Finite fields play a fundamental role in some of the most fascinating applications of modern algebra to the real world. These applications occur in the general area of data communication, a vital concern in our information society. Technological breakthroughs like space and satellite communications and mundane matters like guarding the privacy of information in data banks all depend in one way or another on the use of finite fields. Because of the importance of these applications to communication and information theory, we will present them in greater detail in the following chapters. Chapter 8 discusses applications of finite fields to coding theory, the science of reliable transmission of messages, and Chapter 9 deals with applications to cryptology, the art of enciphering and deciphering secret messages. This chapter is devoted to applications of finite fields within mathematics. These applications are indeed numerous, so we can only offer a selection of possible topics. Section 1 contains some results on the use of finite fields in affine and projective geometry and illustrates in particular their role in the construction of projective planes with a finite number of points and lines. Section 2 on combinatorics demonstrates the variety of applications of finite fields to this subject and points out their usefulness in problems of design of statistical experiments. In Section 3 we give the definition of a linear modular system and show how finite fields are involved in this theory.

Introduction to finite fields and their applications: Theoretical Applications of Finite Fields

Introduction to finite fields and their applications: Factorization of Polynomials

The packing problems of various types have long been of interest in mathematics. Perhaps the most investigated problem is that of packing spheres into n-dimensional Euclidean space. Recently, algebraic codes over finite fields have been used to construct few particularly dense lattice packings in E n . This chapter discusses the problem related to the packing spherical caps on the unit sphere in E n . It highlights the connection between this problem and the equal-energy signaling problem of communications. As the dimensionality of the problem increases and the number of caps packed onto the sphere increases, it becomes important to have a theory that can be used to predict the essential properties of the packing without actually constructing it. This is the role that group representation theory plays. The situation is somewhat similar to that of algebraic coding theory, where the techniques are provided to describe codes with a lower bound on the minimum distance without actually constructing the code. Unfortunately, the development of the application of group representation theory to constructing group codes is still in its infancy.

Group Codes for the Gaussian Channel

The authors review the work of G.D. Forney, Jr., on the algebraic structure of convolutional encoders upon which some new results regarding minimal convolutional encoders rest. An example is given of a basic convolutional encoding matrix whose number of abstract states is minimal over all equivalent encoding matrices. However, this encoding matrix can be realized with a minimal number of memory elements neither in controller canonical form nor in observer canonical form. Thus, this encoding matrix is not minimal according to Forney's definition of a minimal encoder. To resolve this difficulty, the following three minimality criteria are introduced: minimal-basic encoding matrix, minimal encoding matrix, and minimal encoder. It is shown that all minimal-basic encoding matrices are minimal and that there exist minimal encoding matrices that are not minimal-basic. Several equivalent conditions are given for an encoding matrix to be minimal. It is proven that the constraint lengths of two equivalent minimal-basic encoding matrices are equal one by one up to a rearrangement. All results are proven using only elementary linear algebra. >

A linear algebra approach to minimal convolutional encoders

In the paper we study lower bounds on the number of bent functions
that can be obtained by iterative constructions, namely by the
construction proposed by A. Canteaut and P. Charpin in 2003. The
number of bent iterative functions is expressed in terms of sizes of
finite sets and it is shown that evaluation of this number is
closely connected to the problem of decomposing Boolean function
into sum of two bent functions. A new lower bound for the number of
bent iterative functions that is supposed to be asymptotically tight
is given. Applying Monte-Carlo methods the number of bent iterative
functions in $8$ variables is counted. Based on the performed
calculations several hypotheses on the asymptotic value of the
number of all bent functions are formulated.

On the number of bent functions from iterative constructions: lower bounds and hypotheses

In cryptography, the property of randomness in pseudo-random generators is very important to avoid any pattern in output sequences, to provide security against attacks, privacy and anonymity. In this article, the randomness of the family of sequences obtained from the generalized self-shrinking generator is analyzed. Moreover, the characteristics, generalities and relationship between the t-modified self-shrinking generator and the generalized self-shrinking generator are presented. We find that the t-modified self-shrunken sequences can be generated from a generalized self-shrinking generator. Then, an in-depth analysis of randomness focused on the generalized sequences by means of complete and powerful batteries of statistical tests and graphical tools is done, providing a useful vision of the behaviour of these sequences and proving that they are suitable to be used in cryptography.

/pdf/randomness-analysis-for-the-generalized-self-shrinking-ornb5cuerf.pdf

Randomness Analysis for the Generalized Self-Shrinking Sequences

The work of the first and the second authors was partially supported by Spanish grant MTM2011-24858 of the Ministerio de Economia y Competitividad of the Gobierno de Espana. The work of first author was also partially supported by a grant for research students from the Generalitat Valenciana with reference BFPI/2008/138. The work of the third author was partially supported by the research project UMH-Bancaja with reference IPZS01.

https://rua.ua.es/dspace/bitstream/10045/41222/1/MDSArrayCodes-v5f.pdf

A construction of MDS array codes

In this paper we establish some properties of the algebraic normal form of a Boolean function from its support and we propose a method to determine the degree of the Boolean function from its support.

Computing the degree of a Boolean function from its support

In this paper we present a method to construct iteratively new bent functions of $n + 2$ variables from bent functions of $n$ variables using minterms of $n$ variables and minterms of two variables. Also, we provide the number of bent functions of $n+2$ variables that we can obtain with the method here presented.

On the construction of bent functions of $n+2$ variables from bent functions of $n$ variables

We present a method to iteratively construct new bent functions of variables from a bent function of variables and its cyclic shift permutations using minterms of variables and minterms of 2 variables. In addition, we provide the number of bent functions of variables that we can obtain by applying the method here presented, and finally we compare this method with a previous one introduced by us in 2008 and with the Rothaus and Maiorana-McFarland constructions.

/pdf/a-construction-of-bent-functions-of-n-2-variables-from-a-1ph5xjmlah.pdf

A Construction of Bent Functions of n + 2 Variables from a Bent Function of n Variables and Its Cyclic Shifts

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