# Differential uniformity

Given a vectorial Boolean function ${\displaystyle F:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2^{m}}}$, it is called differentially ${\displaystyle \delta }$-uniform if the equation ${\displaystyle F(x+a)-F(x)=b}$ admits at most ${\displaystyle \delta }$ solutions for every non-zero ${\displaystyle a\in \mathbb {F} _{2^{n}}}$ and ${\displaystyle b\in \mathbb {F} _{2^{m}}}$.
This definition can be generalized to the case of functions ${\displaystyle F:\mathbb {F} _{p^{n}}\rightarrow \mathbb {F} _{p^{m}}}$. Functions with the smallest value for ${\displaystyle \delta }$ contribute an optimal resistance to the differential attack.
The smallest possible value is ${\displaystyle \delta =p^{n-m}}$, such functions are called perfect nonlinear (PN) and they exist only for ${\displaystyle p}$ odd and ${\displaystyle m\leq n/2}$. (see also planar functions)
For ${\displaystyle p=2}$ and ${\displaystyle m=n}$ the smallest value is ${\displaystyle \delta =2}$ and such optimal funtions are called almost perfect nonlinear (APN).