scholarly journals Permutation polynomials over Galois fields of characteristic

2021 ◽  
Vol 13 (2) ◽  
pp. 84-86
Author(s):  
Z.L. Dahiru ◽  
A.M. Lawan
Keyword(s):  
Permutation Polynomial ◽  
Permutation Polynomials ◽  
Galois Fields ◽  
Characteristic 2

In this paper, a class of permutation polynomial known as o-polynomial over Galois fields of characteristic 2 was studied. A necessary and sufficients condition for a monomial 𝑥2k to be an o-polynomial over F2t  is given and two results obtained by Gupta and Sharma (2016) were deduced.

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 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 Permutation polynomials in several variables over residue class rings

1987 ◽  
Vol 43 (2) ◽  
pp. 171-175 ◽  
Author(s):  
H. K. Kaiser ◽  
W. Nöbauer
Keyword(s):  
Commutative Ring ◽  
Residue Class ◽  
Polynomial Function ◽  
Permutation Polynomial ◽  
Permutation Polynomials ◽  
Several Variables ◽  
Residue Class Ring ◽  
Class Ring

AbstractThe concept of a permutation polynomial function over a commutative ring with 1 can be generalized to multiplace functions in two different ways, yielding the notion of a k-ary permutation polynomial function (k > 1, k ∈ N) and the notion of a strict k-ary permutation polynomial function respectively. It is shown that in the case of a residue class ring Zm of the integers these two notions coincide if and only if m is squarefree.

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 The number of permutation polynomials of the form f(x) cx over a finite field

1995 ◽  
Vol 38 (1) ◽  
pp. 133-149 ◽  
Author(s):  
Daqing Wan ◽  
Gary L. Mullen ◽  
Peter Jau-Shyong Shiue
Keyword(s):  
Finite Field ◽  
Residue Class ◽  
Projective Planes ◽  
Latin Squares ◽  
Permutation Polynomial ◽  
Permutation Polynomials ◽  
Translation Planes ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Mild Assumption

Let Fq be the finite field of q elements. Let f(x) be a polynomial of degree d over Fq and let r be the least non-negative residue of q-1 modulo d. Under a mild assumption, we show that there are at most r values of c∈Fq, such that f(x) + cx is a permutation polynomial over Fq. This indicates that the number of permutation polynomials of the form f(x) +cx depends on the residue class of q–1 modulo d.As an application we apply our results to the construction of various maximal sets of mutually orthogonal latin squares. In particular for odd q = pn if τ(n) denotes the number of positive divisors of n, we show how to construct τ(n) nonisomorphic complete sets of orthogonal squares of order q, and hence τ(n) nonisomorphic projective planes of order q. We also provide a construction for translation planes of order q without the use of a right quasifield.

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