scholarly journals On multiplicative order of elements in finite fields based on cyclotomic polynomials

2020 ◽  
Vol 26 (2) ◽  
pp. 47-52
Author(s):  
Roman Popovych ◽  
Keyword(s):  
Finite Fields ◽  
Cyclotomic Polynomials ◽  
Multiplicative Order ◽  
Order Of Elements

scholarly journals On the multiplicative order of elements in Wiedemann's towers of finite fields

2015 ◽  
Vol 7 (2) ◽  
pp. 220-225
Author(s):  
R. Popovych
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Field Extensions ◽  
Fermat Numbers ◽  
Multiplicative Order ◽  
Order Of Elements

We consider recursive binary finite field extensions $E_{i+1} =E_{i} (x_{i+1} )$, $i\ge -1$, defined by D. Wiedemann. The main object of the paper is to give some proper divisors of the Fermat numbers $N_{i} $ that are not equal to the multiplicative order $O(x_{i} )$.

MULTIPLICATIVE ORDERS IN ORBITS OF POLYNOMIALS OVER FINITE FIELDS

2017 ◽  
Vol 60 (2) ◽  
pp. 487-493 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Fields ◽  
Algebraic Closure ◽  
Rational Numbers ◽  
Duke Math ◽  
Roots Of Unity ◽  
Multiplicative Order ◽  
Polynomials Over Finite Fields ◽  
Polynomial Orbits

AbstractWe show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J.139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.

Explicit factorizations of cyclotomic polynomials over finite fields

2016 ◽  
Vol 83 (1) ◽  
pp. 197-217 ◽  
Author(s):  
Hongfeng Wu ◽  
Li Zhu ◽  
Rongquan Feng ◽  
Siman Yang
Keyword(s):  
Finite Fields ◽  
Cyclotomic Polynomials ◽  
Polynomials Over Finite Fields

scholarly journals MULTIPLICATIVE ORDER OF GAUSS PERIODS

2010 ◽  
Vol 06 (04) ◽  
pp. 877-882 ◽  
Author(s):  
OMRAN AHMADI ◽  
IGOR E. SHPARLINSKI ◽  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Lower Bound ◽  
Finite Fields ◽  
Normal Bases ◽  
Multiplicative Order ◽  
Gauss Periods

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of von zur Gathen and Shparlinski.

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