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2 edition of construction of self-dual normal polynomials over GF(2) and their applications to the Massey-Omura algorithm found in the catalog.

construction of self-dual normal polynomials over GF(2) and their applications to the Massey-Omura algorithm

Andrew Rae

# construction of self-dual normal polynomials over GF(2) and their applications to the Massey-Omura algorithm

Published by Brunel University, Department of Mathematics and Statistics in Uxbridge .
Written in English

Edition Notes

The Physical Object ID Numbers Statement Andrew Rae, Mahmood Khan Pathan. Series TR/13/90 Contributions Pathan, Mahmood Khan. Pagination 29p. Number of Pages 29 Open Library OL19719810M

Explicit construction and computation of finite fields are emphasized. In particular, the construction of irreducible polynomials and normal basis of finite field is included. A detailed treatment of optimal normal basis and Galoi's rings is included. It is the first time that the galois rings are in book form. Errata(s) Errata. Sample Chapter(s). Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.

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### construction of self-dual normal polynomials over GF(2) and their applications to the Massey-Omura algorithm by Andrew Rae Download PDF EPUB FB2

Algorithm is presented for the construction of self-dual normal polynomials over GF(2) for construction of self-dual normal polynomials over GF book odd degree.

This gives a new constructive proof of the existence of a self-dual basis for odd degree. The use of such polynomials in the Massey-Omura multiplier improves the efficiency and decreases the complexity of construction of self-dual normal polynomials over GF book multiplier.

Gaussian periods are used to locate a normal element of the finite field GF(2e) of odd degree e and an\ud algorithm is presented for the construction of self-dual normal polynomials over GF(2) for any odd degree.\ud This gives a new constructive proof of the existence of a self-dual basis for odd : A Rae and M K Pathan.

SELF-DUAL BASES OF GF(qm) OVER GF(q) 25 Theorem 4. E has a self-dual normal basis over F if and only if either m is odd, or q is even and m ^ 0 (mod 4). The necessity of the condition in Theorem 4 follows, for q odd, trivially from Theorem 1; for q even, it is due to Imamura and Morii [12].

The sufficiency. Recent work of the second author [Int. Number Theory 6, No. 7, – (; Zbl )] has given a construction of self-dual normal bases for extensions of finite fields, whenever. We found the actual construction of polynomials in GF (2^ m) with degree less than or equal to m − 1 and also illustrated how this construction can be done using normal bases.

Construction of N-polynomials over nite elds In this section we establish theorems that will show how Propositions and can be applied to produce N-polynomials over F2s.

Theorem Let P(x) = ∑n i=0 cix i, with P(x) ̸= x an N-polynomial of degree n over F2s such that P(x+1) is a self-reciprocal polynomial over F2s. Also let F(x. It is possible to convert the field such that all the elements are expanded in terms of a self-dual basis rather than the polynomial basis.

For example, suppose we are working in GF(4). The elements x and x 2 comprise a self-dual basis. To convert the field we can do. gf = GaloisField(2, 2, [1, 1, 1]) _sdb([1, 2]).

82 J. Wolfinann The principle of the construction It is well known that q-ary cyclic codes of length n may be seen as principal ideals in a semi-simple algebra, the set of polynomials over GF(q) modulo x" - 1.

self-dual codes of length 10 over these rings. In a similar manner, over GR (5 2, 2), GR (5 3, 2) and GR (7 2, 2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self.

dual. We give a construction of binary doubly even self dual codes as binary images of some principal ideals in a group algebra. In particular, we show how. If I work over $GF(2^4)$, then the condition for the type I optimal normal basis holds, in fact $4+1=5$ is prime and $2$ is a primitive element in $\mathbb{Z}_5$.

construction to study MDS self-dual codes over small Galois rings. Codes over chain rings are of special interest theoretically and practically since most known good codes are from chain rings. In fact, ﬂnding interesting (nontrivial) examples of self-dual codes over Galois rings GR(pr;m) is proposed as.

Fitzgerald and Yucas [3] counted the number of irreducible polynomials of odd degree over GF(2) with the ﬁrst three coeﬃcients prescribed. Niederre-iter [10] obtained the formula for the number of irreducible polynomials over GF(2) with given trace and cotrace to apply to the construction of irreducible polynomials over a binary ﬁeld.

In chapter 11 \Strategies for Polynomial Systems", a variety of di erent applications that depend on solving polynomial systems of equations over nite elds is considered. The author explains the concept of universal maps.

Then he focuses on polynomials over GF(2) and how to reduce their degree. Two. For a suitably chosen initial N -polynomial F 0 (x) ∈ GF (2 s) of degree n, polynomials F k (x) ∈ GF (2 s) of degrees n 2 k are constructed by iteratively applying the transformation x → x + x - 1, and their roots are shown to form a normal basis of GF (2 sn 2 k) over GF (2 s).Cited by: 1.

over a ﬁnite ﬁeld GF(2M) and suggests various ways for implementation. Discrete logarithm problem on the elliptic curves corresponds to scalar multiplication of the point of a given elliptical curve. The point coordinates are M-bit polynomials over GF(2). These polynomials can be represented using standard polynomial basis or so called.

We develop a construction method of isodual codes over G F (q), where q is a prime power; we construct isodual codes over G F (q) of length 2 n + 2 from isodual codes over G F (q) of length this method, we find some isodual codes over G F (q), where q = 2, 3 and 5. In more detail, we obtain binary isodual codes of leng 34, 36, 38, where all these codes of leng   New recursive construction of normal polynomials over finite fields 1 14 free; 1.

Introduction 1 14; 2. Preliminaries 2 15; 3. Irreducibility of Polynomial Compositions 3 16; 4. Some construction of N-polynomials 6 19; References 10 23; Collineation groups strongly irreducible on an oval in a projective plane of odd order 11 24; 1.

Introduction. Codes defined by polynomials over finite fields GF (r) A generic construction of cyclic codes with polynomials. Given any polynomial f (x) over GF (r), we define its associated sequence s ∞ by (5) s i = Tr (f (α i + 1)) for all i ≥ 0. The objective of this paper is to consider the codes C s defined by some monomials and trinomials.

This paper presents procedures for constructing irreducible polynomials over GF(2s)GF(2s) with linearly independent roots (or normal polynomials or N-polynomials). In GF(2 m), when the degree of the result is more than m-1, it needs to be reduced modulo a irreducible can be implemented as bit-shift and XOR.

For example, x 3 +x+1 is an irreducible polynomial and x 4 +x 3 +x+1 ≡ x 2 +x mod (x 3 +x+1). The bit-string representation of x 4 +x 3 +x+1 is and the bit-string representation of x 3 +x+1 is As a consequence the roots of such an mth degree polynomial form a basis of GF(q m) over GF(q).

Such a basis is called a normal basis over GF(q) and the polynomial is called normal over GF(q). Normal bases over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to by: 4.

This chapter is devoted to the problem of constructing irreducible polynomials over a given finite field. Such polynomials are used to implement arithmetic in extension fields and are found in many applications, including coding theory [5], cryptography [13], computer algebra systems [11], multivariate polynomial factorization [21], and parallel polynomial arithmetic [18].Author: Ian F.

Blake, XuHong Gao, Ronald C. Mullin, Scott A. Vanstone, Tomik Yaghoobian. In this paper, a computationally simple and explicit construction of some sequences of normal polynomials and self-reciprocal normal polynomials over finite fields of even characteristic are.

Construction of Irreducible Self-Reciprocal Polynomials In Galois theory it is occasionally useful to remark that for any self-reciprocal f(x) of even degree 2n, x-nf(x) is a polynomial g(y) of degree n in y:= x + 1/x.

Proceeding in the reverse direction we use this quadratic transformation to construct. () Construction of self-dual normal bases and their complexity. Finite Fields and Their Applications() Hamiltonians of quantum systems with positions and momenta in GF(pℓ).Cited by: First, we represent a general element G of GF(28) as a linear polynomial (in y)overGF(24), as G = γ1y + γ0, with multiplication modulo an irreducible polynomial r(y)=y2 + τy+ ν.

All the coeﬃcients are in the 4-bit subﬁeld GF(24). So the pair [γ1,γ0] represents G in terms of a polynomial File Size: KB. "Problems concerning polynomials have impulsed resp. accompanied the development of algebra from its very beginning until today and over the centuries a lot of mathematical gems have been brought to light.

This book presents a few of them, some being classical, but partly probably unknown even to expers, some being quite recently discovered/5(3). Abstract. In this article, an extremely simple and highly regular architecture for finite field multiplier using redundant basis is presented, where redundant basis is a new basis taking advantage of the elegant multiplicative structure of the set of primitive n th roots of unity over F 2 that forms a basis of F 2m over F architecture has an important feature of implementation complexity.

Both parts are then plugged together by inclusion of the linkage file when inheriting from this class. See mial_gf2x for an example.

We illustrate the generic glueing using univariate polynomials over $$\mathop{\mathrm{GF}}(2)$$. Univariate Polynomials over domains and fields; Univariate Polynomials over GF(2) via NTL’s GF2X. Univariate polynomials over number fields. Dense univariate polynomials over $$\ZZ$$, implemented using FLINT.

Dense univariate polynomials over $$\ZZ$$, implemented using NTL. Univariate polynomials over $$\QQ$$ implemented via FLINT. The Euclidean algorithm works for polynomials over any field, in the same way as it does over the classical fields, because polynomial long division-with-remainder algorithm works universally for polynomials that are monic (i.e.

lead coefficient $= 1$). But, over a field, every polynomial is associate to a monic polynomial (multiply it by the inverse of the leading coefficient). The ﬁnite ﬁeld, or Galois ﬁeld, with q = ph elements, p a prime, is denoted GF(q).

The subﬁeld GF(p) of GF(q) is the prime ﬁeld and GF(q) can be considered as an h dimensional vector space over it. For an element a 2 GF(q), the trace of a is given by Tr(a) = a+ap +ap2 +†††+aph•1: This is a linear functional from GF(q) onto its.

GF(p 2) for an odd prime pFor applying the above general construction of finite fields in the case of GF(p 2), one has to find an irreducible polynomial of degree p = 2, this has been done in the preceding p is an odd prime, there are always irreducible polynomials of the form X 2 − r, with r in GF(p).

More precisely, the polynomial X 2 − r is irreducible over GF(p) if. spaces over ﬁnite ﬁelds is associating polynomials to them. This technique was ﬁrst used by Jamison and Bruen, then, followed by several people, became a standard method; nowadays, the contours of a growing theory can be seen already.

The polynomials we use should reﬂect the combinatorial properties of the pointset,File Size: 1MB. I’ll talk more generally about finding irreducible polynomials in F_p[x]. The idea is that the process is similar to the sieve of Eratosthenes.

You list out the monic polynomials degree-by-degree, and cross out those which are products. Since you. The Massey-Omura multiplier of GF(2^{m}) uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are Cited by: Contents Part I: Introduction 1 History of nite elds 2 Finite elds in the th and th centuries.

Get Textbooks on Google Play. Rent and save from the world's largest eBookstore. Read, highlight, and take notes, across web, tablet, and phone. Abstract. The approach for constructing the irreducible polynomials of arbitrary degree n over finite fields which is based on the number of the roots over the extension field is presented.

At the same time, this paper includes a sample to illustrate the specific construction : Yun Song, Zhihui Li. The Massey-Omura multiplier of GF(2^{m}) uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are .Zhegalkin polynomials form one of many possible representations of the operations of Boolean algebra.

Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin inthey are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents.

.In mathematics, particularly in algebra, a field extension is a pair of fields ⊆, such that the operations of E are those of F restricted to this case, F is an extension field of E and E is a subfield of F. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex.