The trace of an optimal normal element and low complexity normal bases
Designs, Codes and Cryptography
Irreducible Quadratic Factors of x(qn+1)/2+ax+bover Fq
Finite Fields and Their Applications
Abelian Groups, Gauss Periods, and Normal Bases
Finite Fields and Their Applications
On the reducibility of some composite polynomials over finite fields
Designs, Codes and Cryptography
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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$, $i\neq j$, 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 $F(x) = c x^{q+1} + d x^q - ax - b$. 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.