Computational challenges and solutions: Prime number generation for enhanced data security.

This review paper explores the critical role of number theory in shaping the foundations and advancements of modern cryptography and cybersecurity. As digital communication and data security become increasingly vital, the mathematical intricacies of number theory—such as prime number generation, modular arithmetic, and elliptic curves—form the backbone of secure encryption protocols like RSA, Diffie-Hellman, and Elliptic Curve Cryptography (ECC). The study employs a structured review methodology, drawing upon a wide range of scholarly publications and verified data up to 2014, including both global and Indian contexts. Key mathematical equations, comparative algorithmic performance, and numerical analyses have been integrated to assess the computational hardness and practical effectiveness of number theory-based cryptosystems. Results indicate that while traditional cryptographic schemes remain effective under classical computation, they face significant vulnerabilities in the face of emerging quantum technologies. The Indian mathematical tradition and its contribution to algorithmic logic further enrich the narrative, underscoring the need for indigenous innovation. This paper identifies major research gaps, particularly in the areas of post-quantum cryptography, algorithmic efficiency for low-resource environments, and provable security models. Future directions suggest hybrid mathematical frameworks and deeper exploration of algebraic structures to build resilient and scalable cryptographic systems. Overall, the paper establishes that number theory is not only a theoretical pursuit but a practical necessity in safeguarding digital infrastructure in an increasingly interconnected world.

Public-key algorithms need prime numbers, any reasonably sized network needs lots of them. Before discussing the mathematics of prime number generation in the article the generation of the large random prime numbers is represented. A new algorithm of generating large prime numbers is also presented.

Generating large prime number is a time consuming problem. It is useful in key generation in network security. The problem of generating prime number is easy to parallelize. Use of high performance computing (HPC) facilities can be applied for getting faster results. Embarrassing parallel pattern is applied to categories of algorithms where concurrency is explicit. We have conducted experiments on sequential and parallel approaches of prime numbers generation. In this paper, our objective is to analyze time taken in generating large prime numbers using multicore cluster. The result could be reusable in estimating time required for key generation in cryptography.

Preface. Notation. Chapter 1 Fundamentals. 1.0 Introduction. 1.1 A Famous Sequence of Numbers. 1.2 The Euclidean ALgorithm. The Oldest Algorithm. Reversing the Euclidean Algorithm. The Extended GCD Algorithm. The Fundamental Theorem of Arithmetic. Two Applications. 1.3 Modular Arithmetic. 1.4 Fast Powers. A Fast Alforithm for ExponentiationPowers of Matrices, Big-O Notation. Chapter 2 Congruences, Equations, and Powers. 2.0 Introduction. 2.1 Solving Linear Congruences. Linear Diophantine Equations in Two Variables. The Conductor. An Importatnt Quadratic Congruence. 2.2 The Chinese Remainder Theorem. 2.3 PowerMod Patterns. Fermat's Little Theorem. More Patterns in Powers. 2.4 Pseudoprimes. Using the Pseudoprime Test. Chapter 3 Euler's Function. 3.0 Introduction. 3.1 Euler's Function. 3.2 Perfect Numbers and Their Relatives. The Sum of Divisors Function. Perfect Numbers. Amicalbe, Abundant, and Deficient Numbers. 3.3 Euler's Theorem. 3.4 Primitive Roots for Primes. The order of an Integer. Primes Have PRimitive roots. Repeating Decimals. 3.5 Primitive Roots for COmposites. 3.6 The Universal Exponent. Universal Exponents. Power Towers. The Form of Carmichael Numbers. Chapter 4 Prime Numbers. 4.0 Introduction. 4.1 The Number of Primes. We'll Never Run Out of Primes. The Sieve of Eratosthenes. Chebyshev's Theorem and Bertrand's Postulate. 4.2 Prime Testing and Certification. Strong Pseudoprimes. Industrial-Grade Primes. Prime Certification Via Primitive Roots. An Improvement. Pratt Certificates. 4.3 Refinements and Other Directions. Other PRimality Tests. Strong Liars are Scarce. Finding the nth Prime. 4.4 A Doszen Prime Mysteries. Chapter 5 Some Applications. 5.0 Introduction. 5.1 Coding Secrets. Tossing a Coin into a Well. The RSA Cryptosystem. Digital Signatures. 5.2 The Yao Millionaire Problem. 5.3 Check Digits. Basic Check Digit Schemes. A Perfect Check Digit Method. Beyond Perfection: Correcting Errors. 5.4 Factoring Algorithms. Trial Division. Fermat's Algorithm. Pollard Rho. Pollard p-1. The Current Scene. Chapter 6 Quadratic Residues. 6.0 Introduction. 6.1 Pepin's Test. Quadratic Residues. Pepin's Test. Primes Congruent to 1 (Mod. 6.2 Proof of Quadratic Reciprocity. Gauss's Lemma. Proof of Quadratic Recipocity. Jacobi's Extension. An Application to Factoring. 6.3 Quadratic Equations. Chapter 7 Continuec Faction. 7.0 Introduction. 7.1 FInite COntinued Fractions. 7.2 Infinite Continued Fractions. 7.3 Periodic Continued Fractions. 7.4 Pell's Equation. 7.5 Archimedes and the Sun God's Cattle. Wurm's Version: Using Rectangular Bulls. The Real Cattle Problem. 7.6 Factoring via Continued Fractions. Chapter 8 Prime Testing with Lucas Sequences. 8.0 Introduction. 8.1 Divisibility Properties of Lucas Sequencese. 8.2 Prime Tests Using Lucas Sequencesse. Lucas Certification. The Lucas-Lehmer Algorithm Explained. Luca Pseudoprimes. Strong Quadratic Pseudoprimes. Primality Testing's Holy Grail. Chapter 9 Prime Imaginaries and Imaginary Primes. 9.0 Introduction. 9.1 Sums of Two Squares. 9.2 The Gaussian Intergers. Complex Number Theory. Gaussian Primes. The Moat Problem. The Gaussian Zoo. 9.3 Higher Reciprocity 325. Appendix A. Maathematica Basics. 1.0 Introduction. A.1 Plotting. A.2 Typesetting. Sending Files By E-Mail. A.3 Types of Functions. A.4 Lists. A.5 Programs. A.6 Solving Equations. A.7 Symbolic Algebra. Appendix B Lucas Certificates Exist. References. Index of Mathematica Objects. Subject Index.

People are used prime numbers in secret codes to secure the message and data. Prime numbers are essential in computer security problem to perform certain operations over the Internet. It can be used for encryption. The larger the prime number used, the intricate it is to decipher the code. So finding the “massive” prime number is a current problem. One of the most widely used applications of prime number in computing is the RSA encryption system. A prime sieve is a fast type of algorithm for finding prime numbers. Sieve of Eratosthenes, Sieve of Sundaram and Sieve of Atkins are the algorithms that can be used to generate prime numbers. This research paper is to study and testing to find out, which algorithm is better used for large prime numbers in terms of time complexity. The prime sieve algorithm is coded in java language and the results are summarized.

Public-key cryptosystems base their security on well-known number-theoretic problems, such as factorisation of a given number n . Hence, prime number generation is an absolute requirement. Many prime number generation techniques have been proposed up-to-date, which differ mainly in terms of complexity, certainty and speed. Pocklington's theorem, if implemented, can guarantee the generation of a true prime. The proposed implementation exhibits low complexity at the expense of long execution time.

Given an integer N, what is the computational complexity of finding all the primes less than N? A modified sieve of Eratosthenes using doubly linked lists yields an algorithm of O A (N) arithmetic complexity. This upper bound is shown to be equivalent to the theoretical lower bound for sieve methods without preprocessing. Use of preprocessing techniques involving space-time and additive-multiplicative tradeoffs reduces this upper bound to O A (N/log logN) and the bit complexity to O B (N logN log log logN). A storage requirement is described using O B (N logN/log logN) bits as well.

A prime number is a natural number that is not divisible without remainder by any other natural number but 1 and itself. Primes are scattered irregularly among the set of natural numbers, and this fact has fascinated mathematicians throughout the centuries. Today, primes are not only a challenge for mathematicians, but are of very practical value – for instance, they play a central role in cryptography. In this chapter the authors explain a very old, but still practical algorithm for computing a table with all prime numbers up to a specified number.

The algebraic structure and distribution of prime numbers remain two of the most fundamental problems in mathematics. The Fundamental Theorem of Arithmetic, proved by Euclid, and Goldbach’s conjecture, while universal in scope with respect to how numbers can be represented multiplicatively or additively, do not provide insights into the structure of primes. Similarly, the definition of a prime −as a number divisible only by 1 and itself− or a sieve algorithm, commonly used to generate primes by successively eliminating multiples, offer no insight into the structure of primes. The powerful and persistent consideration of prime numbers as universal “arithmetic quanta” has not necessitated an equally powerful need for parallel research into a deeper and possibly more insightful explanation of primeness, that is, a better understanding of “why” a number is prime. In this paper, prime and coprime numbers are represented and generated by algebraic expressions. Specifically, given the first <i>n</i> primes, <i>p<sub>1</sub></i>,<i> p<sub>2</sub></i>,…, <i>p<sub>n</sub></i>, sufficient conditions are given for expressing primes greater than <i>p<sub>n</sub></i>, and coprimes with prime factors greater than <i>p<sub>n</sub></i>, as algebraic functions of <i>p<sub>1</sub></i>,<i> p<sub>2</sub></i>,…, <i>p<sub>n</sub></i>. Thus, primality and co-primality are shown to be mathematical properties with inherently evolutionary algebraic characteristics, since larger primes and coprimes can be generated algebraically from smaller ones. The methodology described in the paper can be a useful tool in the study and analysis of the complexity, structure, interrelationships and distribution of primes and coprimes.

An Optimized Intelligent Computational Security Model for Interconnected Blockchain-IoT System & Cities

This paper presents the use of a spreadsheet in generating prime numbers through the Sieve of Eratosthenes. The approach is applied to visualizing frequency and irregularity in the occurrence of prime numbers, distribution of the primes gaps, and the behaviour of the prime counting function π(x).It is argued that these and other aspects of the theory of numbers are more easily represented on a spreadsheet than in a programming language. The spreadsheet used is Excel 4.0 for Macintosh computers.

The massive deployment of unmanned aerial vehicles (UAVs) is growing in various fields in wireless network of 5G and 6G. It supports huge deployment of UAVs that offer secure, reliable, and low-cost data communications. They are widely deployed in surveillance and military applications for collecting sensitive and critical data, transferring the collected data to the data center, procuring high-definition images of disaster areas, surveying natural disaster zones, and other critical missions. Although, the UAV applications are growing, the risk and security factors associated with it are increasing parallelly. So, a need of secure and reliable network communication system for UAV is mandatory. In this chapter, the performance of UAV network system in 6G environment and the technological challenges faced with respect to security and privacy are discussed. Then a possible solution using blockchain and security policies that can be provided for a secure communication in a 6G network is presented.

Hybrid precoding design for secure smart-grid enabled MIMO wireless communications in Industry 5.0

Ensuring reliability and security has been a challenge in modern communication systems. To achieve these challenges, a novel reliable and secure communication system is designed in this paper. Reliability is achieved by constructing a class of error correcting codes called concatenated kernel codes. Security in terms of source authentication is achieved from using graph nature of trellis employing techniques from graph theory namely fundamental cut-set and fundamental circuit. It has been shown that the proposed communication framework achieves the goal of reliability and security considering the channel noise and cryptanalytic attacks. The theoretical basis of the proposed framework is validated and its performance is evaluated through simulations.

Elementary Methods in Number Theory begins with first course in number for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary number theory. In the second and third parts of the book, deep results in number theory are proved using only elementary methods. Part II is about multiplicative number theory, and includes two of the most famous results in mathematics: the Erdos-Selberg elementary proof of the prime number theorem, and Dirichlets theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive number theory: Warings problems for polynomials, Liouvilles method to determine the number of representations of an integer as the sum of an even number of squares, and the asymptotics of partition functions. Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets.