Primitive Elements with Zero Traces

✍ Scribed by Wun-Seng Chou; Stephen D. Cohen

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 167 KB
• Volume: 7
• Category: Article
• ISSN: 1071-5797
• DOI: 10.1006/ffta.2000.0284

✦ Synopsis

Let F

O denote the "nite "eld of order q, a power of a prime p, and n be a positive integer. We resolve completely the question of whether there exists a primitive element of F O L which is such that it and its reciprocal both have zero trace over F O . Trivially, there is no such element when n(5: we establish existence for all pairs (q, n) (n55) except (4, 5), (2, 6), and (3, 6). Equivalently, with the same exceptions, there is always a primitive polynomial P(x) of degree n over F O whose coe$cients of x and of xL\ are both zero. The method employs Kloosterman sums and a sieving technique.