Efficient multiplication in binary fields

The thesis discusses the basics of efficient multiplication in finite fields, especially in binary fields. There are two broad approaches: polynomial representation and normal bases, used in software and hardware implementations, respectively. Due to the advantages of normal bases of low complexity, there is also a brief introduction to constructing optimal normal bases. Furthermore, as irreducible polynomials are of fundamental importance for finite fields, the thesis concludes with some irreducibility test

Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

Let $F/E$ be a finite Galois extension of fields with abelian Galois group $\Gamma$. A self-dual normal basis for $F/E$ is a normal basis with the additional property that $Tr_{F/E}(g(x),h(x))=\delta_{g,h}$ for $g,h\in\Gamma$. Bayer-Fluckiger and Lenstra have shown that when $char(E)\neq 2$, then $F$ admits a self-dual normal basis if and only if $[F:E]$ is odd. If $F/E$ is an extension of finite fields and $char(E)=2$, then $F$ admits a self-dual normal basis if and only if the exponent of $\Gamma$ is not divisible by $4$. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let $K$ be a finite extension of \Q_p, let $L/K$ be a finite abelian Galois extension of odd degree and let \bo_L be the valuation ring of $L$. We define $A_{L/K}$ to be the unique fractional \bo_L-ideal with square equal to the inverse different of $L/K$. It is known that a self-dual integral normal basis exists for $A_{L/K}$ if and only if $L/K$ is weakly ramified. Assuming $p\neq 2$, we construct such bases whenever they exist

Construction of self-dual normal bases and their complexity

Recent work of Pickett has given a construction of self-dual normal bases for extensions of finite fields, whenever they exist. In this article we present these results in an explicit and constructive manner and apply them, through computer search, to identify the lowest complexity of self-dual normal bases for extensions of low degree. Comparisons to similar searches amongst normal bases show that the lowest complexity is often achieved from a self-dual normal basis

Multiplicative Order of Gauss Periods

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.Comment: 9 page