On the Sum of Additive Characters and its Applications over Finite Fields

TL;DR

The paper addresses sums of additive characters over finite fields organized by the $\mathbb{F}_q$-Order, deriving a general formula that mirrors multiplicative-character sums and enabling additive-analytic techniques. By leveraging these sums, it constructs a polynomial Möbius function $\mu(g)$ and a characteristic function for elements with a given $\mathbb{F}_q$-Order, including a robust indicator for $k$-normal elements via additive characters. Key contributions include an explicit sum formula, a polynomial analogue of the Möbius function, and a polynomial-order-based indicator function that generalizes integer identities to the $\mathbb{F}_q$-polynomial setting. These results deepen the connection between additive-character sums and polynomial arithmetic, with potential applications to finite-field constructions and cryptographic primitives.

Abstract

In this paper, we study the sum of additive characters over finite fields, with a focus on those of specified \(\mathbb{F}_q\)-Order. We establish a general formula for these character sums, providing an additive analogue to classical results previously known for multiplicative characters. As an application, we derive a Möbius function \(μ(g)\) for polynomials \(g \in \mathbb{F}_q[x]\), analogous to the integer Möbius function \(μ(n)\), and develop a characteristic function for \(k\)-normal elements. We also generalize several classical identities from the integer setting to the polynomial setting, highlighting the structural parallels between these two domains.

Theorems & Definitions (20)

• Definition 2.1
• Definition 2.2
• Lemma 2.1
• Lemma 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• Remark 3.1