Generalized Bent Functions in Dimension 2

This page provides the classification of Generalized Bent Functions in dimension 2 small primes

## Generalized Bent Function

Let m be a positive integer, p an odd prime. A map from GF(p)^m into GF(p) is bent if the map x->f(x+u) - f(x) is balanced. Clearly, if f is bent then

g(x) = A o f o B (x) + C(x)
is bent for all affine transformation A, B and C.

## Classification of for p=3 m=2

Only two class :
functiondegreeorbit size
XYmd2324
Y^2 + X^22*162
(*) not Maiorana-MacFarland type (d) Dillon Type : 1 (m) Maiorana Type : 1

## Classification of for p=5 m=2

We have 6 class
functiontypedegreeorbit size
X^3Y dm4 375000
X^2 + X^3Y m4 1500000
XY^3 + X^4 d4* 750000
XY m2 7500
XY + X^4 m4 300000
XY + X^3 m3 60000
4 Y^2 + 3 X^2 2* 5000
(*) not Maiorana-MacFarland type (d) Dillon Type : 2 (m) Maiorana Type : 5

## Open problem

According to Xiang-Dong Hou the degree of a bent function in m variables is less or equal to m(p-1)/2 + 1. For m=2 that gives a degree less or equal to p.
• Prove the degree of a bent function is less than p !
• How many class of cubics ? More than one !?
• Does there exist another type of bent functions?

Philippe Langevin Last modification september 2016.