Bent function: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>J. Noel Chiappa
m (Grammar cleanups in intro para)
imported>Andrey Khalyavin
(+normality)
Line 6: Line 6:
:''A boolean function <math>f</math> is bent if and only if every derivative <math>D_uf(x)</math> is balanced for <math>u\neq0</math>.
:''A boolean function <math>f</math> is bent if and only if every derivative <math>D_uf(x)</math> is balanced for <math>u\neq0</math>.
The minimal degree of bent function is <math>2</math> (the function <math>x_1x_2+x_3x_4+\cdots+x_{2n-1}x_{2n}</math> is bent), the maximal degree of bent function of <math>2n</math> variables is <math>n</math>.
The minimal degree of bent function is <math>2</math> (the function <math>x_1x_2+x_3x_4+\cdots+x_{2n-1}x_{2n}</math> is bent), the maximal degree of bent function of <math>2n</math> variables is <math>n</math>.
The maximal dimension of linear space where a bent function of <math>2n</math> variables is constant also equals to <math>n</math>. A bent function which do have such linear space is called '''normal'''. Most constructions of bent functions give normal bent functions.


== Dual bent function ==
== Dual bent function ==

Revision as of 10:52, 18 April 2008

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

A bent function is a boolean function of variables that have nonlinearity equal to . The Walsh-Adamar coefficients of a bent function are equal to ; this gives the alternative definition of bent functions. Bent functions have even number of variables, and achieve the bound of maximal possible nonlinearity. This makes them a good bases for cryptographic stream cyphers. Bent functions are a specific case of plateaued functions.

Main properties

One of the most usefull property distinguishing bent functions uses derivatives:

A boolean function is bent if and only if every derivative is balanced for .

The minimal degree of bent function is (the function is bent), the maximal degree of bent function of variables is .

The maximal dimension of linear space where a bent function of variables is constant also equals to . A bent function which do have such linear space is called normal. Most constructions of bent functions give normal bent functions.

Dual bent function

Signs of Walsh-Adamar coefficients can be transformed to another boolean function of the same number of variables. This function is also bent and is called dual bent function. The dual function to the dual function is the function itself.

Bent function series

Bent function constructions

Bent function enumeration

For the number of bent function very little is know. . Because degree of bent function is bounded by it is easy to show that . This result can be slighly improved but still remain very far from the truth.