scholarly journals On the Polynomial Basis ofGF(2n)Having a Small Number of Trace-One Elements

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Jiantao Wang ◽  
Dong Zheng ◽  
Zheng Huang
Keyword(s):  
Polynomial Basis ◽  
Symmetric Polynomials ◽  
Irreducible Polynomials ◽  
Galois Fields ◽  
New Class ◽  
Polynomial Bases

In the Galois fieldsGF(2n), a polynomial basis with a small number of trace-one elements is desirable for its convenience in computing. To find new irreducible polynomialsg(x)overGF(2)with this property, we research into the auxiliary polynomialf(x)=(x+1)g(x)with roots{1,α1,α2,…,αn}, such that the symmetric polynomialssk=1+α1k+α2k+⋯+αnkare relative to the symmetric polynomials ofg(x). We introduce a new class of polynomials with the number “1” occupying most of the values in itssk. This indicates that the number “0” occupies most of the values of the traces of the elements{α1,α2,…,αn}. This new class of polynomial gives us an indirect way to find irreducible polynomials having a small number of trace-one elements in their polynomial bases.

scholarly journals A new class of irreducible polynomials

2021 ◽  
pp. 1-6
Author(s):  
Jitender Singh ◽  
Sanjeev Kumar
Keyword(s):  
Irreducible Polynomials ◽  
New Class

scholarly journals Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

2021 ◽  
Vol 37 ◽  
pp. 640-658
Author(s):  
Eunice Y.S. Chan ◽  
Robert M. Corless ◽  
Leili Rafiee Sevyeri
Keyword(s):  
Orthogonal Polynomial ◽  
Matrix Polynomial ◽  
Polynomial Basis ◽  
Regular Matrix ◽  
Bernstein Basis ◽  
Monomial Basis ◽  
Matrix Polynomials ◽  
Regular Matrix Polynomial ◽  
Polynomial Bases

We define generalized standard triples $\boldsymbol{X}$, $\boldsymbol{Y}$, and $L(z) = z\boldsymbol{C}_{1} - \boldsymbol{C}_{0}$, where $L(z)$ is a linearization of a regular matrix polynomial $\boldsymbol{P}(z) \in \mathbb{C}^{n \times n}[z]$, in order to use the representation $\boldsymbol{X}(z \boldsymbol{C}_{1}~-~\boldsymbol{C}_{0})^{-1}\boldsymbol{Y}~=~\boldsymbol{P}^{-1}(z)$ which holds except when $z$ is an eigenvalue of $\boldsymbol{P}$. This representation can be used in constructing so-called  algebraic linearizations for matrix polynomials of the form $\boldsymbol{H}(z) = z \boldsymbol{A}(z)\boldsymbol{B}(z) + \boldsymbol{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$. This can be done even if $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$ are expressed in differing polynomial bases. Our main theorem is that $\boldsymbol{X}$ can be expressed using the coefficients of the expression $1 = \sum_{k=0}^\ell e_k \phi_k(z)$ in terms of the relevant polynomial basis. For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases.

scholarly journals An Effective Algorithm for the Synthesis of Irreducible Polynomials over a Galois Fields of Arbitrary Characteristics

2021 ◽  
Vol 20 ◽  
pp. 508-519
Author(s):  
Anatoly Beletsky
Keyword(s):  
Computational Complexity ◽  
High Performance ◽  
Large Degree ◽  
Search Method ◽  
Effective Algorithm ◽  
Irreducible Polynomials ◽  
Computing Systems ◽  
Galois Fields ◽  
Average Performance ◽  
Very High

The known algorithms for synthesizing irreducible polynomials have a significant drawback: their computational complexity, as a rule, exceeds the quadratic one. Moreover, consequently, as a consequence, the construction of large-degree polynomials can be implemented only on computing systems with very high performance. The proposed algorithm is base on the use of so-called fiducial grids (ladders). At each rung of the ladder, simple recurrent modular computations are performers. The purpose of the calculations is to test the irreducibility of polynomials over Galois fields of arbitrary characteristics. The number of testing steps coincides with the degree of the synthesized polynomials. Upon completion of testing, the polynomial is classifieds as either irreducible or composite. If the degree of the synthesized polynomials is small (no more than two dozen), the formation of a set of tested polynomials is carried out using the exhaustive search method. For large values of the degree, the test polynomials are generating by statistical modeling. The developed algorithm allows one to synthesize binary irreducible polynomials up to 2Kbit on personal computers of average performance

scholarly journals Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence

2021 ◽  
Author(s):  
Misha V. Feigin ◽  
Martin A. Hallnäs ◽  
Alexander P. Veselov
Keyword(s):  
Hermite Polynomials ◽  
Higher Order ◽  
Symmetric Polynomials ◽  
Invariant Extension ◽  
Cherednik Algebra ◽  
New Class ◽  
Harmonic Term ◽  
Rational Cherednik Algebra ◽  
Conceptual Explanation ◽  
Moser System

AbstractLassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero–Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations $${\mathcal {A}}$$ A of real hyperplanes with multiplicities admitting the rational Baker–Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call $${\mathcal {A}}$$ A -Hermite polynomials. These polynomials form a linear basis in the space of $${\mathcal {A}}$$ A -quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero–Moser operator with harmonic term. In the case of the Coxeter configuration of type $$A_N$$ A N this leads to a quasi-invariant version of the Lassalle–Nekrasov correspondence and its higher order analogues.

scholarly journals Construction of Irreducible Polynomials in Galois elds, GF(2m) Using Normal Bases

2019 ◽  
pp. 1-15
Author(s):  
Abraham Aidoo ◽  
Kwasi Baah Gyam
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
General Rule ◽  
Galois Field ◽  
Irreducible Polynomials ◽  
Galois Fields ◽  
Normal Bases ◽  
Primitive Polynomials

This thesis is about Construction of Polynomials in Galois fields Using Normal Bases in finite fields. In this piece of work, we discussed the following in the text; irreducible polynomials, primitive polynomials, field, Galois field or finite fields, and the order of a finite field. We found the actual construction of polynomials in GF(2m) with degree less than or equal to m − 1 and also illustrated how this construction can be done using normal bases. Finally, we found the general rule for construction of GF(pm) using normal bases and even the rule for producing reducible polynomials.

On Irreducible Polynomials in Galois Fields

1963 ◽  
Vol 70 (10) ◽  
pp. 1089
Author(s):  
Dennis Travis
Keyword(s):  
Irreducible Polynomials ◽  
Galois Fields

scholarly journals Closed formulas for exponential sums of symmetric polynomials over Galois fields

2018 ◽  
Vol 50 (1) ◽  
pp. 73-98 ◽  
Author(s):  
Francis N. Castro ◽  
Luis A. Medina ◽  
L. Brehsner Sepúlveda
Keyword(s):  
Exponential Sums ◽  
Symmetric Polynomials ◽  
Galois Fields
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