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Computer Engineering and Science

A Quantum Algorithm for Large Prime Factorization

Pan Zheng

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

The quantum factoring algorithm is a new method using the features of quantum waves to factorize integers. In this paper, an introduction of the fundamental principles of quantum factoring algorithms is given together with some relational concepts, followed by an analysis of the features of the quantum factoring algorithm. Finally the prospect of the development of quantum computation is presented.

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The quantum factoring algorithm is a new method using the features of quantum waves to factorize integers. In this paper, an introduction of the fundamental principles of quantum factoring algorithms is given together with some relational concepts, followed by an analysis of the features of the quantum factoring algorithm. Finally the prospect of the development of quantum computation is presented.
One of the most well known mathematically hard problems in number theory is the integer factorization problem, roughly stated that decomposition of a composite number into its prime factors. In modern cryptography, RSA encryption algorithm whose security is based on integer factorization problem is highly practical, widespread and up to date no classical algorithm having polynomial running time for the factorization of large numbers is known. In 1994, Peter Shor proposed an efficient algorithm on quantum computer. In this paper, we  mention about the fundamentals of  Shor's quantum algorithm illustrating a concrete example.
Quantum computing has emerged as an important interdisciplinary field, merging theories in mathematics, physics and computer science. So far, a significant portion of research in quantum computing has focused on the design of quantum algorithms. Quantum Com-puters requires very different algorithms for factorization, which plays an important role in the field of quantum computing and shor’s algo-rithm is used to address integer factorization problem. Shor's algorithm is important because it breaks a widely used public-key cryptography scheme known as RSA. This paper discusses the refinement of quantum factorization algorithm. Keywords: - quantum algorithm, factorization, gcd, prime decomposition, shor’s algorithm.
In 1994 Peter Shor [65] published a factoring algorithm for a quantum computer that finds the prime factors of a composite integer N more efficiently than is possible with the known algorithms for a classical com­ puter. Since the difficulty of the factoring problem is crucial for the se­ curity of a public key encryption system, interest (and funding) in quan­ tum computing and quantum computation suddenly blossomed. Quan­ tum computing had arrived. The study of the role of quantum mechanics in the theory of computa­ tion seems to have begun in the early 1980s with the publications of Paul Benioff [6]' [7] who considered a quantum mechanical model of computers and the computation process. A related question was discussed shortly thereafter by Richard Feynman [35] who began from a different perspec­ tive by asking what kind of computer should be used to simulate physics. His analysis led him to the belief that with a suitable class of "quantum machines" one could imitate any quantum system.
Over the last decade a new way to compute has been defined which, unlike conventional methods, is based on quantum mechanics rather than classical physics. This fundamental change in paradigm can, in principle, solve some important and hitherto intractable problems such as prime factorization of large numbers. This tutorial presentation will review the basics of quantum computation and the design of circuits to implement quantum algorithms. The starting point is the notion of a quantum bit or qubit. Because of the superposition property of quantum states, n qubits can store 2n numbers simultaneously, implying a type of massive parallelism. Furthermore, quantum states allow powerful forms of interaction such as entanglement that have no classical counterparts. Qubits are fragile, however, and are altered by measurement; hence quantum circuits must follow very different rules from classical ones. The differences between quantum and classical logic circuits will be discussed and illustrated. Finally, the physical implementation of quantum devices will be considered, along with the prospects for practical quantum computers.
The field of quantum computing has witnessed remarkable advancements in recent years, particularly in its potential applications to solve computationally hard problems such as integer factorization. a comparative analysis of quantum computing algorithms for integer factorization, focusing on their theoretical foundations, computational complexity, and practical implications. We review prominent algorithms such as Shor's algorithm, which leverages quantum parallelism and period finding to efficiently factor large composite integers, and compare them with classical factorization algorithms like the General Number Field Sieve (GNFS). Through a comprehensive examination of algorithmic strategies, runtime complexities, and quantum circuit implementations, we assess the relative strengths and limitations of quantum computing approaches for integer factorization. Furthermore, we discuss potential challenges and future research directions in harnessing the power of quantum computing to address cryptographic security and algorithmic complexity in the era of post-quantum cryptography.