As quantum computing advances, classical public-key cryptosystems face escalating risks, driving global standardization efforts by NIST, ETSI, and ISO. This paper provides a comprehensive review of recent developments in post-quantum cryptography (PQC), focusing on lattice-based, hash-based, and multivariate approaches. We examine their security foundations, implementation challenges, and applicability across domains such as finance, blockchain, and the Internet of Things (IoT). A core contribution is the proposed Hybrid Cryptographic Framework (HCF), which combines classical and quantum-resistant primitives to enable a secure and interoperable migration path. The framework integrates hybrid key exchange, dual-signature schemes, and PQC-compatible Merkle trees, and is supported by a discrete multi-objective optimization model for selecting algorithms under real-world constraints. The paper also discusses deployment challenges, including hardware acceleration, side-channel resistance, and legacy compatibility. Real-world case studies illustrate how leading institutions are piloting PQC integration in practice. Finally, we propose a strategic roadmap for PQC adoption, incorporating adaptive triggers based on quantum capabilities and evolving threat models. By aligning cryptographic design with international standards and practical system requirements, this work offers guidance for building resilient, quantum-safe infrastructures.

The RSA cryptosystem has been a cornerstone of modern public key infrastructure; however, recent advancements in quantum computing and theoretical mathematics pose significant risks to its security. The advent of fully operational quantum computers could enable the execution of Shor’s algorithm, which efficiently factors large integers and undermines the security of RSA and other cryptographic systems reliant on discrete logarithms. While Grover’s algorithm presents a comparatively lesser threat to symmetric encryption, it still accelerates key search processes, creating potential vulnerabilities. In light of these challenges, there has been an intensified focus on developing quantum-resistant cryptography. Current research is exploring cryptographic techniques based on error-correcting codes, lattice structures, and multivariate public key systems, all of which leverage the complexity of NP-hard problems, such as solving multivariate quadratic equations, to ensure security in a post-quantum landscape. This paper reviews the latest advancements in quantum-resistant encryption methods, with particular attention to the development of robust trapdoor functions. It also provides a detailed analysis of prominent multivariate cryptosystems, including the Matsumoto–Imai, Oil and Vinegar, and Polly Cracker schemes, alongside recent progress in lattice-based systems such as Kyber and Crystals-DILITHIUM, which are currently under evaluation by NIST for potential standardization. As the capabilities of quantum computing continue to expand, the need for innovative cryptographic solutions to secure digital communications becomes increasingly critical.

In the current landscape where quantum algorithms pose a significant threat to conventional digital signature algorithms, code-based digital signature algorithms have emerged as the primary focus of ongoing research in post-quantum cryptography. Digital signatures play a pivotal role in ensuring non-repudiation and authentication, making them an indispensable cryptographic technique. The vulnerability of most digital signature algorithms to quantum attacks have prompted a significant surge in research on code-based digital signature algorithms, which have emerged as a prominent field within post-quantum cryptography. There are generally three distinct approaches to constructing code-based digital signature algorithms: (1) Developing an algorithm that follows the inverse process of the code-based public-key encryption algorithm; (2) Utilizing zero-knowledge identification algorithms in conjunction with the Fiat–Shamir paradigm to formulate a signature algorithm; (3) Constructing a specialized subset of the syndrome space as the foundation for the digital signature algorithm. Chameleon Signature is a non-interactive signature that operates on the hash and signature paradigm, exhibiting comparable efficiency to conventional schemes. Its distinct advantage lies in the fact that the owner of the public key does not necessarily require access to the corresponding secret key within the Chameleon hash algorithm. Notably, Chameleon signatures possess an inherent characteristic of non-transferability, with their validity ascertainable solely by designated recipients. This paper introduces the first Chameleon hash function based on both KKS and HFE schemes, showcasing its superiority over traditional schemes through rank metrics and big fields for enhanced security. The deployment of Chameleon hash functions within hash-and-sign signature schemes introduces a nuanced layer of security and verification flexibility. This study elucidates the implications of integrating Chameleon hash functions into the recipient’s public key infrastructure, highlighting the dual capability it affords authorized parties for secure and adaptable verification processes, alongside mechanisms for the detection of unauthorized alterations.

The RSA cryptosystem has long been a fundamental component of contemporary public key in- frastructure. However, recent developments in quantum computing and mathematical theory have introduced signicant challenges to its security. A fully operational quantum computer would allow for the application of Shor's algorithm, enabling the ecient factoring of large integers, thereby com- promising the integrity of RSA and other cryptographic methods dependent on discrete logarithms. Although Grover's algorithm poses a comparatively smaller threat to symmetric encryption systems, it still presents a vulnerability by accelerating key search processes. The looming threat from quan- tum technologies has driven a surge in research aimed at developing quantum-resistant cryptography. These eorts focus on cryptographic techniques grounded in error-correcting codes, lattice structures, and multivariate public key systems, all of which leverage the complexity of NP-hard problemssuch as solving multivariate quadratic equationsto preserve security in a post-quantum world. This pa- per reviews the current progress in quantum-resistant encryption methods, particularly emphasizing the role of robust trapdoor functions. Additionally, it provides a comprehensive analysis of critical multivariate cryptosystem frameworks, such as Matsumoto-Imai, Oil and Vinegar, and Polly Cracker schemes. It also examines advances in lattice-based systems like Kyber and Crystals-Dilithium, which are undergoing assessment by NIST for potential standardization. As quantum computing continues to evolve, the demand for cutting-edge cryptographic solutions to safeguard digital communications grows increasingly pressing.

Cohen and Lenstra introduced conjectures concerning the distribution of class numbers in quadratic fields, though many of these conjectures remain unproven. This paper investigates the 2-part of class groups in imaginary quadratic fields and examines their alignment with the Cohen-Lenstra heuristics. We provide detailed proofs of key theorems related to ideal decompositions and modular homomorphisms, and we explore the distribution of class groups of imaginary quadratic fields. Our analysis includes constructing imaginary quadratic fields with prescribed 2-class groups and discussing the implications of these findings on the Cohen-Lenstra conjecture.

Cohen and Lenstra introduced conjectures concerning the distribution of class numbers in quadratic fields, though many of these conjectures remain unproven. This paper investigates the 2-part of class groups in imaginary quadratic fields and examines their alignment with the Cohen–Lenstra heuristics. We provide detailed proofs of key theorems related to ideal decompositions and modular homomorphisms, and we explore the distribution of class groups of imaginary quadratic fields. Our analysis includes constructing imaginary quadratic fields with prescribed 2-class groups and discussing the implications of these findings on the Cohen–Lenstra conjecture.