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

2025/06/19 by K., Maithri, R., Vadiraja Bhatta G., P, Indira K.
Computer Science · Mathematics · #Coding theory and cryptography #Analytic Number Theory Research #Cryptography and Residue Arithmetic

paper · pdf · doi:10.48550/arxiv.2506.16081

Abstract

In this paper, we study the sum of additive characters over finite fields, with a focus on those of specified \(\mathbbF_q\)-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 \(g ∈ \mathbbF_q[x]\), 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.

Citations

• r-primitive k-normal elements in arithmetic progressions over finite fields
• Pairs of r-primitive and k-normal elements in finite fields
• Inverses of r-primitive k-normal elements over finite fields
• A note on additive characters of finite fields
• Variations of the Primitive Normal Basis Theorem
• Existence of primitive 1-normal elements in finite fields
• Some notes on the k-normal elements and k-normal polynomials over\n finite fields

Related

• Additive decompositions induced by multiplicative characters over finite fields
• The additive index of polynomials over finite fields
• A note on additive characters of finite fields
• Ax's theorem with an additive character
• Character sums over sparse elements of finite fields