47 research outputs found
Relation between o-equivalence and EA-equivalence for Niho bent functions
Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions. However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials. In the present work we study, for a given o-polynomial, a general construction which provides all possible o-equivalent Niho bent functions, and we considerably simplify it to a form which excludes EA-equivalent cases. That is, we identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions derived from o-equivalence of any given Niho bent function. Furthermore, we determine all pairwise EA-inequivalent Niho bent functions arising from all known o-polynomials via o-equivalence.publishedVersio
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
Relation between o-equivalence and EA-equivalence for Niho bent functions
Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions.
However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials.
In the present work we study, for a given o-polynomial, a general construction which provides all possible o-equivalent Niho bent functions, and we considerably simplify it to a form which excludes EA-equivalent cases.
That is, we identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions derived from o-equivalence of any given Niho bent function. Furthermore, we determine all pairwise EA-inequivalent Niho bent functions arising from all known o-polynomials via o-equivalence
On Some Properties of Quadratic APN Functions of a Special Form
In a recent paper, it is shown that functions of the form
, where and are linear, are a good source for
construction of new infinite families of APN functions. In the present work we
study necessary and sufficient conditions for such functions to be APN
On Inversion in Z_{2^n-1}
In this paper we determined explicitly the multiplicative inverses of the
Dobbertin and Welch APN exponents in Z_{2^n-1}, and we described the binary
weights of the inverses of the Gold and Kasami exponents. We studied the
function \de(n), which for a fixed positive integer d maps integers n\geq 1 to
the least positive residue of the inverse of d modulo 2^n-1, if it exists. In
particular, we showed that the function \de is completely determined by its
values for 1 \leq n \leq \ordb, where \ordb is the order of 2 modulo the
largest odd divisor of d.Comment: The first part of this work is an extended version of the results
presented in ISIT1