Low complexity normal bases for F2mn

AbstractIf C(r) denotes the minimum complexity of a normal basis for F2r, we show that if m > 1, n > 1 are two relatively prime integers, then F2mn has a normal basis of complexity C(m)C(n). Such a normal basis leads to a Massey—Omura multiplier for F2mn which uses C(m)C(n) XOR gates and C(m)C(n) + 1 AND gates per dimension

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

The Gaussian normal basis and its trace basis over finite fields

AbstractIt is well known that normal bases are useful for implementations of finite fields in various applications including coding theory, cryptography, signal processing, and so on. In particular, optimal normal bases are desirable. When no optimal normal basis exists, it is useful to have normal bases with low complexity. In this paper, we study the type k(⩾1) Gaussian normal basis N of the finite field extension Fqn/Fq, which is a classical normal basis with low complexity. By studying the multiplication table of N, we obtain the dual basis of N and the trace basis of N via arbitrary medium subfields Fqm/Fq with m|n and 1⩽m⩽n. And then we determine all self-dual Gaussian normal bases. As an application, we obtain the precise multiplication table and the complexity of the type 2 Gaussian normal basis and then determine all optimal type 2 Gaussian normal bases

Abelian Groups, Gauss Periods, and Normal Bases

AbstractA result on finite abelian groups is first proved and then used to solve problems in finite fields. Particularly, all finite fields that have normal bases generated by general Gauss periods are characterized and it is shown how to find normal bases of low complexity

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

Fast Encoding and Decoding of Gabidulin Codes

Gabidulin codes are the rank-metric analogs of Reed-Solomon codes and have a major role in practical error control for network coding. This paper presents new encoding and decoding algorithms for Gabidulin codes based on low-complexity normal bases. In addition, a new decoding algorithm is proposed based on a transform-domain approach. Together, these represent the fastest known algorithms for encoding and decoding Gabidulin codes.Comment: 5 pages, 1 figure, to be published at ISIT 200

Elliptic periods for finite fields

We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Basis in the second family, the so-called normal elliptic basis are normal basis and allow fast (quasi linear) arithmetic. We prove that all extensions admit models of this kind