Finite Fields

I. INTRODUCTION Discrete transforms play a very important role in Engineer-ing. Particularly significant examples are the well known Dis-crete Fourier Transform (DFT) and the Z Transform [1], which have found many applications in several areas, specially in the field of ...

This paper examines finite field trigonometry as a tool to construct digital trigonometric transforms. In particular, by using properties of k-cosine function over a Galois field, the finite field discrete cosine transform is introduced. The finite field DCT pair in GF(p) is defined, having blocklengths that are divisors of (p+1)/2. A special case is the Mersenne finite field DCT, defined when p is Mersenne prime. In this instance block lengths that are power of two are possible and radix-two fast algorithms can be used to compute the transform.

The book is available as a paperback via Amazon:

https://www.amazon.com/gp/product/199998465X/

https://www.amazon.co.jp/gp/product/199998465X/

https://www.amazon.co.uk/gp/product/199998465X/

https://www.amazon.es/gp/product/199998465X/

This book is primarily intended for undergraduate students in Mathematics, pursuing courses that feature abstract algebra. All aspects of abstract algebra that you would expect to encounter in an undergraduate programme of study are covered, including ring theory, group theory and the beginnings of Galois theory. Unusually for an abstract algebra text, five chapters on linear algebra are also included, making the text a self-contained introduction to undergraduate algebra. The book will be of use throughout your undergraduate studies, and beyond. Very little is assumed by way of previous study.  It is assumed only that you are familiar with basic topics in algebra, such as complex numbers, matrices, and solving systems of linear equations. From these beginnings, you will be led step by step into the study of various algebraic structures, including groups, vector spaces and rings, and structure-preserving mappings between such objects such as homomorphisms of groups and linear transformations of vector spaces. Important theorems and results are presented and discussed at each stage, with proofs that will enable you to understand the logical development of the subject. Mathematical proof is an aspect of the subject which many students find challenging, and care has been taken to ensure that proofs are easy to follow with all steps clearly explained. In the last six chapters, you will be introduced to more novel topics that go beyond the typical undergraduate curriculum. These include exterior algebras, matrix groups and their associated Lie algebras, normed real algebras and Clifford algebras. Any of these topics could form the basis for a final year undergraduate dissertation, or provide a gentle introduction to graduate study in algebra. Throughout the text, the emphasis is upon presenting topics in an accessible manner, with worked examples which will help you to build a firm understanding of each new concept as it is introduced. Each chapter concludes with a comprehensive set of exercises designed to test and consolidate your understanding of the ideas presented. Solutions to the exercises are given at the end of the book. The book is based on lectures given by Michael Butler at the University of Bolton in the U.K. between 1994 and 2019. These were attended by students from an exceptionally wide range of backgrounds. The lecture notes that this book grew from were refined in the light of teaching and in response to feedback from students, over many deliveries of the material. Dr Butler has extensive experience of teaching abstract algebra to undergraduates, and his passion for teaching the subject has shaped this book.

CONTENTS:
1 The Group Axioms and Examples
2 Subgroups and Group Homomorphisms
3 Vector Spaces
4 Subspaces and Linear Transformations
5 The Basis for a Vector Space
6 Eigenvalues and Eigenvectors
7 Inner Product Spaces
8 Cosets and Quotient Groups
9 Group Actions
10 Simple Groups
11 Soluble Groups
12 Fields and their Extensions
13 The Galois Group
14 The Ring Axioms and Examples
15 Subrings, Ideals and Ring Homomorphisms
16 Quotient Rings
17 Integral Domains and Fields
18 Finite Fields
19 Factorisation in an Integral Domain
20 Vector Spaces with Products
21 The Exterior Algebra of a Vector Space
22 Lie Algebras
23 Matrix Groups and their Tangent Spaces
24 Normed Real Algebras
25 Tensor Products and Clifford Algebras.

Uma sinopse sobre séries ortogonais generalizadas (e.g. Legendre-Fourier), com ênfase na representação trigonométrica de Fourier, é introduzida. A apresentação envolve o fenômeno de Gibbs, e critérios para a convergência de séries (condições de Dirichlet, teorema de Fourier, teorema de Fejér). Implicações do “Reino de Fourier” na Engenharia Acústica: consonâncias & dissonâncias, instrumentos musicais.  Apresenta-se a teoria dos tapers de Tukey, a janela de Lanczos e o uso da série de Fourier para modelar fractais determinísticos. A passagem para o contínuo conduz à transformada de Fourier, cujas propriedades são revisadas. O princípio da incerteza de Gabor-Heisenberg é conectado à teoria de Fourier. Cálculos computacionais do espectro conduzem à transformada discreta de Fourier (DFT) e mecanismos para o cálculo eficiente do espectro (algoritmos rápidos). Os resultados de Heidman sobre complexidade multiplicativa para a DFT são apresentados. A análise espectral clássica evolui: a análise moderna com base em wavelets funciona ligada à abordagem Fourier. O teorema da amostragem é discutido, com uma demonstração do tipo “viva Fourier”, assim como o teorema 2BT sobre dimensionalidade de sinais. Aplicações modernas na descontaminação de sinais são argumentadas sob enfoque pragmático. A adaptação da análise de Fourier para sinais estocásticos conduz às séries estocásticas de Fourier, e expansões de Kahunen-Loève. Até a modelagem não-linear para sistemas com base em séries de Volterra é apresentada. Por fim, o reino de Fourier conquista definitivamente o mundo finito e digital, migrando para a transformada de Fourier de corpo finito (transformada de Galois-Fourier).

Digital Transforms have important applications on subjects such as channel coding, cryptography and digital signal processing. In this paper, two Fourier Transforms are considered, the discrete time Fourier transform (DTFT) and the finite field Fourier transform (FFFT). A finite field version of the DTFT is introduced and the FFFT is redefined with a complex kernel, which makes it a more appropriate finite field version of the Discrete Fourier Transform. These transforms can handle FIR and IIR filters defined over finite algebraic structures.

Cryptography is the study of methods of sending messages in disguised form so that only the intended recipients can remove the disguise and read the massage. With widespread availability of computer technology, the rapid growth of wireless communications secured exchange of information has become a challenging encryption/detask. The most basic of the modules in modern cryptography is that of a primitive, which may be regarded as a cryptographic building block which performs one or more desired functions, and may be combined with others to form a cryptographic protocol. The most well-known and perhaps the simplest primitive is encryption, which allows parties to achieve confidential data transmission over an insecure channel. In the present paper we proposed a cryptographic technique using elements of a finite field and logical operators basing on inverse property. To test the efficacy of the proposed scheme the cryptanalysis is performed.

In this paper questions of the width in verbal subgroups of $A_n$ is considered.

Verbal width by set of squares of alternating group is found, it is happened equals to 2. The criterion of squareness in $A_n$ is presented.
The necessary and sufficient conditions when an element of alternating group can be presented as a squares of one element are also found by us. Some necessary conditions to an element $g\in A_n$ being the square in $A_n$ are investigated.

RESUMO: Neste capitulo, a matriz de transformação da Transformada Numérica de Pascal (TNP) é investigada e novas relações baseadas na decomposição desta matriz, por meio do produto de Kronecker de duas matrizes de Pascal, são propostas com aplicações na implementação da TNP.
PALAVRAS –CHAVE: Transformada Numérica de Pascal, Triângulo de Pascal Modular, Corpos Finitos, Produto de Kronecker.
--------------------------------
ABSTRACT: In this chapter, the Pascal Number Theoretic Transform (PNTT) matrix is investigated and new relationships based on the decomposition of this matrix, by means of the Kronecker Product of two Pascal matrices,
are proposed with applications on the PNTT implementation.
KEYWORDS: Pascal Number-Theoretic Transform, Modular Pascal Triangle, Finite Fields, Kronecker Product.

This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also studied in detail by H.M. Cundy and C.F. Parry recently. The main task of the article was to develop an algorithm for creating curves, which pass through triangle centers. During the research, it was noticed that some different triangle centers in distinct triangles coincide. The simplest example: an incenter in a base triangle is an orthocenter in an excentral triangle. This was the key for creating an algorithm. Indeed, we can match points belonging to one curve (base curve) with other points of another triangle. Therefore, we get a new intersting geometrical object. During the research were derived number of new triangle conics and cubics, were considered their properties in Euclidian space. In addition, was discussed corollaries of the obtained theorems in projective geometry, what proves that all of the descovered results could be transfered to the projeticve plane.

In this paper we obtained the formula for the number of irreducible polyno-mials with degree n over finite fields of characteristic two with given trace and subtrace. This formula is a generalization of the result of Cattell et al. (2003) [2].

Методическое пособие по линейной алгебре для студентов специальностей компютерные науки,  системный анализ и дискретная математка. В книге кратнко изложены все разделы линейной алгебры и основы аналитической геометрии. Все темы обеспечены вариантами для проведения практических занятий и самостоятельных работ.

methodical manual on linear algebra for students of specialties computer science, systems analysis and discrete mathematics. The book summarizes all sections of linear algebra and the basics of analytic geometry. All topics are provided with options for practical training and control works.

Substitution boxes with strong cryptographic properties are commonly used in block ciphers to provide the crucial property of nonlinearity. This is important to resist standard attacks such as linear and differential cryptanalysis. A cryptographically-strong s-box must have high nonlinearity, low differential uniformity and high algebraic degree. In this paper, we improve previous s-box construction based on binomial operation on two power functions over the finite field F_(2^8 ). By widening the scope of the power function and introducing new manipulation techniques, we managed to obtain cryptographically-strong s-boxes which are better than the previous construction.

A new approach to multiple-access based on finite field transforms is investigated. These schemes, termed Galois-Division Multiple Access (GDMA), offer compact bandwidth requirements. A new digital transform, the Finite Field Hartley Transform (FFHT) requires dealing with fields of characteristic p (p different to 2). A binary-to-p-ary mapping based on the opportunistic secondary channel is introduced. This allows the use of GDMA in conjunction with available digital systems. The performance of GDMA is also evaluated.

A new class of non-binary error-correcting codes, called Hartley codes, is introduced. The code parity-check matrix, dimension and an upper bound on its minimum distance are obtained from the eigenstructure of the Hartley number theoretic transform. A subclass of codes with rate 1/2, namely the balanced Hartley codes, is presented.

"A new class of non-binary error-correcting codes, called Hartley codes, is introduced. The code parity-check matrix, dimension and an upper bound on its minimum distance are obtained from the eigenstructure of the Hartley number theoretic transform. A subclass of codes with rate 1/2, namely the balanced Hartley codes, is presented.

"

Скуратовького Руслана, всі права захищено ліцензією CC BY: Creative Commons
Attribution.
2
Передмова
Навчальний посібник «Вища математика»
створено автором на основі лекцій, які читав автор для студентів різних університетів зі спеціальностей «Комп’ютерні науки, системний аналіз, прикладна математика і інформаційні системи і технології» протягом багатьох років, а також матеріалів однойменного дистанційного курсу за напрямами 7.0913 (122, 113) «Компютерні науки та обчислювальна техніка», «Прикладна математика».
Цей посібник є складовою навчально-методичного комплексу, який міс-
тить навчальний посібник, практикум, збірник типових розрахункових ро
біт, збірник тестових завдань.
Посібник охоплює матеріал з вищої математики відповідно до програми підготовки бакалаврів інженерно-технічних спеці-
альностей «ІКІТ» за темами: матриці та визначники; системи ліній-
них алгебричних рівнянь; векторна алгебра та лінійні простори; комплекс
ні числа; геометрія прямої і площини; криві та поверхні 2-го порядку.
Посібник містить ретельно відібрані і систематизовані теоретичні
поняття, широкий набір практичних навчальних прикладів, задач для са-
мостійного розв’язання, рисунків, схем тощо.
Застосування відповідних математичних понять окремо проілюстро-
вано на широкому спектрі прикладів економічного та технічного змісту.
Матеріал посібника викладено на двох рівнях: базовому і розширено-
му. Базовий рівень містить означення, формулювання теорем, коментарі до них; методи розв’язання задач, проілюстровані прикладами; застосування математичних понять і методів. Розширений рівень доповнює базовий уточненими формулюваннями, додатковими фактами, доведеннями теорем,
Cкуратовський Руслан “Вища математика” для спец. комп’ютерні
науки і прикладна математика
Автор Руслан Скуратовський 23.07.21р.
Підручник Скуратовького Руслана, всі права захищено ліцензією CC BY: Creative Commons
Attribution.
3
які наведені у тексті дрібним шрифтом і не є обов’язковими. Крім того, посібник містить:
— тест для самоконтролю;
— перелік екзаменаційних питань;
— історичні відомості;
— список використаної і рекомендованої літератури.
Сподіваємось, що цей навчальний посібник буде корисним як для
студентів очної, заочної та дистанційної форм навчання НАУ,
так і для студентів інших навчальних закладів природничого та педагогічного напрямів.