This thesis analyses the assumptions present in security proofs of post-quantum cryptosystems using Koblitz and Menezes' critique to guide the focus points. The classical analysis focusses in particular on isogeny-based and multivariate polynomial cryptosystems. Firstly, we present attacks against hardness assumptions on derivative computational problems arising in protocols for isogeny-based undeniable signature schemes. Secondly, we provide upper bounds on the solving degree of over-determined systems of n+l multivariate polynomials in n variables as an alternative to the degree of regularity, which gives a heuristic upper bound. We finish with an analysis of the resources of a quantum adversary, providing an improved method for quantum gate approximation that gives shorter sequences by a factor of 9/7.
This thesis analyses the assumptions present in security proofs of post-quantum cryptosystems using Koblitz and Menezes' critique to guide the focus points. The classical analysis focusses in particular on isogeny-based and multivariate polynomial cryptosystems. Firstly, we present attacks against hardness assumptions on derivative computational problems arising in protocols for isogeny-based undeniable signature schemes. Secondly, we provide upper bounds on the solving degree of over-determined systems of n+l multivariate polynomials in n variables as an alternative to the degree of regularity, which gives a heuristic upper bound. We finish with an analysis of the resources of a quantum adversary, providing an improved method for quantum gate approximation that gives shorter sequences by a factor of 9/7.
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.
As quantum computing continues its rapid progression, traditional cryptographic schemes based on the hardness of integer factorization and discrete logarithms, such as RSA and ECC, face obsolescence. This chapter explores the number theoretic underpinnings of Post-Quantum Cryptography (PQC), surveying emerging quantum-resistant protocols including lattice-based, code-based, multivariate polynomial, and isogeny-based cryptography. We analyze the mathematical assumptions behind their security, the role of hard problems in number theory (e.g., shortest vector problem, syndrome decoding, supersingular isogeny graphs), and the computational implications of these approaches. Furthermore, the chapter discusses the transition from classical cryptography to PQC within the NIST standardization process and how these cryptographic primitives align with the modern number theoretic and algorithmic challenges. Designed for both researchers and educators, this chapter aims to bridge theory and practice while emphasizing the continuing centrality of number theory in shaping the future of secure communication in the quantum era.
We propose and develop new schemes for post-quantum cryptography based on isogenies over elliptic curves. First we show that ordinary elliptic curves are have less than exponential security against quantum computers. These results were used as the motivation for De Feo, Jao and Pl\\^ut's construction of public key cryptosystems using supersingular elliptic curve isogenies. We extend their construction and show that isogenies between supersingular elliptic curves can be used as the underlying hard mathematical problem for other quantum-resistant schemes. For our second contribution, we propose is an undeniable signature scheme based on elliptic curve isogenies. We prove its security under certain reasonable number-theoretic computational assumptions for which no efficient quantum algorithms are known. This proposal represents only the second known quantum-resistant undeniable signature scheme, and the first such scheme secure under a number-theoretic complexity assumption. Finally, we also propose a security model for evaluating the security of authenticated encryption schemes in the post-quantum setting. Our model is based on a combination of the classical Bellare-Namprempre security model for authenticated encryption together with modifications from Boneh and Zhandry to handle message authentication against quantum adversaries. We give a generic construction based on Bellare-Namprempre for producing an authenticated encryption protocol from any quantum-resistant symmetric-key encryption scheme together with any digital signature scheme or MAC admitting any classical security reduction to a quantum-computationally hard problem. We apply the results and show how we can explicitly construct authenticated encryption schemes based on isogenies.
Public key cryptography plays a pivotal role in maintaining digital security and privacy, with its foundations built on the hardness of certain mathematical problems. However, the emergence of quantum computing threatens the security of current cryptographic systems, making the adoption of post-quantum cryptography (PQC) an urgent necessity. Digital signature schemes are essential components in secure communications, providing authentication, non-repudiation, and data integrity. This paper examines the state of post-quantum digital signature schemes across various cryptographic approaches, including lattice-based, code-based, multivariate polynomial-based, hash-based, symmetric-key-based, and isogeny-based systems, alongside advancements in Multi-Party Computation in the Head (MPC-in-the-head) schemes. The paper then explores application-specific signatures such as blind, undeniable, threshold, chameleon, aggregate, and ring signatures, explaining their main use cases and the existing post-quantum cryptographic classifications that support these constructions individually. Software implementations of selected prominent PQC digital signature schemes were carried out, and their performance was systematically analysed using metrics such as bandwidth consumption, latency, and computational efficiency. These results provide key observations regarding the suitability of different schemes for both resource-constrained and high-performance environments. Subsequently, a detailed literature review of implementations on embedded and hardware platforms is presented, focusing on ARM-based devices and FPGA accelerators to evaluate deployment feasibility beyond general-purpose processors. Based on the software and implementation-level findings, a comparative analysis across software, embedded, and hardware platforms is conducted to highlight architectural influences on performance, scalability, and practicality. Furthermore the cryptanalytic attacks on post-quantum digital signature schemes across various PQC classifications are examined in this paper, taking into account both classical and quantum adversarial models. To evaluate realistic security constraints, implementation-oriented vulnerabilities such as side-channel and fault-based attacks are also examined. In addition, the applicability of post-quantum digital signature schemes within TLS, IoT, and related practical deployment scenarios is analysed. The study also discusses possible directions for further investigation, focusing on the requirement for effective and expandable PQC systems that can satisfy changing cybersecurity requirements. In order to protect sensitive data in the digital era, this study emphasizes the need for implementing quantum-secure systems by fusing theoretical understandings with real-world applications.
The advent of a full-scale quantum computer will severely impact most currently-used cryptographic systems. The most well-known aspect of this impact lies in the computational-hardness assumptions that underpin the security of most current public-key cryptographic systems: a quantum computer can factor integers and compute discrete logarithms in polynomial time, thereby breaking systems based on these problems. However, simply replacing these problems by other which are (believed to be) impervious even to a quantum computer does not completely solve the issue. Indeed, many security proofs of cryptographic systems are no longer valid in the presence of a quantum-capable attacker; while this does not automatically implies that the affected systems would be broken by a quantum computer, it does raises questions on the exact security guarantees that they can provide. This overview document aims to analyze all aspects of the impact of quantum computers on cryptographic, by providing an overview of current quantum-hard computational problems (and cryptographic systems based on them), and by presenting the security proofs that are affected by quantum-attackers, detailing what is the current status of research on the topic and what the expected effects on security are.
This paper explores the design of post-quantum cryptographic algorithms intended to secure data against attacks from future quantum computers. The core strategy involves transitioning away from algorithms reliant on the discrete logarithm problem, which is vulnerable to Shor's algorithm, and towards alternative approaches. These include hash-based cryptography, code-based cryptography, multivariate polynomial cryptography, and isogeny-based cryptography. We detail the mathematical foundations of these approaches, focusing on the security proofs and key generation/decryption processes. The design considerations for each category are outlined, along with the trade-offs between key sizes, performance, and security levels. This work aims to provide a comprehensive overview of the landscape of post-quantum cryptography and contribute to the development of robust algorithms for long-term data protection. The primary goal is to establish a framework for evaluating and selecting suitable post-quantum cryptographic solutions.