scholarly journals The Number of Permutation Polynomials of a Given Degree Over a Finite Field

2002 ◽  
Vol 8 (4) ◽  
pp. 478-490 ◽  
Author(s):  
P Das
Keyword(s):  
Finite Field ◽  
Permutation Polynomials

Several classes of permutation trinomials from Niho exponents over finite fields of characteristic 3

2019 ◽  
Vol 18 (04) ◽  
pp. 1950069
Author(s):  
Qian Liu ◽  
Yujuan Sun
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Coding Theory ◽  
Permutation Polynomials ◽  
Combinatorial Designs ◽  
Dickson Polynomial ◽  
Niho Exponents ◽  
Characteristic 3 ◽  
Permutation Trinomials ◽  
Algebraic Forms

Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. Permutation trinomials attract people’s interest due to their simple algebraic forms and additional extraordinary properties. In this paper, based on a seventh-degree and a fifth-degree Dickson polynomial over the finite field [Formula: see text], two conjectures on permutation trinomials over [Formula: see text] presented recently by Li–Qu–Li–Fu are partially settled, where [Formula: see text] is a positive integer.

scholarly journals On the number of permutation polynomials over a finite field

2016 ◽  
Vol 12 (06) ◽  
pp. 1519-1528
Author(s):  
Kwang Yon Kim ◽  
Ryul Kim ◽  
Jin Song Kim
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Roots Of Unity ◽  
Permutation Polynomials

In order to extend the results of [Formula: see text] in [P. Das, The number of permutation polynomials of a given degree over a finite field, Finite Fields Appl. 8(4) (2002) 478–490], where [Formula: see text] is a prime, to arbitrary finite fields [Formula: see text], we find a formula for the number of permutation polynomials of degree [Formula: see text] over a finite field [Formula: see text], which has [Formula: see text] elements, in terms of the permanent of a matrix. We write down an expression for the number of permutation polynomials of degree [Formula: see text] over a finite field [Formula: see text], using the permanent of a matrix whose entries are [Formula: see text]th roots of unity and using this we obtain a nontrivial bound for the number. Finally, we provide a formula for the number of permutation polynomials of degree [Formula: see text] less than [Formula: see text].

Further results on permutation polynomials of the form (xpm − x + δ)s + L(x) over 𝔽p2m

2016 ◽  
Vol 15 (05) ◽  
pp. 1650098 ◽  
Author(s):  
Guangkui Xu ◽  
Xiwang Cao ◽  
Shanding Xu
Keyword(s):  
Finite Field ◽  
Permutation Polynomials ◽  
Linearized Polynomial

Several classes of permutation polynomials with given form over [Formula: see text] were recently proposed by Tu, Zeng, Li and Helleseth. In this paper, continuing their work, we present more permutation polynomials of the form [Formula: see text] over the finite field [Formula: see text] where [Formula: see text] is a linearized polynomial with coefficients in [Formula: see text].

scholarly journals The factorable core of polynomials over finite fields

1990 ◽  
Vol 49 (2) ◽  
pp. 309-318 ◽  
Author(s):  
S. D. Cohen
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Permutation Polynomials ◽  
Linearized Polynomial ◽  
Polynomials Over Finite Fields ◽  
Value Sets

AbstractFor a polynomial f(x) over a finite field Fq, denote the polynomial f(y)−f(x) by ϕf(x, y). The polynomial ϕf has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over Fq of the form F = Lr, where L is an affine linearized polynomial over Fq, such that f = g(F) for some polynomial g and the part of ϕf which splits completely into linear factors over the algebraic closure of Fq is exactly φF. This illuminates an aspect of work of D. R. Hayes and Daqing Wan on the existence of permutation polynomials of even degree. Related results on value sets, including the exhibition of a class of permutation polynomials, are also mentioned.

A NOTE ON INTERPOLATION OF PERMUTATIONS OF A SUBSET OF A FINITE FIELD

2014 ◽  
Vol 90 (2) ◽  
pp. 213-219 ◽  
Author(s):  
CHRIS CASTILLO ◽  
ROBERT S. COULTER ◽  
STEPHEN SMITH
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Interpolation Formula ◽  
Permutation Polynomials ◽  
Classical Interpolation ◽  
Compositional Inverse

AbstractWe determine several variants of the classical interpolation formula for finite fields which produce polynomials that induce a desirable mapping on the nonspecified elements, and without increasing the number of terms in the formula. As a corollary, we classify those permutation polynomials over a finite field which are their own compositional inverse, extending work of C. Wells.

scholarly journals Permutation polynomials with exponents in an arithmetic progression

1998 ◽  
Vol 57 (2) ◽  
pp. 243-252 ◽  
Author(s):  
Young Ho Park ◽  
June Bok Lee
Keyword(s):  
Finite Field ◽  
Arithmetic Progression ◽  
Necessary Conditions ◽  
Permutation Polynomial ◽  
Permutation Polynomials ◽  
Sufficient And Necessary Conditions

We examine the permutation properties of the polynomials of the type hk, r, s(x) = xr (1 + xs + … + xsk) over the finite field , of characteristic p. We give sufficient and necessary conditions in terms of k and r for hk, r, l(x) to be a permutation polynomial over , for q = p or p2. We also prove that if hk, r, s(x) is a permutation polynomial over , then (k + 1)s = ±1.

scholarly journals On a Conjecture of Carlitz

1987 ◽  
Vol 43 (3) ◽  
pp. 375-384 ◽  
Author(s):  
Wan Daqing
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Permutation Polynomials ◽  
Odd Order ◽  
Equivalent Version ◽  
Exceptional Polynomials

A conjecture of Carlitz on permutation polynomials is as follow: Given an even positive integer n, there is a constant Cn, such that if Fq is a finite field of odd order q with q > Cn, then there are no permutation polynomials of degree n over Fq. The conjecture is a well-known problem in this area. It is easily proved if n is a power of 2. The only other cases in which solutions have been published are n = 6 (Dickson [5]) and n = 10 (Hayes [7]); see Lidl [11], Lausch and Nobauer [9], and Lidl and Niederreiter [10] for remarks on this problem. In this paper, we prove that the Carlitz conjecture is true if n = 12 or n = 14, and give an equivalent version of the conjecture in terms of exceptional polynomials.

scholarly journals Erratum

1995 ◽  
Vol 38 (2) ◽  
pp. i-i
Keyword(s):  
Finite Field ◽  
Mathematical Society ◽  
Permutation Polynomials

Proceedings of the Edinburgh Mathematical SocietyVolume 38 (Series II) Part 1 February 1995Page 133—Paper by Daqing Wan, Gary L. Mullen and Peter Jau-Shyong ShiueDue to a printer's error a mathematical sign was missed in the article heading which should now read:The Number of Permutation Polynomials of the Form f(x) + cx Over a Finite Field

scholarly journals SOME FAMILIES OF PERMUTATION POLYNOMIALS OVER FINITE FIELDS

2008 ◽  
Vol 04 (05) ◽  
pp. 851-857 ◽  
Author(s):  
MICHAEL E. ZIEVE
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Permutation Polynomials ◽  
Polynomials Over Finite Fields ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for a polynomial of the form xr(1 + xv + x2v + ⋯ + xkv)t to permute the elements of the finite field 𝔽q. Our results yield especially simple criteria in case (q - 1)/ gcd (q - 1, v) is a small prime.

Sign in / Sign up

Export Citation Format

Share Document