Isogeny-Based Cryptography for a World with Quantum Computers

Kunal Dey, Sumit Kumar Debnath, Pantelimon Stănică +1 more

Dheerendra Mishra, Rohit Raj Sharma

Gilles Brassard A

Syed Khundmir Azmi

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.

PAULO SÉRGIO DE ANDRADE

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.

Ahmed Farouk, Omar Husain Tarawneh, Mohamed Elhoseny +4 more