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>.

scholarly journals Probabilistic Construction of Normal Basis

1991 ◽  
Vol 20 (361) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen
Keyword(s):  
Finite Field ◽  
Arithmetic Operation ◽  
Irreducible Polynomial ◽  
Field Extension ◽  
Normal Basis ◽  
Expected Time ◽  
Probabilistic Construction

<p>Let GF(q) be the finite field with q elements. A normal basis polynomial f in GF(q)[x] of degree n is an irreducible polynomial, whose roots form a (normal) basis for the field extension (GF(q^n) : GF(q). We show that a normal basis polynomial of degree <em>n</em> can be found in expected time O(n^(2 + varepsilon) . log(q) + n^(3 + varepsilon) $, when an arithmetic operation and the generation of a random constant in the field GF(q) cost unit time.</p><p> </p><p>Given some basis B = alpha_1, alpha_2,..., alpha_n for the field extension GF(qn) : GF(q) together with an algorithm for multiplying two elements in the B-representation in time O(n^beta), we can find a normal basis for this extension and express it in terms of B in expected time O(n^(1 + beta + varepsilon) € log(q) + n^(3 + varepsilon).</p>

scholarly journals Normal Bases on Galois Ring Extensions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 702
Author(s):  
Aixian Zhang ◽  
Keqin Feng
Keyword(s):  
Signal Processing ◽  
Finite Field ◽  
Block Cipher ◽  
Field Extension ◽  
Galois Ring ◽  
Normal Basis ◽  
Galois Rings ◽  
Ring Extension ◽  
Galois Fields ◽  
Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

scholarly journals On Monomorphisms and Subfields

2019 ◽  
Vol 27 (2) ◽  
pp. 133-137
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Irreducible Polynomial ◽  
Field Extension ◽  
Isomorphic Copy ◽  
Field Extensions ◽  
The Third ◽  
Fourth Part ◽  
Roots Of Polynomials ◽  
Classical Proof ◽  
Do So

Summary This is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/<p> as the desired field extension E [5], [3], [4]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/<p>. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ < p > as sets, so F is not a subfield of F [X]/<p>, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ : F → F [X]/<p> and show that the translated polynomial ϕ (p) has a root over F [X]/<p>. Because F is not a subfield of F [X]/<p> we construct in this second part the field (E \ ϕF )∪F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/<p> and therefore consider F as a subfield of F [X]/<p>”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar’s representations of 𝕑n, 𝕈 and 𝕉 we have 𝕑n ∩ 𝕑n[X] = ∅, 𝕈 ∩ 𝕈 [X] = ∅ and 𝕉 ∩ 𝕉 [X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/<p> with the canonical monomorphism ϕ : F → F [X]/<p>. Together with the first part this gives - for fields F with F ∩ F [X] = ∅ - a field extension E of F in which p ∈ F [X]\F has a root.

Polynomial index in discrete valuation rings

2019 ◽  
Vol 56 (2) ◽  
pp. 260-266
Author(s):  
Mohamed E. Charkani ◽  
Abdulaziz Deajim
Keyword(s):  
Lower Bound ◽  
Valuation Ring ◽  
Irreducible Polynomial ◽  
Discrete Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Adic Valuation

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

scholarly journals Polynomial expressions of p-ary auction functions

2019 ◽  
Vol 13 (2) ◽  
pp. 69-80
Author(s):  
Shizuo Kaji ◽  
Toshiaki Maeno ◽  
Koji Nuida ◽  
Yasuhide Numata
Keyword(s):  
Homomorphic Encryption ◽  
Target Function ◽  
Fully Homomorphic Encryption ◽  
Degree Polynomial ◽  
Prime Field ◽  
Polynomial Expression ◽  
The Common ◽  
Finite Prime Field ◽  
Maximum Minimum ◽  
Unique Polynomial

Abstract One of the common ways to design secure multi-party computation is twofold: to realize secure fundamental operations and to decompose a target function to be securely computed into them. In the setting of fully homomorphic encryption, as well as some kinds of secret sharing, the fundamental operations are additions and multiplications in the base field such as the field {\mathbb{F}_{2}} with two elements. Then the second decomposition part, which we study in this paper, is (in theory) equivalent to expressing the target function as a polynomial. It is known that any function over the finite prime field {\mathbb{F}_{p}} has a unique polynomial expression of degree at most {p-1} with respect to each input variable; however, there has been little study done concerning such minimal-degree polynomial expressions for practical functions. This paper aims at triggering intensive studies on this subject, by focusing on polynomial expressions of some auction-related functions such as the maximum/minimum and the index of the maximum/minimum value among input values.

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.

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