2011/09/22 by Aleksandr Tuxanidy, Qiang Wang, Tuxanidy, Aleksandr +1
Computer Science · Mathematics · #11T06 #12D05 #12E20 #94A55 #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.NT #msc:11T06 #msc:12D05 #msc:12E20 #msc:94A55
paper · pdf · doi:10.48550/arxiv.1109.4693
24 pages
arxiv created 2011/09/22 · openalex publication_date 2011/09/22 · arxiv updated 2011/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let q = ps be a power of a prime number p and let \mathbbFq be the finite field with q elements. In this paper we obtain the explicit factorization of the cyclotomic polynomial Φ2nr over \mathbbFq where both r ≥ 3 and q are odd, gcd(q,r) = 1, and n∈ ℕ. Previously, only the special cases when r = 1, 3, 5 had been achieved. For this we make the assumption that the explicit factorization of Φr over \mathbbFq is given to us as a known. Let n = p1e1p2e2... pses be the factorization of n ∈ ℕ into powers of distinct primes pi, 1≤ i ≤ s. In the case that the orders of q modulo all these prime powers piei are pairwise coprime we show how to obtain the explicit factors of Φn from the factors of each Φpiei. We also demonstrate how to obtain the factorization of Φmn from the factorization of Φn when q is a primitive root modulo m and gcd(m,n) = gcd(ϕ(m),\ordn(q)) = 1. Here ϕ is the Euler's totient function, and \ordn(q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over \mathbbFq and generalize a result due to Varshamov (1984) \citeVarshamov.