{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-classification-of-s-boxes-generated-by","title":"A classification of S-boxes generated by Orthogonal Cellular Automata","arxiv_id":"2303.05228","date":"2023-03-09","proceeding":null,"authors":["Luca Mariot","Luca Manzoni"],"abstract":"Most of the approaches published in the literature to construct S-boxes via Cellular Automata (CA) work by either iterating a finite CA for several time steps, or by a one-shot application of the global rule. The main characteristic that brings together these works is that they employ a single CA rule to define the vectorial Boolean function of the S-box. In this work, we explore a different direction for the design of S-boxes that leverages on Orthogonal CA (OCA), i.e. pairs of CA rules giving rise to orthogonal Latin squares. The motivation stands on the facts that an OCA pair already defines a bijective transformation, and moreover the orthogonality property of the resulting Latin squares ensures a minimum amount of diffusion. We exhaustively enumerate all S-boxes generated by OCA pairs of diameter $4 \\le d \\le 6$, and measure their nonlinearity. Interestingly, we observe that for $d=4$ and $d=5$ all S-boxes are linear, despite the underlying CA local rules being nonlinear. The smallest nonlinear S-boxes emerges for $d=6$, but their nonlinearity is still too low to be used in practice. Nonetheless, we unearth an interesting structure of linear OCA S-boxes, proving that their Linear Components Space (LCS) is itself the image of a linear CA, or equivalently a polynomial code. We finally classify all linear OCA S-boxes in terms of their generator polynomials.","url_abs":"https://arxiv.org/abs/2303.05228v1","url_pdf":"https://arxiv.org/pdf/2303.05228v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-classification-of-s-boxes-generated-by","repo_url":"https://github.com/rymoah/orthogonal-ca-sboxes","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}