ExplicitN-polynomials of 2-power degree over finite fields, I

Abstract

Abstract.For an odd prime power q the infinite field GF(q 2~ ) = U,_>0 GF(q2~ ) is explicitly presented by a sequence (f,,),,_>l of N-polynomials. This means that, for a suitably chosen initial polynomial fl, the defining polynomials f, ~ GF(q ) [x ] of degrees 2 n are constructed by iteration of the transformation of variable x ~+ x + 1/x and have linearly independent roots over GF(q). In addition, the sequences are trace-compatible in the sense that the relative traces map the corresponding roots onto each other. In this first paper the case q ---1 (mod 4) is considered and the case q ---3 (rood 4) will be dealt with in a second paper. This specific construction solves a problem raised by A. Scheerhorn in [11].

Construction of self-reciprocal normal polynomials over finite fields of even characteristic

“…Now, by (10) and observing that P (x) is an irreducible polynomial of degree n over F 2 s , we obtain α + 1 = (α + 1)…”

“…Some results regarding constructions of special sequences (F k (x)) k≥0 of normal polynomials over F q can be found in [2,4,6,7,9] and [10,11]. All constructions are considered as computationally easy and explicit.…”

Completely Normal Elements in Iterated Quadratic Extensions of Finite Fields

“…If f 1 is quadratic, then f k is irreducible of degree 2 k . Given an extension F q n /F q of finite fields, in order to show that an element Ͱ ʦ F q n is a normal element Meyn [5] considers F q n as a module over the ring…”

“…In the case where n is a power of 2, and q ϵ 1 (mod 4) Cohen [3] has given a recursive construction of an irreducible polynomial of degree n, and Meyn [5] has shown that the roots of these polynomials are normal elements. McNay [4] has extended Cohen's results to the case where q ϵ 3 (mod 4).…”

A Relation between Self-Reciprocal Transformation and Normal Basis over Odd Characteristic Field

“…This paper considers a relation between self-reciprocal transformation and normal bases over non-binary field, in other words, this paper extends the binary case result to nonbinary cases. In the previous work [13], the base extension degree m needs to be a power of 2. Thus, it is critically limited.…”