In mathematics, the supersingular isogeny graphs are a class of expander graphs that arise in computational number theory and have been applied in elliptic-curve cryptography.
Their vertices represent supersingular elliptic curves over finite fields and their edges represent isogenies between curves.
A supersingular isogeny graph is determined by choosing a large prime number
, and considering the class of all supersingular elliptic curves defined over the finite field
The vertices in the supersingular isogeny graph represent these curves (or more concretely, their j-invariants, elements of
[4] One proposal for a cryptographic hash function involves starting from a fixed vertex of a supersingular isogeny graph, using the bits of the binary representation of an input value to determine a sequence of edges to follow in a walk in the graph, and using the identity of the vertex reached at the end of the walk as the hash value for the input.
The security of the proposed hashing scheme rests on the assumption that it is difficult to find paths in this graph that connect arbitrary pairs of vertices.
[6] However, a leading variant of supersingular isogeny key exchange was broken in 2022 using non-quantum methods.