Hard Homogeneous Spaces

By J. Couveignes. IACR Cryptology ePrint Archive, 2006

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Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies (2011, background): “Every isogeny of degree greater than 1 can be factored into a composition of isogenies of prime degree over ¯ F q [6].”
Constructing elliptic curve isogenies in quantum subexponential time (2010, background): “Although quantum attacks are known against several cryptographic protocols of an algebraic nature [12, 19, 34], until now there has been no nontrivial quantum algorithm for constructing isogenies.”
An Efficient Key Recovery Attack on SIDH (2023, unknown): “In particular, it cannot be adjusted in an obvious way to attack primitives that do not reveal this information, such as CRS/CSIDH [10], [39], [7] and SQISign [12], and the general supersingular isogeny path problem remains unaﬀected [44].”
Post-quantum cryptography (2008, unknown): “We do not describe isogeny-based cryptography [20, 21, 22], for example, and we do not discuss Kuperberg’s algorithm [23].”