Ivi062: Created page with "Given a vectorial Boolean function $F:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^m}$ , it is called differentially $\delta$ -uniform if the equation $F..."$

2019-09-05T09:52:22Z

Created page with "Given a vectorial Boolean function <math>F:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^m}</math>, it is called differentially <math>\delta</math>-uniform if the equation <math>F..."

New page

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

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

The smallest possible value is <math>\delta=p^{n-m}</math>, such functions are called perfect nonlinear (PN) and they exist only for <math>p</math> odd and <math>m\le n/2</math>. (see also [[Commutative Presemifields and Semifields|planar functions]])

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

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

Differential uniformity - Revision history

Ivi062: Created page with "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..."

Ivi062: Created page with "Given a vectorial Boolean function $F:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^m}$ , it is called differentially $\delta$ -uniform if the equation $F..."$