An upper bound for the number of functions satisfying the strict avalanche criterion

The strict avalanche criterion was introduced by Webster and Tavares while studying some cryptographic functions. We say that a binary function $f(x)$, $x \in V_n$, satisfies this criterion if replacing any coordinate of the vector $x$ by its complement changes the values of $f(x)$ exactly in a half of cases. In this paper we establish an upper bound for the number of such functions for $n$ large enough.