Primitive Roots in Cubic Extensions of Finite Fields

Donald Mills () and Gavin McNay ()
Additional contact information
Donald Mills: U.S. Military Academy, Department of Mathematical Sciences

A chapter in Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, 2002, pp 239-250 from Springer

Abstract: Abstract N. Katz has shown that the absolute value of sums of the form $$\sum\limits_{b \in {F_q}} X \left( {\theta + b} \right)$$ , F q the finite field of q elements, χ a nontrivial multiplicative character of $${F_{{q^n}}}$$ , and θ a F q -generator of $${F_{{q^n}}}$$ , is bounded from above by $$\left( {n - 1} \right)\sqrt q$$ . We use this result in conjunction with a sieve due to S. Cohen to show the following for n = 3: For any prime power q and any F q -generator θ of $${F_{{q^n}}}$$ , there exists a primitive element of the form aθ + b ∈ $${F_{{q^n}}}$$ for some a, b ∈ F q , a ≠ 0. We discuss an application of these primitive sums in their use as pseudorandom vector generators, and conclude by discussing the harder problem of guaranteeing the existence of such roots when a is forced to be 1.

Keywords: Finite Field; Prime Power; Primitive Element; Primitive Root; Primitive Polynomial (search for similar items in EconPapers)
Date: 2002
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-59435-9_19

Ordering information: This item can be ordered from
http://www.springer.com/9783642594359

DOI: 10.1007/978-3-642-59435-9_19

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().