## Cryptology ePrint Archive: Report 2021/1488

SuperSolver: accelerating the Delfs-Galbraith algorithm with fast subfield root detection

Maria Corte-Real Santos and Craig Costello and Jia Shi

Abstract: We give a new algorithm for finding an isogeny from a given supersingular elliptic curve $E/\mathbb{F}_{p^2}$ to a subfield elliptic curve $E'/\mathbb{F}_p$, which is the bottleneck step of the Delfs-Galbraith algorithm for the general supersingular isogeny problem. Our core ingredient is a novel method of rapidly determining whether a polynomial $f \in L[X]$ has any roots in a subfield $K \subset L$, while crucially avoiding expensive root-finding algorithms. In the special case when $f=\Phi_{\ell,p}(X,j) \in \mathbb{F}_{p^2}[X]$, i.e. when $f$ is the $\ell$-th modular polynomial evaluated at a supersingular $j$-invariant, this provides a means of efficiently determining whether there is an $\ell$-isogeny connecting the corresponding elliptic curve to a subfield curve. Together with the traditional Delfs-Galbraith walk, inspecting many $\ell$-isogenous neighbours in this way allows us to search through a larger proportion of the supersingular set per unit of time. Though the asymptotic $\tilde{O}(p^{1/2})$ complexity of our improved algorithm remains unchanged from that of the original Delfs-Galbraith algorithm, our theoretical analysis and practical implementation both show a significant reduction in the runtime of the subfield search. This sheds new light on the concrete hardness of the general supersingular isogeny problem, the foundational problem underlying isogeny-based cryptography.

Category / Keywords: public-key cryptography / Isogeny-based cryptography, supersingular isogeny problem, Delfs-Galbraith algorithm.

Date: received 9 Nov 2021, last revised 11 Nov 2021

Contact author: maria santos 20 at ucl ac uk, craigco at microsoft com, janeshi99 at gmail com

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2021/1488

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