Multiplicative complexity of bijective 4×4 S-boxes

Multiplicative complexity of S-box is the minimum number of 2-input AND-gates required to implement the S-box in AND, XOR, NOT logic. We show that under an affine equivalence there is only a single class of bijective n×n S-boxes with multiplicative complexity 1. Furthermore, we show that each bijective 4×4 S-box has multiplicative complexity at most 5. Finally, we refine the bounds on the multiplicative complexity of each affine class of bijective 4×4 S-boxes.