The Supersingular Isogeny Problem in Time and Memory $p^{1/3+o(1)}$, Unconditionally

Given a supersingular elliptic curve $E/\mathbb{F}_{p^2}$, the $\mathsf{OneEnd}$ problem asks for a non-scalar endomorphism of $E$. By known reductions, solving this problem also solves the supersingular endomorphism ring and isogeny problems. Wesolowski obtained exponent $1/3$ under an assumption on the factorization of a small degree, whereas the previous unconditional exponent was $2/5$. We give a Las Vegas algorithm, analyzed without a smoothness heuristic, with expected time and memory [ p^{1/3}\exp\bigl(O(\sqrt{\log p,\log\log p})\bigr) = p^{1/3+o(1)}. ] The algorithm fixes in advance a family of degrees that are products of small primes. Known counting results provide many isogenies of these degrees from curves to their Frobenius conjugates, and a collision estimate shows that the isogenies occur on sufficiently many distinct curves for a random walk to reach one of them. From such a curve, the algorithm splits a degree into two parts, enumerates two lists of shorter isogenies, and matches their targets to obtain an isogeny to the conjugate, whose composition with Frobenius gives the required endomorphism.

Source: arXiv:2609.22018v1 - http://arxiv.org/abs/2609.22018v1 PDF: https://arxiv.org/pdf/2609.22018v1 Original Link: http://arxiv.org/abs/2609.22018v1