On the construction of primitive elements and primitive normal bases in a finite field

2011 ◽  
pp. 1-14 ◽  
Author(s):  
S.A. Stepanov ◽  
I.E. Shparlinskiy
Keyword(s):  
Finite Field ◽  
Primitive Elements ◽  
Normal Bases

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 Linear combinations of primitive elements of a finite field

2018 ◽  
Vol 51 ◽  
pp. 388-406 ◽  
Author(s):  
Stephen D. Cohen ◽  
Tomás Oliveira e Silva ◽  
Nicole Sutherland ◽  
Tim Trudgian
Keyword(s):  
Finite Field ◽  
Primitive Elements ◽  
Linear Combinations

scholarly journals Mean value theorems for a class of density-like arithmetic functions

2020 ◽  
pp. 1-15
Author(s):  
Lucas Reis
Keyword(s):  
Finite Field ◽  
Arithmetic Function ◽  
Field Extension ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Mean Value Theorem ◽  
Primitive Elements ◽  
Mean Values ◽  
Mean Value Theorems ◽  
Nonzero Mean

This paper provides a mean value theorem for arithmetic functions [Formula: see text] defined by [Formula: see text] where [Formula: see text] is an arithmetic function taking values in [Formula: see text] and satisfying some generic conditions. As an application of our main result, we prove that the density [Formula: see text] (respectively, [Formula: see text]) of normal (respectively, primitive) elements in the finite field extension [Formula: see text] of [Formula: see text] are arithmetic functions of (nonzero) mean values.

Construction of MDS Matrices Based on the Primitive Elements of the Finite Field

2021 ◽  
Author(s):  
Wu You ◽  
Dong Xin-feng ◽  
Wang Jin-bo ◽  
Zhang Wen-zheng
Keyword(s):  
Finite Field ◽  
Primitive Elements

scholarly journals Existence results for primitive elements in cubic and quartic extensions of a finite field

2018 ◽  
Vol 88 (316) ◽  
pp. 931-947 ◽  
Author(s):  
Geoff Bailey ◽  
Stephen D. Cohen ◽  
Nicole Sutherland ◽  
Tim Trudgian
Keyword(s):  
Finite Field ◽  
Existence Results ◽  
Primitive Elements

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.

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