Equivalent class of boolean functions using their truth tables

Abstract

Let Γ, φ be the set of all boolean and affine functions on $F_{2^{n}}$, respectively. Since φ is a subgroup of group Γ, then we can construct the equivalent classes of boolean functions, Γ / φ with $|Γ / φ|=2^{2^{n}-n-1}$, where $|Γ|=2^{2^n}$ and $|φ|=2^{n+1}$. Each representative element of each equivalent class is called function generator. The important reason to define the function generator is that all boolean functions in a equivalent class have again properties of that function generator. That is, all elements of each equivalent class have the same nonlinearity so it is easy to prove that if a function generator is bent function or SAC function, then all boolean functions in that equivalent class are bent functions or SAC functions. So we can know properties of all boolean functions to investigate only function generators. The set of function generators can be very various. In this paper, One form of the set of function generator is introduced using the truth tables of function generations.