Advances and Applications in Discrete Mathematics
Volume 3, Issue 1, Pages 1 - 46
(January 2009)
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-BENT FUNCTIONS
Laurent Poinsot (France)
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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 G
to
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.