Quantum computing furnishes exponential speed up over classical computing in specific areas. For example, Shor’s algorithm can factor two numbers in a polynomial time complexity. Thus, many encryption algorithms that rely on large number factorization are potentially vulnerable to quantum computers. In order to address this, the National Institute of Standard and Test (NIST) has organized a competition to evaluate several post quantum cryptography (PQC) algorithms, that are secure from the attacks from quantum computers. Several of these lattice-based PQC encryption algorithms are based on Learning With Errors (LWE) computation. Conversely, LWE is the heaviest computation in a classical computer, which incurs significant portion of the latency overhead for the entire encryption algorithm. In this paper, we design an optimized quantum circuit for LWE computation. The proposed quantum circuit does not need any ancillary qubits and scales efficiently and easily if there are more qubits available on a higher qubit quantum computer.
Quantum computing furnishes exponential speed up over classical computing in specific areas. For example, Shor’s algorithm can factor two numbers in a polynomial time complexity. Thus, many encryption algorithms that rely on large number factorization are potentially vulnerable to quantum computers. In order to address this, the National Institute of Standard and Test (NIST) has organized a competition to evaluate several post quantum cryptography (PQC) algorithms, that are secure from the attacks from quantum computers. Several of these lattice-based PQC encryption algorithms are based on Learning With Errors (LWE) computation. Conversely, LWE is the heaviest computation in a classical computer, which incurs significant portion of the latency overhead for the entire encryption algorithm. In this paper, we design an optimized quantum circuit for LWE computation. The proposed quantum circuit does not need any ancillary qubits and scales efficiently and easily if there are more qubits available on a higher qubit quantum computer.
The imminent threat posed by the realization of large-scale, fault-tolerant quantum computers (QCs), capable of leveraging Shor's algorithm to break foundational classical public-key cryptosystems like RSA and ECC, necessitates an immediate and comprehensive shift to Post-Quantum Cryptography (PQC). This urgency is intensified by the "Harvest Now, Decrypt Later" (HNDL) threat, which compromises long-term data confidentiality. The NIST PQC Standardization process has established Lattice-Based Cryptography as the leading family for new standards. This paper presents a comprehensive comparative analysis of major PQC schemes built upon the foundational mathematical problem, Learning with Errors (LWE). We systematically evaluate these LWE-based schemes (CRYSTALS-Kyber, CRYSTALS-Dilithium) by balancing three critical dimensions: cryptographic security (against classical and quantum attacks), efficiency metrics (key size, signature size, and operational speed), and implementation feasibility across diverse hardware environments. Our findings provide essential insights and a structured framework to guide engineers, policy-makers, and organizations in making informed decisions for the strategic and secure migration to quantum-resistant standards in the post-quantum transition era.
Quantum computing has advanced rapidly in recent decades, driven by developments across the technology stack, including quantum error-correcting codes and efficient quantum algorithms. Among these, quantum arithmetic circuits serve as fundamental building blocks for various promising algorithms. Despite their crucial role, the design of quantum arithmetic circuits faces challenges arising from the no-cloning theorem, qubit limitations, and circuit depth constraints, which significantly impact the efficiency of large-scale quantum computing. We provide an overview of quantum arithmetic circuits in the context of public-key cryptanalysis, with particular emphasis on optimization strategies such as measurement-based uncomputation and conditionally clean ancilla. We review state-of-the-art designs for essential arithmetic operations in public-key cryptanalysis such as addition, multiplication, and modular exponentiation. We also present an overview of the techniques used for fault-tolerant runtime and resource estimation in quantum cryptanalysis. In brief, this chapter emphasizes strategies for designing resource-efficient quantum arithmetic circuits, providing a basis for realistic evaluations of quantum cryptanalytic capabilities.
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.
The threats are existential to the traditional cryptosystems of public keys, including RSA, Diffie-Hellman, and elliptic curve cryptography (ECC), due to the rapid progress in quantum computing. Such algorithms as the Shor and Grover algorithms prove that as soon as large scale quantum computers are achieved, these classical cryptographic protocols will not provide any privacy and integrity to online communication. This has created an emergency to find post-quantum cryptography (PQC), or quantum-safe cryptography, which will resist attacks by both classical and quantum attackers. The paper is a review of the latest research and development in post-quantum cryptographic algorithms and their implementation in current security systems. We organize quantum-safe methods into significant organizations such as lattice-based, hash-based, multivariate polynomial-based and code-based encrypted communications. Each type has been evaluated critically based on computational efficiency, key sizes, speed of encryption/decryption and security proofs. The survey also examines the current standardization activities happening in the world and especially the U.S. National Institute of Standards and Technology (NIST) PQC competition, which lists candidate algorithms that have advanced to standard adoption. Besides this, the paper addresses hybrid cryptographic designs that use both classical and post-quantum bases in ensuring a seamless migration in the transition phase. Internet security protocols like TLS, VPNs and blockchain consensus as applications of PQC are examined in detail. Lastly, we pinpoint open research issues, which include creating lightweight PQC to support resource-constrained IoT devices, balancing performance and strong security assurances, and interoperability between the new PQC algorithms and existing infrastructures. Through detailed survey we inform the future of researchers, practitioners, and policymakers on the preparation of the post-quantum era of secure digital communications.
The imminent arrival of quantum computing has accelerated the need for cryptographic systems resistant to quantum attacks. Such attacks exploit the vulnerability in private and public key encryption systems, where the public key is derived from the private key, which could be refactored from the public key. To address this issue, the National Institute of Standards and Technology (NIST) launched a global competition in 2016 to create quantum-resistant algorithms. CRYSTALS-Kyber, a lattice-based algorithm focused on the learning with errors (LWE) problem, was selected for standardization. This work introduces RKyber, a variant that instead targets the learning with rounding (LWR) problem, simplifying computations by using deterministic errors rather than random noise. Both algorithms were executed 1000 times, showing that RKyber offers improved speed at the cost of some security.
The swift advancement of quantum computing devices holds the potential to create robust machines that can tackle an extensive array of issues beyond the scope of conventional computers. Consequently, quantum computing machines create new risks at a velocity and scale never seen before, especially with regard to encryption. Lattice-based cryptography is regarded as post-quantum cryptography’s future and a competitor to a quantum computer attack. Thus, there are several advantages to lattice-based cryptographic protocols, including security, effectiveness, reduced energy usage and speed. In this work, we study the learning with errors (LWE) problem and the cryptosystems that are based on the LWE problem and, in addition, we present a new efficient variant of LWE cryptographic scheme.