Let Fqt be the finite field with qt elements and let F*qt be its multiplicative group. We study the diagonal equation axq−1 + byq−1 = c, where a,b and c ∈ F*qt. This equation can be written as xq−1+αyq−1 = β, where α, β ∈ F ∗ q t . Let Nt(α, β) denote the number of solutions (x,y) ∈ F*qt × F*qt of xq−1 + αyq−1 = β and I(r; a, b) be the number of monic irreducible polynomials f ∈ Fq[x] of degree r with f(0) = a and f(1) = b. We show that Nt(α, β) can be expressed in terms of I(r; a, b), where r | t and a, b ∈ F*q are related to α and β. A recursive formula for I(r; a, b) will be given and we illustrate this by computing I(r; a, b) for 2 ≤ r ≤ 4. We also show that N3(α, β) can be expressed in terms of the number of monic irreducible cubic polynomials over Fq with prescribed trace and norm. Consequently, N3(α, β) can be expressed in terms of the number of rational points on a certain elliptic curve. We give a proof that given any a, b ∈ F*q and integer r ≥ 3, there always exists a monic irreducible polynomial f ∈ Fq[x] of degree r such that f(0) = a and f(1) = b. We also use the result on N2(α, β) to construct a new family of planar functions.
| Identifer | oai:union.ndltd.org:USF/oai:scholarcommons.usf.edu:etd-1038 |
| Date | 29 May 2009 |
| Creators | Sze, Christopher |
| Publisher | Scholar Commons |
| Source Sets | University of South Flordia |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Graduate Theses and Dissertations |
| Rights | default |
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