12,660 research outputs found
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
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
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
Some new results on permutation polynomials over finite fields
Permutation polynomials over finite fields constitute an active research area
and have applications in many areas of science and engineering. In this paper,
four classes of monomial complete permutation polynomials and one class of
trinomial complete permutation polynomials are presented, one of which confirms
a conjecture proposed by Wu et al. (Sci. China Math., to appear. Doi:
10.1007/s11425-014-4964-2). Furthermore, we give two classes of trinomial
permutation polynomials, and make some progress on a conjecture about the
differential uniformity of power permutation polynomials proposed by Blondeau
et al. (Int. J. Inf. Coding Theory, 2010, 1, pp. 149-170).Comment: 21 pages. We have changed the title of our pape
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
New Classes of Permutation Binomials and Permutation Trinomials over Finite Fields
Permutation polynomials over finite fields play important roles in finite
fields theory. They also have wide applications in many areas of science and
engineering such as coding theory, cryptography, combinatorial design,
communication theory and so on. Permutation binomials and trinomials attract
people's interest due to their simple algebraic form and additional
extraordinary properties. In this paper, several new classes of permutation
binomials and permutation trinomials are constructed. Some of these permutation
polynomials are generalizations of known ones.Comment: 18 pages. Submitted to a journal on Aug. 15t
Linear Projections of the Vandermonde Polynomial
An n-variate Vandermonde polynomial is the determinant of the n x n matrix
where the ith column is the vector (1, x_i, x_i^2, ...., x_i^{n-1})^T.
Vandermonde polynomials play a crucial role in the theory of alternating
polynomials and occur in Lagrangian polynomial interpolation as well as in the
theory of error correcting codes. In this work we study structural and
computational aspects of linear projections of Vandermonde polynomials.
Firstly, we consider the problem of testing if a given polynomial is linearly
equivalent to the Vandermonde polynomial. We obtain a deterministic polynomial
time algorithm to test if the polynomial f is linearly equivalent to the
Vandermonde polynomial when f is given as product of linear factors. In the
case when the polynomial f is given as a black-box our algorithm runs in
randomized polynomial time. Exploring the structure of projections of
Vandermonde polynomials further, we describe the group of symmetries of a
Vandermonde polynomial and show that the associated Lie algebra is simple.Comment: Submitted to a conferenc
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
Involutions Restricted by 3412, Continued Fractions, and Chebyshev Polynomials
We study generating functions for the number of involutions, even
involutions, and odd involutions in subject to two restrictions. One
restriction is that the involution avoid 3412 or contain 3412 exactly once. The
other restriction is that the involution avoid another pattern or
contain exactly once. In many cases we express these generating
functions in terms of Chebyshev polynomials of the second kind.Comment: 30 page
Orthogonal Polynomials and Fluctuations of Random Matrices
In this paper we establish a connection between the fluctuations of Wishart
random matrices, shifted Chebyshev polynomials, and planar diagrams whose
linear span form a basis for the irreducible representations of the annular
Temperly-Lieb algebra
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