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 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.

On permutation polynomials over finite fields of characteristic 2

2016 ◽  
Vol 15 (07) ◽  
pp. 1650133 ◽  
Author(s):  
Rohit Gupta ◽  
R. K. Sharma
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Permutation Polynomial ◽  
Permutation Polynomials ◽  
Polynomials Over Finite Fields ◽  
Characteristic 2 ◽  
Polynomial Formula

Let [Formula: see text] denotes the finite field of order [Formula: see text] where [Formula: see text] A permutation polynomial [Formula: see text] over [Formula: see text] with [Formula: see text] and [Formula: see text] such that for each [Formula: see text] is a permutation polynomial satisfying [Formula: see text] is called a o-polynomial. In this paper, we determine all o-polynomials up to degree [Formula: see text].

scholarly journals CONSTRUCTING PERMUTATION POLYNOMIALS OVER FINITE FIELDS

2013 ◽  
Vol 89 (3) ◽  
pp. 420-430 ◽  
Author(s):  
XIAOER QIN ◽  
SHAOFANG HONG
Keyword(s):  
Finite Fields ◽  
Open Problem ◽  
Symmetric Polynomial ◽  
Permutation Polynomial ◽  
Permutation Polynomials ◽  
Elementary Symmetric Polynomial ◽  
Polynomials Over Finite Fields

AbstractIn this paper, we construct several new permutation polynomials over finite fields. First, using the linearised polynomials, we construct the permutation polynomial of the form ${ \mathop{\sum }\nolimits}_{i= 1}^{k} ({L}_{i} (x)+ {\gamma }_{i} ){h}_{i} (B(x))$ over ${\mathbf{F} }_{{q}^{m} } $, where ${L}_{i} (x)$ and $B(x)$ are linearised polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalise a result of Marcos by constructing permutation polynomials of the forms $xh({\lambda }_{j} (x))$ and $xh({\mu }_{j} (x))$, where ${\lambda }_{j} (x)$ is the $j$th elementary symmetric polynomial of $x, {x}^{q} , \ldots , {x}^{{q}^{m- 1} } $ and ${\mu }_{j} (x)= {\mathrm{Tr} }_{{\mathbf{F} }_{{q}^{m} } / {\mathbf{F} }_{q} } ({x}^{j} )$. This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form ${L}_{1} (x)+ {L}_{2} (\gamma )h(f(x))$ over ${\mathbf{F} }_{{q}^{m} } $, which extends a result of Kyureghyan.

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 Image encryption based on permutation polynomials over finite fields

2020 ◽  
Vol 50 (3) ◽  
Author(s):  
Jianhua Wu ◽  
Hai Liu ◽  
Xishun Zhu
Keyword(s):  
Image Encryption ◽  
Finite Fields ◽  
High Efficiency ◽  
Symmetric Matrix ◽  
Permutation Polynomial ◽  
Encryption Algorithm ◽  
Permutation Polynomials ◽  
Polynomials Over Finite Fields ◽  
Intrinsic Nonlinearity ◽  
Image Encryption Algorithm

In this paper, we propose an image encryption algorithm based on a permutation polynomial over finite fields proposed by the authors. The proposed image encryption process consists of four stages: i) a mapping from pixel gray-levels into finite field, ii) a pre-scrambling of pixels’ positions based on the parameterized permutation polynomial, iii) a symmetric matrix transform over finite fields which completes the operation of diffusion and, iv) a post-scrambling based on the permutation polynomial with different parameters. The parameters used for the polynomial parameterization and for constructing the symmetric matrix are used as cipher keys. Theoretical analysis and simulation demonstrate that the proposed image encryption scheme is feasible with a high efficiency and a strong ability of resisting various common attacks. In addition, there are not any round-off errors in computation over finite fields, thus guaranteeing a strictly lossless image encryption. Due to the intrinsic nonlinearity of permutation polynomials in finite fields, the proposed image encryption system is nonlinear and can resist known-plaintext and chosen-plaintext attacks.

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.

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 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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