Explicit factorization ofX2mpn−1over a finite field

EXPLICIT FACTORIZATION OF x^{2^{a}p^{b}r^{c}}-1 OVER A FINITE FIELD

“…(1) the explicit factorization of f (x) over F. Factoring polynomials over finite fields is very important for the study of the algebraic structure of finite fields, and is also usefull for information security and coding theory. So that research on the explicit factorization of polynomials over finite fields is a classical topic of mathematics(e.g.see [2], [3], [4], [7], [9], [10]). Although the decomposition of polynomials over finite fields have some algorithms, such as the famous Berlekamp algorithm, see [6], Chapter 4, etc., but these algorithms are only effective for a small field.…”

“…Meyn in [7] generalized the main results in [2] and obtained the explicit factorization of x 2 m + 1 over F q by proposing a shorter approach, where q is a prime power with q ≡ 3(mod 4). In [10], B.Chen et al obtained the explicit factorization of x 2 m p n − 1, where m, n are positive integers and p|q − 1. and the irreducible factors of x 2 m p n − 1 over F q are either binomials or trinomials. Irreducible binomials and trinomials over a finite field are extensively used in some fast algorithms implementations.…”

On the number of k-normal elements over finite fields

“…The relationship between cyclotomic polynomials Q 2 m r and Q r was given in [15] where r and q are odd. The work in [3] covers a complete factorization of Q n where n = 2 m p k , p being an odd prime such that q ≡ 1 (mod p). Lastly, in [17], the authors studied the relationship between Q p m r and Q r , where r is odd.…”

“…In this example we will consider the finite field F 109 3 6 over F 109 . In this case we have q − 1 = 4 • 33…”

Further factorization of x − 1 over a finite field

“…Applying this result, they presented a general idea to factorize cyclotomic polynomials over finite fields. Some explicit factorizations of certain cyclotomic polynomials or Dickson polynomials can be found in [1,2,3,4,5,6,7,11,12], etc.…”