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Journal of Al-Nahrain University-Science

A Comparative Analysis of Shor’s Algorithm and Factorization Methods

Alaa Khudhair Abbas, Saba Hussein Rashid, Saba Alaa Abdulwahhab, Imad Fakhri Taha Alshaikhli

Published Jun 15, 2026• 0 citations
Publisher

Abstract

Large integer factorization has long been a significant problem in computer science and mathematics. With the advent of Shor's algorithm, a polynomial-time factoring technique created for quantum computers, an effective solution to this problem became possible. Using quantum mechanics concepts. Qantum computing appears to be a novel approach to solving these complex problems. Classical computers are currently using the General Number Field Sieve (GNFS), Trial Division, the Elliptic Curve Method (EC),Fermat's Factorization, and the Quadratic Sieve(QS) are some of the best classical factoring techniques. However, this study examines these conventional approaches to determine their implications for cryptographic security, analyzing their weaknesses, usefulness, strengths, and difficulties. This paper alsodiscusses how Shor's algorithm affects cybersecurity because of its exponential speed advantage over traditional techniques. This paper also uses Shor’s algorithm get Q*P in using quantum facilities to break the algorithm like RSA algorithm

DOI
10.22401/anjs.29.2.13

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Large integer factorization has long been a significant problem in computer science and mathematics. With the advent of Shor's algorithm, a polynomial-time factoring technique created for quantum computers, an effective solution to this problem became possible. Using quantum mechanics concepts. Qantum computing appears to be a novel approach to solving these complex problems. Classical computers are currently using the General Number Field Sieve (GNFS), Trial Division, the Elliptic Curve Method (EC),Fermat's Factorization, and the Quadratic Sieve(QS) are some of the best classical factoring techniques. However, this study examines these conventional approaches to determine their implications for cryptographic security, analyzing their weaknesses, usefulness, strengths, and difficulties. This paper alsodiscusses how Shor's algorithm affects cybersecurity because of its exponential speed advantage over traditional techniques. This paper also uses Shor’s algorithm get Q*P in using quantum facilities to break the algorithm like RSA algorithm
Factoring large integers has been one of the most difficult problems in the history of mathematics and computer science. There was no efficient solution of this problem until Shor's algorithm emerged. Shor's algorithm is a polynomial time factoring algorithm which works on a quantum computer. Quantum computing is a new paradigm of computing that uses quantum mechanical phenomena in solving problems. The computers we are using right now are called classical computer. The most efficient classical factoring algorithm is General Number Field Sieve (GNFS). GNFS also cannot factor integers in polynomial time. In this paper, we compared these two algorithms in factoring integer in a standalone system.
Quantum integer factorization is a groundbreaking field that explores the use of quantum algorithms, notably Shor's algorithm, to efficiently factorize large integers into their prime components. Its potential implications on modern encryption techniques, such as the Rivest-Shamir-Adleman (RSA) algorithm, are of considerable interest. This paper's central aim is to contrast the computational time and implementation of the Integer Factorization Problem (IFP) between classical and quantum computers. The inner workings of Quantum Shor's algorithm are broken down step by step, and the estimated processing time is contrasted with the classical General Number Field Sieve (GNFS) approach. Given the exponential speedup provided by quantum computing, this comparison helps assess Shor's algorithm's possible advantage over GNFS in factoring large numbers. Direct comparisons with GNFS are currently difficult due to Shor's algorithm's reliance on large-scale quantum computer development for practical execution speeds. This paper focuses on solving integer factorization up to2n2^{n}2nbits.
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.
One of the most critical components of modern cryptography and thus cybersecurity is the ability to factor large integers quickly and efficiently. RSA encryption, one of the most used types, is based largely on the assumption that factoring for large numbers is computationally infeasible for humans and computers alike. However, with quantum computers, people can use an algorithm like Shor’s algorithm to perform the same task exponentially faster than any normal device ever could. This investigation will go into the strength and vulnerability of RSA encryption using the power of factorization in an age of quantum computers.We start by looking at the foundations of both classical and quantum factoring with greater detail at number field sieve (NFS) and Shor’s. We examine the mathematical background of each topic and the associated algorithms. We conclude with theoretical analysis and experimental simulations that address the difficulty and implications of the above-mentioned algorithms in cryptography. The final thing that I will be discussing is where quantum computing is at present and how this could pose a threat to the current type of cryptographic systems, we use every day. I will be mentioning how we need post-quantum cryptography and how people are currently creating algorithms that are designed to be attack-resistant even to large-scale quantum computers. This investigation has shown the changing dynamics of cybersecurity in the quantum era and helps us understand the challenges and the need to innovate the current cryptographic systems.
The advent of quantum computing poses a fundamental threat to classical cryptographic systems, particularly RSA encryption, which relies on the computational limit of prime factorization. This paper presents a comparative analysis of four integer factorization approaches-a selfcurated brute-force classical implementation, optimized Trial Division, Pollard's Rho algorithm, and Shor's algorithmevaluated across a dataset of increasing semiprime inputs representative of scaled-down RSA encryption keys. Through Python-based implementations and simulations, we empirically measure and plot runtime complexity for classical methods while modeling theoretical runtimes for Shor's algorithm, given hardware limitations. Our findings reveal the exponential scaling limitations of classical algorithms, and contrast them with the polynomial-time behavior of Shor's algorithm. Despite being simulated, Shor's consistently flatter runtime trendline demonstrated its theoretical superiority and the critical urgency of post-quantum cryptography. This study not only visualizes the performance divide between classical and quantum paradigms but also underscores the transformational potential of quantum computing in modern cryptanalysis.
The present study explores the groundbreaking possibilities of Shor's algorithm in cracking RSA encryption, hence enhancing the security consequences of this extensively employed cryptographic protocol. Due to Shor's algorithm's exponential speed advantage over classical algorithms, RSA encryption—a popular public key cryptosystem—is vulnerable to quantum attacks. To keep its security, RSA depends on the difficulty of factoring huge semi-primes. RSA's security presumptions are called into question by Shor's algorithm's quick factorization on a quantum computer, which has prompted research into post-quantum cryptography solutions to guarantee secure communication in the quantum age. Even with parallel computation, the speed and storage capacity of classical computing are constrained. A key idea in quantum computing is quantum parallelism, which allows quantum systems to carry out several computations at once. In contrast to classical portions. The scope of this paper is to understand the fundamentals of Quantum Computing, the current state of Shor’s algorithm implementation, and efficiency, particularly its application in compromising public key cryptosystems like RSA. The project aims to achieve how factorization is done through classical methods and further, how is this done through Shor’s algorithm and QFT techniques.