相关论文: A New Criterion on Normal Bases of Finite Field Ex…
In this paper, a method for constructing a near optimal normal basis for algebraic extensions of a finite field is described. In each extension, except for the squares of basis elements, the product of two distinct normal basis elements can…
Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. We prove that there exists a primitive element of $\mathbb{F}_{q^n}$ that produces a completely normal basis…
The notion of normal elements for finite fields extension has been generalized as k-normal elements by Huczynska et al. [3]. The number of k-normal elements for a fixed finite field extension has been calculated and estimated [3], and…
Let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a prime power and, for each integer $n\ge 1$, let $\mathbb F_{q^n}$ be the unique $n$-degree extension of $\mathbb F_q$. The $\mathbb F_q$-orders of an element in…
Normal bases in finite fields constitute a vast topic of large theoretical and practical interest. Recently, $k$-normal elements were introduced as a natural extension of normal elements. The existence and the number of $k$-normal elements…
In this paper, we give decision criteria for normal binomial difference polynomial ideals in the univariate difference polynomial ring F{y} to have finite difference Groebner bases and an algorithm to compute the finite difference Groebner…
We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse…
The main result of this paper is that if E is a field extension of finite odd degree over a real field Q, and if E is a repeated radical extension of Q, then every intermediate field is also a repeated radical extension of Q. This paper…
In this paper, we study several definitions of generalized rank weights for arbitrary finite extensions of fields. We prove that all these definitions coincide, generalizing known results for extensions of finite fields.
Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…
This is an introduction to the theory of normal bases of finite fields. The first few chapters cover a wide range of topics on the theory of normal bases of finite fields. Most standard definitions and results, including proofs are given.…
A primitive completely normal element for an extension $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ of Galois fields is a generator of the multiplicative group of $\mathbb{F}_{q^n}$, which simultaneously is normal over every intermediate field of that…
Let F be a field complete for a real valuation. It is a standard result in valuation theory that a finite extension of F admits a valuation basis if and only if it is without defect. We show that even otherwise, one can construct bases in…
Recently, Widmer introduced a new sufficient criterion for the Northcott property on the finiteness of elements of bounded height in infinite algebraic extensions of number fields. We provide a simplification of Widmer's criterion when the…
Binary field extensions are fundamental to many applications, such as multivariate public key cryptography, code-based cryptography, and error-correcting codes. Their implementation requires a foundation in number theory and algebraic…
Finite hypergeometric functions are functions of a finite field ${\bf F}_q$ to ${\bf C}$. They arise as Fourier expansions of certain twisted exponential sums and were introduced independently by John Greene and Nick Katz in the 1980's.…
In this paper, we give a finiteness criterion for the solutions of the sequence of semi-$q$-decomposable form equations and inequalities, where the semi-$q$-decomposable form is factorized into a family of $q$ nonconstant homogeneous…
Most algorithms constructing bases of finite-dimensional vector spaces return basis vectors which, apart from orthogonality, do not show any special properties. While every basis is sufficient to define the vector space, not all bases are…
Let $q=p^k$ be a prime power, let $\mathbb{F}_q$ be a finite field and let $n\geq2$ be an integer. This note investigates the existence small primitive normal elements in finite field extensions $\mathbb{F}_{q^n}$. It is shown that a small…
We are interested in extending normal bases of $\mathbf{F}_{\!2^n}/\mathbf{F}_{\!2}$ to bases of $\mathbf{F}_{\!2^{nd}}/\mathbf{F}_{\!2}$ which allow fast arithmetic in $\mathbf{F}_{\!2^{nd}}$. This question has been recently studied by…