Prime Number Factorization Research Articles

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.

Factorization of large tetra and penta prime numbers on IBM quantum processor

The factorization of large digit integers in polynomial time is a challenging computational task to decipher. The development of Shor’s algorithm sparked a new resolution for solving the factorization problem. However, putting Shor’s algorithm into use in real-world situations presents major difficulties. The algorithm largely depends on the availability of large-scale, fault-tolerant quantum computers, which are not available at present. The need for qubit coherence and error correction makes the algorithm susceptible to noise and decoherence, hindering its practical realization. Therefore, exploring alternative quantum factorization algorithms and investing in quantum computing hardware advancements are vital steps toward overcoming these drawbacks and harnessing the full potential of quantum computing for factorization tasks. This article explores an alternative method of converting the factorization problem into an optimization problem using appropriate analytic algebra. The generalized Grover’s protocol is used to increase the amplitude of the necessary states and, in turn, help in the execution of the quantum factorization of tetra and penta primes as a proof of concept for different integers, including 875, 1 269 636 549 803, and 4375, using three and four qubits of IBMQ Perth (a seven-qubit processor). The fidelity of the quantum factorization protocol with the IBMQ Perth qubits was near unity. A generalization of the method is provided at the end for implementing factorization problems in various cases.

Prime number factorization with light beams carrying orbital angular momentum

We point out a link between orbital angular momentum (OAM) carrying light beams and number theory. The established link makes it possible to formulate and implement a simple and ultrafast protocol for prime number factorization by employing OAM endowed beams that are modulated by a prime number sieve. We are able to differentiate factors from non-factors of a number by simply measuring the on-axis intensity of light in the rear focal plane of a thin lens focusing on a source beam. The proposed protocol solely relies on the periodicity of the OAM phase distribution, and hence, it is applicable to fully as well as partially coherent fields of any frequency and physical nature—from optical or x-ray to matter waves—endowed with OAM. Our experimental results are in excellent agreement with our theory. We anticipate that our protocol will trigger new developments in optical cryptography and information processing with OAM beams.

Orbit design of satellite quantum key distribution constellations in different ground stations networks

In the field of Cryptography, Quantum Key Distribution (QKD) is an application of Quantum Information theory that obtained a great deal of attention in recent years. It allows to establish secret keys between two or more parties, in a much safer way than that implemented by classic cryptography (based on discrete logarithms and factorization of prime numbers). The most promising way of realizing a QKD network (especially over great distances) in the near future is by a constellation of satellites. This paper considers the problem of optimizing the orbits of the satellites in order to maximize the minimum key length shared in a network of ground stations over a fixed amount of time. Different networks of stations are considered and the influence of their geographical disposition on the design and the performance index is highlighted. The networks considered are: a global constellation, a regional European constellation, and two in which there are groups of stations in two different narrow bands of latitude. The effect of Inter-satellite links is then taken into account and how, in some cases, they can improve the performances. Finally the daily performance of the considered constellations are analyzed.

Prime number factorization using a spinor Bose–Einstein condensate-inspired topological quantum computer

Inspired by non-abelian vortex anyons in spinor Bose–Einstein condensates, we consider the quantum double \(\mathcal {D}(\mathbb {Q}_8)\) anyon model as a platform to carry out a particular instance of Shor’s factorization algorithm. We suggest that the \(\mathcal {D}(\mathbb {Q}_8)\) anyon model could be realized by a specific low-temperature phase of a spin-2 Bose–Einstein condensate. We provide the excitation spectrum for this model, as well as the fusion rules, braid group representations, and a circuit architecture that facilitates the computation. All necessary quantum gates, less one, can be compiled exactly for this hybrid topological quantum computer. The required non-topological gate could be implemented using measurement based protocols. To analyze the effect of decoherence on the non-topological gate, a noise model based on stochastic unitary rotations is considered. The computational potential of this quantum double anyon model is similar to that of the Majorana fermion-based Ising anyon model, thus offering a complementary future platform for topological quantum computation.

Revival and Expansion of the Theory of Coherent Lattices.

An effective way to design structured coherent wave interference patterns that builds on the theory of coherent lattices, is presented. The technique combines prime number factorization in the complex plane with moiré theory to provide a robust way to design structured patterns with variable spacing of intensity maxima. In addition, the proposed theoretical framework facilitates an elegant computation of previously unexplored high-order superlattices both for the periodic and quasiperiodic case. A number of beam configurations highlighting prime examples of patterns for lattices with three-, four-, and fivefold symmetry are verified in a multibeam interference experiment.

Implementation of Shor’s Quantum Factoring Algorithm Using ProjectQ Framework

Prime number factorization is a problem in computer science where the solution to that problem takes super-polynomial time classically. Shor’s quantum factoring algorithm is able to solve the problem in polynomial time by harnessing the power of quantum computing. The implementation of the quantum algorithm itself is not detailed by Shor in his paper. In this paper, an approach and experiment to implement Shor’s quantum factoring algorithm are proposed. The implementation is done using Python and a quantum computer simulator from ProjectQ. The testing and evaluation are completed in two computers with different hardware specifications. User time of the implementation is measured in comparison with other quantum computer simulators: ProjectQ and Quantum Computing Playground. This comparison was done to show the performance of Shor’s algorithm when simulated using different hardware. There is a 33% improvement in the execution time (user time) between the two computers with the accuracy of prime factorization in this implementation is inversely proportional to the number of qubits used. Further improvements upon the program that has been developed for this paper is its accuracy in terms of finding the factors of a number and the number of qubits used, as previously mentioned.

Collaboration of RSA Algorithm Using EM2B Key with Word Auto Key Encryption Cryptography Method in Encryption of SQL Plaintext Database

RSA algorithm is a modern cryptographic algorithm and is often used for data security, and until now no one has been able to solve it. This algorithm uses two keys, public key and private key, the degree of difficulty of this algorithm lies in the factorization of large prime numbers. EM2B key algorithm function is to change the primary key become a new key into ASCII characters. This algorithm has the increament key algorithm function to change the length of the key characters are same as the plaintext. The Wake method is a stream cipher algorithm uses a 128 bit key and a 256 X 32 bit table. The strength of the RSA algorithm is the bit length used, and the strength of the WAKE method also lies in the bit length used and many rotation that occur. While the strength of the EM2B key algorithm lies in the increament key. After testing, the combination of three alogritmas consists of the RSA algorithm using the EM2B key and WAKE method and a combination of two algorithms with WAKE method using the EM2B key, the encryption results are better and safer than using only one WAKE algorithm.

Residue number system arithmetic based on integrated nanophotonics.

The residue number system (RNS) enables dimensionality reduction of an arithmetic problem by representing a large number as a set of smaller integers, where the number is decomposed by prime number factorization. These reduced problem sets can then be processed independently and in parallel, thus improving computational efficiency and speed. Here, we show an optical RNS hardware representation based on integrated nanophotonics. The digit-wise shifting in RNS arithmetic is expressed as spatial routing of an optical signal in 2×2 hybrid photonic-plasmonic switches. Here, the residue is represented by spatially shifting the input waveguides relative to the routers' outputs, where the moduli are represented by the number of waveguides. By cascading the photonic 2×2 switches, we design a photonic RNS adder and a multiplier forming an all-to-all sparse directional network. The advantage of this photonic arithmetic processor is the short (10's ps) computational execution time given by the optical propagation delay through the integrated nanophotonic router. Furthermore, we show how photonic processing in-the-network leverages the natural parallelism of optics such as wavelength-division-multiplexing in this RNS processor. A key application for such a photonic RNS engine is the functional analysis of convolutional neural networks.

Modifications in RSA through Positive Justification of Random Components

There is a surge of interest today in Wireless Networks both for organizations and individuals. It’s emerging as general consensus that the current generation as well as next generation will have more of wireless connectivity. However, wireless networks have many security problems. Effective Wireless Security policies are required as wireless networks have been found to be relatively easy to break in by the hackers. The various techniques are being used for counteracting security risks. The most prevalent algorithm warranting security is RSA .RSA, a public - key cryptography algorithm is based on the factorization of large prime numbers. Presently the use of 512 bits keys are considered as insecure keys after the successful attack and by implementing the General Number Field Sieve (GNFS) algorithm .It has still got gaps of security that needs to be handled. In this, modifications has been proposed in RSA 512 bit and modified RSA through positive justification of alphabets through appending in the plain text and prime numbers in cipher text through multiplication to make the communication more secure. As the large prime numbers are not easily perishable, this modification will provide reliability over the network .This paper proposes a modification in classical RSA and modified RSA in which random numbers are appended in cipher text leading to improvement in previous results. The

A Path Tracing Scheme for All-Optical Packet-Switched Networks

A novel optical scheme is proposed to perform path tracing in an all-optical packet-switched network. Prime-number-encoded tag is employed as the distinct network node or network link identifier. As the optical data packet traverses every network node, the path trace is recorded and updated optically, through prime number multiplication via an optical delay circuit encoder. Hence, all traversed network nodes along the path of a received optical data packet can be retrieved and identified at the destination node, via simple prime number factorization. Possible looping of optical data packets in the network can also be detected. The scheme can facilitate the connection management in an optical packet-switched network.

Complexity of inverting the Euler function

Given an integer n n , how hard is it to find the set of all integers m m such that φ ( m ) = n \varphi (m) = n , where φ \varphi is the Euler totient function? We present a certain basic algorithm which, given the prime number factorization of n n , in polynomial time “on average” (that is, ( log ⁡ n ) O ( 1 ) (\log n)^{O(1)} ), finds the set of all such solutions m m . In fact, in the worst case this set of solutions is exponential in log ⁡ n \log n , and so cannot be constructed by a polynomial time algorithm. In the opposite direction, we show, under a widely accepted number theoretic conjecture, that the Partition Problem, an NP-complete problem, can be reduced in polynomial (in the input size) time to the problem of deciding whether φ ( m ) = n \varphi (m) = n has a solution, for polynomially (in the input size of the Partition Problem) many values of n n (where the prime factorizations of these n n are given). What this means is that the problem of deciding whether there even exists a solution m m to φ ( m ) = n \varphi (m) = n , let alone finding any or all such solutions, is very likely to be intractable. Finally, we establish close links between the problem of inverting the Euler function and the integer factorization problem.

Moyennes de certaines fonctions multiplicatives sur les entiers friables

Article Moyennes de certaines fonctions multiplicatives sur les entiers friables was published on November 12, 2003 in the journal Journal für die reine und angewandte Mathematik (volume 2003, issue 564).

Fast Searches with Nuclear Magnetic Resonance Computers

If a computer could be built from quantum elements, which can exist in superpositions of many states rather than the 1s and 0s of conventional binary logic, then important hard problems could be solved rapidly. One example is the prime number factorization problem at the heart of modern encryption: given the product of two large primes, find the primes themselves. Another important problem is exhaustive search: given an unstructured list of items, quickly find the one that satisfies a certain property. A pair of Research Commentaries in this issue discuss some important new results in which simple quantum computers have been constructed with liquids probed by nuclear magnetic resonance techniques. The first, by L. K. Grover of Bell Labs, describes the theoretical situation and the implementation of an algorithm that he designed for exhaustive search. The second, by J. A. Jones of Oxford University, UK, discusses the recent experimental approaches to quantum computing with nuclear magnetic resonance.

The Advantages of Superposition

If a computer could be built from quantum elements, which can exist in superpositions of many states rather than the 1s and 0s of conventional binary logic, then important hard problems could be solved rapidly. One example is the prime number factorization problem at the heart of modern encryption: Given the product of two large primes, find the primes themselves. Another important problem is exhaustive search: Given an unstructured list of items, quickly find the one that satisfies a certain property. A pair of Research Commentaries in this issue discuss some important new results in which simple quantum computers have been constructed with liquids probed by nuclear magnetic resonance techniques. The first, by L. K. Grover of Bell Labs, describes the theoretical situation and the implementation of an algorithm that he designed for exhaustive search. The second, by J. A. Jones of Oxford University, UK, discusses the recent experimental approaches to quantum computing with nuclear magnetic resonance.

An accurate computation of the hypergeometric distribution function

The computation of the cumulative hypergeometric distribution function is of interest to many researchers who are working in the computational sciences and related areas. Presented here is a new method for computing this function that applies prime number factorization to the factorials. We also apply cancellation to the numerator and denominator to reduce the computational complexity of the initial, the tail end, or weighted probabilities to achieve maximum accuracy. The new method includes two algorithms, one using recursion and the other using iteration. These two algorithms are machine independent; precision is arbitrary, subject to storage limitation. The development of the algorithms is discussed, and some test results and the comparison of these two algorithms are given. To implement both algorithms, we use the Ada programming language that is an American National Standard Institute standardized language. The language has special features such as exception handling and tasks . Exception handling is used to make programming easier and to prevent overflow or underflow conditions during the execution of the program. Tasks are used to compute the numerator and denominator concurrently, and to maximize the possible number of integer multiplications in the numerator and denominator. All of the computations can be done on currently available machines, and the time consumed by these computations remains reasonably small.

A canonical ordering of polybenzenes and polymantanes using a prime number factorization technique

The correlation described over twenty years ago by Matula between the prime factorization of integers and the class of alkanes is re-examined with a view to explaining the probable reason why there have, to date, been no major extensions of this idea. By considering the class of alkanes as a one-dimensional one-parameter system, a new perspective on the method is gained that is amenable to extension, but in a different direction than originally anticipated. With this new perspective, the classes of polybenzenes and polymantanes are seen to be the representatives of two- and three-dimensional one-parameter systems, respectively. A “nomenclature”, comparable to one that Matula used for alkanes, is created that gives a unique canonical name to all possible combinations of either polybenzene or polymantane modules. Such a “nomenclature” contains a “built-in” means of positioning the molecule in the field of interest in accordance with arbitrary pre-selected criteria, such as Patterson's rules, and also coding that indicates symmetries inherent in the structure of this “molecule”.