Prime Research Articles

Multiscale Sieve for Smart Prime Generation and Application in Info-Security, IoT and Blockchain

The huge computational cost required to test whether a number is prime and the inefficiency of the known sieving algorithms for extremely large inputs have posed significant challenges in computational number theory. Traditional deterministic prime generation methods struggle to maintain performance when the input sizes increase exponentially. In this work, we show that, through multiscale distribution and deterministic prime number generation, it is possible to create a multiscale sieve with drastically better performance than the deterministic algorithms known to date, providing a more efficient solution for large-scale prime number generation, demonstrated by several benchmarks that highlight the potential of our approach. Consequently, we can gain some advantages in cryptography and in info-security, such as in IoT and blockchain environments.

On positive sequences of reals whose block sequence has an asymptotic distribution function

In this paper we study the properties of the unbounded sequence $0 < y_1 \le y_2 \le y_3 \le \cdots$ of positive reals having asymptotic distribution function of the form $x^\lambda$. As a consequence, we immediately get information on the asymptotic behavior of the power means of order $r>0$ of function values of some arithmetic functions, e.g., the first $n$ prime numbers or the values of the prime counting function.

An Enhanced RSA Algorithm to Counter Repetitive Ciphertext Threats Empowering User-centric Security

The technology-driven modern age emphasizes the security and privacy of communication. Through this paper, we delve deep into the need for user-centric security within cloud-based environments. The need for enhancement in encryption arises due to the increasing cases of data breaches and insider threats in cloud-based environments recently. The focus is laid upon the use of RSA encryption, end-to-end encryption, and anonymous messaging to address security-based concerns. The primary focus of this research is to develop a comprehensive security system to ensure the confidentiality and authenticity of text-based messages shared in the cloud. The proposed improved RSA algorithm, as suggested, incorporates three prime numbers in the key generation process. To address repetitive ciphertext, the proposed algorithm involves adding the index of each character in the plaintext string to the character’s integer value before encryption. Conversely, during decryption, the same index is subtracted. This proposed algorithm has been utilized in a practical scenario, specifically in the implementation of a chat application. This paper presents a proof-of-concept for the proposed enhanced version of the RSA algorithm, accompanied by a thorough comparison and analysis of computational times across various bit lengths. Increase in data security at a cost of minor increase in computation time was observed through this research.

Двоїстий алгоритм пошуку простих чисел на відрізках великих розмірностей

A matrix model of subsequence of natural numbers with a multiplicative basis from the first prime numbers is proposed. The matrix model is a square matrix, where vector columns are arithmetic progressions with difference and number of progressions equal to product of base elements. By crossing out arithmetic progressions with first terms which are multiples of base elements, we obtain a symmetric sparse matrix that contains all the prime numbers of subsequence of natural numbers, except for the base ones, which increases the density of prime numbers in sparse matrices. Sparse matrices are not formed clearly, only the vector of first terms of arithmetic progressions is formed. Properties of sparse matrices have been proven. Formulas that speed up the calculation of compound numbers in arithmetic progressions have been derived, the structures of vector elements of first terms of arithmetic progressions have been determined, the connectivity of symmetric parts of sparse matrices has been investigated. With the expansion of the base the number of pairs of elements with the difference equal to the power of two («twins», «fours», etc.) increases in the vector of first terms. This is a necessary condition for the existence of constants for which linear equations of two variables can have an infinite set of solutions in prime numbers. The irregularity of distribution of prime numbers in subsequences of natural numbers is related to the structure of elements of the vector of first terms. An algorithm for finding prime numbers on segments of large dimensions with a parallel calculation process has been built. The proposed algorithm is binary to algorithms for sifting subsequences of natural numbers by prime divisors. In these algorithms, it is not possible to parallelize the calculation process, since screening procedure requires storing the numerical information from the preceding steps (vector model of array processing). The binary algorithm calculates compound numbers in each pair of arithmetic progressions with symmetric first terms simultaneously, using only vector of the first terms of arithmetic progressions, which makes possible processing of large-dimensional arrays

Small‐scale distribution of linear patterns of primes

AbstractLet be a system of linear forms with finite complexity. In their seminal paper, Green and Tao showed the following prime number theorem for values of the system : where are the corresponding local densities. In this paper, we demonstrate limits to equidistribution of these primes on small scales; we show the analog to Maier's result on primes in short intervals. In particular, we show that for all , there exist such that for sufficiently large, there exist boxes of sidelengths at least such that

A New Cryptographic Key Planning Algorithm Based on Blum Blum Shub

The Blum Blum Shub (BBS) algorithm is one of the known powerful pseudo random number generators. This algorithm can be used for key generation. BBS is basically based on the product of two large prime numbers and a seed value. The selection of these values is a critical issue. In this study, a new approach is proposed to overcome this problem. In the proposed approach, a prime number pool is first created. At this point, the user sets a start and end value. The primes in this range are generated and stored in an array. Then, two primes are randomly selected from this prime number pool with chaotic maps. The positions of these prime numbers in the array are recorded. The seed value is taken as the sum of the positions of these two primes. In other words, the parameters to be selected will be randomly selected in the ranges that the user will enter at that moment. In this study, two random bit sequences were obtained in this way. These sequences are 1 million bits long. NIST SP 800-22 tests were applied to these sequences and the sequences successfully completed all tests.

No Sum (NS) Sequence: A Tool for Quantum-Safe Cryptography

Quantum computing is emerging as a promising technology that can solve the world’s most complex and computationally intensive problems, including cryptography. Conventional cryptography is now facing a severe threat because of the speed of computation offered by quantum computers. There is a need for the standardization of novel quantum-resistant public-key cryptographic algorithms. In this article, the No-Sum (NS) sequence-based public key cryptosystem is proposed, increasing the computational resource requirement against the quantum computer-assisted bruteforce attack. In this paper, we demonstrate an elevation of security of the public key cryptosystem with NS sequence. The proposed methodology works in two phases; in the first phase, the conventional RSA algorithm is used to share the first element and offset parameter between the receiver and sender. In the second phase, prime numbers are generated from the NS sequence and used for encryption/ decryption of messages/ciphers. This paper illustrates the potential of the NS sequence in developing a quantum-resistant cryptosystem, contributing to the advancement of secure communication in the quantum computing era.

Integer partitions detect the primes

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer n ≥ 2 is prime if and only if [Formula: see text]where the [Formula: see text] are MacMahon's well-studied partition functions. More generally, for MacMahonesque partition functions [Formula: see text] we prove that there are infinitely many such prime detecting equations with constant coefficients, such as [Formula: see text].

Varieties over Q¯$\overline{\mathbb {Q}}$ with infinite Chow groups modulo almost all primes

AbstractLet be the Fermat cubic curve over . In 2002, Schoen proved that the group is infinite for all primes . We show that is infinite for all prime numbers . This gives the first example of a smooth projective variety over such that is infinite for all but at most finitely many primes . A key tool is a recent theorem of Farb–Kisin–Wolfson, whose proof uses the prismatic cohomology of Bhatt–Scholze.

The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves

Abstract We show that if F is $\mathbb{Q}$ or a multiquadratic number field, $p\in\left\{{2,3,5}\right\}$ , and $K/F$ is a Galois extension of degree a power of p, then for elliptic curves $E/\mathbb{Q}$ ordered by height, the average dimension of the p-Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K-rank of elliptic curves $E/\mathbb{Q}$ for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F. The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p, as $E/\mathbb{Q}$ varies, the difference in dimension between the Galois fixed space in the p-Selmer group of $E/K$ and the p-Selmer group of $E/F$ has bounded average.

DNA and its Mosaic Structure

We consider that the DNA is an information processing system, receiving, registering and transferring information. In DNA we encounter four different nucleotides, which may be considered as the “letters” of an alphabet. Each distinct DNA base may be represented equally well by a specific number and a DNA strand by a unique Gödel number G, where G is a product of prime numbers raised to appropriate powers. We studied the statistical distribution of g, the logarithm of G, in the case of equal probability for the four DNA bases. In the present work we extend our analysis in two different directions We consider unequal probabilities for the DNA bases We raise the issue of scaling. For relatively small DNA strands we use the prime number theorem and we obtain a Poisson distribution for the average g. The Poisson distribution depends upon a parameter λ, which connects together the DNA pattern and the prime number density. Different values for λ lead to a mosaic structure of DNA. Our work indicates a strong link between DNA structure, its representation by Gödel’s numbers and the distribution of the prime numbers.

Multicellular artificial neural network-type architectures demonstrate computational problem solving.

Here, we report a modular multicellular system created by mixing and matching discrete engineered bacterial cells. This system can be designed to solve multiple computational decision problems. The modular system is based on a set of engineered bacteria that are modeled as an 'artificial neurosynapse' that, in a coculture, formed a single-layer artificial neural network-type architecture that can perform computational tasks. As a demonstration, we constructed devices that function as a full subtractor and a full adder. The system is also capable of solving problems such as determining if a number between 0 and 9 is a prime number and if a letter between A and L is a vowel. Finally, we built a system that determines the maximum number of pieces of a pie that can be made for a given number of straight cuts. This work may have importance in biocomputer technology development and multicellular synthetic biology.

An Approach for Cipher Communication Utilizing the Concepts of Elliptic Curve Cryptography

Message encryption is a process of protection of the original data in transmission. Several cryptosystems were designed in the history for different environments. Public key or Asymmetric cryptosystem has become more popular in recent era where the key space consists both public and private keys. Elliptic curve cryptography dominates public key cryptosystems like Diffie Hellman key exchange and RSA. ECC is more powerful and alternative technique of RSA, with a simple replacement of exponent calculations of large prime numbers to mathematical representation of Elliptic curve points. Besides the computation ECC works with shorter key length than RSA. The present paper describes an encryption algorithm using elliptic curves over finite fields. Here a group of authentic users transmit.

الرسوم البيانية الموجهة لمصفوفات المرافق لفروبنيوس ذات البعد 2

In this work, we investigate Frobenius companion matrices with entries from the ring of integers modulo a prime number. We introduce a mapping 𝝍 to construct a directed graph, where vertices represent these matrices and edges are defined by 𝝍. Our analysis focuses on the structure and properties of this directed graph, including vertex degrees,cycles, and connected components, in relation to the eigenvalues of the matrices،

Minimal zero entropy subshifts can be unrestricted along any sparse set

Abstract We present a streamlined proof of a result essentially presented by the author in [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28(4) (2008), 1291–1322], namely that for every set $S = \{s_1, s_2, \ldots \} \subset \mathbb {N}$ of zero Banach density and finite set A, there exists a minimal zero-entropy subshift $(X, \sigma )$ so that for every sequence $u \in A^{\mathbb {Z}}$ , there is $x_u \in X$ with $x_u(s_n) = u(n)$ for all $n \in \mathbb {N}$ . Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the polynomial Sarnak conjecture reported by Eisner [A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris353(7) (2015), 569–572] which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł [Prime number theorem for analytic skew products. Ann. of Math. (2)199 (2024), 591–705] and by Lian and Shi [A counter-example for polynomial version of Sarnak’s conjecture. Adv. Math.384 (2021), Paper no. 107765] and shows that no similar result can hold under only the assumptions of minimality and zero entropy.

Distribution of the prime numbers

Distribution of the prime numbers

A class of constacyclic codes of length 2pˢ over Fp^m[u,v] / ⟨u², v², uv − vu⟩

Let p be an odd prime number. This paper investigates λ-constacyclic codes of length 2ps over the ring Rᵤ,ᵥ = Fpm[u,v] / ⟨u², v², uv − vu⟩, where λ = ψ + ρu + ηv + τuv, with ψ, ρ, η, τ ∈ Fpm, ψ ∈ F*pm, and ρ, η not both zero. When λ = μ² for some μ ∈ Rᵤ,ᵥ, the structures of all λ-constacyclic codes of length 2ps over Rᵤ,ᵥ are determined and can be expressed as the sum of a (−μ)-constacyclic code and a μ-constacyclic code of length ps. When λ is not a square, we classify these codes into four distinct types, provide the number of codewords for each type, and give the details of their dual codes.

Electronic Fourier–Galois Spectrum Analyzer for the Field GF(31)

A scheme for the Fourier–Galois spectrum analyzer for the field GF(31) is proposed. It is shown that this analyzer allows for solving a wide enough range of problems related to image processing, in particular those arising in the course of experimental studies in the field of physical chemistry. Such images allow digital processing when divided into a relatively small number of pixels, which creates an opportunity to use Galois fields of relatively small size. The choice of field GF(31) is due to the fact that the number 31 is a Mersenne prime number, which considerably simplifies the algorithm of calculating the Fourier–Galois transform in this field. The proposed scheme of the spectrum analyzer is focused on the use of threshold sensors, at the output of which signals corresponding to binary logic are formed. Due to this fact, further simplification of the proposed analyzer scheme is achieved. The constructiveness of the proposed approach is proven using digital modeling of electronic circuits. It is concluded that when solving applied problems in which an image can be divided into a relatively small number of pixels, it is important to take into account the specificity of particular Galois fields used for their digital processing.

Prime number factorization and degree of coherence of speckled light beams.

We discover a connection between a Gauss sum of number theory and the degree of coherence (DOC) of the field in a transverse plane of structured speckled light beams. We theoretically demonstrate and experimentally validate that prime number factorization can be achieved by manipulating the source beam's DOC in Young's double-slit experiment. The determination of whether a number can be factored is based solely on the visibility of the resulting interference patterns. Our findings offer new insights into information encryption and decryption, data compression, etc.

Calculation of Nielsen periodic numbers on infra-solvmanifolds

Recently, a formula for computing the Nielsen periodic numbers NFn(f) and NPn(f) of self maps f on infra-nilmanifolds and infra-solvmanifolds of type (R) was found. In this paper, we extend this formula to the case of general infra-solvmanifolds. We show that infra-solvmanifolds are essentially reducible to the GCD and essentially toral, and determine conditions under which NFn(f)=N(fn). We show that the prime Nielsen-Jiang periodic number NPn(f) of a self map f on an infra-solvmanifold M can be calculated by Nielsen numbers of lifts of suitable iterates of f to an NR-solvmanifold that finitely covers M.