Decoding CSIDH

A Guide to Isogeny-Based Cryptography

Master thesis by Joris Perrenet · Mathematical Institute, Leiden University · Supervised by Dr. S.A. Arpin

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Abstract

In today's digital age, private communication is anchored in robust encryption techniques. As the horizon of quantum computing draws nearer, current encryption methods may soon become vulnerable to quantum attacks, pointing to the need for post-quantum cryptography. The CSIDH encryption scheme, grounded in isogeny-based cryptography, emerges as a compelling candidate for such post-quantum cryptography. Understanding CSIDH requires a solid grasp of finite fields, number fields, and particularly elliptic curves. This thesis strives to break down these topics to make the innovative CSIDH approach more accessible, encouraging further exploration in the field of isogeny-based cryptography to ensure digital privacy for years to come.

Topics covered

• Elliptic curves over finite fields and their group law
• Isogenies, supersingular curves, and the Frobenius endomorphism
• The ideal class group and its action on supersingular curves
• The CSIDH key-exchange protocol, derived step by step
• Worked examples in Sage demonstrating the protocol on small primes

Citation

Perrenet, L. J. (2024). Decoding CSIDH: A Guide to Isogeny-Based Cryptography [Master's thesis, Leiden University]. https://jorisperrenet.com/MasterThesis/decoding-csidh.pdf