Character Sums, Primitive Elements, and Powers in Finite Fields

2009

In this talk I will present estimates on incomplete character sums in finite fields, with special emphasize on the non-prime case. Some of the results are of the same strength as Burgess celebrated theorem for prime fields. The improvements are mainly based on arguments from arithmetic combinatorics providing new bounds on multiplicative energy and an improved amplification strategy. In particular, we improve on earlier work of Davenport-Lewis and Karacuba.

Chinese Annals of Mathematics, Series B, 2016

Let Fq be a finite field of characteristic p. In this paper, by using the index sum method the authors obtain a sufficient condition for the existence of a primitive element α ∈ Fqn such that α + α −1 is also primitive or α + α −1 is primitive and α is a normal element of Fqn over Fq.

ArXiv, 2017

In this note we construct a series of small subsets containing a non-d-th power element in a finite field by applying certain bounds on incomplete character sums. Precisely, let $h=\lfloor q^{\delta}\rfloor>1$ and $d\mid q^h-1$. If $q^h-1$ has a prime divisor $r$ with $r=O((h\log q)^c)$, then there is a constant $0 0$ shows that there exists an explicit subset of cardinality $q^{1-d}=O(\log^{2+\epsilon'}(q^h))$ containing a non-quadratic element in $\mathbb{F}_{q^h}$. On the other hand, the choice of $h=2$ shows that for any odd prime power $q$, there is an explicit subset of cardinality $O(\sqrt{q})$ containing a non-quadratic element in $\mathbb{F}_{q^2}$, essentially improving a $O(q)$ construction by Coulter and Kosick \cite{CK}. In addition, we obtain a similar construction for small sets containing a primitive element. The construction works well provided $\phi(q^h-1)$ is very small, where $\phi$ is the Euler's totient function.

Mathematics of Computation, 2015

We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime p.

arXiv: Number Theory, 2020

In this paper we generalize the results of Sharma, Awasthi and Gupta (see \cite{SAG}). We work over a field of any characteristic with $q = p^k$ elements and we give a sufficient condition for the existence of a primitive element $\alpha \in \mathbb{F}_{p^k}$ such that $f(\alpha)$ is also primitive in $\mathbb{F}_{p^k}$, where $f(x) \in \mathbb{F}_{p^k}(x)$ is a quotient of polynomials with some restrictions. We explicitly determine the values of $k$ for which such a pair exists for $p=2,3,5$ and $7$.

Finite Fields and Their Applications, 2018

We examine linear sums of primitive roots and their inverses in finite fields. In particular, we refine a result by Li and Han, and show that every p > 13 has a pair of primitive roots a and b such that a + b and a −1 + b −1 are also primitive roots mod p.

Mathematics of Computation, 2015

We prove that for all q > 61, every non-zero element in the finite field F q can be written as a linear combination of two primitive roots of F q. This resolves a conjecture posed by Cohen and Mullen.

arXiv (Cornell University), 2022

In this paper, we explore the existence of m-terms arithmetic progressions in F q n with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for m ≥ 4 and concrete results for m ∈ {2, 3}, where the complete list of exceptions when the common difference belongs to F * q is obtained. The proofs combine character sums, sieve estimates, and computational arguments using the software SageMath.

Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, 2002

N. Katz has shown that the absolute value of sums of the form b∈Fq χ(θ+ b), Fq the finite field of q elements, χ a nontrivial multiplicative character of Fqn , and θ a Fq-generator of Fqn , is bounded from above by (n−1) √ q. We use this result in conjunction with a sieve due to S. Cohen to show the following for n = 3: For any prime power q and any Fq-generator θ of F q 3 , there exists a primitive element of the form aθ + b ∈ F q 3 for some a, b ∈ F q , a = 0. We discuss an application of these primitive sums in their use as pseudorandom vector generators, and conclude by discussing the harder problem of guaranteeing the existence of such roots when a is forced to be 1.

Acta Arithmetica, 2002

Finite Fields and Their Applications, 1998

HAL (Le Centre pour la Communication Scientifique Directe), 2013

Mappings of finite fields play an important role in many applications like coding theory, combinatorics, cryptology or finite geometry. In this article we survey recent progress on classification and explicit constructions of almost perfect nonlinear, bent, crooked mappings and those having a linear structure. We present the switching method, which proved itself as a powerful tool for constructing mappings satisfying additive properties. We describe main open challenges in this research area.

Finite Fields and Their Applications, 2022

In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterizations of permutation polynomials.

International Journal of Mathematical Education in Science and Technology, 2016

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Bulletin of the Brazilian Mathematical Society, New Series, 2021

Let F q n be a finite field with q n elements, and let m 1 and m 2 be positive integers. Given polynomials f 1 (x), f 2 (x) ∈ F q [x] with deg(f i (x)) ≤ m i , for i = 1, 2, and such that the rational function f 1 (x)/f 2 (x) belongs to a certain set which we define, we present a sufficient condition for the existence of a primitive element α ∈ F q n , normal over F q , such that f 1 (α)/f 2 (α) is also primitive.