12 research outputs found
A conjecture on permutation trinomials over finite fields of characteristic two
In this paper, by analyzing the quadratic factors of an -th degree
polynomial over the finite field \ftwon, a conjecture on permutation
trinomials over \ftwon[x] proposed very recently by Deng and Zheng is
settled, where and is a positive integer with
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
On a Class of Permutation Trinomials in Characteristic 2
Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form
, where is even and
. They found sufficient conditions on for to be
a permutation polynomial (PP) of and they conjectured that the
sufficient conditions are also necessary. The conjecture has been confirmed by
Bartoli using the Hasse-Weil bound. In this paper, we give an alternative
solution to the question. We also use the Hasse-Weil bound, but in a different
way. Moreover, the necessity and sufficiency of the conditions are proved by
the same approach
Two types of permutation polynomials with special forms
Let be a power of a prime and be a finite field with
elements. In this paper, we propose four families of infinite classes of
permutation trinomials having the form over
, and investigate the relationship between this type of
permutation polynomials with that of the form . Based on
this relation, many classes of permutation trinomials having the form
without restriction on over
are derived from known permutation trinomials having the form
A family of permutation trinomials in
Let and consider a prime power . We completely characterize
permutation polynomials of of the type . In particular, using
connections with algebraic curves over finite fields, we show that the already
known sufficient conditions are also necessary
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
On the constructions of -cycle permutations
Any permutation polynomial is an -cycle permutation. When is a
specific small positive integer, one can obtain efficient permutations, such as
involutions, triple-cycle permutations and quadruple-cycle permutations. These
permutations have important applications in cryptography and coding theory.
Inspired by the AGW Criterion, we propose criteria for -cycle
permutations, which mainly are of the form . We then propose
unified constructing methods including recursive ways and a cyclotomic way for
-cycle permutations of such form. We demonstrate our approaches by
constructing three classes of explicit triple-cycle permutations with high
index and two classes of -cycle permutations with low index
Finding compositional inverses of permutations from the AGW criterion
Permutation polynomials and their compositional inverses have wide
applications in cryptography, coding theory, and combinatorial designs.
Motivated by several previous results on finding compositional inverses of
permutation polynomials of different forms, we propose a general method for
finding these inverses of permutation polynomials constructed by the AGW
criterion. As a result, we have reduced the problem of finding the
compositional inverse of such a permutation polynomial over a finite field to
that of finding the inverse of a bijection over a smaller set. We demonstrate
our method by interpreting several recent known results, as well as by
providing new explicit results on more classes of permutation polynomials in
different types. In addition, we give new criteria for these permutation
polynomials being involutions. Explicit constructions are also provided for all
involutory criteria.Comment: 24 pages. Revision submitte
Determination of a Class of Permutation Trinomials in Characteristic Three
Let , where . In a series of recent papers by several authors, sufficient
conditions on and were found for to be a permutation polynomial
(PP) of and, in characteristic , the sufficient conditions
were shown to be necessary. In the present paper, we confirm that in
characteristic 3, the sufficient conditions are also necessary. More precisely,
we show that when , is a PP of if
and only if and is a square in
.Comment: 31 page
Permutation polynomials and complete permutation polynomials over
Motivated by many recent constructions of permutation polynomials over
, we study permutation polynomials over in
terms of their coefficients. Based on the multivariate method and resultant
elimination, we construct several new classes of sparse permutation polynomials
over , , . Some of them are complete
mappings.Comment: 31 page