Introduction to finite fields and their applications: Preface

(1992). Matrices: Methods and Applications. Journal of the Operational Research Society: Vol. 43, No. 12, pp. 1185-1185.

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Matrices: Methods and Applications

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

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Hilbert's tenth problem

This paper proposes a color image encryption algorithm based on a cloud model Fibonacci chaotic system, as well as a matrix convolution operation that can protect image content effectively and safely. The algorithm combines the cloud model with the generalized Fibonacci, creating a new complex chaotic system that realizes the dynamic random variation of chaotic sequences. The chaotic sequence is used to scramble the pixel coordinates of the mosaic images of the R, G, and B components of the color image. Then, the chaotic sequence value is used as a matrix convolution cloud algorithm that alternately updates the input value of the matrix convolution operation and the pixel value to obtain the permutation transformation of the original pixel value. Finally, the pixel values of the replacement and cloud model Fibonacci chaotic sequence and the pixel values of the front (rear) adjacent pixel points are subjected to a two-way exclusive XOR operation. Realizing the change of the arbitrary pixel value causes a chain transformation of the pixel values of all of the pixel points, and sequentially generates an encrypted image. Experiments show that the histogram of the encrypted image is smoother and adjacent pixels of the image have low correlation. In addition, this algorithm can resist attack experiments such as differential attack, select plaintext attack and noise attack and provides high encryption security, high anti-interference, and strong robustness. The dynamic chaotic system is used to realize the color image encryption of the dynamic key, and the encryption algorithm has higher security and the validity of the algorithm.

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Color Image Encryption Algorithm Based on Dynamic Chaos and Matrix Convolution

The cyclic groups via the Pascal matrices and the generalized Pascal matrices

In this paper, we define the quaternion-Pell sequence and derive the generating matrix of the sequence. Then we produce the semigroups using the multiplicative orders of the generating matrix of th...

The quaternion-Pell sequence

In this paper, we define the recurrence sequences using the Circulant matrices which are obtained from the characteristic polynomial of the Fibonacci sequence, and then, we give miscellaneous properties of these sequences. In addition, we consider the cyclic groups which are generated by the generating matrices and the auxiliary equations of the defined recurrence sequences, and then, we study the orders of these cyclic groups. Furthermore, we extend the defined sequences to groups. Finally, we obtain the lengths of the periods of the extended sequences in the polyhedral groups (2, 3, 3) and (n, 2, 2) as applications of the results obtained.

The Fibonacci-Circulant Sequences and their Applications

In this article, we define the complex-type k-Fibonacci numbers and then give the relationships between the k-step Fibonacci numbers and the complex-type k-Fibonacci numbers. Also, we obtain miscel...

The complex-type k-Fibonacci sequences and their applications

A k-nacci sequence in a finite group is a sequence of group elements x0,x1,…,xn,… for which, given an initial (seed) set x0,x1,…,xj-1, each element is defined by It is important to note that the Fibonacci length depends on the chosen generating n-tuple for a group. The binary polyhedral groups have been studied recently by C.M. Campbell and P.P. Campbell for their Fibonacci lengths. In this paper, we obtain the period of k-nacci sequences in the binary polyhedral groups 〈2,2,2〉, 〈n,2,2〉, 〈2,n,2〉 and 〈2,2,n〉 for any n > 2.

Ömür Deveci

Papers

The cyclic groups via the Pascal matrices and the generalized Pascal matrices

The quaternion-Pell sequence

The Fibonacci-Circulant Sequences and their Applications

The complex-type k-Fibonacci sequences and their applications

On the k-Nacci Sequences in Finite Binary Polyhedral Groups