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

Low complexity normal elements over finite fields of characteristic two

In this paper, we extend previously known results on the complexities of normal elements. Using algorithms that exhaustively test field elements, we are able to provide the distribution of the complexity of normal elements for binary fields with degree extensions up to 39. We also provide current results on the smallest known complexity for the remaining degree extensions up to 512 by using a combination of constructive theorems and known exact values. We give an algorithm to exhaustively search field elements by using Gray codes, which allows us to reuse previous computations. We compare this with a standard method. We analyze this algorithm and show both experimentally and asymptotically t