attackers to account for content-dependent, non-analytic transformations.

Improved AI System Capabilities

The integrated system can:

• Encrypt images with permutation-only stages that achieve avalanche-effect sensitivity (NPCR > 97%) without needing diffusion stages.

• Generate unique, non-transferable encryption keys per image, making known-plaintext attacks computationally infeasible (probability 10−631000).

• Adaptively choose iteration counts based on image complexity, reducing average processing time to 1.5 seconds for 5 iterations on 512×512 images.

• Detect and localize single-pixel tampering in images via triangulation-based geometric hashing.

• Provide a novel defense against adversarial attacks by making input transformations content-dependent and unpredictable.

• Operate in reference-based security scenarios (e.g., secure image sharing with a trusted receiver) where classical content-independent permutations fail.

Pixel permutation is a fundamental tool in image processing, image encryption, and data hiding (including watermarking and steganography) that rearranges pixels without changing their values. A common assumption in the literature is that permutation alone cannot create differential sensitivity; changing one pixel merely relocates that pixel in the output, producing no avalanche effect. This paper challenges this by introducing the Triangular Content-Aware Permutation (TCA) algorithm. The method extracts edge points using Canny and applies Delaunay Triangulation to edges and corners, creating a unique partition. Since triangulation is highly sensitive to image geometry, changing a single pixel alters the edge map, resulting in a completely different triangulation and global permutation pattern. Unlike classical dimension-based permutations and advanced content-aware methods (2025-2026), which lack differential sensitivity, TCA increases NPCR from near-zero to 97.10% solely through pixel relocation. Experiments on 50 images show that TCA, with an average of 14.81 iterations, achieves NPCR = 97.10% and UACI = 20.06%, proving pure permutation can create significant differential sensitivity. Conventional methods maintain near-zero NPCR. The iteration threshold varies from 6.4 to 30.7 based on content complexity. Low PSNR (11.93 dB) and near-zero correlation (10-3) confirm superior statistical performance. Although slower than classical methods due to triangulation, this is a deliberate trade-off for stronger security. Given the non-analytic, content-dependent nature of the pattern, TCA is ideal for reference-based encryption, fragile watermarking, and non-blind steganography.