149 research outputs found
Metrical properties of the set of bent functions in view of duality
In the paper, we give a review of metrical properties of the entire set of bent functions and its significant subclasses of self-dual and anti-self-dual bent functions. We present results for iterative construction of bent functions in n + 2 variables based on the concatenation of four bent functions and consider related open problem proposed by one of the authors. Criterion of self-duality of such functions is discussed. It is explored that the pair of sets of bent functions and affine functions as well as a pair of sets of self-dual and anti-self-dual bent functions in n > 4 variables is a pair of mutually maximally distant sets that implies metrical duality. Groups of automorphisms of the sets of bent functions and (anti-)self-dual bent functions are discussed. The solution to the problem of preserving bentness and the Hamming distance between bent function and its dual within automorphisms of the set of all Boolean functions in n variables is considered
Systematic Constructions of Bent-Negabent Functions, 2-Rotation Symmetric Bent-Negabent Functions and Their Duals
Bent-negabent functions have many important properties for their application
in cryptography since they have the flat absolute spectrum under the both
Walsh-Hadamard transform and nega-Hadamard transform. In this paper, we present
four new systematic constructions of bent-negabent functions on
and variables, respectively, by modifying the truth tables of two
classes of quadratic bent-negabent functions with simple form. The algebraic
normal forms and duals of these constructed functions are also determined. We
further identify necessary and sufficient conditions for those bent-negabent
functions which have the maximum algebraic degree. At last, by modifying the
truth tables of a class of quadratic 2-rotation symmetric bent-negabent
functions, we present a construction of 2-rotation symmetric bent-negabent
functions with any possible algebraic degrees. Considering that there are
probably no bent-negabent functions in the rotation symmetric class, it is the
first significant attempt to construct bent-negabent functions in the
generalized rotation symmetric class
On the Equivalence of Quadratic APN Functions
Establishing the CCZ-equivalence of a pair of APN functions is generally
quite difficult. In some cases, when seeking to show that a putative new
infinite family of APN functions is CCZ inequivalent to an already known
family, we rely on computer calculation for small values of n. In this paper we
present a method to prove the inequivalence of quadratic APN functions with the
Gold functions. Our main result is that a quadratic function is CCZ-equivalent
to an APN Gold function if and only if it is EA-equivalent to that Gold
function. As an application of this result, we prove that a trinomial family of
APN functions that exist on finite fields of order 2^n where n = 2 mod 4 are
CCZ inequivalent to the Gold functions. The proof relies on some knowledge of
the automorphism group of a code associated with such a function.Comment: 13 p
A note on constructions of bent functions from involutions
Bent functions are maximally nonlinear Boolean functions. They are important
functions introduced by Rothaus and studied rstly by Dillon and next by many researchers
for four decades. Since the complete classication of bent functions seems
elusive, many researchers turn to design constructions of bent functions. In this note,
we show that linear involutions (which are an important class of permutations) over
nite elds give rise to bent functions in bivariate representations. In particular, we
exhibit new constructions of bent functions involving binomial linear involutions whose
dual functions are directly obtained without computation
- …