Advances in quantum computing have heightened concerns about information security in the post-quantum era. While traditional cryptographic schemes rooted in number- theoretic assumptions (e.g., integer factorization, elliptic curve discrete logarithm, and discrete logarithm problems) remain foundational to modern security infrastructures, their vulnerabilities in quantum computing environments have become apparent. To address this, researchers have shifted focus toward quantum-resistant mathematical problems. Among these, isogeny-based cryptography has emerged as a strong candi- date within the five primary post-quantum cryptographic approaches, garnering sig- nificant academic interest. Recent progress in isogeny-based cryptographic schemes includes SIDH (though recent cryptanalysis by Castryck and Decru has compromised its security), CSIDH, and SQISign. Despite progress, the inherent structural characteristics of isogenies impose limitations on design flexibility compared to other post-quantum alternatives. The goal of the thesis is to develop isogeny-based cryptographic protocols with ad- vanced functionalities. We develop isogeny-based blind signatures based on techniques from CSIDH, CSI-FiSh, and other works. We also develop ring signature schemes us- ing SQISign. Finally, we develop a verifiable encryption scheme (based on CSIDH) and bit commitment scheme (based on the Charles–Goren–Lauter isogeny-based hash function). For both schemes, we give ways to prove to a verifier that the encrypted/ committed bits satisfy a given Boolean equation.
Advances in quantum computing have heightened concerns about information security in the post-quantum era. While traditional cryptographic schemes rooted in number- theoretic assumptions (e.g., integer factorization, elliptic curve discrete logarithm, and discrete logarithm problems) remain foundational to modern security infrastructures, their vulnerabilities in quantum computing environments have become apparent. To address this, researchers have shifted focus toward quantum-resistant mathematical problems. Among these, isogeny-based cryptography has emerged as a strong candi- date within the five primary post-quantum cryptographic approaches, garnering sig- nificant academic interest. Recent progress in isogeny-based cryptographic schemes includes SIDH (though recent cryptanalysis by Castryck and Decru has compromised its security), CSIDH, and SQISign. Despite progress, the inherent structural characteristics of isogenies impose limitations on design flexibility compared to other post-quantum alternatives. The goal of the thesis is to develop isogeny-based cryptographic protocols with ad- vanced functionalities. We develop isogeny-based blind signatures based on techniques from CSIDH, CSI-FiSh, and other works. We also develop ring signature schemes us- ing SQISign. Finally, we develop a verifiable encryption scheme (based on CSIDH) and bit commitment scheme (based on the Charles–Goren–Lauter isogeny-based hash function). For both schemes, we give ways to prove to a verifier that the encrypted/ committed bits satisfy a given Boolean equation.
Quantum cryptography enhances security and addresses vulnerabilities present in traditional cryptography, thereby providing better protection against potential risks posed by quantum computing. As quantum computing progresses, the feasibility of quantum cryptanalysis increases, necessitating the development of various post-quantum cryptography techniques. These techniques include cryptography that relies on lattices, codes, hash functions, multivariate polynomials, and isogenies, all aimed at ensuring stronger security measures. This paper focuses specifically on isogeny-based cryptography, a novel branch of cryptography that utilizes isogenies between elliptic curves to construct cryptographic tools. These schemes are known for their speed, efficiency, and resistance to quantum attacks. Our research delves into the latest advancements in isogeny-based cryptography, exploring the challenges and opportunities for future investigation. We firmly believe that isogeny-based cryptography holds great potential in shaping the future of cryptography, and we encourage fellow researchers to delve into this captivating field.
The rise of quantum computing poses a significant threat to contemporary public-key cryptography, which relies on problems like integer factorization and the discrete logarithm, both of which are solvable in polynomial time by Shor's algorithm. This has prompted a global effort to develop and standardize post-quantum cryptographic primitives that are resistant to attacks by both classical and quantum computers. Among the various families of post-quantum candidates, isogeny-based cryptography has emerged as a particularly compelling, albeit challenging, alternative. This paper explores the mathematical foundations of elliptic curve isogenies and their application as a basis for quantum-resistant cryptographic protocols. We delve into the structure of supersingular elliptic curves and the associated isogeny graphs, which form the basis for the conjectured hardness of the underlying computational problems. The Supersingular Isogeny Diffie-Hellman (SIDH) protocol and its derivative, Supersingular Isogeny Key Encapsulation (SIKE), are examined in detail as principal examples of this cryptographic paradigm. We analyze their key advantages, most notably their exceptionally small key sizes, which offer significant benefits for constrained environments. However, the security landscape of isogeny-based cryptography has been dramatically reshaped by recent cryptanalytic breakthroughs that completely broke the SIKE protocol. This paper analyzes the nature of these attacks, discussing the vulnerabilities they exploited and the profound implications for the field. We conclude by surveying the current state of isogeny-based cryptography in the aftermath of these developments, outlining the proposed countermeasures, alternative designs, and the open research questions that define the new frontier for this promising area of post-quantum cryptography.
The impending arrival of large-scale quantum computers poses a significant threat to modern public-key cryptography, which is largely based on the hardness of integer factorization and the discrete logarithm problem. Elliptic Curve Cryptography (ECC), a cornerstone of current digital security, is particularly vulnerable to attacks by quantum algorithms, notably Shor's algorithm. This has catalyzed the development of Post-Quantum Cryptography (PQC), a new generation of cryptographic systems designed to be secure against both classical and quantum adversaries. Among the various PQC families, isogeny-based cryptography has emerged as a compelling, albeit complex, candidate. This paper explores the reimagination of elliptic curve principles for a post-quantum world through the lens of isogeny graphs. We provide a comprehensive overview of the transition from classical ECC to isogeny-based schemes, detailing the mathematical foundations of elliptic curve isogenies and the construction of supersingular isogeny graphs. The core of our analysis focuses on how the problem of finding a secret isogeny path between two curves in this graph forms the basis for quantum-resistant cryptographic protocols. We review the evolution of seminal protocols such as Supersingular Isogeny Diffie-Hellman (SIDH) and its variants, discussing their initial promise. Crucially, the paper also addresses the recent groundbreaking cryptanalytic attacks on SIDH, which have reshaped the landscape of isogeny-based cryptography. We analyze the implications of these attacks and the ongoing research to develop new, more secure isogeny-based schemes like SQISign. By comparing the performance metrics (key size, performance, security) of isogeny-based systems with other leading PQC candidates, such as those based on lattices and codes, this paper offers a balanced perspective on their potential role in future cryptographic standards.
The emergence of quantum computing poses a significant threat to the security of currently deployed public-key cryptosystems, including those based on elliptic curves (ECC). Shor's algorithm can solve the underlying integer factorization and discrete logarithm problems in polynomial time, rendering established cryptographic standards obsolete. This has catalyzed the development of post-quantum cryptography (PQC), a new generation of algorithms resistant to attacks by both classical and quantum computers. Among the most promising PQC candidates are systems based on supersingular isogeny graphs. This paper provides a comprehensive analysis of the supersingular isogeny paradigm as a quantum-secure evolution of elliptic curve cryptography. It begins by introducing the mathematical foundations of elliptic curves and isogenies, which are structure-preserving maps between them. The core of isogeny-based cryptography lies in the conjectured hardness of finding an isogeny path between two given supersingular elliptic curves in the vast isogeny graph. This is fundamentally different from the discrete logarithm problem that underpins traditional ECC. We review the foundational Supersingular Isogeny Diffie-Hellman (SIDH) key exchange protocol and its subsequent realization as the Supersingular Isogeny Key Encapsulation (SIKE) mechanism, which was a candidate in the NIST PQC standardization process. We discuss its advantages, such as small key sizes and perfect forward secrecy, which made it an attractive alternative. However, the paper also presents a balanced view by thoroughly examining the devastating cryptanalytic breakthroughs of 2022 that rendered the original SIDH/SIKE schemes insecure. These attacks, which exploit auxiliary point information, underscore the dynamic nature of PQC research. Despite this setback, the isogeny-based field remains active, with ongoing research into alternative constructions like CSIDH and other variants that are not vulnerable to the known attacks. We conclude that while the initial promise of SIDH/SIKE was broken, the underlying mathematical problem of finding isogenies remains a compelling basis for quantum-secure cryptography, representing a paradigm of active research rather than a discarded concept.
Recent advances in quantum computing have made existing public key cryptosystems vulnerable to enormous security threats. Therefore, numerous efforts have been exploring post-quantum cryptographic techniques to address the emergence of quantum computing. We focus on one promising post-quantum cryptography known as “isogeny-based cryptography,” first by reviewing some concepts of the elliptic curve isogeny, and then, we present three major methods for constructing isogeny-based difficult problems. Then, we present some existing isogeny-based cryptographic primitives, such as signature and key exchange, and analyze their advantages and disadvantages. Finally, we discuss a few major challenges of isogeny research that, we hope, will attract more attention to isogeny-based 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.