Boolean and Vectorial Plateaued Functions and APN Functions

Boolean plateaued functions and vectorial functions with plateaued components play a significant role in cryptography, sequences for communications, and the related combinatorics and designs. Our knowledge on them is not at a level corresponding to their importance. We introduce new characterizations of plateaued Boolean functions. We give the characterizations of vectorial functions whose components are all plateaued (with possibly different amplitudes), that we simply call plateaued, by means of the value distributions of their derivatives (we characterize similarly those functions whose components are partially bent) and autocorrelation functions, and of the power moments of their Walsh transform. This allows us to derive several characterizations of almost perfect nonlinear (APN) functions in this framework. We prove that all the main results known for quadratic APN functions extend to plateaued functions, allowing the study of their APN-ness to be simplified. We show that if, additionally, the component functions are all unbalanced, this study is still simpler: the APN-ness of such functions depends only on their value distribution. This allows proving, for instance, that any plateaued (n, n)-function, n even, having similar value distribution as the APN power functions, is APN, and has the same extended Walsh spectrum as the APN Gold functions. As by-products, we obtain a few other new results. For instance, any plateaued function in even dimension, which is Carlet-Charpin-Zinoviev (CCZ)-equivalent to a Gold or Kasami APN function, is necessarily extended affine (EA)-equivalent to it.