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 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].

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.

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.

scholarly journals On some permutation polynomials over finite fields

2005 ◽  
Vol 2005 (16) ◽  
pp. 2631-2640 ◽  
Author(s):  
Amir Akbary ◽  
Qiang Wang
Keyword(s):  
Finite Field ◽  
Recurrence Relation ◽  
Finite Fields ◽  
Permutation Polynomials ◽  
Initial Values ◽  
Polynomials Over Finite Fields

Letpbe prime,q=pm, andq−1=7s. We completely describe the permutation behavior of the binomialP(x)=xr(1+xes)(1≤e≤6) over a finite fieldFqin terms of the sequence{an}defined by the recurrence relationan=an−1+2an−2−an−3(n≥3) with initial valuesa0=3,a1=1, anda2=5.

scholarly journals PERMUTATION POLYNOMIALS OF DEGREE 8 OVER FINITE FIELDS OF ODD CHARACTERISTIC

2019 ◽  
Vol 101 (1) ◽  
pp. 40-55 ◽  
Author(s):  
XIANG FAN
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Computer Algebra ◽  
Personal Computer ◽  
Computer Algebra System ◽  
Computer Work ◽  
Permutation Polynomials ◽  
Polynomial Equations ◽  
Manual Analysis ◽  
Odd Order

We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form.

Dickson Polynomials Over Finite Fields and Complete Mappings

1987 ◽  
Vol 30 (1) ◽  
pp. 19-27 ◽  
Author(s):  
Gary L. Mullen ◽  
Harald Niederreiter
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Permutation Polynomials ◽  
Dickson Polynomials ◽  
Complete Mapping ◽  
Dickson Polynomial ◽  
Special Cases ◽  
Polynomials Over Finite Fields ◽  
Complete Mappings ◽  
Special Case

AbstractDickson polynomials over finite fields are familiar examples of permutation polynomials, i.e. of polynomials for which the corresponding polynomial mapping is a permutation of the finite field. We prove that a Dickson polynomial can be a complete mapping polynomial only in some special cases. Complete mapping polynomials are of interest in combinatorics and are defined as polynomials f(x) over a finite field for which both f(x) and f(x) + x are permutation polynomials. Our result also verifies a special case of a conjecture of Chowla and Zassenhaus on permutation polynomials.

scholarly journals On permutation polynomials over finite fields

1987 ◽  
Vol 10 (3) ◽  
pp. 535-543 ◽  
Author(s):  
R. A. Mollin ◽  
C. Small
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Permutation Polynomial ◽  
Permutation Polynomials ◽  
Polynomials Over Finite Fields ◽  
One To One ◽  
Necessary And Sufficient

A polynomialfover a finite fieldFis called a permutation polynomial if the mappingF→Fdefined byfis one-to-one. In this paper we consider the problem of characterizing permutation polynomials; that is, we seek conditions on the coefficients of a polynomial which are necessary and sufficient for it to represent a permutation. We also give some results bearing on a conjecture of Carlitz which says essentially that for any even integerm, the cardinality of finite fields admitting permutation polynomials of degreemis bounded.

scholarly journals The compositional inverse of a class of permutation polynomials over a finite field

2002 ◽  
Vol 65 (3) ◽  
pp. 521-526 ◽  
Author(s):  
Robert S. Coulter ◽  
Marie Henderson
Keyword(s):  
Finite Field ◽  
Permutation Polynomials ◽  
New Class ◽  
Compositional Inverse

A new class of bilinear permutation polynomials was recently identified. In this note we determine the class of permutation polynomials which represents the functional inverse of the bilinear class.

scholarly journals Constructions and Necessities of Some Permutation Polynomials over Finite Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Xiaogang Liu
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Coding Theory ◽  
Combinatorial Design ◽  
Permutation Polynomials ◽  
Science And Engineering ◽  
Polynomials Over Finite Fields

Let F q denote the finite field with q elements. Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, and combinatorial design. The study of permutation polynomials has a long history, and many results are obtained in recent years. In this paper, we obtain some further results about the permutation properties of permutation polynomials. Some new classes of permutation polynomials are constructed, and the necessities of some permutation polynomials are studied.

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