Encryption schemes attempt to provide a means for entities to communicate confidentially over a public channel. Such schemes have been studied for centuries, and their use has become widespread. However, developments in the area of quantum computing indicate that many of the public key cryptosystems currently in use could easily be broken if large-scale quantum computers become technologically feasible. Further, encrypted messages captured and stored in the past could also be decrypted by a future quantum attacker. Since large-scale quantum computers may one day be developed, We need to prepare for that eventuality by analysing modern cryptosystems with respect to attacks with a quantum computer. In this thesis, submitted in fulfilment of the requirements for the degree of Master of Mathematics in Combinatorics & Optimisation at the University of Waterloo, we examine the quantum strengths and vulnerabilities of several of the most popular classical public key cryptosystems.
Encryption schemes attempt to provide a means for entities to communicate confidentially over a public channel. Such schemes have been studied for centuries, and their use has become widespread. However, developments in the area of quantum computing indicate that many of the public key cryptosystems currently in use could easily be broken if large-scale quantum computers become technologically feasible. Further, encrypted messages captured and stored in the past could also be decrypted by a future quantum attacker. Since large-scale quantum computers may one day be developed, We need to prepare for that eventuality by analysing modern cryptosystems with respect to attacks with a quantum computer. In this thesis, submitted in fulfilment of the requirements for the degree of Master of Mathematics in Combinatorics & Optimisation at the University of Waterloo, we examine the quantum strengths and vulnerabilities of several of the most popular classical public key cryptosystems.
Public-key cryptography is indispensable in maintaining the security and integrity of digital data. The most widely used current public-key cryptography is based on the integer factorization problem and the elliptic-curve discrete logarithm problem, which are vulnerable against an adversary with large-scale quantum computers. Fortunately, post-quantum cryptography (PQC) can provide security against both classical and quantum adversaries. Due to rapid advancement in quantum computer development, the transition from classical public-key cryptography to PQC has become imperative. A watershed moment in this transition is the recent publication of a set of PQC schemes by the National Institute of Standards and Technology (NIST). Although it is a significant step, the research and development in PQC is quite immature compared to several decades-old classical public-key cryptographic schemes. Therefore, several open problems, such as physical attack analysis and their countermeasures, application-specific modifications, lightweight implementations for resource-constrained devices, integration into different secure protocols, etc., need to be addressed before the widespread deployment of PQC in real-world applications. This dissertation aims to address some of these problems in order to bridge the gap between the theory and practice of PQC.
Cryptography is a critical technology in the modern computing industry, but the security of many cryptosystems relies on the difficulty of mathematical problems such as integer factorization and discrete logarithms. Large quantum computers can solve these problems efficiently, enabling the effective cryptanalysis of many common cryptosystems using such algorithms as Shor’s and Grover’s. If data integrity and security are to be preserved in the future, the algorithms that are vulnerable to quantum cryptanalytic techniques must be phased out in favor of quantum-proof cryptosystems. While quantum computer technology is still developing and is not yet capable of breaking commercial encryption, these steps can be taken immediately to ensure that the impending development of large quantum computers does not compromise sensitive data.
The evolution of quantum computing challenges the security of existing cryptosystems for which classical machines are also inefficient. This paper analyses the quantum cryptanalysis techniques that exploit vulnerabilities in classical cryptographic schemes. Based on a comprehensive review of attacker knowledge and quantum techniques, hierarchical classification is proposed to map the classical equivalent quantum attack designs with significant advancement and implications. A comparative complexity analysis highlights the computational advantages of quantum algorithms over classical counterparts, and quantum resource requirements are additionally evaluated to find practical limitations and set severe benchmarks. This work provides critical insights and concerns to develop a robust cryptosystem and it emphasizes the need to transition to quantum-resilient cryptographic methods.
Public-key encryption allows anyone to encrypt messages that only the intended recipient can decrypt. It is an essential component of most network security protocols; however, the algorithms in widespread use are inherently vulnerable to attacks by quantum computers, should they become a practical reality. This thesis looks at public-key encryption from computationally hard problems based on high-dimensional lattices, believed to resist classical and quantum attacks. On those foundations, it proposes three post-quantum public-key encryption schemes, respectively featuring richer access control, tighter reductionist security proofs, and stronger confidentiality in the presence of external leakage.
Quantum computing offers major computational advances but threatens modern public-key cryptography. Classical algorithms such as RSA, Diffie–Hellman (DH), and Elliptic Curve Cryptography (ECC) are vulnerable to quantum attacks, particularly Shor’s algorithm. As large-scale quantum capabilities emerge, post-quantum cryptography (PQC) has become essential to ensure future data confidentiality, integrity, and authentication. This paper discusses the need to replace classical cryptography, explores quantum-safe solutions, and examines challenges in large-scale migration. PQC is critical across government, critical infrastructure, finance, healthcare, telecommunications, IoT, autonomous vehicles, and 6G networks. A key concern is “harvest-now, decrypt-later” attacks, where encrypted data is stored today for future quantum decryption. The study analyzes classical cryptographic vulnerabilities and reviews major PQC families: lattice-based, hash-based, code-based, multivariate-based, and isogeny-based schemes, highlighting the ongoing NIST standardization efforts. It proposes a migration framework including quantum-readiness assessment, algorithm selection, hybrid implementation, and performance evaluation. Results show that although PQC introduces higher computational complexity, optimized implementations can support real-time applications with reasonable overhead. Among PQC approaches, lattice-based schemes appear most mature and balanced in terms of security and key size. The paper concludes that quantum-safe cryptography is a necessary evolution requiring continuous monitoring, adaptable systems, and alignment with emerging standards.
The use of public key cryptosystems range from securely encrypting emails and files to creating digital signatures for non-repudiation. The security of public key cryptosystems is based on computationally "unsolvable" complex mathematical problems. This tends to change with the introduction of quantum computers that undoubtedly pose a threat to the current schemes. In response to that extensive research has been conducted that resulted in several cryptosystems that are believed to be quantum resistant. This thesis presents a concise overview of multiple hash-based signature schemes and provides a comparative description and analysis of them. The comparisons are based on different signature scheme properties, such as key sizes, signature sizes and security level provided. The proposed quantum resistant schemes are also compared to the standard schemes used today, such as RSA and ECDSA.