# Bent Functions

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

# Background and Definition

The covering radius bound states that the nonlinearity $nl(F)$ of any $(n,m)$ -function $F$ satisfies

$nl(F)\leq 2^{n-1}-2^{n/2-1}.$ 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 $(n,m)$ -function is at most $n/2$ .

An $(n,m)$ -function is bent if and only if all of its derivatives $D_{a}F$ for $a\neq 0$ are balanced. In this sense, bent functions are also referred to as perfect nonlinear (PN) functions.

Bent (PN) $(n,m)$ -functions exist only for $n$ even and $m\leq n/2$ . Conversely, for any pair of integers $(n,m)$ satisfying this hypothesis, there exists a bent $(n,m)$ -function.

1. 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.