The new system, called 3DSLCM-CNN, separates these two roles. Confusion, the scrambling of pixel positions, is handled by a modified and extended version of the sinusoidal logistic function, pushed into three dimensions so that it produces three coupled random sequences. Diffusion, which changes actual pixel values, relies on a key map generated from a separate one-dimensional logistic sinusoidal function and then compressed by a deep network with dilated convolution before it is applied at the bit level.

The three-dimensional map at the heart of the confusion stage extends the classic one-dimensional logistic map, a staple of chaos-based cryptography since the 1970s, by adding two more state variables and three control parameters. The initial values range from zero to one, two control parameters lie between zero and one, and the third ranges from 3.5 to 4, specified to fifteen decimal places. In the reported experiments the initial values were set to 0.20, 0.557 and 0.653, with control parameters 0.0125, 0.0257 and 3.8691.

To verify that the map genuinely behaves chaotically, the team computed Lyapunov exponents, which quantify whether nearby trajectories of a dynamical system diverge or converge over time. The x-sequence produced a positive exponent of 0.568, while the y and z sequences yielded small negative values of −0.0126 and −0.01187; with respect to the other two control parameters the x-sequence exponents were 0.47 and 0.585. Bifurcation analysis showed period-doubling up to a control parameter value of 3.5, with bifurcation points near 2.96 and 3.97, supporting the conclusion that the system enters a chaotic regime.

Randomness was further checked with the standard NIST statistical test suite. The authors generated one thousand sequences of one million bits each and report that the sequence passed all fifteen tests, with pass proportions in the range of roughly 0.9 to 1 and p-values lying between zero and one. These statistical checks concern the generated sequences themselves rather than any clinical or physical outcome, but they are the customary evidence that a chaotic source is suitable for cryptographic scrambling.

The encryption pipeline itself unfolds in five steps over four rounds of confusion and one diffusion. First, the plaintext image, sized 256 by 256 in the experiments, is scrambled by row-column shuffling according to predefined rules that split the image into sixteen horizontal blocks of varying heights and permute them in a fixed pattern. The scrambled image is then sliced into odd and even row images, each 128 by 256, reshaped into 256 by 128 forms, and scrambled with the first chaotic map derived from the three-dimensional system before being recombined into a single 256 by 256 image.

Diffusion follows at the bit level: the confused image and the key map are both converted to eight-bit binary form and combined with the XOR operation, altering pixel values rather than merely their positions. The key map begins as 262,144 random values from the one-dimensional logistic sinusoidal function, normalized to the 0-to-255 intensity range and reshaped to 512 by 512. A CNN with dilated convolution, using a three-by-three kernel with stride two, padding two and dilation rate two, then reduces it to 256 by 256 so it matches the plaintext dimensions. Because the network is used purely as a fixed size-reduction operator, no training or testing is required. Two further rounds of confusion with the second and third chaotic maps complete the encryption, and decryption simply reverses the entire process.

Performance was measured on standard test images including Bridge, Baboon, Peppers and a color Baboon, drawn from freely available databases such as the SIPI image database, the MAMe dataset and the Google Landmarks dataset. Encrypted images reached information entropy values of 7.9969 to 7.9971 on the theoretical maximum of 8 bits per pixel, indicating near-uniform pixel distributions. Histograms of the encrypted images appeared flat, and a chi-square test on one encrypted image gave a statistic of 281.921875, below the ideal threshold of 293.2478, with a p-value of 0.118633, quantitatively confirming the uniformity. Peak signal-to-noise ratios between plaintext and cipher images fell between 8.9251 and 9.6889 dB, below the 10 dB level the authors cite as desirable for encryption quality.

Resistance to differential attacks was assessed with the standard NPCR and UACI metrics, which compare two ciphertexts encrypted from plaintexts differing in a single pixel. Correlation coefficients between adjacent pixels in the encrypted images, measured horizontally, vertically and diagonally, all sat near zero, in contrast to the near-one values of natural plaintext images. Against occlusion attacks, in which portions of the ciphertext are cropped away, decrypted images remained recognizable even with half the encrypted data destroyed, with PSNR values of 14.5923 for 25 percent cropping and 10.1334 for 75 percent cropping, and structural similarity scores declining predictably as cropping increased. The system also recovered plaintext information from ciphertexts contaminated with up to 50 percent salt-and-pepper noise.

Key space analysis counted eight independent secret parameters, three initial values and three control parameters for the three-dimensional map plus one control parameter and one initial value for the key-generating sequence, each specified to fifteen decimal places. That yields a theoretical key space of roughly 10 to the power 120, equivalent to about 2 to the power 399 or 399 bits, well above the commonly cited lower bound of 2 to the power 100. Sensitivity tests showed that changing any initial value in the fifteenth decimal place, or perturbing the CNN kernel weights, produced complete decryption failure. The authors also report robustness against chosen-plaintext and known-plaintext attacks, with adjacent-pixel correlations in encrypted all-black and all-white test images of 0.00530 and 0.00509 respectively, and an encryption time of 1.325 seconds per 256-by-256 grayscale image on an Intel i5 machine with 8 GB of RAM, faster than the comparison schemes listed. The team acknowledges one limitation: the predefined shuffling rules are static and deterministic, and they plan to explore dynamic rules in future work. In comparative tables against recent methods, the proposed scheme scored comparably on NPCR, UACI, entropy and correlation, with the authors noting that the differences among recent schemes are small, while its distinctive contribution remains the decoupling of the diffusion key from the chaotic map through the CNN-generated key map.

Subject of Research: A new image encryption method using a 3D sinusoidal logistic chaotic map and a CNN-based key map for diffusion.

Article Title: 3DSLCM-CNN: 3D sinusoidal logistic chaotic map and CNN for image encryption

Article References: Mahapriya, C., Thanikaiselvan, V., & Thomas, S. B. (2026). 3DSLCM-CNN: 3D sinusoidal logistic chaotic map and CNN for image encryption. Frontiers in Artificial Intelligence, 9, Article 1919612. https://doi.org/10.3389/frai.2026.1919612

Image Credits: AI Generated

DOI: 10.3389/frai.2026.1919612

Keywords: image encryption, chaotic map, cryptography, CNN, dilated convolution, diffusion, confusion, NPCR, UACI, entropy, key space, differential attack