Finite Abelian Groups, Elliptic Curves, Blockchain with Hashing and Graphs

Introductory Essentials to Finite Abelian Groups

2025-09-11T09:31:04+00:00

This study deals with the properties of finite abelian groups. We first start by stating the fundamental theorems such as those of Cayley, Lagrange, Poincar´e, and the three isomorphism theorems. Then, we move on to the first and second Sylow theorems. Finally, we discuss some applications of finite Abelian groups, and we include fifty exercises on this topic, some of which belong to the ”Agregation”.

Introductory Essentials to Finite Fields

2025-09-11T09:35:08+00:00

This study starts with questions about the construction of extension, splitting and finite fields. Then, we move to cyclotomic polynomials as we need some of their properties in the final part of our proof to Wedderburn’s theorem. We also discuss the cyclicity of the multiplicative group over a finite field, and then we present results on the factorization of polynomials over a finite field. Finally, the chapter includes 25 exercises on this topic, some of which belong to ”Agregation.

Introductory Essentials to Elliptic Curves

2025-09-11T09:38:35+00:00

This study starts with some generalities on elliptic curves through the definitions of theWeierstrass equation and groups laws. Then, we provide our proof to Hasse’s theorem and we discuss its consequences. Finally, we discuss some applications of elliptic curves and their contributions to signatures, and the chapter introduces seventeen exercises on this topic.

Introductory Essentials to Blockchain, Hashing and Graphs

2025-09-11T09:43:04+00:00

The present study explains the reasons behind decentralization and introduces the basic concepts of blockchain technology with clear references to its key components, such as consensus mechanisms and peer-to-peer networks. An explanation of hashing is treated with different examples of classical algorithms, as this is very important in blockchain signatures. A brief introduction to graphs is also presented since they are used to develop decentralization and counterattacks. Finally, we introduce 16 exercises on these two elements.

Introductory Essentials to Python Programming

2025-09-11T09:48:08+00:00

This study presents the introductory concepts of programming in Python language, from the basics of algorithmics, declarations of variables, syntaxes of loops and functions, to different structures, namely strings, sets, tuples, dictionaries, lists and arrays. Finally, the paper introduce 10 exercises on this topic.

Computational Techniques in Finite Group Theory: Python Implementations and Applications

2025-09-11T09:51:29+00:00

This study introduces algorithmic schemes in Python for computing some basic substructures of finite groups, namely subgroups cosets and generators, as well as some examples of cyclic groups. For this, paper provide 12 examples to motivate the reader to become familiar with programming on this topic.

A Python Programming Initiative for Elliptic Curves over Finite Fields

2025-09-11T09:55:37+00:00

This study introduces algorithmic schemes in Python for computing some concepts in elliptic curves, namely point addition, Cayley’s table, Diffie-Hellman protocol, and Schoof’s problem. For this, we provide 10 examples to motivate the reader to become familiar with programming on this topic.

A Python Programming Initiative for Hash Construction through the Example of SHA-2

2025-09-11T09:59:00+00:00

As opposed to the different methods used in the preprocessing and concatenation steps associated with Secure Hash Algorithm 1 (SHA-1), we treat here the case of Secure Hash Algorithm 2 (SHA-2) that is still unbreakable up to the present. In addition, before its implementation in Python, we first explain the difference between the two approaches.

Advancing Blockchain Security Using Graph Theory: A Python Programming Perspective

2025-09-11T10:02:44+00:00

This study develops codes for visualizing decentralized graphs through consensus methods, namely mining pool, stake pool, wealth and developer concentrations, as well as simulating topological; Domain Name System (DNS), Distributed Denialof- Service (DDoS), phishing, 51% and eclipse attack types.