<p>Final S-box.</p>

While numerous methods for S-box generation exist, there is a continuous pursuit of designs that offer a superior and balanced profile against various cryptanalytic attacks. This paper presents a new, two-step approach to the construction of highly nonlinear S-boxes, with group-theoretic underpinnings.

During the first step, we use a parameterized permutation representation of the triangle group on the projective line over the finite field , denoted by , to produce an initial S-box with a nonlinearity of . The second stage is the main innovation, in which we use a refinement process via the action of the permutation group on this original S-box. The refinement increases the average nonlinearity to 112 and yields differential uniformity 4, linear approximation probability 0.0625, strong bit-independence behavior, and strict avalanche characteristics close to the ideal value.