The security of current public-key cryptosystems relies on the hardness of factoring large integers or solving discrete logarithm problems. However, these mathematical problems can be solved in polynomial time using a quantum computer. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. In this expository article, we give an overview of recent research on post-quantum cryptography. In particular, we describe the construction and security of multivariate polynomial cryptosystems and lattice-based cryptosystems, which are the main candidates of post-quantum cryptography.
The security of current public-key cryptosystems relies on the hardness of factoring large integers or solving discrete logarithm problems. However, these mathematical problems can be solved in polynomial time using a quantum computer. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. In this expository article, we give an overview of recent research on post-quantum cryptography. In particular, we describe the construction and security of multivariate polynomial cryptosystems and lattice-based cryptosystems, which are the main candidates of post-quantum cryptography.
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.
The advent of quantum computing poses a significant threat to current cryptographic systems, particularly public-key algorithms such as RSA and ECC, which are vulnerable to quantum attacks like Shor's algorithm. Post-Quantum Cryptography (PQC) aims to develop and standardize quantum-resistant cryptographic algorithms to secure digital communications in the quantum era. This paper examines the current landscape of PQC, focusing on the evaluation and standardization efforts led by organizations such as the National Institute of Standards and Technology (NIST). We analyze the major families of quantum-resistant algorithms, including lattice-based, hash-based, code-based, multivariate, and isogeny-based cryptography, discussing their security guarantees, performance trade-offs, and implementation challenges. Furthermore, we review NIST's PQC standardization process, highlighting selected finalists and alternate candidates, and explore the ongoing transition challenges in adopting these algorithms across industries. Finally, we discuss future directions for PQC research, including hybrid cryptographic systems and the need for continued cryptanalysis to ensure long-term security. This work provides a comprehensive overview of the progress and challenges in deploying quantum-resistant cryptography to safeguard against emerging quantum threats.
Information security is a fundamental and urgent issue in the digital transformation era. Cryptographic techniques and digital signatures have been applied to protect and authenticate relevant information. However, with the advent of quantum computers and quantum algorithms, classical cryptographic techniques have been in danger of collapsing because quantum computers can solve complex problems in polynomial time. Stemming from that risk, researchers worldwide have stepped up research on post-quantum algorithms to resist attack by quantum computers. In this review paper, we survey studies in recent years on post-quantum cryptography (PQC) and provide statistics on the number and content of publications, including a literature overview, detailed explanations of the most common methods so far, current implementation status, implementation comparisons, and discussion on future work. These studies focused on essential public cryptography techniques and digital signature schemes, and the US National Institute of Standards and Technology (NIST) launched a competition to select the best candidate for the expected standard. Recent studies have practically implemented the public key encryption/key encapsulation mechanism (PKE/KEM) and digital signature schemes on different hardware platforms and applied various optimization measures based on other criteria. Along with the increasing number of scientific publications, the recent trend of PQC research is increasingly evident and is the general trend in the cryptography industry. The movement opens up a promising avenue for researchers in public key cryptography and digital signatures, especially on algorithms selected by NIST.
The development of large quantum computers will have dire consequences for cryptography. Most of the symmetric and asymmetric cryptographic algorithms are vulnerable to quantum algorithms. Grover's search algorithm gives a square root time boost for the searching of the key in symmetric schemes like AES and 3DES. The security of asymmetric algorithms like RSA, Diffie Hellman, and ECC is based on the mathematical hardness of prime factorization and discrete logarithm. The best classical algorithms available take exponential time. Shor's factoring algorithm can solve the problems in polynomial time. Major breakthroughs in quantum computing will render all the present-day widely used asymmetric cryptosystems insecure. This paper analyzes the vulnerability of the classical cryptosystems in the context of quantum computers discusses various post-quantum cryptosystem families, discusses the status of the NIST post-quantum cryptography standardization process, and finally provides a couple of future research directions in this field.
The need and emergence of quantum computing under National Quantum Mission threatens the cyber-security of existing cryptographic protocols, especially public-key schemes built on integer factorization and discrete logarithms. Lattice-based cryptography has gained prominence as a robust candidate for post-quantum cryptography—offering security against quantum attacks while enabling efficient operations, parallelization, and novel functionalities. This paper analyzes the mathematical foundations of lattices, explores hardness assumptions underlying widely studied lattice problems, highlights prominent cryptographic constructions, analyzes performance trade-offs, and reviews standardization efforts. This paper concludes by identifying ongoing issues and future directions for lattice-based cryptographic systems in a quantum-secure digital world. The algorithm has been implemented in Python.
The development of large quantum computers will have dire consequences for cryptography. Most of the symmetric and asymmetric cryptographic algorithms are vulnerable to quantum algorithms. Grover’s search algorithm gives a square root time boost for the searching of the key in symmetric schemes like AES and 3DES. The security of asymmetric algorithms like RSA, Diffie Hellman, and ECC is based on the mathematical hardness of prime factorization and discrete logarithms. The best classical algorithms available take exponential time. Shor’s factoring algorithm can solve the problems in polynomial time. Major breakthroughs in quantum computing will render all the present-day widely used asymmetric cryptosystems insecure. This paper analyzes the vulnerability of the classical cryptosystems in the context of quantum computers, discusses various post-quantum cryptosystem families, discusses the status of the NIST post-quantum cryptography standardization process, provides a performance comparison of post-quantum cryptography (PQC) algorithms on different platforms, and finally provides a couple of future research directions in this field.