An iterative construction of almost perfect nonlinear functions

Vectorial Boolean functions $F$ and $G$ are equivalent if $\forall a\neq 0\,\forall b\,[\exists x(F(x)\oplus F(x\oplus a)=b)\Leftrightarrow\exists x(G(x)\oplus G(x\oplus a)=b)]$. It is proved that every class of equivalent almost perfect nonlinear (APN) functions in $n$ variables contains $2^{2n}$ different functions. An iterative procedure is proposed for constructing APN functions in $n+1$ variables from two APN and two Boolean functions in $n$ variables satisfying some conditions. Computer experiment show that among functions in small variables there are many functions satisfying these conditions.