Differential uniformity

Given a vectorial Boolean function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2^{m}}} , it is called differentially $\delta$-uniform if the equation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(x+a)-F(x)=b} admits at most $\delta$ solutions for every non-zero Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a\in \mathbb {F} _{2^{n}}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b\in \mathbb {F} _{2^{m}}} .

This definition can be generalized to the case of functions Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \delta =p^{n-m}} , such functions are called perfect nonlinear (PN) and they exist only for $p$ odd and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m\leq n/2} . (see also planar functions)

For Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p=2} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m=n} the smallest value is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \delta =2} and such optimal funtions are called almost perfect nonlinear (APN).

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