2024/05/16 by Leandro Cagliero, Cagliero, Leandro, Allen Herman +3
Computer Science · #Cellular Automata and Applications #Chaos-based Image/Signal Encryption #Coding theory and cryptography #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2405.10159
openalex publication_date 2024/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a prime number p, we consider the tower of finite fields Fp=L-1⊂ L0⊂ L1⊂⋯, where each step corresponds to an Artin-Schreier extension of degree p, so that for i≥ 0, Li=Li-1[ci], where ci is a root of Xp-X-ai-1 and ai-1=(c-1⋯ ci-1)p-1, with c-1=1. We extend and strengthen to arbitrary primes prior work of Popovych for p=2 on the multiplicative order of the given generator ci for Li over Li-1. In particular, for i≥ 0, we show that O(ci)=O(ai), except only when p=2 and i=1, and that O(ci) is equal to the product of the orders of cj modulo Lj-1^×, where 0≤ j≤ i if p is odd, and i≥ 2 and 1≤ j≤ i if p=2. We also show that for i≥ 0, the Gal(Li/Li-1)-conjugates of ai form a normal basis of Li over Li-1. In addition, we obtain the minimal polynomial of c1 over Fp in explicit form.