Welcome to the resource topic for 2023/1448
Title:
The supersingular endomorphism ring problem given one endomorphism
Authors: Arthur Herlédan Le Merdy, Benjamin Wesolowski
Abstract:Given a supersingular elliptic curve E and a non-scalar endomorphism \alpha of E, we prove that the endomorphism ring of E can be computed in classical time about \text{disc}(\mathbb{Z}[\alpha])^{1/4} , and in quantum subexponential time, assuming the generalised Riemann hypothesis. Previous results either had higher complexities, or relied on heuristic assumptions.
Along the way, we prove that the Primitivisation problem can be solved in polynomial time (a problem previously believed to be hard), and we prove that the action of smooth ideals on oriented elliptic curves can be computed in polynomial time (previous results of this form required the ideal to be powersmooth, i.e., not divisible by any large prime power).
Following the attacks on SIDH, isogenies in high dimension are a central ingredient of our results.
ePrint: https://eprint.iacr.org/2023/1448
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