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A DETERMINISTIC ALGORITHM FOR integer factorization

MATHEMATICS OF COMPUTATION 2016年第300期85卷 2065-2069页

作者： Hiary, Ghaith A. Ohio State Univ Dept Math 231 W 18th Ave Columbus OH 43210 USA

A deterministic algorithm for factoring n using n(1/3+o(1)) bit operations is presented. The algorithm tests the divisibility of n by all the integers in a short interval at once rather than integer by integer as in t... 详细信息

关键词： integer factorization algorithm continued fraction

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Variational Quantum Computation integer factorization Algorithm

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INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS 2023年第11期62卷 1-20页

作者： Zhang, Xinglan Zhang, Feng Beijing Univ Technol Fac Informat Technol Beijing 100124 Peoples R China Beijing Key Lab Trusted Comp Beijing 100124 Peoples R China

The integer factorization problem is a major challenge in the field of computer science, and Shor's algorithm provides a promising solution for this problem. However, Shor's algorithm involves complex modular exponentiation computation, which leads to the construction of complicated quantum circuits. Moreover, the precision of continued fraction computations in Shor's algorithm is influenced by the number of qubits, making it difficult to implement the algorithm on Noisy Intermediate-Scale Quantum (NISQ) computers. To address these issues, this paper proposes variational quantum computation integer factorization (VQCIF) algorithm based on variational quantum algorithm (VQA). Inspired by classical computing, this algorithm utilizes the parallelism of quantum computing to calculate the product of parameterized quantum states. Subsequently, the quantum multi-control gate is used to map the product satisfying pq = N onto an auxiliary qubit. Then the variational quantum circuit is adjusted by the optimizer, and it is possible to obtain a prime factor of the integer N with a high probability. While maintaining generality, VQCIF has a simple quantum circuit structure and requires only 2n + 1 qubits. Furthermore, the time complexity is exponentially accelerated. VQCIF algorithm is implemented using the Qiskit framework, and tests are conducted on factorization instances to demonstrate its feasibility.

关键词： integer factorization Shor's algorithm Variational quantum algorithm Variational quantum computation integer factorization algorithm Qiskit

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integer factorization on the GPU

Integer Factorization on the GPU

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作者： Podhorský, Jiří Brno University of Technology

Tato práce pojednává o faktorizaci, tedy rozkladu složených čísel na prvočísla a možnostech její paralelizace. Dále shrnuje nejznámější algoritmy pro faktorizaci a ne... 详细信息

Tato práce pojednává o faktorizaci, tedy rozkladu složených čísel na prvočísla a možnostech její paralelizace. Dále shrnuje nejznámější algoritmy pro faktorizaci a nejznámější platformy pro implementaci těchto algoritmů na grafické kartě. Hlavní část práce se zaobírá návrhem a implementací hardwarové akcelerace současného nejrychlejšího algoritmu na grafické kartě s využitím frameworku OpenCL. Následně je v práci uvedeno srovnání rychlostí akcelerovaného algoritmu implementovaného v rámci této práce s ostatními nejznámějšími verzemi algoritmů pro faktorizaci, zpracovávané sériově. Na závěr je v práci diskutována délka klíče algoritmu RSA potřebná pro bezpečný provoz bez možnosti jejího prolomení v reálném časovém intervalu.

关键词： CUDA OpenCL celočíselná faktorizace Kvadratické prosívání Obecné prosívání číselného pole Lenstrova faktorizace eliptické křivky Pollardova p-1 faktorizace Pollardova rho faktorizace Fermatova factorizace. CUDA OpenCL integer factorization Qua

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integer factorization for RSA cryptosystem under a PVM environment

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COMPUTER SYSTEMS SCIENCE AND ENGINEERING 2007年第1-2期22卷 25-35页

作者： Wu, C-L Lou, D-C Chang, T-J Chen, S-Y Chinese Air Force Inst Technol Dept Aviat & Commun Elect Kaohsiung 82047 Taiwan Natl Def Univ Natl Chung Cheng Univ Dept Elect Engn Tao Yuan 33509 Taiwan

The security of some of these cryptosystems Such as the Rivest-Shamir-Adelman (RSA) cryptosystem depends on the difficulty of integer factorization problem. In recent years, with the computation capability brought by modern Cluster Computing technique, the integer factorization has become much easier than before. We here use the Cluster Computing technique and fast integer factorization algorithms to show the computation power and factorization capability. This paper will incorporate the Miller-Rabin primality test method and several famous factorization algorithms, including the Pollard p-1 method, elliptic Curve method (ECM), and multiple polynomial quadratic sieve (MPQS) algorithms in the parallel virtual machine (PVM) environment. Based on the experimental results, we can find Out we can improve integer factorization performance by using modern Cluster computing technique, moreover, as some of the algorithms are very well operated and suited to parallel computing environment, our integer factorization implementation achieves additional performance improvement. We believe that the PC cluster computing technology has ushered in low-cost commodity supercomputing as a new parallel computing era, aupercomputer is no longer the only Solution to solve complex problems in the future.

关键词： parallel processing cluster computation parallel virtual machine public-key cryptosystem integer factorization primality proving

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A Study on the Limitations of Evolutionary Computation and other Bio-inspired Approaches for integer factorization

A Study on the Limitations of Evolutionary Computation and o...

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International Conference on Soft Computing and Software Engineering (SCSE)

作者： Mishra, Mohit Gupta, Vaibhav Chaturvedi, Utkarsh Shukla, K. K. Yampolskiy, R. V. Indian Inst Technol BHU Varanasi Dept Comp Sci & Engn Varanasi 221005 Uttar Pradesh India Univ Louisville Dept Comp Engn & Comp Sci Louisville KY 40292 USA

integer factorization is a vital number theoretic problem frequently finding application in public-key cryptography like RSA encryption systems, and other areas like Fourier transform algorithm. The problem is computationally intractable because it is a one-way mathematical function. Due to its computational infeasibility, it is extremely hard to find the prime factors of a semi prime number generated from two randomly chosen similar sized prime numbers. There has been a recently growing interest in the community with regards to evolutionary computation and other alternative approaches to solving this problem as an optimization task. However, the results still seem to be very rudimentary in nature and there's much work to be done. This paper emphasizes on such approaches and presents a critic study in details. The paper puts forth criticism and ideas in this aspect. (C) 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://***/lienses/by-nc-nd/4.0). Peer-review under responsibility of organizing committee of the 7th Scientific-Technical Conference Material Problems in Civil Engineering

关键词： Evolutionary Computation Swarm Intelligence integer factorization

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A Method of integer factorization 2nd

A Method of Integer Factorization

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2nd International Conference on Security and Privacy (ICSP)

作者： Pan, Zhizhong Li, Xiao Chinese Acad Sci Inst Informat Engn State Key Lab Informat Secur Beijing 100093 Peoples R China Univ Chinese Acad Sci Sch Cyber Secur Beijing 100093 Peoples R China

ISBN: (纸本)9783030905538;9783030905521

Suppose that we want to factor integer where N = pq, p, q are two distinct odd primes. Then we can reduce the problem of integer factorization to computing the generators of the Mordell-Weil group of ENr : y(2) = x(3) - Nrx, where r is a suitable integer with (r, N) = 1. We consider the family of elliptic curves E-Nr. As r varies in Z, we get two results. Firstly, we estimate the probability of r(ENr) >= 1. Secondly, we estimate probability that the points on E-Nr can factor N. Finally we conduct some experiments to illustrate our method.

关键词： Elliptic curve integer factorization Analytic rank

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Comparative Study on integer factorization Algorithm-Pollard's RHO and Pollard's P-1 2

Comparative Study on Integer Factorization Algorithm-Pollard...

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2nd International Conference on Computing for Sustainable Global Development (INDIACom)

作者： Sarnaik, Sonal Bhakkad, Rupali Desai, Chitra MIT E MCA Dept Aurangabad Maharashtra India Dept Comp Sci NDA Khadakwasla India

ISBN: (纸本)9789380544168

The difficulty of solving any cryptographic algorithm is often based on integer factorization or Discrete Logarithm or both at a same time. Most secure public key cryptographic algorithm is base on integer factorization have gained their security level because of the fact that there exist no known deterministic polynomial time algorithm for finding the factors of given composite number. This paper is focused to integer factorization problem. Here we have outlined Pollard's rho algorithm and Pollard's p-1 algorithm. The two algorithms are implemented in MuPad and have been executed on some set of numbers to arrive at comparative conclusion.

关键词： integer factorization Pollard rho Pollard P-1

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Zero-Knowledge Proofs Technique using integer factorization for Analyzing Robustness in Cryptography 3

Zero-Knowledge Proofs Technique using Integer Factorization ...

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3rd International Conference on Computing for Sustainable Global Development (INDIACom)

作者： Sah, Chitranjan Prasad Jha, Kanhaiya Nepal, Sushil Kathmandu Univ Sch Engn Dhulikhel Nepal Kathmandu Univ Sch Sci Dhulikhel Nepal

ISBN: (纸本)9789380544199

We have proved that zero-knowledge proofs technique using integer factorization problem has big-oh O(tau(1/4)) for factoring integers algorithm given by rho in comparison with Henry for discrete logarithm problem that is tau+tau/lg tau. Also, we have positively presented covariance between our result and Henry which clearly implies the input variables used for both functions tend to show similar behavior.

关键词： Covariance integer factorization Non-interactive Oblivious Transfer Zero-Knowledge Proofs

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Experimental Analysis of integer factorization Methods Using Lattices 1

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19th International Workshop on Security on Advances in Information and Computer Security (IWSEC)

作者： Sato, Arata Auzemery, Aurelien Katayama, Akira Yasuda, Masaya Rikkyo Univ Tokyo Japan Univ Limoges Limoges France

ISBN: (数字)9789819777372

ISBN: (纸本)9789819777365;9789819777372

Since 1991, Schnorr has proposed methods using lattices for solving the integer factorization problem whose hardness supports the security of the RSA cryptosystem. In 2022, Yan et al. proposed a modification of Schnorr's lattices and reported numerical experiments by an optimization method using quantum computation. After that, Yamaguchi et al. reported that they succeeded in factoring RSA-type composite numbers of at most 55 bits based on Yan et al.'s modification, using classical annealing calculation. In this paper, we report experimental results of integer factorization methods using lattices in classical computing. Specifically, we analyze the structure of Schnorr's lattices to select suitable parameters and apply existing lattice algorithms for finding smooth relations in the difference-of-squares method. We report the running time of integer factorization using lattices for RSA-type composite numbers of at most 90 bits and the success probability of finding a smooth relation from a lattice. We also discuss the time complexity of integer factorization methods using lattices based on experimental results.

关键词： integer factorization lattice problems lattice algorithms

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A New Heuristic Algorithm based on Molecular Geometry Optimization and Its Application to the integer factorization Problem 1

A New Heuristic Algorithm based on Molecular Geometry Optimi...

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2014 International Conference on Soft Computing & Machine Intelligence (ISCMI 2014)

作者： Mishra, Mohit Chaturvedi, Utkarsh Shukla, K. K. Indian Inst Technol BHU Varanasi Dept Comp Sci & Engn Varanasi Uttar Pradesh India

ISBN: (纸本)9781479942312

In this paper, the intractable problem of finding the prime factors of an integer has been represented as the integer programming problem to be solved using nature inspired heuristics. integer factorization is a one-way mathematical function and because of its computational intractability, it is frequently used in public key cryptography, for example in RSA encryption systems. Since integer factorization can be represented in the form of a discrete optimization task, more specifically as integer programming problem, various optimization tools can be utilized for cryptanalysis. In this contribution, we approach this problem using a heuristic algorithm designed with concepts derived from computational chemistry that involves energy minimization of a molecular geometry of a crystal. We observe that computational chemistry can provide a great insight into such problems of unknown dynamics. Future work remains to optimize the algorithm for scalability.

关键词： Decision Problem Function Problem integer factorization integer Programming Molecular Geometry Opitmization Nature-inspired Computing

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