Computational Results on Gowers and Norms of Known S-Boxes

The th order Gowers norm of a Boolean function is a measure of its resistance to rth order approximations. Gowers, Green, and Tao presented the connection between Gowers uniformity norm of a function and its correlation with polynomials of a certain degree. Gowers and norms measure the resistance of a Boolean function against linear and quadratic approximations, respectively. This paper presents computational results on the Gowers norms of some known 4, 5, and 6-bit S-Boxes used in the present-day block ciphers. It is observed that there are S-Boxes having the same algebraic degree, differential uniformity, and linearity, but their Gowers norms values are different. Equivalently, they possess different strengths against linear and quadratic cryptanalysis.