Abstract: A
function from a finite Abelian group Gand
with values in the unit circle Tof
the complex field is called bent
if
its Fourier transform (i.e.,
the
decomposition of fin
the basis of characters of G)
has a constant magnitude equals to the number of elements of G.
In this contribution we define a modulo 2
notion
of characters by allowing the characters of an elementary finite Abelian p-group
Gto
take their values in the multiplicative group of the roots of the unity in the finite field with
elements
rather than in the complex roots of the unity T.
We show that this kind of characters forms an orthogonal basis of the -vector
space of functions from Gto
that
permits us to define a modulo 2
version
of the Fourier transform (as a decomposition of a vector in this basis of
characters). We show that many classical properties of the Fourier transform
still hold for this characteristic 2
version.
In particular, we can define an appropriate notion of bent functions, called -bent
functions,
with respect to this Fourier transform. Finally we construct a class of -bent
functions and we also study their relations with classical and group action
versions of perfect nonlinearity.
Keywords and phrases: bent functions, perfect nonlinearity, finite Abelian groups, theory of characters, Fourier transform and group actions.
