Quantum computing promises revolutionary advances in cryptography, particularly through Shor's algorithm, which can factor large integers in polynomial time and threaten traditional RSA encryption. Through simulation experiments, our research compares Shor's algorithm implementation in ideal versus noisy quantum environments. Under ideal conditions, we factored semiprimes up to N=299 with reasonable resources, but with noise introduction, this ceiling dropped to approximately N=95, beyond which computational costs increased exponentially. The performance gap is striking: ideal simulations factored N=91 in 4 seconds, while noisy simulations required 2,600 seconds—a 650-fold increase. Our hybrid quantum-classical approach showed that noise simulation required modified circuit architectures, reducing gate count growth from 50 to 42.5 gates for comparable test numbers. Qubit requirements followed a step-function pattern in both environments, scaling from 12 qubits for N=15 to 21 qubits for N=91-95, while circuit depth increased from 11 to 17-19 units, with slightly shallower circuits in noisy conditions. In our paper, we propose a hybrid implementation strategy that addresses hardware limitations, present empirical findings on the impact of quantum noise on factoring efficiency, and evaluate the cryptographic security implications under realistic conditions. Our findings underscore the need for fault-tolerant quantum computing and error correction, especially as optimizations suggest factoring 2048-bit RSA keys would require approximately 20 million qubits running for 8 hours. This highlights the urgency in transitioning to post-quantum cryptographic algorithms like CRYSTALS-Dilithium and CRYSTALS-Kyber being standardized by NIST.
Quantum computing promises revolutionary advances in cryptography, particularly through Shor's algorithm, which can factor large integers in polynomial time and threaten traditional RSA encryption. Through simulation experiments, our research compares Shor's algorithm implementation in ideal versus noisy quantum environments. Under ideal conditions, we factored semiprimes up to N=299 with reasonable resources, but with noise introduction, this ceiling dropped to approximately N=95, beyond which computational costs increased exponentially. The performance gap is striking: ideal simulations factored N=91 in 4 seconds, while noisy simulations required 2,600 seconds—a 650-fold increase. Our hybrid quantum-classical approach showed that noise simulation required modified circuit architectures, reducing gate count growth from 50 to 42.5 gates for comparable test numbers. Qubit requirements followed a step-function pattern in both environments, scaling from 12 qubits for N=15 to 21 qubits for N=91-95, while circuit depth increased from 11 to 17-19 units, with slightly shallower circuits in noisy conditions. In our paper, we propose a hybrid implementation strategy that addresses hardware limitations, present empirical findings on the impact of quantum noise on factoring efficiency, and evaluate the cryptographic security implications under realistic conditions. Our findings underscore the need for fault-tolerant quantum computing and error correction, especially as optimizations suggest factoring 2048-bit RSA keys would require approximately 20 million qubits running for 8 hours. This highlights the urgency in transitioning to post-quantum cryptographic algorithms like CRYSTALS-Dilithium and CRYSTALS-Kyber being standardized by NIST.
Quantum computing has raised significant concerns regarding the security of classical cryptographic systems, particularly RSA, as it depends on the computational difficulty of factoring large semiprime numbers.In this study, we demonstrate the practical application of Shor's algorithm by successfully factorizing a semiprime integer, up to 30 bits (857830637 = 29167 X 29411) using the QuantumRings simulator.Our work highlights the effectiveness of Shor's algorithm in solving the factorization problem in polynomial time, which will significantly improve over the classical method for a large number.The results suggest that with advances in quantum hardware, such as increasing qubit counts and improving error correction, breaking larger RSA keys (e.g., RSA-2048) may soon become feasible.This seriously threatens current cryptographic systems, emphasizing the need to adopt post-quantum cryptography.Our findings aim to raise awareness among researchers, policymakers, and industry leaders of the importance of preparing for a quantum-safe future.
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 rapid advancement of quantum computing threatens to undermine the security foundations of modern cryptographic systems, particularly those based on integer factorization and discrete logarithm problems. Algorithms such as Shor’s and Grover’s promise exponential or quadratic speedups, rendering RSA, ECC, and certain symmetric schemes vulnerable within the next decade. This paper presents a simulated evaluation of classical cryptographic algorithms against quantum attacks, followed by a performance assessment of post-quantum cryptographic (PQC) schemes—specifically lattice-based, hash-based, and code-based algorithms. A testing framework was developed using a quantum simulator (Qiskit) and a classical benchmarking environment to model potential attack timelines and measure computational efficiency. Results demonstrate that RSA-2048 can be theoretically broken within 8 hours on a 4000-qubit fault-tolerant quantum processor, while ECC-P256 succumbs in less than 4 hours. In contrast, lattice-based schemes such as CRYSTALS-Kyber resisted quantum simulation attacks, maintaining equivalent classical security levels with only a 27% performance overhead. The study also analyzes the trade-offs between key size, encryption/decryption speed, and resistance to quantum attacks. Our findings emphasize the urgent need for migration to NIST-recommended PQC algorithms and provide a decision-support matrix for selecting suitable replacements in government and enterprise systems.
The advent of large-scale quantum computers poses an existential threat to the cryptographic algorithms that underpin modern digital security. Shor's algorithm enables quantum computers to factor large integers and compute discrete logarithms in polynomial time, rendering RSA, Elliptic Curve Cryptography (ECC), and Diffie-Hellman key exchange fundamentally insecure. Meanwhile, Grover's algorithm provides a quadratic speedup for brute-force search, effectively halving the security strength of symmetric ciphers and hash functions. This paper presents a comprehensive investigation into the impact of quantum computing on both asymmetric and symmetric cryptographic algorithms. We analyze the computational resources required by quantum attacks on current standards, evaluate the timeline of quantum threats based on hardware roadmaps from leading quantum computing companies, and conduct a systematic performance comparison of NIST-standardized post-quantum cryptographic (PQC) algorithms. Our experimental evaluation covers key generation time, encapsulation/decapsulation latency, signature generation/verification speed, key sizes, and bandwidth overhead across lattice-based, code-based, hash-based, and multivariate cryptographic schemes. Results indicate that lattice-based schemes, particularly CRYSTALS-Kyber and CRYSTALS-Dilithium, offer the most favorable balance of security and performance for practical deployment. We also propose a phased migration framework for transitioning existing cryptographic infrastructure to quantum-resistant alternatives, addressing compatibility, performance, and operational challenges.
Quantum computers pose a fundamental threat to widely deployed public-key cryptosystems, such as RSA and ECC, by enabling efficient integer factorization using Shor's algorithm. Theoretical resource estimates suggest that 2048-bit RSA keys could be broken using Shor's algorithm with fewer than a million noisy qubits. Although such machines do not yet exist, the availability of smaller, cloud-accessible quantum processors and open-source implementations of Shor's algorithm raises the question of what key sizes can realistically be factored with today's platforms. In this work, we experimentally investigate Shor's algorithm on several cloud-based quantum computers using publicly available implementations. Our results reveal a substantial gap between the capabilities of current quantum hardware and the requirements for factoring cryptographically relevant integers. In particular, we observe that circuit constructions still need to be highly specific for each modulus, and that machine fidelities are unstable, with high and fluctuating error rates.
This paper explores a quantum-inspired technique for factoring semi-prime integers, a problem central to the security of RSA cryptography. While Shor's algorithm offers a polynomial-time solution on a quantum computer, posing a significant threat to RSA, current quantum hardware limitations hinder its practical implementation. This study simulates Shor's algorithm classically, replacing the quantum period-finding subroutine—the source of the quantum speedup—with a classical method to analyze its performance. The experiments factored semi-primes ranging from 2 to 8 digits, demonstrating that while the technique works, the execution time increases exponentially with the size of the number, escalating from seconds for small integers to an order of minutes for an 8-digit number. This highlights the inefficiency of classical period-finding compared to the quantum approach. The results validate the theoretical framework of Shor's algorithm but underscore the impracticality of classical simulations for cryptographic-scale integers. This work also explores the necessity of advancing quantum computing to realize scalable factorization while classical methods remain confined to pedagogical or small-scale applications. The study emphasizes the urgency of hybrid quantum-classical optimizations and reinforces Shor's algorithm as pivotal for future cryptographic vulnerability assessments and quantum-ready solutions.