15,183 research outputs found
A New Approach to Permutation Polynomials over Finite Fields, II
Let be a prime and a power of . For , let be the polynomial defined by the functional equation
. When is
a permutation polynomial (PP) of ? This turns out to be
a challenging question with remarkable breath and depth, as shown in the
predecessor of the present paper. We call a triple of positive integers
{\em desirable} if is a PP of . In the
present paper, we find many new classes of desirable triples whose
corresponding PPs were previously unknown. Several new techniques are
introduced for proving a given polynomial is a PP.Comment: 47 pages, 3 table
New Constructions of Permutation Polynomials of the Form over
Permutation polynomials over finite fields have been studied extensively
recently due to their wide applications in cryptography, coding theory,
communication theory, among others. Recently, several authors have studied
permutation trinomials of the form over
, where , and are
integers. Their methods are essentially usage of a multiplicative version of
AGW Criterion because they all transformed the problem of proving permutation
polynomials over into that of showing the corresponding
fractional polynomials permute a smaller set , where
. Motivated by these results,
we characterize the permutation polynomials of the form
over such that
is arbitrary and is also an arbitrary prime power.
Using AGW Criterion twice, one is multiplicative and the other is additive, we
reduce the problem of proving permutation polynomials over
into that of showing permutations over a small subset of a proper subfield
, which is significantly different from previously known
methods. In particular, we demonstrate our method by constructing many new
explicit classes of permutation polynomials of the form
over . Moreover, we can explain
most of the known permutation trinomials, which are in [6, 13, 14, 16, 20, 29],
over finite field with even characteristic.Comment: 29 pages. An early version of this paper was presented at Fq13 in
Naples, Ital
New Permutation Trinomials Constructed from Fractional Polynomials
Permutation trinomials over finite fields consititute an active research due
to their simple algebraic form, additional extraordinary properties and their
wide applications in many areas of science and engineering. In the present
paper, six new classes of permutation trinomials over finite fields of even
characteristic are constructed from six fractional polynomials. Further, three
classes of permutation trinomials over finite fields of characteristic three
are raised. Distinct from most of the known permutation trinomials which are
with fixed exponents, our results are some general classes of permutation
trinomials with one parameter in the exponents. Finally, we propose a few
conjectures
A new class of permutation trinomials constructed from Niho exponents
Permutation polynomials over finite fields are an interesting subject due to
their important applications in the areas of mathematics and engineering. In
this paper we investigate the trinomial
over the finite field , where is an odd prime and
with being a positive integer. It is shown that when or ,
is a permutation trinomial of if and only if is even.
This property is also true for more general class of polynomials
, where is a
nonnegative integer and . Moreover, we also show that for
the permutation trinomials proposed here are new in the sense that
they are not multiplicative equivalent to previously known ones of similar
form.Comment: 17 pages, three table
Linear Permutations and their Compositional Inverses over
The use of permutation polynomials has appeared, along to their compositional
inverses, as a good choice in the implementation of cryptographic systems.
Hence, there has been a demand for constructions of these polynomials which
coefficients belong to a finite field. As a particular case of permutation
polynomial, involution is highly desired since its compositional inverse is
itself. In this work, we present an effective way of how to construct several
linear permutation polynomials over as well as their
compositional inverses using a decomposition of
based on its primitive idempotents. As a consequence, an immediate construction
of involutions is presented
A note on the permutation behaviour of the polynomial
Let and a positive integer. In this short note, we present a class
of permutation polynomials over . We also present a
generalization.Comment: 5 page
Permutation polynomials over from rational functions
Let denote the set of -th roots of unity in
. We construct permutation polynomials over
by using rational functions of any degree that induce
bijections either on or between and . In particular, we generalize results from Zieve
New Permutation Trinomials From Niho Exponents over Finite Fields with Even Characteristic
In this paper, a class of permutation trinomials of Niho type over finite
fields with even characteristic is further investigated. New permutation
trinomials from Niho exponents are obtained from linear fractional polynomials
over finite fields, and it is shown that the presented results are the
generalizations of some earlier works
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
A Class of Permutation Trinomials over Finite Fields
Let be a prime power and , where
. We prove that is a permutation polynomial of if and only if one of the following occurs: (i) is even and
; (ii) and .Comment: 18 page
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