Affine Equivalence Classes Of Vector Boolean Functions And EA Equivalence Classes Count

Vectorial Boolean functions play an important role in cryptography,sequences and coding theory.Affine transformations of vectorial Boolean functions preserve many cryptographic properties of the functions,and it is of great theoretical importance and application value to classify vectorial Boolean functions by equivalence relations.Very recently,an explicit formula for the number of affine equivalent classes of q-ary functions and an asymptotic formula for the number of EA-equivalent classes of Boolean funcitons have been obtained by Hou.In this thesis,we analyze the relationship between affine equivalent classes and group actions,and give an description of the set of fixed points on the vectorial q-ary function.We obtain the explicit formula for the number of affine equivalence classes of vectorial Boolean functions with the help of Burnside's Lemma and the results of the affine group AGL(n,F_q).And we investigate the matrix representation of??AGL(n,F₂)acting on Boolean functions,give an alternative expression for the set of fixed points.We obtain an asymptotic formula for the number of EA-equivalent classes of vectorial Boolean funcitons based on the Kronecker product.