Normal and Self-Dual Normal Bases from Factorization of $cx^{q + 1} + dx^q - ax - b$

Abstract

The present paper is interested in a family of normal bases, considered by Sidel’nikov [Math. USSR-Sb., 61 (1988), pp. 485–494], with the property that all the elements in a basis can be obtained from one element by repeatedly applying to it a linear fractional function of the form $\varphi ( x ) = ( ax + b )/( cx + d ),a,b,c,d \in F_q $. Sidel’nikov proved that the products for such a basis $\{ \alpha _i \}$ are of the form $\alpha _i \alpha _j = e_{i - j} \alpha _i + e_{j - i} \alpha _j + \gamma ,i = j$, where $e_k ,\gamma \in F_q $. It is shown that every such basis can be formed by the roots of an irreducible factor of $F( x ) = cx^{q + 1} + dx^q - ax - b$. The following are constructed: (a) a normal basis of $F_{q^n } $ over $F_q $ with complexity at most $3n - 2$ for each divisor n of $q - 1$ and for $n = p$, where p is the characteristic of $F_q$; (b) a self-dual normal basis of $F_{q^n } $ over $F_q $ for $n = p$ and for each odd divisor n of $q - 1$ or $q + 1$. When $n = p$, the self-dual normal basis constructed of $F_{q^p } $ over $F_q$ also has complexity at most $3p - 2$. In all cases, the irreducible polynomials and the multiplication tables are given explicitly.