A classification of S-boxes generated by orthogonal cellular automata

Abstract Most of the approaches published in literature to construct S-boxes via Cellular Automata (CA) work by either iterating a finite CA for several time steps, or one-shot application global rule. The main characteristic that brings together these works is they employ single rule define vectorial Boolean function S-box. In this work, we explore different direction design leverages on Orthogonal (OCA), i.e. pairs rules giving rise orthogonal Latin squares. motivation stands facts an OCA pair already defines bijective transformation, and moreover orthogonality property resulting squares ensures minimum amount diffusion. We exhaustively enumerate all generated diameter $$4 \le d 6$$ 4 ≤ d 6 , measure their nonlinearity. Interestingly, observe $$d=4$$ = $$d=5$$ 5 are linear, despite underlying local being nonlinear. smallest nonlinear emerges $$d=6$$ but nonlinearity still too low be used practice. Nonetheless, unearth interesting structure linear S-boxes, proving Linear Components Space itself image CA, equivalently polynomial code. finally classify terms generator polynomials.