On the Algebraic Immunities and Higher Order Nonlinearities of Vectorial Boolean Functions

We recall the two different notions of algebraic immunity of vectorial functions: the basic algebraic immunity and the graph algebraic immunity. We introduce a new one, the component algebraic immunity, which helps studying the two others. We study the tightness of the bounds on the two first notions and prove bounds between them three. We recall the known bounds on the r-th order nonlinearity of a vectorial function, given its basic algebraic immunity. We recall why the basic algebraic immunity is not a relevant parameter when the number of output bits of the vectorial function is not small enough and/or when the S-box is used in a block cipher. This leads us to showing bounds on the r-th order nonlinearity, given the graph algebraic immunity, that we deduce from similar bounds involving the component algebraic immunity.