scholarly journals Counting Irreducible Polynomials of Degree r over Fqn and Generating Goppa Codes Using the Lattice of Subfields of Fqnr

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Kondwani Magamba ◽  
John A. Ryan
Keyword(s):  
Irreducible Polynomial ◽  
Goppa Codes ◽  
Irreducible Polynomials ◽  
Goppa Code

The problem of finding the number of irreducible monic polynomials of degree r over Fqn is considered in this paper. By considering the fact that an irreducible polynomial of degree r over Fqn has a root in a subfield Fqs of Fqnr if and only if (nr/s,r)=1, we show that Gauss’s formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields of Fqnr . We also use the lattice of subfields of Fqnr to determine if it is possible to generate a Goppa code using an element lying in a proper subfield of Fqnr.

scholarly journals Constructing irreducible polynomials with prescribed level curves over finite fields

2001 ◽  
Vol 27 (4) ◽  
pp. 197-200
Author(s):  
Mihai Caragiu
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomial ◽  
Irreducible Polynomials ◽  
Level Curves ◽  
Curves Over Finite Fields ◽  
Irreducibility Criterion ◽  
Absolutely Irreducible Polynomial

We use Eisenstein's irreducibility criterion to prove that there exists an absolutely irreducible polynomialP(X,Y)∈GF(q)[X,Y]with coefficients in the finite fieldGF(q)withqelements, with prescribed level curvesXc:={(x,y)∈GF(q)2|P(x,y)=c}.

INVARIANTS OF DEFECTLESS IRREDUCIBLE POLYNOMIALS

2010 ◽  
Vol 09 (04) ◽  
pp. 603-631 ◽  
Author(s):  
RON BROWN ◽  
JONATHAN L. MERZEL
Keyword(s):  
Irreducible Polynomial ◽  
Newton Polygon ◽  
Extension Field ◽  
Irreducible Polynomials ◽  
Technical Result ◽  
Valued Field ◽  
Minimal Pairs ◽  
Simple Characterization ◽  
Polynomial Extensions

Defectless irreducible polynomials over a Henselian valued field (F, v) have been studied by means of strict systems of polynomial extensions and complete (also called "saturated") distinguished chains. Strong connections are developed here between these two approaches and applications made to both. In the tame case in which a root α of an irreducible polynomial f generates a tamely ramified extension of (F, v), simple formulas are given for the Krasner constant, the Brink separant and the diameter of f. In this case a (best possible) result is given showing that a sufficiently good approximation in an extension field K of F to a root of a defectless polynomial f over F guarantees the existence of an exact root of f in K. Also in the tame case a (best possible) result is given describing when a polynomial is sufficiently close to a defectless polynomial so as to guarantee that the roots of the two polynomials generate the same extension fields. Another application in the tame case gives a simple characterization of the minimal pairs (in the sense of N. Popescu et al.). A key technical result is a computation in the tame case of the Newton polygon of f(x+α). Invariants of defectless polynomials are discussed and the existence of defectless polynomials with given invariants is proven. Khanduja's characterization of the tame polynomials whose Krasner constants equal their diameters is generalized to arbitrary defectless polynomials. Much of the work described here will be seen not to require the hypothesis that (F, v) is Henselian.

scholarly journals Hamming distance from irreducible polynomials over $\mathbb {F}_2$

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Gilbert Lee ◽  
Frank Ruskey ◽  
Aaron Williams
Keyword(s):  
Empirical Study ◽  
Hamming Distance ◽  
Irreducible Polynomial ◽  
Constant Term ◽  
Irreducible Polynomials ◽  
International Audience ◽  
Insight Into

International audience We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irreducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we prove that the number of degree $n$ polynomials of Hamming distance one from a randomly chosen set of $\lfloor 2^n/n \rfloor$ odd density polynomials, each of degree $n$ and each with non-zero constant term, is asymptotically $(1-e^{-4}) 2^{n-2}$, and this appears to be inconsistent with the numbers for irreducible polynomials. We also conjecture that there is a constant $c$ such that every polynomial has Hamming distance at most $c$ from an irreducible polynomial. Using exhaustive lists of irreducible polynomials over $\mathbb{F}_2$ for degrees $1 ≤ n ≤ 32$, we count the number of polynomials with a given Hamming distance to some irreducible polynomial of the same degree. Our work is based on this "empirical" study.

scholarly journals EXTENDED SUBCLASS OF CYCLIC SEPARABLE GOPPA CODES

2016 ◽  
Vol 15 (5) ◽  
pp. 6776-6784
Author(s):  
Ajay Sharma ◽  
O. P. VINOCHA
Keyword(s):  
Cyclic Codes ◽  
Goppa Codes ◽  
Goppa Code

In 2013[4]a new subclass of cyclic Goppa code with Goppa polynomial of degree 2 is presented by Bezzateev and Shekhunova. They proved that this subclass contains all cyclic codes of considered length. In the present work we consider a Goppa polynomial of degree three and proved that the subclass generated by this polynomial represent a cyclic, reversible and separable Goppa code.

scholarly journals Parallel Construction of Irreducible Polynomials

1991 ◽  
Vol 20 (358) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen
Keyword(s):  
Irreducible Polynomial ◽  
Field Extension ◽  
Normal Basis ◽  
Arithmetic Circuits ◽  
Irreducible Polynomials ◽  
Prime Field ◽  
Parallel Construction ◽  
Finite Prime Field

Let arithmetic pseudo-<strong>NC</strong>^k denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field GF(p) of depth O(log^k (n + p)) and size polynomial in (n + p). We show that the problem of constructing an irreducible polynomial of specified degree over GF(p) belongs to pseudo-<strong>NC</strong>^2.5. We prove that the problem of constructing an irreducible polynomial of specified degree over GF(p) whose roots are guaranteed to form a normal basis for the corresponding field extension pseudo-<strong>NC</strong>^2 -reduces to the problem of factor refinement. We show that factor refinement of polynomials is in arithmetic <strong>NC</strong>^3. Our algorithm works over any field and compared to other known algorithms it does not assume the ability to take <em>p</em>'th roots when the field has characteristic <em>p</em>.

ON LIFTINGS OF POWERS OF IRREDUCIBLE POLYNOMIALS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250222 ◽  
Author(s):  
ANUJ BISHNOI ◽  
SANJEEV KUMAR ◽  
SUDESH K. KHANDUJA
Keyword(s):  
Residue Field ◽  
Sufficient Conditions ◽  
Irreducible Polynomial ◽  
Simple Extension ◽  
Canonical Homomorphism ◽  
Irreducible Polynomials ◽  
Transcendental Extension ◽  
Simple Transcendental Extension ◽  
Irreducibility Criteria

Let v be a henselian valuation of arbitrary rank of a field K with valuation ring Rv having maximal ideal Mv. Using the canonical homomorphism from Rv onto Rv/Mv, one can lift any monic irreducible polynomial with coefficients in Rv/Mv to yield monic irreducible polynomials over Rv. Popescu and Zaharescu extended this approach and introduced the notion of lifting with respect to a residually transcendental prolongation w of v to a simple transcendental extension K(x) of K. As it is well known, the residue field of such a prolongation w is [Formula: see text], where [Formula: see text] is the residue field of the unique prolongation of v to a finite simple extension L of K and Y is transcendental over [Formula: see text] (see [V. Alexandru, N. Popescu and A. Zaharescu, A theorem of characterization of residual transcendental extension of a valuation, J. Math. Kyoto Univ.28 (1988) 579–592]). It is known that a lifting of an irreducible polynomial belonging to [Formula: see text] with respect to w, is irreducible over K. In this paper, we give some sufficient conditions to ensure that a given polynomial in K[x] satisfying these conditions which is a lifting of a power of some irreducible polynomial belonging to [Formula: see text] with respect to w, is irreducible over K. Our results extend Eisenstein–Dumas and generalized Schönemann irreducibility criteria.

scholarly journals A Novel Construction of Perfect Strict Avalanche Criterion S-box using Simple Irreducible Polynomials

2020 ◽  
Vol 7 (1) ◽  
pp. 10-22 ◽  
Author(s):  
Alamsyah Alamsyah
Keyword(s):  
Irreducible Polynomial ◽  
Test Results ◽  
Irreducible Polynomials ◽  
A Value ◽  
Strict Avalanche Criterion ◽  
Main Components ◽  
Algebraic Technique ◽  
Selection Of

An irreducible polynomial is one of the main components in building an S-box with an algebraic technique approach. The selection of the precise irreducible polynomial will determine the quality of the S-box produced. One method for determining good S-box quality is strict avalanche criterion will be perfect if it has a value of 0.5. Unfortunately, in previous studies, the strict avalanche criterion value of the S-box produced still did not reach perfect value. In this paper, we will discuss S-box construction using selected irreducible polynomials. This selection is based on the number of elements of the least amount of irreducible polynomials that make it easier to construct S-box construction. There are 17 irreducible polynomials that meet these criteria. The strict avalanche criterion test results show that the irreducible polynomial p17(x) =x8 + x7 + x6 + x + 1 is the best with a perfect SAC value of 0.5. One indicator that a robust S-box is an ideal strict avalanche criterion value of 0.5

scholarly journals Goppa codes over Edwards curves

2021 ◽  
Author(s):  
Giuseppe Filippone
Keyword(s):  
Singular Points ◽  
Goppa Codes ◽  
Goppa Code ◽  
Edwards Curves ◽  
Geometric Goppa Code ◽  
Generating Matrix ◽  
Edwards Curve

Abstract Given an Edwards curve, we determine a basis for the Riemann-Roch space of any divisor whose support does not contain any of the two singular points. This basis allows us to compute a generating matrix for an algebraic-geometric Goppa code over the Edwards curve.

On the extensions of discrete valuations in number fields

2019 ◽  
Vol 69 (5) ◽  
pp. 1009-1022
Author(s):  
Abdulaziz Deajim ◽  
Lhoussain El Fadil
Keyword(s):  
Number Field ◽  
Irreducible Polynomial ◽  
Number Fields ◽  
Irreducible Polynomials

Abstract Let K be a number field defined by a monic irreducible polynomial F(X) ∈ ℤ [X], p a fixed rational prime, and νp the discrete valuation associated to p. Assume that F(X) factors modulo p into the product of powers of r distinct monic irreducible polynomials. We present in this paper a condition, weaker than the known ones, which guarantees the existence of exactly r valuations of K extending νp. We further specify the ramification indices and residue degrees of these extended valuations in such a way that generalizes the known estimates. Some useful remarks and computational examples are also given to highlight some improvements due to our result.

Construction of irreducible polynomials over finite fields

2021 ◽  
Author(s):  
P. L. Sharma ◽  
Ashima
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomial ◽  
Irreducible Polynomials ◽  
Polynomials Over Finite Fields

Irreducible polynomials over finite fields and their applications have been quite well studied. Here, we discuss the construction of the irreducible polynomials of degree [Formula: see text] over the finite field [Formula: see text] for a given irreducible polynomial of degree [Formula: see text]. Furthermore, we construct the irreducible polynomials of degree [Formula: see text] over the finite field [Formula: see text] for a given irreducible polynomial of degree [Formula: see text] by using the method of composition of polynomials with some conditions on coefficients and degree of a given irreducible polynomial.

Sign in / Sign up

Export Citation Format

Share Document