In this paper the (effective) bias of certain generalised linear approximations to the S-box are considered. Whereas, in the literature, the cryptanalyst typically restricts this search to linear approximations over Z2, we here consider linear approximations over Z4 and, more generally still, consider approximations which are linear in the sense that they can be completely factorised into the t...

We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functi...