1,733 research outputs found
On q-ary Bent and Plateaued Functions
We obtain the following results. For any prime the minimal Hamming
distance between distinct regular -ary bent functions of variables is
equal to . The number of -ary regular bent functions at the distance
from the quadratic bent function is
equal to for . The Hamming distance
between distinct binary -plateaued functions of variables is not less
than and the Hamming distance between distinctternary
-plateaued functions of variables is not less than
. These bounds are tight.
For we prove an upper bound on nonlinearity of ternary functions in
terms of their correlation immunity. Moreover, functions reaching this bound
are plateaued. For analogous result are well known but for large it
seems impossible. Constructions and some properties of -ary plateaued
functions are discussed.Comment: 14 pages, the results are partialy reported on XV and XVI
International Symposia "Problems of Redundancy in Information and Control
Systems
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
A new class of three-weight linear codes from weakly regular plateaued functions
Linear codes with few weights have many applications in secret sharing
schemes, authentication codes, communication and strongly regular graphs. In
this paper, we consider linear codes with three weights in arbitrary
characteristic. To do this, we generalize the recent contribution of Mesnager
given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present
a new class of binary linear codes with three weights from plateaued Boolean
functions and their weight distributions. We next introduce the notion of
(weakly) regular plateaued functions in odd characteristic and give
concrete examples of these functions. Moreover, we construct a new class of
three-weight linear -ary codes from weakly regular plateaued functions and
determine their weight distributions. We finally analyse the constructed linear
codes for secret sharing schemes.Comment: The Extended Abstract of this work was submitted to WCC-2017 (the
Tenth International Workshop on Coding and Cryptography
Dual and Hull code in the first two generic constructions and relationship with the Walsh transform of cryptographic functions
We contribute to the knowledge of linear codes from special polynomials and
functions, which have been studied intensively in the past few years. Such
codes have several applications in secret sharing, authentication codes,
association schemes and strongly regular graphs.
This is the first work in which we study the dual codes in the framework of
the two generic constructions; in particular, we propose a Gram-Schmidt
(complexity of ) process to compute them explicitly. The
originality of this contribution is in the study of the existence or not of
defining sets , which can be used as ingredients to construct the dual code
for a given code in the context of the second
generic construction. We also determine a necessary condition expressed by
employing the Walsh transform for a codeword of to belong in the
dual. This achievement was done in general and when the involved functions are
weakly regularly bent. We shall give a novel description of the Hull code in
the framework of the two generic constructions. Our primary interest is
constructing linear codes of fixed Hull dimension and determining the (Hamming)
weight of the codewords in their duals
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