On the properties of vectorial functions with plateaued components and their consequences on APN functions

Abstract

[This is an extended abstract of paper [15], which has been submitted to a journal] Boolean plateaued functions and vectorial functions with plateaued components, that we simply call plateaued, play a significant role in cryptography, but little is known on them. We give here, without proofs, new characterizations of plateaued Boolean and vectorial functions, by means of the value distributions of derivatives and of power moments of the Walsh transform. This allows us to derive several characterizations of APN functions in this framework, showing that all the main results known for quadratic APN functions extend to plateaued functions. Moreover, we prove that the APN-ness of those plateaued vectorial functions whose component functions are unbalanced depends only on their value distribution. This proves that any plateaued (n, n)-function, n even, having same value distribution as APN power functions, is APN and has same extendedWalsh spectrum as the APN Gold functions.