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Journal of Cybersecurity and Privacy

A Novel Hybrid Quantum Circuit for Integer Factorization: End-to-End Evaluation in Simulation and Real Quantum Hardware

Jesse Van Griensven Thé, Victor Oliveira Santos, Bahram Gharabaghi

Published Apr 10, 2026• 0 citations
Publisher

Abstract

The literature indicates that the qubit requirements for factoring RSA-2048 remain on the order of 1 million, under commonly assumed architectures and error-correction models, leaving a substantial gap between current resource estimates and near-term practical feasibility. Reducing this requirement to the low thousands qubit regime therefore remains an important open research objective. This work proposes a hybrid classical-quantum algorithm using a classical modular exponentiation subroutine with a Quantum Number Theoretic Transform (QNTT) circuit, to increase the speed and reduce the number of quantum components, including gates and qubits, to factor integer numbers, which serve as keys in cryptographic methods, like RSA and ECC, when compared with Shor’s algorithm. Several composite numbers, the result of multiplication of two primes, were validated through both simulation and real quantum hardware by benchmarking the full Shor pipeline on simulation and on a real IBM quantum computer. In simulation, the proposed Jesse–Victor–Gharabaghi (JVG) algorithm achieved substantial practical reductions in computational resources, decreasing runtime from 174.1 s to 5.4 s, memory usage from 12.5 GB to 0.27 GB, and quantum gate counts by approximately 99%. Because Shor and JVG use different register sizes for the same composite N, the reported gate/depth reductions should be interpreted as end-to-end quantum-resource budgets to factor the same N, rather than a per-qubit or transform-only efficiency claim. On quantum hardware, JVG reduced the required runtime from 67.8 s to 2 s, and the quantum gate counts by over 98%. Projection for RSA-2048 indicates that the JVG algorithm significantly outperforms Shor’s approach, requiring a projected quantum runtime of 11 hours for a factorization under identical scaling assumptions. The results from these evaluations support JVG as a more hardware-compatible and robust noise-tolerant substitute for Shor’s framework, offering a viable path toward practical quantum integer factorization on near-term Noisy Intermediate-Scale Quantum (NISQ) devices.

DOI
10.3390/jcp6020071

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The literature indicates that the qubit requirements for factoring RSA-2048 remain on the order of 1 million, under commonly assumed architectures and error-correction models, leaving a substantial gap between current resource estimates and near-term practical feasibility. Reducing this requirement to the low thousands qubit regime therefore remains an important open research objective. This work proposes a hybrid classical-quantum algorithm using a classical modular exponentiation subroutine with a Quantum Number Theoretic Transform (QNTT) circuit, to increase the speed and reduce the number of quantum components, including gates and qubits, to factor integer numbers, which serve as keys in cryptographic methods, like RSA and ECC, when compared with Shor’s algorithm. Several composite numbers, the result of multiplication of two primes, were validated through both simulation and real quantum hardware by benchmarking the full Shor pipeline on simulation and on a real IBM quantum computer. In simulation, the proposed Jesse–Victor–Gharabaghi (JVG) algorithm achieved substantial practical reductions in computational resources, decreasing runtime from 174.1 s to 5.4 s, memory usage from 12.5 GB to 0.27 GB, and quantum gate counts by approximately 99%. Because Shor and JVG use different register sizes for the same composite N, the reported gate/depth reductions should be interpreted as end-to-end quantum-resource budgets to factor the same N, rather than a per-qubit or transform-only efficiency claim. On quantum hardware, JVG reduced the required runtime from 67.8 s to 2 s, and the quantum gate counts by over 98%. Projection for RSA-2048 indicates that the JVG algorithm significantly outperforms Shor’s approach, requiring a projected quantum runtime of 11 hours for a factorization under identical scaling assumptions. The results from these evaluations support JVG as a more hardware-compatible and robust noise-tolerant substitute for Shor’s framework, offering a viable path toward practical quantum integer factorization on near-term Noisy Intermediate-Scale Quantum (NISQ) devices.
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 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.
Shor's algorithm has seriously challenged information security based on public key cryptosystems. However, to break the widely used RSA-2048 scheme, one needs millions of physical qubits, which is far beyond current technical capabilities. Here, we report a universal quantum algorithm for integer factorization by combining the classical lattice reduction with a quantum approximate optimization algorithm (QAOA). The number of qubits required is O(logN/loglog N), which is sublinear in the bit length of the integer NNN, making it the most qubit-saving factorization algorithm to date. We demonstrate the algorithm experimentally by factoring integers up to 48 bits with 10 superconducting qubits, the largest integer factored on a quantum device. We estimate that a quantum circuit with 372 physical qubits and a depth of thousands is necessary to challenge RSA-2048 using our algorithm. Our study shows great promise in expediting the application of current noisy quantum computers, and paves the way to factor large integers of realistic cryptographic significance.
Shor’s algorithm stands as a breakthrough in quantum computing due to its potential to factor large integers exponentially quicker than classical algorithms. However, implementing and evaluating this algorithm on real quantum computer hardware remains exciting due to qubit limitations, gate noise, and hardware constraints. This research presents a comprehensive performance evaluation of Shor’s algorithm using simulated quantum backends provided by Qiskit. A flexible and generic implementation is proposed, allowing dynamic input of integers to be factored, with randomized co-prime selection and automated circuit generation. The algorithm is tested on various semiprime numbers, such as 15, 21, and 35, using IBM’s Aer simulator. A major contribution of this work is the circuit-level analysis conducted both before and after transpilation. Metrics such as gate counts, circuit depth, and simulator runtime are extracted to assess scalability and resource requirements. High-resolution plots of the pre-transpiled circuits are saved to visualize algorithmic complexity, while post-transpilation metrics inform future quantum hardware feasibility. The output measurement distributions are analyzed to estimate periodicity and derive correct factors. The proposed implementation is compared with existing fixed-instance Shor demonstrations to highlight its flexibility and extensibility. Experimental results show consistent success in factor retrieval and provide valuable insight into circuit growth and complexity under realistic constraints. This analysis lays the groundwork for future adaptation to NISQ hardware and contributes to understanding Shor’s algorithm from both computational and architectural perspectives.
Factoring large integers has long been a computationally difficult problem in classical computing, forming the foundation of widely used encryption methods like RSA. In 1994, Shor’s quantum algorithm introduced a revolutionary method for factoring integers exponentially faster than classical algorithms, laying the groundwork for quantum cryptography. Despite numerous attempts to enhance Shor’s algorithm over the last three decades, significant breakthroughs remained elusive until Regev’s discovery in 2023. Regev introduced a novel, multi-dimensional version of the quantum factoring algorithm, achieving a substantial improvement by reducing the required number of quantum gates to O(n^2 lo g⁡n )(n^(3/2) log⁡n ), compared to Shor’s originalO(n^2 lo g⁡n )original O(n^2 log⁡n ) gate complexity. Although Regev’s approach offers a significant speedup, it comes with increased qubit requirements and relies on an unproven number theory assumption. This paper presents an overview of Regev’s factoring algorithm, including a review of Shor’s work for context, followed by an examination of key recent developments and follow-up research. These include efforts to reduce the qubit count, improve error resilience, generalizegeneralise Regev’s algorithm to related problems, and validate the number theory assumption. This review serves as an accessible entry point for researchers interested in the rapidly evolving field of quantum computing and factoring algorithms.