Bent functions, semi-bent functions and o-polynomials

- FENS
- Bent functions, semi-bent functions and o-polynomials

Bent functions, semi-bent functions and

o-polynomials

with S. Mesnager

Boolean functions are called bent when they lie at maximal distance to

the rst order Reed-Muller code RM(1; n) (n even). This distance is called

the nonlinearity of the function. Bent functions are characterized by the fact

that their Walsh transform takes values 2n=2. play roles not only in coding

theory but also in cryptography (where they can be used to design balanced

functions with high nonlinearity), designs, dierence sets in elementary Abelian

2-groups... Their study has been initiated in the 70's by Dillon and Rothaus in

parallel with the design of the DES. One of the classes of bent Boolean functions

introduced by John Dillon in his thesis is family H. While this class corresponds

to a nice original construction of bent functions in bivariate form, Dillon could

exhibit in it only functions which already belonged to the well-known Maiorana-

McFarland class. After noticing that H can be extended to a slightly larger class

that we shall denote by H, we shall observe that the bent functions constructed

via Niho power functions, which four examples are known, due to Dobbertin et

al. and to Leander-Kholosha, are the univariate form of the functions of class

H. Their restrictions to the vector spaces uF2n=2 , u 2 F?

2n, are linear. We shall

answer to the open question raised by Dobbertin et al. on whether the duals

of the Niho bent functions introduced in the paper are Niho bent as well. The

fact that this Niho function also belongs to the Maiorana-McFarland class will

bring us back to the problem of knowing whether H (or H) is a subclass of the

Maiorana-McFarland completed class. We shall then show that the condition

for a function in bivariate form to belong to class H is equivalent to the fact

that a polynomial directly related to its denition is an o-polynomial (a notion

from discrete geometry) and deduce several new cases of bent functions in H

which are potentially new bent functions and probably not ane equivalent to

Maiorana-McFarland functions.

Semi-bent functions in n variables (n even) are those Boolean functions

whose Walsh transforms take values in f0;2n=2g. They do not have optimal

nonlinearity but, contrarily to bent functions, they can be balanced, which is

interesting from cryptographic viewpoint. We shall show how obtaining semi-

bent functions from a function belonging to class H and a function from the

so-called PSap class of bent functions.

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