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Journal articles on the topic 'Prime factorization'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

KNOPFMACHER, ARNOLD, and FLORIAN LUCA. "ON PRIME-PERFECT NUMBERS." International Journal of Number Theory 07, no. 07 (2011): 1705–16. http://dx.doi.org/10.1142/s1793042111004447.

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We prove that the Diophantine equation [Formula: see text] has only finitely many positive integer solutions k, p1, …, pk, r1, …, rk, where p1, …, pk are distinct primes. If a positive integer n has prime factorization [Formula: see text], then [Formula: see text] represents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equation are designated as prime-perfect numbers.

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2

Liu, Jinwang, Tao Wu, Dongmei Li, and Jiancheng Guan. "On Zero Left Prime Factorizations for Matrices over Unique Factorization Domains." Mathematical Problems in Engineering 2020 (April 22, 2020): 1–3. http://dx.doi.org/10.1155/2020/1684893.

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In this paper, zero prime factorizations for matrices over a unique factorization domain are studied. We prove that zero prime factorizations for a class of matrices exist. Also, we give an algorithm to directly compute zero left prime factorizations for this class of matrices.

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3

Papadakis, Ioannis N. M. "Algebraic Representation of Primes by Hybrid Factorization." Mathematics and Computer Science 9, no. 1 (2024): 12–25. http://dx.doi.org/10.11648/j.mcs.20240901.12.

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The representation of integers by prime factorization, proved by Euclid in the Fundamental Theorem of Arithmetic −also referred to as the Prime Factorization Theorem− although universal in scope, does not provide insight into the algebraic structure of primes themselves. No such insight is gained by summative prime factorization either, where a number can be represented as a sum of up to three primes, assuming Goldbach’s conjecture is true. In this paper, a third type of factorization is introduced, called hybrid prime factorization, defined as the representation of a number as sum −or differe

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4

Mahato, Prabhat, and Aayush Shah. "A Review of Prime Numbers, Squaring Prime Pattern, Different Types of Primes and Prime Factorization Analysis." International Journal for Research in Applied Science and Engineering Technology 11, no. 7 (2023): 2036–43. http://dx.doi.org/10.22214/ijraset.2023.54904.

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Abstract: The study of prime numbers and their properties has always been an intriguing and fascinating topic for mathematicians. Primes can be considered the “basic building blocks,” the atoms, of the natural numbers. They play a significant role in number theory. Also, prime numbers, in this current world of computers and digitalization, have paramount significance for the computer programmers and scientists to tackle relevant real-life problems. Since long time, many studies and researches have been conducted regarding prime numbers pattern. In this paper, a squaring prime pattern is presen

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MANN, A., M. REVZEN, and J. ZAK. "THE PHYSICS OF FACTORIZATION." International Journal of Quantum Information 04, no. 01 (2006): 173–80. http://dx.doi.org/10.1142/s0219749906001670.

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The Ndistinct prime numbers that make up a composite number M allow its bi-partitioning into pairs of two relatively prime factors. Each such pair defines a pair of conjugate representations. An example of such pairs of conjugate representations, each of which spans the M-dimensional space, are the kq representations, which are the most natural representations for periodic systems. Here, we emphasize their relevance to factorizations: the number of prime numbers that make up M relates directly to the number of conjugate pairs of kq representations. It is also shown how Schwinger's factorizatio

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MEYEROVITCH, TOM. "Direct topological factorization for topological flows." Ergodic Theory and Dynamical Systems 37, no. 3 (2015): 837–58. http://dx.doi.org/10.1017/etds.2015.67.

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This paper considers the general question of when a topological action of a countable group can be factored into a direct product of non-trivial actions. In the early 1980s, D. Lind considered such questions for $\mathbb{Z}$-shifts of finite type. In particular, we study direct factorizations of subshifts of finite type over $\mathbb{Z}^{d}$ and other groups, and $\mathbb{Z}$-subshifts which are not of finite type. The main results concern direct factors of the multidimensional full $n$-shift, the multidimensional $3$-colored chessboard and the Dyck shift over a prime alphabet. A direct factor

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WOLF, Marc, and François WOLF. "On the Factorization of Numbers of the Form X^2+c." Transactions on Machine Learning and Artificial Intelligence 10, no. 4 (2022): 59–77. http://dx.doi.org/10.14738/tmlai.104.12959.

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We study the factorization of the numbers N=X^2+c, where c is a fixed constant, and this independently of the value of gcd⁡(X,c). We prove the existence of a family of sequences with arithmetic difference (Un,Zn) generating factorizations, i.e. such that: (Un)^2+c= ZnZn+1. The different properties demonstrated allow us to establish new factorization methods by a subset of prime numbers and to define a prime sieve. An algorithm is presented on this basis and leads to empirical results which suggest a positive answer to Landau's 4th problem.

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8

Marchei, Daniele, and Emanuela Merelli. "RNA secondary structure factorization in prime tangles." BMC Bioinformatics 23, S6 (2022): 345. https://doi.org/10.1186/s12859-022-04879-5.

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<strong>Background: </strong>Due to its key role in various biological processes, RNA secondary structures have always been the focus of in-depth analyses, with great efforts from mathematicians and biologists, to find a suitable abstract representation for modelling its functional and structural properties. One contribution is due to Kauffman and Magarshak, who modelled RNA secondary structures as mathematical objects <i>constructed</i> in link theory: <i>tangles of the Brauer Monoid</i>. In this paper, we extend the tangle-based model with its minimal prime factorization, useful to analyze p

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9

Rengkung, Matthew Evans Audric, and Arya Wicaksana. "RSA Prime Factorization on IBM Qiskit." Journal of Internet Services and Information Security 13, no. 2 (2023): 203–10. http://dx.doi.org/10.58346/jisis.2023.i2.013.

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The advancement of quantum computing in recent years poses severe threats to the RSA public-key cryptosystem. The RSA cryptosystem fundamentally relies its security on the computational hardness of number theory problems: prime factorization (integer factoring). Shor’s quantum factoring algorithm could theoretically answer the computational problem in polynomial time. This paper contributes to the experiment and demonstration of Shor’s quantum factoring algorithm for RSA prime factorization using IBM Qiskit. The performance of the quantum program is evaluated based on user time and the success

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Overmars, Anthony, and Sitalakshmi Venkatraman. "New Semi-Prime Factorization and Application in Large RSA Key Attacks." Journal of Cybersecurity and Privacy 1, no. 4 (2021): 660–74. http://dx.doi.org/10.3390/jcp1040033.

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Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these

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Oral, Kursat Hakan, Unsal Tekir, and Ahmet Goksel Agargun. "Weakly unique factorization modules." Tamkang Journal of Mathematics 41, no. 3 (2010): 245–52. http://dx.doi.org/10.5556/j.tkjm.41.2010.729.

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In this work we give the definition of weakly prime element of a module. Therefore we give a new definition of factorization in a module, which is called weakly factorization. So we call a module weakly unique factorization module if all elements have a weakly factorization which is unique. We give the relation between weakly prime elements and weakly prime submodules. Then we characterize such weakly unique factorization modules.

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Papadakis, Ioannis. "On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation." Mathematics and Computer Science 9, no. 5 (2024): 96–113. https://doi.org/10.11648/j.mcs.20240905.12.

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An analysis, based on different mathematical approaches, of the binary Goldbach conjecture −which states that every even integer s≥6 is the sum of two odd primes, called Goldbach primes− is presented. Each approach leads to a different reformulation of this conjecture, thus contributing unique insights into the structure, properties and distribution of prime numbers. The above-mentioned reformulations are based on the following distinct, interrelated and complementary approaches: projection, optimization, hybrid prime factorization, prime symmetry and analytic approximation. Additionally, it i

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Kawuwung, Westy B., Bonefasius Yanwar Boy, and Epiphani I. Y. Palit. "PENINGKATAN PEMAHAMAN KONSEP FAKTORISASI PRIMA DAN APLIKASINYA MENGGUNAKAN MEDIA KREATIVITAS SISWA BAGI GURU SD NEGERI ENTROP JAYAPURA PROVINSI PAPUA." JURNAL PENGABDIAN PAPUA 4, no. 3 (2020): 102–5. http://dx.doi.org/10.31957/.v4i3.1374.

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The aim of this service activity is to increase the understanding of Entrop Elementary School teachers about one of the basic concepts of mathematics, namely prime factorization, its application in solving mathematical problems, as well as creative aids that can be used in teaching the concept to students. The method used is the explanation of mathematical concepts about factors, prime factors, and prime factorization of a number with examples of its application in solving mathematical problems. After that, examples of teaching aids which can be used to explain prime factorization to students

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14

Rao, K. S. Mallikarjuna. "Extremality of Prime Factorization." Resonance 26, no. 12 (2021): 1643–48. http://dx.doi.org/10.1007/s12045-021-1276-z.

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15

Wagstaff, S. S., and Hideo Wada. "Computers and Prime Factorization." Mathematics of Computation 53, no. 187 (1989): 451. http://dx.doi.org/10.2307/2008382.

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16

Omollo, Richard, and Arnold Okoth. "Factorization Algorithm for Semi-primes and the Cryptanalysis of Rivest-Shamir-Adleman (RSA) Cryptography." Asian Journal of Research in Computer Science 17, no. 6 (2024): 85–95. http://dx.doi.org/10.9734/ajrcos/2024/v17i6458.

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This paper introduces a new factoring algorithm called Anorld’s Factorization Algorithm that utilizes semi-prime numbers and their implications for the cryptanalysis of the Rivest-Shamir-Adleman (RSA) cryptosystem. While using the concepts of number theory and algorithmic design, we advance a novel approach that notably enhances the efficiency of factoring large semi-prime numbers compared to other algorithms that have been developed earlier. In our approach, we propose a three-step algorithm that factorizes relatively large semi-primes in polynomial time. We have introduced factorization up t

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17

Zhang, Xingyi, Yunyun Niu, Linqiang Pan, and Mario J. Pérez-Jiménez. "Linear Time Solution to Prime Factorization by Tissue P Systems with Cell Division." International Journal of Natural Computing Research 2, no. 3 (2011): 49–60. http://dx.doi.org/10.4018/jncr.2011070105.

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Prime factorization is useful and crucial for public-key cryptography, and its application in public-key cryptography is possible only because prime factorization has been presumed to be difficult. A polynomial-time algorithm for prime factorization on a quantum computer was given by P. W. Shor in 1997. In this work, it is considered as a function problem, and in the framework of tissue P systems with cell division, a linear-time solution to prime factorization problem is given on biochemical computational devices – tissue P systems with cell division, instead of computational devices based on

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18

Rahul, Arora. "BEHAVIOUR OF RATIONAL PRIMES IN THE RING OF ALGEBRAIC INTEGERS." International Journal of Applied and Advanced Scientific Research 2, no. 2 (2017): 62–64. https://doi.org/10.5281/zenodo.842726.

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We know that the primes in Z (hereafter referred as rational primes) are irreducible in Z i.e they don’t have proper factorization. If R is any factorization domain such that Z is properly contained in R then are these rational primes also irreducible in R? The answer to this question in general is No. For example, 13 is prime in Z but 13 is not prime in Z[i] as we can write 13 as: 13 = (2 + 3i)(2 – 3i) where both 2 + 3i & 2 – 3i are irreducible (rather non units) in Z[i]. In this paper we will see how the rational primes spilt in the ring of algebraic integers.

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19

Rahul, Arora. "SPLITTING OF RATIONAL PRIMES IN THE RING OF ALGEBRAIC INTEGERS." International Journal of Current Research and Modern Education 2, no. 2 (2017): 28–29. https://doi.org/10.5281/zenodo.842730.

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We know that the primes in Z (hereafter referred as rational primes) are irreducible in Z i.e they don’t have proper factorization. If R is any factorization domain such that Z is properly contained in R then are these rational primes also irreducible in R? The answer to this question in general is No. For example, 2 is prime in Z but 2 is not prime in Z[i] as we can write 2 as: 2 = (1 + i)(1 – i) where both 1 + i & 1 – i are irreducible (rather non units) in Z[i]. In this paper we will see how the rational primes spilt in the ring of algebraic integers.

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20

Hamiss, Kabenge. "A Simple Algorithm for Prime Factorization and Primality Testing." Journal of Mathematics 2022 (December 15, 2022): 1–10. http://dx.doi.org/10.1155/2022/7034529.

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We propose a new simple and faster algorithm to factor numbers based on the nature of the prime numbers contained in such composite numbers. It is well known that every composite number has a unique representation as a product of prime numbers. In this study, we focus mainly on composite numbers that contain a product of prime numbers that are greater than or equal to 5 which are of the form 6 k + 1 or 6 k + 5 . Therefore, we use the condition that every prime or composite P of primes greater than or equal to 5 satisfies P 2 ≡ 1 mod 24 . This algorithm is very fast especially when the differen

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21

James, Joseph. "UNIQUE FACTORIZATION FERMAT'S LAST THEOREM BEAL'S CONJECTURE." Journal of Progressive Research in Mathematics 10, no. 1 (2016): 1434–39. https://doi.org/10.5281/zenodo.3976651.

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In this paper the following statememt of Fermat\rq{}s Last Theorem is proved.  If  $x, y, z$ are positive integers$\pi$ is an odd prime and  $z^\pi=x^\pi+y^\pi, x, y, z$ are all even. Also, in this paper, is proved (Beal\rq{}s conjecture): The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z, $ with $\xi, \mu, \nu$ primes at least $3.

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22

Dogondaji, A., B. Sani, A. Ibrahim, and S. Abubakar. "Cryptanalysis Attacks on Multi Prime Power Modulus Through Analyzing Prime Difference." International Journal of Science for Global Sustainability 9, no. 1 (2023): 9. http://dx.doi.org/10.57233/ijsgs.v9i1.405.

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The Security of Rivest, Shamir and Adleman Cryptosystem known as RSA and its variants rely on the difficulty of integer factorization problem. This paper presents a short decryption exponent attack on RSA variant based on the key equation where prime difference was carefully analyzed and came up with an approximation of as which enabled us to obtain an improved bound that led to the polynomial time factorization of the variant .

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23

Ştefănescu, Doru. "Polynomials, Constructivity and Randomness." JUCS - Journal of Universal Computer Science 2, no. (5) (1996): 396–409. https://doi.org/10.3217/jucs-002-05-0396.

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We discuss some effective characterizations of the prime elements in a polynomial ring and polynomial factorization techniques. We emphasize that some factorization methods are probabilistic, their efficiency justifies the experimental trend in mathematics. The possibility of an effective version of Hilbert's irreducibility theorem and the probabilistic techniques of Berlekamp will be also discussed. Finally, bounds on the heights of integer polynomials are used as tools for improving polynomial factorizations. 1 C. Calude (ed.). The Finite, the Unbounded and the Infinite, Proceedings of the S

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张, 泰滺. "Prime Factorization of Sperner Theory." Advances in Applied Mathematics 04, no. 04 (2015): 357–64. http://dx.doi.org/10.12677/aam.2015.44044.

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Overmars, Anthony, and Sitalakshmi Venkatraman. "Mathematical Attack of RSA by Extending the Sum of Squares of Primes to Factorize a Semi-Prime." Mathematical and Computational Applications 25, no. 4 (2020): 63. http://dx.doi.org/10.3390/mca25040063.

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The security of RSA relies on the computationally challenging factorization of RSA modulus N=p1 p2 with N being a large semi-prime consisting of two primes p1and p2, for the generation of RSA keys in commonly adopted cryptosystems. The property of p1 and p2, both congruent to 1 mod 4, is used in Euler’s factorization method to theoretically factorize them. While this caters to only a quarter of the possible combinations of primes, the rest of the combinations congruent to 3 mod 4 can be found by extending the method using Gaussian primes. However, based on Pythagorean primes that are applied i

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Hanoymak, Turgut, and Cihan Kayak. "Another Approach to Factoring by Continued Fractions." Turkish Journal of Mathematics and Computer Science 17, no. 1 (2025): 33–46. https://doi.org/10.47000/tjmcs.1569163.

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The problem of prime factorization is particularly important in fields such as cryptography, where it plays a crucial role, especially in the security of public key cryptosystems like RSA. There are numerous factorization algorithms that have been developed over time, each with varying levels of complexity. These algorithms have played a crucial role in fields like mathematics and cryptography, where prime factorization remains a key challenge. In this study, the continued fraction method one of the factorization methods, is examined. To highlight the importance of the continued fraction facto

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27

KHANDUJA, SUDESH K., and MUNISH KUMAR. "ON A THEOREM OF DEDEKIND." International Journal of Number Theory 04, no. 06 (2008): 1019–25. http://dx.doi.org/10.1142/s1793042108001833.

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Let K = ℚ(θ) be an algebraic number field with θ in the ring AK of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ℚ of rational numbers. For a rational prime p, let [Formula: see text] be the factorization of the polynomial [Formula: see text] obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct monic irreducible polynomials over ℤ/pℤ. Dedekind proved that if p does not divide [AK : ℤ[θ]], then the factorization of pAK as a product of powers of distinct prime ideals is given by [Formula: see text], with 𝔭i = pAK + gi(θ)AK, a

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28

Tekir, Ünsal. "A Note on Dedekind and ZPI Modules." Algebra Colloquium 13, no. 01 (2006): 41–45. http://dx.doi.org/10.1142/s1005386706000071.

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Let R be a domain. A non-zero R-module M is called a Dedekind module if every submodule N of M such that N ≠ M either is prime or has a prime factorization N=P1P2… PnN*, where P1, P2, … Pn are prime ideals of R and N* is a prime submodule in M. When R is a ring, a non-zero R-module M is called a ZPI module if every submodule N of M such that N ≠ M either is prime or has a prime factorization. The purpose of this paper is to introduce interesting and useful properties of Dedekind and ZPI modules.

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29

Arieska, Sarlin, Dewi Rahimah, and Effie efrida Muchlis. "Kesalahan Siswa dalam Menyelesaikan Soal Uraian pada Materi Operasi Hitung Bilangan Bulat di Kelas V SD Negeri 88 Kota Bengkulu." Jurnal Penelitian Pembelajaran Matematika Sekolah (JP2MS) 2, no. 2 (2018): 166–70. https://doi.org/10.33369/jp2ms.2.2.166-170.

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This study aims to find out the mistake of students in solving math problems on the material integer arithmetic operations. This research is a descriptive research. The subjects were 25 students of class V SDN 88 Bengkulu City Odd Semester Academic Year 2017/2018. Data collection techniques used in this research that tests. Analysis of the data in this study was done descriptively. The results showed the students make the mistake of using the basic competence to determine the prime factors the Commission and the FPB, and resolve issues related to arithmetic operations, the Commission and the F

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30

Babindamana, Regis Freguin, Gilda Rech Bansimba, and Basile Guy Richard Bossoto. "Lattice Points on the Fermat Factorization Method." Journal of Mathematics 2022 (January 28, 2022): 1–18. http://dx.doi.org/10.1155/2022/6360264.

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In this paper, we study algebraic properties of lattice points of the arc on the conics x 2 − d y 2 = N especially for d = 1 , which is the Fermat factorization equation that is the main idea of many important factorization methods like the quadratic field sieve, using arithmetical results of a particular hyperbola parametrization. As a result, we present a generalization of the forms, the cardinal, and the distribution of its lattice points over the integers. In particular, we prove that if N − 6 ≡ 0 mod 4 , Fermat’s method fails. Otherwise, in terms of cardinality, it has, respectively, 4, 8

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31

Berend, D., and G. Kolesnik. "Prime-power factorization of binomial coefficients." Acta Arithmetica 193, no. 1 (2020): 49–74. http://dx.doi.org/10.4064/aa180814-10-4.

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32

Olberding, Bruce. "Factorization into Prime and Invertible Ideals." Journal of the London Mathematical Society 62, no. 2 (2000): 336–44. http://dx.doi.org/10.1112/s0024610700001319.

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33

Zalaket, Joseph, and Joseph Hajj-Boutros. "Prime factorization using square root approximation." Computers & Mathematics with Applications 61, no. 9 (2011): 2463–67. http://dx.doi.org/10.1016/j.camwa.2011.02.027.

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Khivintsev, Yuri, Mojtaba Ranjbar, David Gutierrez, et al. "Prime factorization using magnonic holographic devices." Journal of Applied Physics 120, no. 12 (2016): 123901. http://dx.doi.org/10.1063/1.4962740.

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Peter Rehm, Hans. "Prime factorization of integral Cayley octaves." Annales de la faculté des sciences de Toulouse Mathématiques 2, no. 2 (1993): 271–89. http://dx.doi.org/10.5802/afst.767.

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Wan-Ying, Wang, Shang Bin, Wang Chuan, and Long Gui-Lu. "Prime Factorization in the Duality Computer." Communications in Theoretical Physics 47, no. 3 (2007): 471–73. http://dx.doi.org/10.1088/0253-6102/47/3/019.

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Rahmeh, Samer, and Adam Neumann. "HUBO & QUBO and Prime Factorization." International Journal of Bioinformatics and Intelligent Computing 3, no. 1 (2024): 45–69. http://dx.doi.org/10.61797/ijbic.v3i1.301.

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This document details the methodology and steps taken to convert Higher Order Unconstrained Binary Optimization (HUBO) models into Quadratic Unconstrained Binary Optimization (QUBO) models. The focus is primarily on prime factorization problems; a critical and computationally intensive task relevant in various domains including cryptography, optimization, and number theory. The conversion from Higher-Order Binary Optimization (HUBO) to Quadratic Unconstrained Binary Optimization (QUBO) models is crucial for harnessing the capabilities of advanced computing methodologies, particularly quantum c

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Omollo, Richard, and Arnold Okoth. "Large Semi Primes Factorization with Its Implications to RSA Cryptosystems." BOHR International Journal of Smart Computing and Information Technology 3, no. 1 (2020): 1–8. http://dx.doi.org/10.54646/bijscit.011.

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RSA’s strong cryptosystem works on the principle that there are no trivial solutions to integer factorization. Furthermore, factorization of very large semi primes cannot be done in polynomial time when it comes to the processing power of classical computers. In this paper, we present the analysis of Fermat’s Last Theorem and Arnold’s Theorem. Also highlighted include new techniques such as Arnold’s Digitized Summation Technique (A.D.S.T.) and a top-to-bottom, bottom-to-top approach search for the prime factors. These drastically reduce the time taken to factorize large semi primes as for the

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Balinskiy, Michael, and Alexander Khitun. "Prime factorization using coupled oscillators with positive feedback." AIP Advances 12, no. 4 (2022): 045307. http://dx.doi.org/10.1063/5.0086563.

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Prime factorization is a procedure of determining the prime factors of a given number N that requires super-polynomial time for conventional digital computers. In this work, we describe an approach to prime factorization using coupled oscillators with positive feedback. The approach includes several steps, where some of the steps are accomplished on a general type computer, and some steps are accomplished using coupled oscillators. We present experimental data on finding the primes of N = 817. The experiment is performed on a system of two coupled active ring oscillators. Each of the oscillato

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40

Candra, Ade, Mohammad Andri Budiman, and Dian Rachmawati. "On Factoring The RSA Modulus Using Tabu Search." Data Science: Journal of Computing and Applied Informatics 1, no. 1 (2017): 30–37. http://dx.doi.org/10.32734/jocai.v1.i1-65.

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It is intuitively clear that the security of RSA cryptosystem depends on the hardness of factoring a very large integer into its two prime factors. Numerous studies about integer factorization in the field of number theory have been carried out, and as a result, lots of exact factorization algorithms, such as Fermat’s factorization algorithm, quadratic sieve method, and Pollard’s rho algorithm have been found. The factorization problem is in the class of NP (non-deterministic polynomial time). Tabu search is a metaheuristic in the field of artificial intelligence which is often used to solve N

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GIBSON, ANDREW. "Factorization of homotopies of nanophrases." Mathematical Proceedings of the Cambridge Philosophical Society 152, no. 1 (2011): 55–90. http://dx.doi.org/10.1017/s0305004111000612.

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AbstractHomotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. For any homotopy given by a composite homotopy data triple we define a complete invariant of nanophrases. This invariant is used to show that equivalence of nanophrases under such a homotopy can be calculated just by using the homotopies given by its prime factors.

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Cesati, Marco. "A New Idea for RSA Backdoors." Cryptography 7, no. 3 (2023): 45. http://dx.doi.org/10.3390/cryptography7030045.

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This article proposes a new method to inject backdoors in RSA (the public-key cryptosystem invented by Rivest, Shamir, and Adleman) and other cryptographic primitives based on the integer factorization problem for balanced semi-primes. The method relies on mathematical congruences among the factors of the semi-primes based on a large prime number, which acts as a “designer key” or “escrow key”. In particular, two different backdoors are proposed, one targeting a single semi-prime and the other one a pair of semi-primes. This article also describes the results of tests performed on a SageMath i

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Liu, Jinwang, Dongmei Li, and Licui Zheng. "Minor Prime Factorization for n-D Polynomial Matrices over Arbitrary Coefficient Field." Complexity 2018 (October 8, 2018): 1–9. http://dx.doi.org/10.1155/2018/6235649.

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In this paper, we investigate two classes of multivariate (n-D) polynomial matrices whose coefficient field is arbitrary and the greatest common divisor of maximal order minors satisfy certain condition. Two tractable criterions are presented for the existence of minor prime factorization, which can be realized by programming and complexity computations. On the theory and application, we shall obtain some new and interesting results, giving some constructive computational methods for carrying out the minor prime factorization.

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44

Kritsanapong, Somsuk. "The new integer factorization algorithm based on fermat's factorization algorithm and euler's theorem." International Journal of Electrical and Computer Engineering (IJECE) 10, no. 2 (2020): 1469–76. https://doi.org/10.11591/ijece.v10i2.pp1469-1476.

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Although, Integer Factorization is one of the hard problems to break RSA, many factoring techniques are still developed. Fermat’s Factorization Algorithm (FFA) which has very high performance when prime factors are close to each other is a type of integer factorization algorithms. In fact, there are two ways to implement FFA. The first is called FFA-1, it is a process to find the integer from square root computing. Because this operation takes high computation cost, it consumes high computation time to find the result. The other method is called FFA-2 which is the different technique to

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45

Karakuş, Ali, and Merve Gökçe. "Factorization of Modules on Commutative Rings." Journal of Basic and Applied Research International 30, no. 6 (2024): 67–72. http://dx.doi.org/10.56557/jobari/2024/v30i68940.

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In this article, the factorization of a torsion module structure is examined and definitions and theorems related to the uniform factorization part in modules are given. Then, the prime sub-modules of the modules that can be factorized by a single method are examined and basic definitions and theorems are given. We have also studied the module elements in written form as the product of the unreducible elements of the ring with the unreducible elements of the modules with the help of weak prime units defined on the torsion modules.

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46

Aminudin, Aminudin, and Eko Budi Cahyono. "A Practical Analysis of the Fermat Factorization and Pollard Rho Method for Factoring Integers." Lontar Komputer : Jurnal Ilmiah Teknologi Informasi 12, no. 1 (2021): 33. http://dx.doi.org/10.24843/lkjiti.2021.v12.i01.p04.

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The development of public-key cryptography generation using the factoring method is very important in practical cryptography applications. In cryptographic applications, the urgency of factoring is very risky because factoring can crack public and private keys, even though the strength in cryptographic algorithms is determined mainly by the key strength generated by the algorithm. However, solving the composite number to find the prime factors is still very rarely done. Therefore, this study will compare the Fermat factorization algorithm and Pollard rho by finding the key generator public key

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47

Somsuk, Kritsanapong. "The new integer factorization algorithm based on Fermat’s Factorization Algorithm and Euler’s theorem." International Journal of Electrical and Computer Engineering (IJECE) 10, no. 2 (2020): 1469. http://dx.doi.org/10.11591/ijece.v10i2.pp1469-1476.

Full text

Abstract:

Although, Integer Factorization is one of the hard problems to break RSA, many factoring techniques are still developed. Fermat’s Factorization Algorithm (FFA) which has very high performance when prime factors are close to each other is a type of integer factorization algorithms. In fact, there are two ways to implement FFA. The first is called FFA-1, it is a process to find the integer from square root computing. Because this operation takes high computation cost, it consumes high computation time to find the result. The other method is called FFA-2 which is the different technique to find p

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48

Trifina, Lucian, and Daniela Tarniceriu. "When Is the Number of True Different Permutation Polynomials Equal to 0?" Mathematics 7, no. 11 (2019): 1018. http://dx.doi.org/10.3390/math7111018.

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In this paper, we have obtained the prime factorization form of positive integers N for which the number of true different fourth- and fifth-degree permutation polynomials (PPs) modulo N is equal to zero. We have also obtained the prime factorization form of N so that the number of any degree PPs nonreducible at lower degree PPs, fulfilling Zhao and Fan (ZF) sufficient conditions, is equal to zero. Some conclusions are drawn comparing all fourth- and fifth-degree permutation polynomials with those fulfilling ZF sufficient conditions.

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Wagstaff, S. S., and Hans Riesel. "Prime Numbers and Computer Methods for Factorization." Mathematics of Computation 48, no. 177 (1987): 439. http://dx.doi.org/10.2307/2007903.

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Olberding, Bruce. "Factorization into prime and invertible ideals II." Journal of the London Mathematical Society 80, no. 1 (2009): 155–70. http://dx.doi.org/10.1112/jlms/jdp017.

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