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Isogeny-based cryptography is a branch of cryptography whose protocols are based on the hardness of finding structured isogenies between (typically supersingular) elliptic curves. Isogeny-based protocols are believed to be quantum-safe, and so they are candidates for use in the age of quantum computers.
Professor LeGrow's work primarily deals with protocols built on the ideas of Commutative Supersingular Isogeny Diffie-Hellman (CSIDH), an isogeny-based key establishment protocol which uses the action of the ideal class group on the collection of elliptic curves with given Fp-rational endomorphism ring. The CSIDH framework is very flexibile, and many CSIDH-based protocols have since been constructed.
Among other projects, Professor LeGrow has worked on optimization of algorithms for computing the class group action in the context of CSIDH, fault attacks on CSIDH and their countermeasures, quantum cryptanalysis of CSIDH, and design of CSIDH-based exotic signatures.