The emergence of quantum computers threatens current cryptographic systems, and NIST is preparing for the post-quantum era through the post-quantum cryptography (PQC) contest. CRYSTALS-Kyber is a lattice-based cipher suite that is used as a PQC standard. Lattice-based cryptography is considered quantum-safe for quantum computing because a quantum algorithm that can more efficiently solve the lattice problem of lattice-based cryptography compared to a classic algorithm has not been reported as yet. In this paper, we present quantum circuits tailored to NTT and inverse NTT, employed for optimized polynomial multiplication within CRYSTALS-Kyber. The proposed quantum circuits operate at Z3329[X]/(X256+1), which are the parameters of CRYSTALS-Kyber. We provide an in-depth description of the NTT/InvNTT quantum circuit’s operation and subsequently assess and analyze the quantum resources necessary for these functions. The NTT/InvNTT quantum circuits comprise four unique sub-functions, with the InvNTT additionally incorporating Barrett reduction. To the best of our knowledge, this represents the inaugural implementation of the CRYSTALS-Kyber NTT/InvNTT quantum circuits. We anticipate that our findings will aid in analyzing the security strengths of quantum computers for lattice-based cryptography.
The emergence of quantum computers threatens current cryptographic systems, and NIST is preparing for the post-quantum era through the post-quantum cryptography (PQC) contest. CRYSTALS-Kyber is a lattice-based cipher suite that is used as a PQC standard. Lattice-based cryptography is considered quantum-safe for quantum computing because a quantum algorithm that can more efficiently solve the lattice problem of lattice-based cryptography compared to a classic algorithm has not been reported as yet. In this paper, we present quantum circuits tailored to NTT and inverse NTT, employed for optimized polynomial multiplication within CRYSTALS-Kyber. The proposed quantum circuits operate at Z3329[X]/(X256+1), which are the parameters of CRYSTALS-Kyber. We provide an in-depth description of the NTT/InvNTT quantum circuit’s operation and subsequently assess and analyze the quantum resources necessary for these functions. The NTT/InvNTT quantum circuits comprise four unique sub-functions, with the InvNTT additionally incorporating Barrett reduction. To the best of our knowledge, this represents the inaugural implementation of the CRYSTALS-Kyber NTT/InvNTT quantum circuits. We anticipate that our findings will aid in analyzing the security strengths of quantum computers for lattice-based cryptography.
Powerful new computers based on the principles of quantum mechanics could crack the codes that safeguard our information today. The National Institute of Standards and Technology is running a competition to find more secure encryption methods to address this future challenge. CRYSTALS-Kyber is a messaging scramble Algorithm built with a complex structure called a lattice. This design is considered secure against powerful quantum computers because there is no known shortcut for them to break the code efficiently, unlike some other classical encryption Algorithms. In this paper, we introduce primitive information about quantum characteristics, computation, and main gate circuits, as well as the quantum theories for the design of post-quantum cryptosystems. Then, we listed the main key players in the quantum field and their achievements. We also compared the programming platforms used in developing programs in a quantum environment. Finally, we focused on cryptography applications and showed some of the mathematics background required to build knowledge for post-quantum cryptography algorithms, especially the CRYSTALS-Kyber algorithms and its core block Number Theoretic Transformation (NTT).
As quantum computing advances, modern cryptographic standards face an existential threat, necessitating a transition to post-quantum cryptography (PQC). The National Institute of Standards and Technology (NIST) has selected CRYSTALS-Kyber and CRYSTALS-Dilithium as standardized PQC algorithms for secure key exchange and digital signatures, respectively. This study conducts a comprehensive performance analysis of these algorithms by benchmarking execution times across cryptographic operations such as key generation, encapsulation, decapsulation, signing, and verification. Additionally, the impact of AVX2 optimizations is evaluated to assess hardware acceleration benefits. Our findings demonstrate that Kyber and Dilithium achieve efficient execution times, outperforming classical cryptographic schemes such as RSA and ECDSA at equivalent security levels. Beyond technical performance, the real-world deployment of PQC introduces challenges in telecommunications networks, where large-scale infrastructure upgrades, interoperability with legacy systems, and regulatory constraints must be addressed. This paper examines the feasibility of PQC adoption in telecom environments, highlighting key transition challenges, security risks, and implementation strategies. Through industry case studies, we illustrate how telecom operators are integrating PQC into 5G authentication, subscriber identity protection, and secure communications. Our analysis provides insights into the computational trade-offs, deployment considerations, and standardization efforts shaping the future of quantum-safe cryptographic infrastructure.
The security of classical public-key cryptosystems, which operate based on the computational intractability of integer factorization and discrete logarithm problems, is profoundly threatened by the rapid development of quantum computing. Post-Quantum Cryptography (PQC) has been proposed to improve security against quantum attacks. Lattice-based cryptography is widely regarded as one of the most promising PQC method because of its proven security, efficient implementation, and functional versatility, and it has achieved significant progress in both algorithmic design and engineering implementation in recent years. This paper presents a systematic survey of the status of research on lattice-based cryptographic algorithms. First, the fundamental theory of lattices and their core hard problems are discussed. Next, the technical characteristics and performance of encryption and key encapsulation mechanisms, digital signature schemes, and key exchange protocols constructed based on lattice hard problems are reviewed and compared. Subsequently, the principles and optimization paths of two key technologies for lattice cryptography implementation, namely polynomial multiplication using the fast Number Theoretic Transform (NTT) and pre-image sampling algorithms, are discussed. Finally, considering the PQC standardization process of the National Institute of Standards and Technology (NIST), an outlook on future research directions for lattice-based cryptography is presented. The research directions include the development of more "succinct" fundamental hard problems, strengthening security proofs, improving algorithm efficiency, and countering novel attacks. This review serves as a reference for promoting the practical deployment and further research of lattice-based cryptography.
Quantum computing threatens current public-key cryptographic systems, necessitating a transition to quantum-resistant alternatives. This paper analyzes how Shor's algorithm enables quantum computers to efficiently break RSA and Elliptic Curve Cryptography by solving their underlying mathematical problems, demonstrating these systems cannot be salvaged through parameter increases. We evaluate lattice-based cryptography as a post-quantum solution, examining its foundation on high-dimensional lattice problems such as the Shortest Vector Problem and Learning With Errors. Through analysis of NIST-standardized algorithms CRYSTALS-Kyber and CRYSTALS-Dilithium, we assess the security assumptions, computational efficiency, and implementation considerations of lattice-based schemes. Our findings reveal that while quantum computing poses an existential threat to current cryptographic infrastructure, lattice-based cryptography provides a viable quantum-resistant alternative with strong security guarantees and practical performance characteristics.
The advance of quantum computing poses significant risks to traditional cryptographic systems, including RSA and ECC, due to algorithms such as Shor's, prompting an urgent need for robust post-quantum cryptography (PQC) solutions. This paper presents a systematic review of PQC algorithms, encompassing lattice-based, code-based, multivariate polynomial, hash-based, and supersingular isogeny-based cryptography. By evaluating recent publications and NIST standardization efforts, we assess these approaches in terms of quantum resistance, computational efficiency, scalability, and implementation challenges. Lattice-based schemes, notably CRYSTALS-Kyber, excel in quantum-secure encryption and performance, whereas codebased systems like McEliece are hindered by large key sizes. Hash-based signatures, such as SPHINCS+, and isogeny-based protocols, including SIDH, provide promising alternatives but require further optimization to reduce computational overhead. Our analysis highlights the potential of hybrid PQC methods and lightweight hardware implementations for securing emerging domains like IoT, blockchain, and financial systems. The review serves as a foundational resource for researchers and practitioners, advocating the adoption of quantum-resistant cryptography to preserve data confidentiality, integrity, and authenticity in the quantum era.
The advent of quantum computing poses a significant existential threat to current public-key cryptographic systems, most notably RSA and Elliptic Curve Cryptography (ECC), which rely on the presumed hardness of integer factorization and discrete logarithm problems. Shor's algorithm, a quantum algorithm, can efficiently solve these problems, rendering classical asymmetric primitives insecure. This paper addresses the imperative need for post-quantum cryptography (PQC) by exploring lattice-based constructs as robust alternatives for asymmetric primitives. We provide a comprehensive overview of the fundamental mathematical problems underlying lattice-based cryptography, such as the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Learning With Errors (LWE), and Ring Learning With Errors (RLWE). The paper surveys prominent lattice-based schemes, including NTRU, CRYSTALS-Kyber, and CRYSTALS-Dilithium, analyzing their security foundations, performance characteristics, and the extent to which they offer a viable "post-quantum RSA." We delve into the methodology of constructing these primitives, discuss their resilience against both classical and quantum attacks, and evaluate their practical implications for key sizes, computational efficiency, and deployment challenges. Our analysis highlights that lattice-based cryptography, particularly the NIST-standardized schemes, presents a strong candidate for securing digital communications in the quantum era, offering provable security rooted in worst-case to average-case reductions for lattice problems.