13 research outputs found
Landscape Boolean Functions
In this paper we define a class of Boolean and generalized Boolean functions
defined on with values in (mostly, we consider
), which we call landscape functions (whose class containing generalized
bent, semibent, and plateaued) and find their complete characterization in
terms of their components. In particular, we show that the previously published
characterizations of generalized bent and plateaued Boolean functions are in
fact particular cases of this more general setting. Furthermore, we provide an
inductive construction of landscape functions, having any number of nonzero
Walsh-Hadamard coefficients. We also completely characterize generalized
plateaued functions in terms of the second derivatives and fourth moments.Comment: 19 page
A complete characterization of plateaued Boolean functions in terms of their Cayley graphs
In this paper we find a complete characterization of plateaued Boolean
functions in terms of the associated Cayley graphs. Precisely, we show that a
Boolean function is -plateaued (of weight ) if and only
if the associated Cayley graph is a complete bipartite graph between the
support of and its complement (hence the graph is strongly regular of
parameters ). Moreover, a Boolean function is
-plateaued (of weight ) if and only if the associated
Cayley graph is strongly -walk-regular (and also strongly
-walk-regular, for all odd ) with some explicitly given
parameters.Comment: 7 pages, 1 figure, Proceedings of Africacrypt 201
A NOTE ON SEMI-BENT BOOLEAN FUNCTIONS
We show how to construct semi-bent Boolean functions from PSap-
like bent functions. We derive innite classes of semi-bent functions in even
dimension having multiple trace terms
A note on semi-bent functions with multiple trace terms and hyperelliptic curves
Semi-bent functions with even number of variables are a class of important Boolean
functions whose Hadamard transform takes three values. In this note we are interested
in the property of semi-bentness of Boolean functions defined on the Galois field (n
even) with multiple trace terms obtained via Niho functions and two Dillon-like functions
(the first one has been studied by Mesnager and the second one have been studied very
recently by Wang, Tang, Qi, Yang and Xu). We subsequently give a connection between the
property of semi-bentness and the number of rational points on some associated hyperelliptic
curves. We use the hyperelliptic curve formalism to reduce the computational complexity in
order to provide a polynomial time and space test leading to an efficient characterization of
semi-bentness of such functions (which includes an efficient characterization of the hyperbent
functions proposed by Wang et al.). The idea of this approach goes back to the recent work
of Lisonek on the hyperbent functions studied by Charpin and Gong
ON DILLON\u27S CLASS H OF BENT FUNCTIONS, NIHO BENT FUNCTIONS AND O-POLYNOMIALS
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. We first notice that H can be extended to a slightly larger
class that we denote by H. We 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 also characterize the bent functions whose restrictions to the
uF2n=2 \u27s are affine. We answer to the open question raised by Dobbertin et al. in JCT A 2006
on whether the duals of the Niho bent functions introduced in the paper are Niho bent as well,
by explicitely calculating the dual of one of these functions. We observe that this Niho function
also belongs to the Maiorana-McFarland class, which brings us back to the problem of knowing
whether H (or H) is a subclass of the Maiorana-McFarland completed class. We 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 definition is an o-polynomial and we deduce eight new cases
of bent functions in H which are potentially new bent functions and most probably not affine
equivalent to Maiorana-McFarland functions
Generalizations of Bent Functions. A Survey
Bent functions (Boolean functions with extreme nonlinearity properties) are actively studied for their numerous applications in cryptography, coding theory, and other fields. New statements of problems lead to a large number of generalizations of the bent functions many of which remain little known to the experts in Boolean functions. In this article, we offer a systematic survey of them