Differential uniformity

Given a vectorial Boolean function $F:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2^{m}}$, it is called differentially $\delta$-uniform if the equation $F(x+a)-F(x)=b$ admits at most $\delta$ solutions for every non-zero $a\in \mathbb {F} _{2^{n}}$ and $b\in \mathbb {F} _{2^{m}}$.

This definition can be generalized to the case of functions $F:\mathbb {F} _{p^{n}}\rightarrow \mathbb {F} _{p^{m}}$. Functions with the smallest value for $\delta$ contribute an optimal resistance to the differential attack.

The smallest possible value is $\delta =p^{n-m}$, such functions are called perfect nonlinear (PN) and they exist only for $p$ odd and $m\leq n/2$. (see also planar functions)

For $p=2$ and $m=n$ the smallest value is $\delta =2$ and such optimal funtions are called almost perfect nonlinear (APN).

Differential uniformity is invariant under affine, EA- and CCZ-equivalence.