Nikolay: Created page with "= Background and Definition = The covering radius bound states that the nonlinearity $nl(F)$ of any $(n,m)$ -function $F$ satisfies
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2019-02-10T18:03:10Z

Created page with "= Background and Definition = The covering radius bound states that the nonlinearity <math>nl(F)</math> of any <math>(n,m)</math>-function <math>F</math> satisfies <div>..."

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= Background and Definition =

The covering radius bound states that the [[nonlinearity]] <math>nl(F)</math> of any <math>(n,m)</math>-function <math>F</math> satisfies

<div><math> nl(F) \le 2^{n-1} - 2^{n/2 - 1}.</math></div>

A function is called ''bent'' if it achieves this bound with equality.

= Properties =

Since nonlinearity is a CCZ-invariant, CCZ-equivalence (and therefore also EA-equivalence and affine equivalence) preserves the property of a function of being bent.

The algebraic degree of any bent <math>(n,m)</math>-function is at most <math>n/2</math>.

An <math>(n,m)</math>-function is bent if and only if all of its derivatives <math>D_aF</math> for <math>a \ne 0 </math> are balanced. In this sense, bent functions are also referred to as ''perfect nonlinear (PN)'' functions.

Bent (PN) <math>(n,m)</math>-functions exist only for <math>n</math> even and <math>m \le n/2</math><ref name="nybergNonlinearity">Nyberg K. Perfect nonlinear S-boxes. InWorkshop on the Theory and Application of of Cryptographic Techniques 1991 Apr 8 (pp. 378-386). Springer, Berlin, Heidelberg.</ref>. Conversely, for any pair of integers <math>(n,m)</math> satisfying this hypothesis, there exists a bent <math>(n,m)</math>-function.

Bent Functions - Revision history

Nikolay: Created page with "= Background and Definition = The covering radius bound states that the nonlinearity nl(F) of any (n,m)-function F satisfies ..."

Nikolay: Created page with "= Background and Definition = The covering radius bound states that the nonlinearity $nl(F)$ of any $(n,m)$ -function $F$ satisfies
..."