Annaby, M. H., M. E. Mahmoud, H. A. Abdusalam, H. A. Ayad, and M. A. Rushdi, 2D Representations of 3D Point Clouds Via the Stereographic Projection with Encryption Applications, , vol. 30, issue 4, pp. 173, 2024. AbstractWebsite

Using the topological equivalence between the Riemann sphere $$\mathbb {S}$$and the extended complex plane $$\overline{\mathbb {C}} = \mathbb {C} \cup \{\infty \}$$, where $$\mathbb {C}$$is the field of complex numbers, we establish 2D-bijective representations of 3D point clouds. Points of 3D point clouds are mapped into the Riemann sphere $$\mathbb {S}$$, and a stereographic projection is implemented to map the points into the complex plane $$\mathbb {C}$$. The way the 3D objects are mapped into $$\mathbb {S}$$may be varied for various applications. To prove the accuracy and efficiency of the proposed 2D representation of 3D objects, we apply this correspondence to 3D point cloud encryption. We utilize chaotic permutations, chaotic circuits, and Latin cubes in addition to the stereographic projection representation to construct our scheme. The permutation steps using chaotic maps and Latin cubes are carried out on the object data points in both $$\mathbb {S}$$and $$\overline{\mathbb {C}}$$, while the chaotic circuits are applied to 2D projections of the 3D objects. To the best of our knowledge, no earlier work employed stereographic projections for 3D object encryption. Experimental simulations of this method show high encryption strength and strong confusion and diffusion properties based on quantitative and statistical measures.

Annaby, M. H., M. E. Mahmoud, H. A. Abdusalam, H. A. Ayad, and M. A. Rushdi, On 3D encryption schemes based on chaotic permutations and rotations with geometric stability, , vol. 300, pp. 171680, 2024. AbstractWebsite

Three-dimensional point-cloud data has been enormously abundant with the emergence of 3D data acquisition, processing, and visualization technologies. Encryption algorithms have been recently introduced to ensure secure storage and communication for this type of data. Maintaining algorithmic correctness and the geometric stability are still key challenges towards the construction of reliable, trustful, and practical ciphers of 3D point clouds. To address these challenges, Jolfaei et al. (2015) proposed a 3D object encryption algorithm along with geometric notions of dimensional and spatial stability. However, these notions are not consistent, and the geometric stability and correctness of that algorithm are not guaranteed as we show through counterexamples. In this paper, we introduce two enhanced ciphers with correctness, reversibility, and geometric stability guarantees. These ciphers employ chaotic permutations with hyperchaotic maps of highly complex behavior for enhanced security. The permutation scheme ensures the creation of consistent solvable equations for the decryption stage. Also, an enhanced 3D point rotation scheme is proposed to ensure geometric stability. The soundness and significance of the proposed ciphers are demonstrated by rigorous mathematical proofs. As well, extensive experimentation and comparisons against state-of-the-art methods are demonstrated through similarity analysis based on the Hausdorff and Euclidean distances, analysis of the sensitivity to plaintext and key perturbations, and analysis of the robustness to statistical attacks.

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