Composite field GF(((2 2 ) 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)2) 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.