# Bent Functions

# Background and Definition

The covering radius bound states that the nonlinearity of any -function satisfies

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 -function is at most .

An -function is bent if and only if all of its derivatives for are balanced. In this sense, bent functions are also referred to as *perfect nonlinear (PN)* functions.

Bent (PN) -functions exist only for even and ^{[1]}. Conversely, for any pair of integers satisfying this hypothesis, there exists a bent -function.

- ↑ 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.