The definition of affine equivalence is that 2 $n$-variable Boolean functions are affine equivalent if there exist affine permutations $A$ of ${F}^n_2$ such that $g(x)=f(A(x))$.
What do affine permutations mean here? And how do they apply to the definition of affine equivalence?
It would be great if anyone could share their understanding of affine equivalence. Thank you so much for your time!
In the context of finite fields, a boolean function is a mapping $$F:\mathbb{F}_{2^n}\rightarrow \mathbb{F}_2.$$
Such a function is linear (the representation below is sometimes called a linearized polynomial) if it can be expressed as $$ F(x)=\sum_{i=0}^{n-1} a_i x^{2^i},\quad a_i \in \mathbb{F}_{2^n}. $$ If $F(0)=a_0=0,$ then the function is called linear, otherwise it is called affine.
There are families of such functions which give permutations, and those are the ones referred to in this case.
Claude Carlet has chapters on boolean functions and vector boolean functions you can find with a google search.