On the Linear Components Space of S-boxes Generated by Orthogonal Cellular Automata

We investigate S-boxes defined by pairs of Orthogonal Cellular Automata (OCA), motivated the fact that such CA always define bijective vectorial Boolean functions, and could thus be interesting for design block ciphers. In particular, we perform an exhaustive search all nonlinear OCA diameter $$d=4$$ $$d=5$$ , which generate size $$6\times 6$$ $$8\times 8$$ respectively. Surprisingly, these turn out to linear, they are not useful confusion layers in However, a closer inspection reveals very structure. Indeed, remark linear components space OCA-based found our themselves kernels CA, or, equivalently, polynomial codes. finally classify codes obtained observe that, most cases, actually correspond cyclic code with generator $$X^{b}+1$$ where $$b=d-1$$ . Although findings rule possibility using good ciphers, give nonetheless some insights theoretical characterization pairs, is still open question general.