Hidden Stabilizers, the Isogeny To Endomorphism Ring Problem and the Cryptanalysis of pSIDH
The Isogeny to Endomorphism Ring Problem (IsERP) asks to compute the endomorphism ring of the codomain of an isogeny between supersingular curves in characteristic p given only a representation for this isogeny, i.e. some data and an algorithm to evaluate this isogeny on any torsion point. This problem plays a central role in isogeny-based cryptography; it underlies the security of pSIDH protocol (ASIACRYPT 2022) and it is at the heart of the recent attacks that broke the SIDH key exchange. Prior to this work, no efficient algorithm was known to solve IsERP for a generic isogeny degree, the hardest case seemingly when the degree is prime. In this paper, we introduce a new quantum polynomial-time algorithm to solve IsERP for isogenies whose degrees are odd and have O(log(log p)) many prime factors. As main technical tools, our algorithm uses a quantum algorithm for computing hidden Borel subgroups, a group action on supersingular isogenies from EUROCRYPT 2021, various algorithms for the Deuring correspondence and a new algorithm to lift arbitrary quaternion order elements modulo an odd integer N with O(log(log p)) many prime factors to powersmooth elements. As a main consequence for cryptography, we obtain a quantum polynomial-time key recovery attack on pSIDH. The technical tools we use may also be of independent interest.