**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 :
function | degree | orbit size |

XY | md | 2 | | 324 |

Y^2 + X^2 | 2* | 162 |

(*) not Maiorana-MacFarland type
(d) Dillon Type : 1
(m) Maiorana Type : 1
## Classification of for p=5 m=2

We have 6 class
function | type | degree | orbit size |

X^3Y | dm | 4 | 375000 |

X^2 + X^3Y | m | 4 | 1500000 |

XY^3 + X^4 | d | 4* | 750000 |

XY | m | 2 | 7500 |

XY + X^4 | m | 4 | 300000 |

XY + X^3 | m | 3 | 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.