Algebraic geometry in cryptography: Secure post-quantum schemes using isogenies and elliptic curves

Syed Khundmir Azmi *

Independent Researcher, USA.
 
Research Article
International Journal of Science and Research Archive, 2023, 10(02), 1509-1517.
Article DOI: 10.30574/ijsra.2023.10.2.0965
Publication history: 
Received on 14 October 2023; revised on 19 November 2023; accepted on 26 November 2023
 
Abstract: 
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.
 
Keywords: 
Post-Quantum Cryptography; Elliptic Curves; Isogeny Cryptography; Quantum Resistance; Key Exchange; Cryptographic Schemes
 
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Copyright © 2023 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution Liscense 4.0