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A survey on isogeny-based cryptographic protocols

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Abstract

As the threat of quantum computing looms over classical cryptographic schemes, the search for post-quantum cryptographic solutions has gained significant attention. IBC has emerged as a promising candidate, offering a robust foundation for secure communications in a quantum-resistant manner. This survey provides a comprehensive overview of isogeny-based protocols, their underlying mathematics, security properties, and current research trends. IBC is founded upon the notion of isogenies, which are algebraic mappings between ECs. These protocols rely on the presumed hardness of the isogeny problem, where the challenge is to compute an isogeny between two ECs with specific properties. One of the most notable isogeny-based schemes is the Supersingular Isogeny Diffie–Hellman (SIDH) protocol, which forms the basis for many subsequent developments. This survey begins by exploring the mathematical foundations of isogeny-based cryptography, delving into the EC theory, and explaining the essential components of SIDH. We then discuss the security assumptions and threat models associated with isogeny-based protocols, including resistance to quantum attacks and potential vulnerabilities. Furthermore, we examine the practicality and efficiency of isogeny-based schemes, considering factors like key sizes, computational requirements, and performance trade-offs. We briefly explain mathematical terms related to isogeny-based cryptosystems, such as isogeny, ordinary EC, supersingular EC, and isogeny graph. We analyze the security of these protocols against quantum and classical computers. The survey also reviews recent advancements and variations of isogeny-based protocols, including SIKE (Supersingular Isogeny Key Encapsulation), CSIDH (Commensurate Isogeny Diffie–Hellman), and more. We highlight their strengths, weaknesses, and potential use cases in various cryptographic applications. Additionally, we provide an overview of the standardization efforts and adoption status of isogeny-based protocols within the cryptographic community and industry. We discuss ongoing research challenges, open problems, and the prospects of IBC in a post-quantum era. In conclusion, this survey offers a comprehensive insight into the world of isogeny-based protocols, shedding light on their mathematical foundations, security features, practical considerations, and potential future developments.

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  1. Bhaskar Mondal and Rishi Kumar Jha contributed equally to this work.

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  1. Department of Mathematics, National Institute of Technology Patna, Patna, Bihar, 800005, India

    Sweta Mishra & Rishi Kumar Jha

  2. Department of Computer Science and Engineering, National Institute of Technology Patna, Patna, Bihar, 800005, India

    Bhaskar Mondal

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SM drafted the main manuscript text, Dr. BM prepared all the figures and did the proofreading and Dr. RKJ contributed by arranging the systematic review throughout. All authors equally contributed to the scientific work and reviewed the manuscript.

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Mishra, S., Mondal, B. & Jha, R.K. A survey on isogeny-based cryptographic protocols. Wireless Netw (2025). https://doi.org/10.1007/s11276-025-03906-6

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