Tomik Yaghoobian

Applications of finite fields

1 Introduction to Finite Fields and Bases.- 2 Factoring Polynomials over Finite Fields.- 3 Construction of Irreducible Polynomials.- 4 Normal Bases.- 5 Optimal Normal Bases.- 6 The Discrete Logarithm Problem.- 7 Elliptic Curves over Finite Fields.- 8 Elliptic Curve Cryptosystems.- 9 Introduction to Algebraic Geometry.- 10 Codes From Algebraic Geometry.- Appendix - Other Applications.

Hermitian codes as generalized Reed-Solomon codes

Hermitian codes obtained from Hermitian curves are shown to be concatenated generalized Reed-Solomon codes. This interpretation of Hermitian codes is used to investigate their structure. An efficient encoding algorithm is given for Hermitian codes. A new general decoding algorithm is given and applied to Hermitian codes to give a decoding algorithm capable of decoding up to the full error correcting capability of the code.

Two new decoding algorithms for Reed-Solomon codes

The subject of decoding Reed-Solomon codes is considered. By reformulating the Berlekamp and Welch key equation and introducing new versions of this key equation, two new decoding algorithms for Reed-Solomon codes will be presented. The two new decoding algorithms are significant for three reasons. Firstly the new equations and algorithms represent a novel approach to the extensively researched problem of decoding Reed-Solomon codes. Secondly the algorithms have algorithmic and implementation complexity comparable to existing decoding algorithms, and as such present a viable solution for decoding Reed-Solomon codes. Thirdly the new ideas presented suggest a direction for future research. The first algorithm uses the extended Euclidean algorithm and is very efficient for a systolic VLSI implementation. The second decoding algorithm presented is similar in nature to the original decoding algorithm of Peterson except that the syndromes do not need to be computed and the remainders are used directly. It has a regular structure and will be efficient for implementation only for correcting a small number of errors. A systolic design for computing the Lagrange interpolation of a polynomial, which is needed for the first decoding algorithm, is also presented.

The Discrete Logarithm Problem

Let G be a finite cyclic group, and let a be a generator for G. Then

Construction of Irreducible Polynomials

G = \{ {\alpha ^i}|0 \leqslant i\# G\}

Elliptic Curves over Finite Fields

, where #G is the order of G. The discrete logarithm (logarithm) of an element β to the base α in G is an integer x such that α x = β. If x is restricted to the interval 0 ≤ x < #G then the discrete logarithm of β to the base α is unique. We typically write x = log α β.

Codes From Algebraic Geometry

This introductory chapter contains some basic results on bases for finite fields that will be of interest or use throughout the book. The concentration is on the existence of certain types of bases, their duals and their enumeration. There has been considerable activity in this area in the past decade and while many of the questions are resolved, a few of the important ones remain open. The presentation here tries to complement that of Lidl and Niederreiter [21] although there is some unavoidable overlap. For a more extensive treatment of the topics covered in this chapter, we recommend the recent book by D. Jungnickel [15]. For the remainder of this section some basic properties of the trace and norm functions are recalled.

Elliptic Curve Cryptosystems

This chapter is devoted to the problem of constructing irreducible polynomials over a given finite field. Such polynomials are used to implement arithmetic in extension fields and are found in many applications, including coding theory [5], cryptography [13], computer algebra systems [11], multivariate polynomial factorization [21], and parallel polynomial arithmetic [18].