Composite field GF(((2²)²)²) Advanced Encryption Standard (AES) S-box with algebraic normal form representation in the subfield inversion

In this study, the authors categorise all of the feasible constructions for the composite Galois field GF(((2²)²)²) Advanced Encryption Standard (AES) S-box into four main architectures by their field representations and their algebraic properties. For each of the categories, a new optimisation scheme which exploits algebraic normal form representation followed by a sub-structure sharing optimisation is presented. This is performed by converting the subfield GF((2²) inversion into several logical expressions, which will be in turn reduced using a common sub-expression elimination algorithm. The authors show that this technique can effectively reduce the total area gate count as well as the critical path gate count in composite field AES S-boxes. The resulting architecture that achieves maximum reduction in both total area coverage and critical path gate count is found and reported. The hardware implementations of the authors proposed AES S-boxes, along with their performance and cost are presented and discussed.