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.
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.
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.
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 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 rapid growth of quantum computing poses a critical threat to classical cryptographic systems. That is due in large part to the fact that they represent efficient solutions for some of the most challenging mathematical problems such as integer factoring or finding discrete logarithms. Schemes currently in wide use including RSA, Diffie–Hellman, and elliptic curve cryptography are particularly susceptible to being attacked by quantum computers, using algorithms like Shor's Algorithm and Grover's Algorithm. The advent of post-quantum cryptography (PQC) provides a promising potential answer to providing longterm security from adversaries using quantum computing capabilities. This paper presents a comprehensive state-of-the-art review of various PQC techniques currently being researched, including lattice-based methods, code-based methods, multivariate methods, hash-based methods, and isogeny-based methods. key challenges that will need to be addressed include computational overhead, large key sizes and implementation complexity. Furthermore the study reviews the current standardization efforts by the National Institute of Standards and Technology related to PQC and discusses some of the very specific practical considerations that need to be considered before widely deploying these techniques. The paper concludes by a complete list of existing research challenges and proposed future work that will assist in the development of efficient and scalable systems that will resist attacks from quantum computers.
In this article, we discuss how algebraic geometry, especially the isogenies and elliptic curves, have been used to construct secure post-quantum cryptographic systems. Since quantum computing is a very big threat to conventional cryptographic techniques, it is essential to develop new strategies that would enhance protection of data in quantum age. Algebraic geometry provides such solutions as the elliptic curve cryptography and the protocols based on isogenic, which is also resistant to the attacks of quantum algorithms. The article explores the mathematical basis of these cryptographic techniques, their effectiveness, scalability and security. It is noteworthy that isogeny-based cryptography schemes such as the Supersingular Isogeny Diffie-Hellman (SIDH) and Supersingular Isogeny Key Encapsulation (SIKE) prove that isogeny-based cryptography is capable of secure key exchange and encapsulation. The results indicate that algebraic geometry is not only enhancing the cryptography systems, but also opening up a reasonable channel in which strong post-quantum systems can be created. The future of secure communication in a quantum-driven world has profound implications to the methods.
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.