: Isogeny graphs in cryptography

Speaker: Dr Luca de Feo

Date/Time: 26-Feb-2015, 17:00 UTC

Venue: MPEB 1.03

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Abstract

Elliptic curves are well known in cryptography : given an elliptic curve, we can take points on it, add them, form a group, and ultimately implement group-based protocols such as Diffie-Helmann key exchange.

Much less known is the graph structure of elliptic curves connected by their morphisms. These graphs are called *isogeny graphs*. Classical number theory attributes these graphs to two very different families : the *ordinary* case, and the *supersingular* case. Both families have very interesting properties : they both form expander graphs, they show some regularity, and a level structure. However, they are fundamentally different. Ordinary graphs, also known as *isogeny volcanoes*, are infinite and have a very rigid level structure, tied to the index of their endomorphism ring. Supersingular graphs are finite, regular, and only exhibit a level structure for a special subgraph.

Both families have been used in cryptography in various creative ways : constructing provably secure hash functions, quantum-resistant key exchange, key escrow schemes, zero-knowledge proofs... This talk will give a review of the main properties and the most interesting protocols and cryptanalysis based on isogeny graphs.

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Please see http://defeo.lu/

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