This is the discussion thread for the fifth session of season three of The Isogeny Club , where Péter presents his talk titled: Hidden stabilizers, the isogeny to endomorphism ring problem and the cryptanalysis of pSIDH
Let E be a supersingular elliptic curve with known endomorphism and let φ be an isogeny whose domain is E. When the isogeny has a smooth degree, then one can easily compute the endomorphism ring of the codomain. How about when the isogeny degree is not smooth? In this case we cannot write down our isogeny efficiently, but we have a representation of the isogeny which allows us to evaluate the isogeny efficiently at any point. The SIDH attacks do not seem to help here as the isogeny degree is not smooth and its kernel is defined over a very large extension.
The key idea to tackle this problem is to consider a non-abelian group action on the set of fixed degree isogenies. In this talk, we describe this method in detail and convince cryptographers that non-abelian hidden subgroup problems can be very useful in cryptography.