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Journal of Student Research

Implementation of Shor's Algorithm and Its Demonstrated Quantum Efficiency

Chenxuan Wang

Published Feb 29, 2024• 0 citations
Publisher

Abstract

The unique principles of quantum physics introduce an element of uncertainty into our previously deterministic world. Quantum computing, derived from quantum physics, enables the representation of a qubit in both |0> and |1> states, thus encoding more information and unleashing a realm of new possibilities. This paper conducts an analysis of the classical RSA encryption protocol and its vulnerability to the quantum algorithm—Shor's algorithm. Furthermore, this paper rigorously establishes the mathematical theories required for the analysis. Finally, the algorithm is thoroughly explored, and the potential efficiencies of quantum computing are discussed. As the study of quantum computing deepens, its application is poised to surpass that of traditional computers in various fields, leaving a profound impact.

DOI
10.47611/jsrhs.v13i1.6348

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The unique principles of quantum physics introduce an element of uncertainty into our previously deterministic world. Quantum computing, derived from quantum physics, enables the representation of a qubit in both |0> and |1> states, thus encoding more information and unleashing a realm of new possibilities. This paper conducts an analysis of the classical RSA encryption protocol and its vulnerability to the quantum algorithm—Shor's algorithm. Furthermore, this paper rigorously establishes the mathematical theories required for the analysis. Finally, the algorithm is thoroughly explored, and the potential efficiencies of quantum computing are discussed. As the study of quantum computing deepens, its application is poised to surpass that of traditional computers in various fields, leaving a profound impact.
Cryptography is crucial for protecting private information. However, the emergence of quantum computing poses a significant threat to the security of classical cryptographic systems. Traditional algorithms such as RSA can be vulnerable to quantum attacks. Shor's quantum algorithm can break RSA encryption in polynomial time. This article examines the RSA cryptosystem, the impact of quantum computing on cryptography, the challenges of quantum computing, and its applications in various fields. The article also presents an experiment comparing traditional and Shor factorization methods for semi-prime numbers. Although the experiment does not involve factoring extremely large numbers, it demonstrates the simplicity of implementing algorithms. By conducting experiments and analysing the results, computer science students can better understand the principles of quantum computing, its capabilities, and limitations, and develop practical skills.
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.
The ongoing development of Quantum Computing (QC) is a breakthrough paradigm that holds the power to revolutionize the way we process information and solve complex problems today. Quantum Computation works on the fundamental principles of Superposition and Entanglement, enabling Quantum Computers to Leverage the power of Physics in Computing. QC has the potential to revolutionize fields such as Drug Development, Financial Modeling, Optimization, and Cryptography. Shor’s algorithm, a quantum algorithm used to factorize large numbers, poses significant threats to RSA, a widely used public key cryptosystem. RSA relies on the difficulty of factoring large semi-primes to keep its security. Shor's algorithm's ability to factorize quickly on a quantum computer undermines RSA's security assumptions, necessitating the exploration of post-quantum cryptographic solutions to ensure secure communication in the quantum era. The goal of this paper is to assess the present state of Shor's algorithm implementation and efficiency, specifically its application in the context of compromising public key cryptosystems like RSA.
The ongoing development of Quantum Computing (QC) is a breakthrough paradigm that holds the power to revolutionize the way we process information and solve complex problems today. Quantum Computation works on the fundamental principles of Superposition and Entanglement, enabling Quantum Computers to Leverage the power of Physics in Computing. QC has the potential to revolutionize fields such as Drug Development, Financial Modeling, Optimization, and Cryptography. Shor’s algorithm, a quantum algorithm used to factorize large numbers, poses significant threats to RSA, a widely used public key cryptosystem. RSA relies on the difficulty of factoring large semi-primes to keep its security. Shor's algorithm's ability to factorize quickly on a quantum computer undermines RSA's security assumptions, necessitating the exploration of post-quantum cryptographic solutions to ensure secure communication in the quantum era. The goal of this paper is to assess the present state of Shor's algorithm implementation and efficiency, specifically its application in the context of compromising public key cryptosystems like RSA.
Quantum Computing is a prominent word in this era as it allows computation to be performed in no time. The motive of using Quantum Computing (QC) is that even exponentially large number of problems can be solved using it which was earlier difficult with the classical computing. Conventional methods are based on usage of bits which consist of 0's and 1's while QC works with qubits. The main issue is that conventional computing has issue of storage as well as computation even when parallel computation is performed on it. Concept of quantum parallelism allows the computation to be performed in exponentially very low time as compared to conventional method. This paper will discuss about Quantum Computing Algorithms and how Shor's algorithm is able to break RSA algorithms is discussed. Entanglement and superposition of qubits helps fast computation. The demonstration of the applicability has been evaluated based on Computation time, storage capacity, accuracy, confidentiality, efficiency, integrity, and availability. Among various algorithms Shor's technique is able to break various encryption algorithm with more supremacy as compared to conventional computing methods. In nutshell, paper will discuss about various QC algorithms and will illustrate how shor's algorithm is able to crack RSA.
Quantum computing can be regarded as a disruptive enabling technology that can pose a threat to traditional cryptography as it invented new approaches to encrypt data. This paper will compare the level of security that accompanies quantum algorithms as well as the level of efficiency. Specifically, we examine the impact of efficient quantum computing on traditional cryptography and propose techniques against quantum invasions. Four quantum algorithms are analyzed for their efficiency and security benefits in the post-quantum era: Grover’s Algorithm, Shor’s Algorithm, The Quantum Key Distribution (QKD), Lattice-based cryptography. The results reveal that Grover’s algorithm improves search time complexity from O(n) to O(√n), which is advantageous over large data sets. As it will be clear shortly, Shor’s algorithm challenges RSA encryption because it outperforms classical algorithms in factoring large integers. QKD presents secure key exchange protocols that cannot be breached notwithstanding the attempts made using quantum computing. Lattice-based schemes are quantum-resistant alternatives based on the security of certain hard mathematical problems. Based on our experimental results, computational efficiency can be improved, but new cryptographic systems will have to be developed for the mitigation of quantum-related vulnerabilities. This research shows the importance of quantum-resistant cryptographic techniques to ensure data security in the emerging quantum computing era.