Using the theory of cyclotomy to factor cyclotomic polynomials over finite fields

Abstract

Abstract. We examine the problem of factoring the rth cyclotomic polynomial, Φr(x), over Fp, r and p distinct primes. Given the traces of the roots of Φr(x) we construct the coefficients of Φr(x) in time O(r 4 ). We demonstrate a deterministic algorithm for factoring Φr(x) in time O((r 1/2+ log p) 9 ) when Φr(x) has precisely two irreducible factors. Finally, we present a deterministic algorithm for computing the sum of the irreducible factors of Φr(x) in time O(r 6 ).

On explicit factors of cyclotomic polynomials over finite fields

“…We make the assumption that the explicit factorization of Φ r is given to us as a known. When q = p and r is an odd prime (distinct from p) one may use for instance the results due to Stein (2001) [20] to compute the factors of Φ r efficiently. We achieve our result by applying the theory of composed products as well as by using, and refining in some cases, some of the techniques and results in [26] now generalized for arbitrary odd number r > 1.…”

“…In this section and the following we make the assumption that the explicit factorization of Φ r is given to us as a known. One may use for instance the results due to Stein (2001) to compute the factors of Φ r efficiently when q = p and r is an odd prime distinct to p. First, we need the following well known theorem concerning the factorization of Φ 2 n when q ≡ 1 (mod 4) which follows from Theorems 2.47 and 3.35 in [15]. Theorem 3.10 ( [15]).…”

Some subgroups of a finite field and their applications for obtaining explicit factors

“…In [16], Wang and Wang obtained the explicit factorization of Φ 2 k 5 (x) for q ≡ ±2 (mod 5). When q and r are distinct odd primes, Stein [13] computed the factors of Φ r (x) from the traces of the roots of Φ r (x) over prime field F q . Assuming that the explicit factors of Φ r (x) are known, Tuxanidy and Wang [14] obtained the irreducible factors of Φ 2 n r (x) over F q , where r > 1 is an arbitrary odd integer.…”

Explicit factorization of $x^{2^nd}-1$ over a finite field

“…Wang and Wang [15] gave the explicit factorization of Φ 2 n 5 (x). Stein [12] obtained the factors of Φ r (x) when q and r are distinct odd primes. Further, assuming the explicit factors of Φ r (x) are known for arbitrary odd integer r > 1, Tuxanidy and Wang [13] obtained the irreducible factors of Φ 2 n r (x) over F q .…”