Research Proposal: Analyzing the Vulnerability of RSA Encryption in Quantum Computing Environments

Introduction

Background and Significance

The digital infrastructure of the modern global economy relies heavily on public-key cryptography to secure communications, financial transactions, and data integrity. At the core of this cryptographic framework is the RSA (Rivest-Shamir-Adleman) algorithm, which has served as the de facto standard for asymmetric encryption for decades. The security of RSA is predicated on the computational intractability of integer factorization; specifically, classical computers require sub-exponential or exponential time to factorize the product of two large prime numbers. However, the emergence of quantum computing represents a profound paradigm shift that threatens to dismantle this foundational security assumption. Unlike classical bit-based computing, quantum computing leverages the principles of quantum mechanics—namely, superposition and entanglement—to perform complex computations at unprecedented speeds. In 1994, mathematician Peter Shor introduced a quantum algorithm capable of solving the integer factorization problem in polynomial time. Consequently, the advent of a sufficiently powerful, fault-tolerant quantum computer executing Shor’s algorithm would render current RSA key lengths obsolete, exposing sensitive digital communications to severe retroactive and proactive interception (Sharma et al., 2020).

Research Objectives

The primary objective of this research is to comprehensively analyze the vulnerability thresholds of standard RSA encryption protocols when subjected to simulated quantum computing environments. Specifically, the study aims to mathematically and empirically model the application of Shor’s algorithm against varying RSA key sizes to determine the exact inflection point of cryptographic failure. Secondary objectives include evaluating the practical feasibility of executing these quantum attacks, considering the current and projected constraints of Noisy Intermediate-Scale Quantum (NISQ) devices, such as qubit coherence times, gate fidelity, and error correction overhead. By rigorously modeling these parameters, this research seeks to provide a definitive timeline and resource estimation for the quantum compromise of RSA. Ultimately, the significance of addressing these vulnerabilities lies in facilitating a preemptive and structured migration toward Post-Quantum Cryptography (PQC), ensuring the continuous security of global communication architectures before cryptanalytically relevant quantum computers (CRQCs) become a reality (Kayode, 2025).

Literature Review

Overview of RSA Encryption

The RSA cryptosystem, introduced in 1977, remains ubiquitous in establishing secure channels over insecure networks, forming the backbone of protocols such as TLS/SSL, SSH, and PGP. The mathematical foundation of RSA relies heavily on number theory. The key generation process begins with the selection of two distinct, large prime numbers, $p$ and $q$. Their product forms the modulus $N = p \cdot q$, which dictates the key length. The public exponent $e$ and the private exponent $d$ are derived using Carmichael’s totient function or Euler’s totient function $\phi(N) = (p-1)(q-1)$, ensuring that the encryption and decryption functions are mathematical inverses of one another. An extensive review of classical cryptanalysis literature indicates that despite relentless scrutiny, RSA has proven resilient against classical attacks, provided that adequately large key sizes (e.g., 2048-bit or 4096-bit) are employed. Classical factoring methods, such as the General Number Field Sieve (GNFS), exhibit sub-exponential time complexities, making the brute-force derivation of the private key practically impossible with conventional supercomputers (Riesel, 1985).

Quantum Computing and Shor’s Algorithm

Quantum computing diverges fundamentally from classical computing architecture. While classical systems process discrete bits (0 or 1), quantum systems utilize quantum bits, or qubits, which exist in multidimensional state spaces until measured. This unique property allows quantum algorithms to evaluate vast numbers of computational paths simultaneously. Shor’s algorithm leverages these properties to execute prime factorization exponentially faster than the GNFS. The algorithm utilizes the Quantum Fourier Transform (QFT) to perform period finding—a classically intractable task—reducing the factorization problem to calculating the period of a modular exponential function. Previous theoretical studies indicate that factoring a 2048-bit RSA modulus requires exactly $2n$ qubits and $O(n^3)$ quantum logic gates. However, contemporary literature regarding the application of Shor’s algorithm highlights a significant disparity between theoretical requirements and physical reality. Current studies emphasize that executing such an algorithm necessitates robust Quantum Error Correction (QEC), inflating the physical qubit requirement into the millions. This literature review synthesizes these findings, establishing a baseline for the resource estimations required to execute successful cryptographic attacks (Ahmed, 2024).

Methodology

Research Design

This study will employ a mixed-methods research design, combining rigorous theoretical analysis with advanced computational simulation. Due to the unavailability of fault-tolerant, large-scale quantum computers, the empirical component will rely on simulating quantum environments using state-of-the-art frameworks, such as IBM’s Qiskit and Google’s Cirq. The design will involve scaling down the RSA modulus to manageable bit lengths (e.g., 8-bit to 32-bit integers) that can be factored using available quantum hardware and high-performance classical quantum simulators. By systematically executing Shor’s algorithm on these scaled-down moduli, the research will map the algorithmic behavior and resource consumption. The criteria for evaluating RSA encryption vulnerability will be based on three primary metrics: asymptotic time complexity scaling, circuit depth requirements, and logical-to-physical qubit ratios necessary to achieve a computational fidelity threshold exceeding 99% (Kayode, 2025).

Data Collection and Analysis

Data collection will occur systematically through iterative executions of the quantum period-finding circuits across varying modulus sizes. The data points will include the number of required quantum gates (specifically T-gates and Toffoli gates), the overall circuit depth, error rates induced by environmental decoherence, and the algorithmic success probability per execution cycle. To analyze the impact of Shor’s algorithm, the study will utilize extrapolation models to project the empirical data gathered from small-scale simulations onto standard cryptographic key sizes (e.g., RSA-1024, RSA-2048). Statistical techniques, including multivariate regression analysis and Monte Carlo simulations, will be employed to model the probabilistic nature of quantum measurements and to establish confidence intervals regarding the estimated timeline for RSA deprecation (Devitt et al., 2004).

Expected Outcomes

Anticipated Findings

The research is expected to quantitatively demonstrate the precise resource threshold at which RSA encryption transitions from theoretically vulnerable to practically obsolete. It is anticipated that the findings will reveal a non-linear scaling in the physical qubit requirements due to the immense overhead introduced by quantum error correction codes, particularly surface codes. By projecting these requirements against the current trajectory of quantum hardware development, the study will likely forecast a critical window of vulnerability (often referred to as “Y2Q”). These findings will carry profound implications for current cryptographic practices, underscoring the immediate inadequacy of simply increasing RSA key lengths (e.g., migrating from 2048-bit to 4096-bit), as such measures only offer a marginal, linear delay against an exponentially powerful quantum attack. This will unequivocally highlight the urgent necessity for the rapid standardization and deployment of quantum-resistant encryption methods (Seeburrun et al., 2024).

Contribution to the Field

This study will provide a significant contribution to the disciplines of quantum computer science and cybersecurity by bridging the gap between theoretical quantum algorithms and practical cryptographic risk assessment. By delivering highly accurate, empirical extrapolations of Shor’s algorithm within realistic, noisy quantum environments, the research will advance the academic understanding of cryptographic vulnerabilities. Furthermore, these findings will serve as a crucial reference point for industry leaders, government agencies, and standards bodies (such as NIST) in establishing definitive timelines for the deprecation of legacy public-key infrastructure. Ultimately, this work will catalyze future research and development in post-quantum cryptography, providing a scientifically rigorous justification for the extensive financial and infrastructural investments required to secure the next generation of global communication networks (Ajala et al., 2024).

Conclusion

Summary of Key Points

In summary, this research proposal outlines a comprehensive framework for investigating the critical vulnerabilities of RSA encryption when subjected to the transformative capabilities of quantum computing. By focusing on the mechanics and resource demands of Shor’s algorithm, the study contextualizes the impending threat to the mathematical foundations of modern digital security. The proposed methodology leverages both theoretical analysis and advanced quantum simulation to gather empirical data on circuit depth, qubit requirements, and error correction overhead. The anticipated contributions include providing precise vulnerability thresholds and realistic timelines, thereby emphasizing the urgent need to transition away from classical public-key infrastructure (Rama Krishna et al., 2025).

Future Research Directions

The findings of this study will naturally pave the way for several vital avenues of future research. Chief among these is the performance and optimization of Post-Quantum Cryptography (PQC) algorithms, particularly lattice-based and hash-based cryptographic systems, as direct replacements for RSA. Future research must evaluate the computational overhead, key sizes, and energy consumption of these PQC algorithms when integrated into existing classical network protocols. Additionally, exploring the implementation of “hybrid encryption schemes”—where classical algorithms like RSA are combined with PQC algorithms to provide a transitional security layer—will be critical. Ongoing research in these domains is imperative to ensure a seamless, secure, and resilient transition to a post-quantum digital ecosystem (Pasupuleti, 2025).

References

Ahmed, D. (2024). Quantum Computing Algorithms for Integer Factorization: A Comparative Analysis. Modern Dynamics: Mathematical Progressions, 1(1), 6–9. https://doi.org/10.36676/mdmp.v1.i1.02
Ajala, O., Arinze, C., Ofodile, O., Okoye, C., & Daraojimba, A. (2024). Exploring and reviewing the potential of quantum computing in enhancing cybersecurity encryption methods. Magna Scientia Advanced Research and Reviews, 10(1), 321–329. https://doi.org/10.30574/msarr.2024.10.1.0038
Devitt, S., Fowler, A., & Hollenberg, L. (2004). Robustness of Shor’s algorithm. In arXiv (Cornell University). Technische Universitat Dresden. https://doi.org/10.48550/arxiv.quant-ph/0408081
Kayode, B. (2025). Qiskit-Based Simulation of Quantum Attacks Against RSA Encryption. International Journal of Latest Technology in Engineering Management & Applied Science, 14(5), 592–596. https://doi.org/10.51583/ijltemas.2025.140500061
Pasupuleti, M. K. (2025). Post-Quantum Cryptography: Algorithms and Implementation Challenges. International Journal of Academic and Industrial Research Innovations(IJAIRI), 05(06), 234–243. https://doi.org/10.62311/nesx/rphcrcscrbc4
Rama Krishna, M. S., Krishnarao, P., & Srinubabu, M. (2025). Evaluating the Impact of Quantum Algorithms on Modern Cybersecurity Mechanisms. INTERNATIONAL JOURNAL OF SCIENTIFIC RESEARCH IN ENGINEERING AND MANAGEMENT, 09(06), 1–8. https://doi.org/10.55041/ijsrem.ncft016
Riesel, H. (1985). Prime Numbers and Cryptography (pp. 223–236). Birkhauser Boston. https://doi.org/10.1007/978-1-4757-1089-2_6
Seeburrun, K., Veerabudren, K., Sharma, M., & Bekaroo, G. (2024). Demystifying Cryptography: An Experimental Study of Classical and Quantum Cryptography. 1–7. https://doi.org/10.1109/elecom63163.2024.10892169
Sharma, M., Choudhary, V., Bhatia, R. S., Malik, S., Raina, A., & Khandelwal, H. (2020). Leveraging the power of quantum computing for breaking RSA encryption. Cyber-Physical Systems, 7(2), 73–92. https://doi.org/10.1080/23335777.2020.1811384