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 Normal Bases on Galois Ring Extensions

2018 ◽  
Author(s):  
Zhang Aixian ◽  
Feng Keqin
Keyword(s):  
Finite Field ◽  
Field Extension ◽  
Galois Ring ◽  
Normal Basis ◽  
Ring Extension ◽  
Normal Bases ◽  
Ring Extensions

In this paper we study the normal bases for Galois ring extension ${{R}} / {Z}_{p^r}$ where ${R}$ = ${GR}$(pr, n). We present a criterion on normal basis for ${{R}} / {Z}_{p^r}$ and reduce this problem to one of finite field extension $\overline{R} / \overline{Z}_{p^r}=F_{q} / F_{p}  (q=p^n)$ by Theorem 1. We determine all optimal normal bases for Galois ring extension.

Symbol-Pair Distances of Repeated-Root Negacyclic Codes of Length 2s over Galois Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 581-600
Author(s):  
Hai Q. Dinh ◽  
Hualu Liu ◽  
Roengchai Tansuchat ◽  
Thang M. Vo
Keyword(s):  
Finite Field ◽  
Galois Ring ◽  
Maximum Distance ◽  
Galois Rings ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Negacyclic Codes ◽  
Singleton Bound ◽  
Maximum Distance Separable ◽  
Special Case

Negacyclic codes of length [Formula: see text] over the Galois ring [Formula: see text] are linearly ordered under set-theoretic inclusion, i.e., they are the ideals [Formula: see text], [Formula: see text], of the chain ring [Formula: see text]. This structure is used to obtain the symbol-pair distances of all such negacyclic codes. Among others, for the special case when the alphabet is the finite field [Formula: see text] (i.e., [Formula: see text]), the symbol-pair distance distribution of constacyclic codes over [Formula: see text] verifies the Singleton bound for such symbol-pair codes, and provides all maximum distance separable symbol-pair constacyclic codes of length [Formula: see text] over [Formula: see text].

On the Complexity of the Dual Bases of the Gaussian Normal Bases

2015 ◽  
Vol 22 (spec01) ◽  
pp. 909-922
Author(s):  
Alok Mishra ◽  
Rajendra Kumar Sharma ◽  
Wagish Shukla
Keyword(s):  
Finite Field ◽  
Normal Basis ◽  
Dual Basis ◽  
Normal Bases ◽  
Gaussian Normal Basis

In this paper, we study the complexity of the dual bases of the Gaussian normal bases of type (n, t), for all n and t = 3, 4, 5, 6, of 𝔽qn over 𝔽q and provide conditions under which the complexity of the Gaussian normal basis of type (n, t) is equal to the complexity of the dual basis over any finite field.

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.

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 Self-Dual Normal Basis of a Galois Ring

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Irwansyah ◽  
Intan Muchtadi-Alamsyah ◽  
Aleams Barra ◽  
Ahmad Muchlis
Keyword(s):  
Free Module ◽  
Galois Ring ◽  
Normal Basis ◽  
Galois Rings ◽  
Orthogonal Matrices

LetR′=GR(ps,psml)andR=GR(ps,psm)be two Galois rings. In this paper, we show how to construct normal basis in the extension of Galois rings, and we also define weakly self-dual normal basis and self-dual normal basis forR′overR, whereR′is considered as a free module overR. Moreover, we explain a way to construct self-dual normal basis using particular system of polynomials. Finally, we show the connection between self-dual normal basis forR′overRand the set of all invertible, circulant, and orthogonal matrices overR.

Sign in / Sign up

Export Citation Format

Share Document