文摘
In this paper, we use a linear algebra point of view to describe the derivatives and higher order derivatives over \(\mathbb {F}_{2^n}\). On one hand, this new approach enables us to prove several properties of these functions, as well as the functions that have these derivatives. On the other hand, we provide a method to construct all of the higher order derivatives in given directions. We also demonstrate some properties of the higher order derivatives and their decomposition as a sum of functions with 0-linear structure. Moreover, we introduce a criterion and an algorithm to realize discrete antidifferentiation of vectorial Boolean functions. This leads us to define a new equivalence of functions, that we call differential equivalence, which links functions that share the same derivatives in directions given by some subspace. Finally, we discuss the importance of finding 2-to-1 functions.
