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.
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.
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 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.
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.
In this paper, we have discussed isogeny-based cryptography as a rather appealing option for post-quantum security. We provide analysis of current protocols like SIDH (Supersingular Isogeny Diffie-Hellman) and SIKE (Supersingular Isogeny Key Encapsulation) setting up fundamental issues with their security and performance. In this work we introduce a novel isogeny pathfinding algorithm that employs graph theory as well as probabilistic models in order to lower its computational complexity. The results show significant improvement in both efficiency and security, strengthening the case for the use of isogeny based systems in a quantum-resilient cryptographic space.
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.
The emergence of quantum computing is a serious threat to the principles of modern crypto-logical mathematics. The classical public-key cryptography, including Rivest-Shamir-Adleman (RSA), Elliptic Curve Cryptography (ECC), and Diffie-Hellman are very safe against classical adversaries yet become susceptible to quantum algorithms, including the Shor algorithm, which can easily solve the underlying mathematical problems. This weakness is jeopardizing confidentiality and integrity of the world-wide systems of communication, money transactions, and essential structures, and requires the immediate creation of safe alternatives. The current literature on post-quantum cryptography (PQC) has covered various classes of algorithm, such as lattice-based, code-based, multivariate schemes, and isogeny-based schemes. Although lattice-based approaches like Nth Degree Truncated Polynomial Ring Unit (NTRU) and FrodoKEM can be highly theoretically resistant, they can be computationally and memory-intensive. Cryptosystems based on code such as Classic McEliece are secure but not practicable because they have huge public key sizes. Multivariate schemes like Rainbow, and isogeny-based schemes like Super singular Isogeny Key Encapsulation (SIKE) have been attacked recently by cryptanalytic attacks, and broken completely. Such restrictions open the necessity to find more efficient and secure quantum-resistant solutions.This research study postulates a hybrid post-quantum cryptographic (PQC) system that combines CRYSTALS-Kyber for encryption with CRYSTALS-Dilithium for digital signatures both of which are approved through the National Institute of Standards and Technology (NIST) PQC standardization program. The model will be such that it ensures quantum resistance but at the same time it will be scalable, efficient and compatible with the existing infrastructures. The test outcomes indicate that the hybrid PQC model is 35-45 percent faster to encrypt and decrypt, signature performance is enhanced by up to 40 percent, and the throughput is 20 times higher (98.7 Mbps) than the same base PQC scheme including Nth Degree Truncated Polynomial Ring Unit (NTRU), McEliece and Rainbow. These enhancements affirm that the suggested model provides a viable trade-off between the security, speed, and efficiency in terms of key size, which makes it a viable and a sound base to ensure secure communication in the quantum age.