The Fourier Entropy-Influence conjecture holds for a log-density 1 class of cryptographic Boolean functions

We consider the Fourier Entropy-Influence (FEI) conjecture in the context of cryptographic Boolean functions. We show that the FEI conjecture is true for the functions satisfying the strict avalanche criterion, which forms a subset of asymptotic log–density 1 in the set of all Boolean functions. Further, we prove that the FEI conjecture is satisfied for plateaued Boolean functions, monomial algebraic normal form (with the best involved constant), direct sums, as well as concatenations of Boolean functions. As a simple consequence of these general results we find that each affine equivalence class of quadratic Boolean functions contains at least one function satisfying the FEI conjecture. Further, we propose some “leveled” FEI conjectures.