157 research outputs found
bent functions constructed from finite pre-quasifield spreads
Bent functions are of great importance in both mathematics and information
science. The class of bent functions was introduced by
Dillon in 1974, but functions belonging to this class that can be explicitly
represented are only the functions, which
were also constructed by Dillon after his introduction of the
class. In this paper, a technique of using finite
pre-quasifield spread from finite geometry to construct
bent functions is proposed. The constructed functions
are in similar styles with the functions.
To explicitly represent them in bivariate forms, the main task is to compute
compositional inverses of certain parametric permutation polynomials over
finite fields of characteristic 2. Concentrated on the Dempwolff-M\"uller
pre-quasifield, the Knuth pre-semifield and the Kantor pre-semifield, three new
subclasses of the class are obtained. They are the
only sub-classes that can be explicitly constructed more than 30 years after
the subclass was introduced.Comment: 14page
On constructing complete permutation polynomials over finite fields of even characteristic
In this paper, a construction of complete permutation polynomials over finite
fields of even characteristic proposed by Tu et al. recently is generalized in
a recursive manner. Besides, several classes of complete permutation
polynomials are derived by computing compositional inverses of known ones.Comment: Stupid mistakes in previous versions are correcte
New constructions of quaternary bent functions
In this paper, a new construction of quaternary bent functions from
quaternary quadratic forms over Galois rings of characteristic 4 is proposed.
Based on this construction, several new classes of quaternary bent functions
are obtained, and as a consequence, several new classes of quadratic binary
bent and semi-bent functions in polynomial forms are derived. This work
generalizes the recent work of N. Li, X. Tang and T. Helleseth
A general construction of permutation polynomials of the form over \F_{2^{2m}}
Recently, there has been a lot of work on constructions of permutation
polynomials of the form over the finite field
\F_{2^{2m}}, especially in the case when is of the form
(Niho exponent). In this paper, we further investigate permutation polynomials
with this form. Instead of seeking for sporadic constructions of the parameter
, we give a general sufficient condition on such that
permutes \F_{2^{2m}}, that is, , where is
any integer. This generalizes a recent result obtained by Gupta and Sharma who
actually dealt with the case . It turns out that most of previous
constructions of the parameter are covered by our result, and it yields
many new classes of permutation polynomials as well
Complete permutation polynomials induced from complete permutations of subfields
We propose several techniques to construct complete permutation polynomials
of finite fields by virtue of complete permutations of subfields. In some
special cases, any complete permutation polynomials over a finite field can be
used to construct complete permutations of certain extension fields with these
techniques. The results generalize some recent work of several authors
The compositional inverse of a class of bilinear permutation polynomials over finite fields of characteristic 2
A class of bilinear permutation polynomials over a finite field of
characteristic 2 was constructed in a recursive manner recently which involved
some other constructions as special cases. We determine the compositional
inverses of them based on a direct sum decomposition of the finite field. The
result generalizes that in [R.S. Coulter, M. Henderson, The compositional
inverse of a class of permutation polynomials over a finite field, Bull.
Austral. Math. Soc. 65 (2002) 521-526].Comment: 17 page
A remark on algebraic immunity of Boolean functions
In this correspondence, an equivalent definition of algebraic immunity of
Boolean functions is posed, which can clear up the confusion caused by the
proof of optimal algebraic immunity of the Carlet-Feng function and some other
functions constructed by virtue of Carlet and Feng's idea.Comment: 7 page
Linearized polynomials over finite fields revisited
We give new characterizations of the algebra
formed by all linearized polynomials over the
finite field after briefly surveying some known ones. One
isomorphism we construct is between and the
composition algebra
. The other
isomorphism we construct is between and the
so-called Dickson matrix algebra . We also
further study the relations between a linearized polynomial and its associated
Dickson matrix, generalizing a well-known criterion of Dickson on linearized
permutation polynomials. Adjugate polynomial of a linearized polynomial is then
introduced, and connections between them are discussed. Both of the new
characterizations can bring us more simple approaches to establish a special
form of representations of linearized polynomials proposed recently by several
authors. Structure of the subalgebra which
are formed by all linearized polynomials over a subfield of
where are also described.Comment: 30 page
More characterizations of generalized bent function in odd characteristic, their dual and the gray image
In this paper, we further investigate properties of generalized bent Boolean
functions from to , where is an odd prime and is a
positive integer. For various kinds of representations, sufficient and
necessary conditions for bent-ness of such functions are given in terms of
their various kinds of component functions. Furthermore, a subclass of gbent
functions corresponding to relative difference sets, which we call
-bent functions, are studied. It turns out that -bent
functions correspond to a class of vectorial bent functions, and the property
of being -bent is much stronger then the standard bent-ness. The dual
and the generalized Gray image of gbent function are also discussed. In
addition, as a further generalization, we also define and give
characterizations of gbent functions from to for a
positive integer with
A new proof to complexity of dual basis of a type I optimal normal basis
The complexity of dual basis of a type I optimal normal basis of
over was determined to be or
according as is even or odd, respectively, by Z.-X. Wan and K. Zhou in
2007. We give a new proof to this result by clearly deriving the dual of a type
I optimal normal basis with the aid of a lemma on the dual of a polynomial
basis.Comment: 7 page
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