𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Factorization of the Cyclotomic Polynomialx2n+ 1 over Finite Fields

✍ Scribed by Helmut Meyn


Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
152 KB
Volume
2
Category
Article
ISSN
1071-5797

No coin nor oath required. For personal study only.

✦ Synopsis


The aim of this note is to show that the (well-known) factorization of the 2 nΟ©1 th cyclotomic polynomial x 2 n Ο© 1 over GF(q) with q Ο΅ 1 (mod 4) can be used to prove the (more complicated) factorization of this polynomial over GF(q) with q Ο΅ 3 (mod 4).


πŸ“œ SIMILAR VOLUMES


The Factorization of Dickson Polynomials
✍ Wun-Seng Chou πŸ“‚ Article πŸ“… 1997 πŸ› Elsevier Science 🌐 English βš– 257 KB

Let T n (x, a) Κ¦ GF(q)[x] be a Dickson polynomial over the finite field GF(q) of either the first kind or the second kind of degree n in the indeterminate x and with parameter a. We give a complete description of the factorization of T n (x, a) over GF(q).

The rank of sparse random matrices over
✍ Johannes BlΓΆmer; Richard Karp; Emo Welzl πŸ“‚ Article πŸ“… 1997 πŸ› John Wiley and Sons 🌐 English βš– 182 KB πŸ‘ 2 views

Let M be a random n = n -matrix over GF q such that for each entry M in i j w x Ž . M and for each nonzero field element ␣ the probability Pr M s ␣ is pr q y 1 , where i j ## Ž . p slog n y c rn and c is an arbitrary but fixed positive constant. The probability for a Ž . matrix entry to be zero

Characterising the Linear Complexity of
✍ Peter A. Hines πŸ“‚ Article πŸ“… 1998 πŸ› Elsevier Science 🌐 English βš– 307 KB

We give a complete resolution to a conjecture regarding the characterisation of linear complexities of span 1 de Bruijn sequences over nonprime finite fields. This contrasts with results for prime fields, where the characterisation is equivalent to an open question concerning permutation polynomials

On the Multiplicative Orders of Ξ³ and Ξ³+
✍ Igor Shparlinski πŸ“‚ Article πŸ“… 2001 πŸ› Elsevier Science 🌐 English βš– 261 KB

Using bounds of character sums we show that one of the open questions about the possible relation between the multiplicative orders of and # \ has a negative answer. In fact we show that in some sense the multiplicative orders of these elements are independent.