A Search Strategy to Optimize the Affine Variant Properties of S-Boxes

Affine transformations are an often used tool in symmetric key cryptography. They are mostly known as a way of removing fixed points in S-boxes, as for instance in the AES S-box. In general, affine transformations do not have an influence on most cryptographic properties, since those properties are affine invariant; affine transformations only change the representation of the S-box. Because of that, there is not much research on what would be the best affine transformation in terms of usability in practical scenarios. With this research, we try to close that gap; we concentrate on several cryptographic properties and one implementation property that are variable under various affine transformations. To provide experimental validations, we concentrate on affine transformations in S-boxes of three sizes, namely, 4x4, 5x5, and 8x8. Our results indicate that it is possible to optimize one or more of the considered properties. Finally, although we experiment with only a handful of properties, our methodology is of a general nature and could be used for other cryptographic properties that are affine variant.