  In this paper we study the relationship between vectorial (Boolean) functions
and cyclic codes in the context of algebraic attacks. We first derive a direct
link between the annihilators of a vectorial function (in univariate form) and
certain $2^{n}$-ary cyclic codes (which we prove that they are LCD codes)
extending results due to R{\o}njom and Helleseth. The knowledge of the minimum
distance of those codes gives rise to a lower bound on the algebraic immunity
of the associated vectorial function. Furthermore, we solve an open question
raised by Mesnager and Cohen. We also present some properties of those cyclic
codes (whose generator polynomials determined by vectorial functions) as well
as their weight enumerator. In addition we generalize the so-called algebraic
complement and study its properties.
