The supersingular isogeny problem in genus 2 and beyond - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Year :

The supersingular isogeny problem in genus 2 and beyond

Craig Costello
  • Function : Author
  • PersonId : 987506

Abstract

Let $A/\overline{\mathbb{F}}_p$ and $A'/\overline{\mathbb{F}}_p$ be supersingular principally polarized abelian varieties of dimension $g>1$. For any prime $\ell \ne p$, we give an algorithm that finds a path $\phi \colon A \rightarrow A'$ in the $(\ell, \dots , \ell)$-isogeny graph in $\widetilde{O}(p^{g-1})$ group operations on a classical computer, and $\widetilde{O}(\sqrt{p^{g-1}})$ calls to the Grover oracle on a quantum computer. The idea is to find paths from $A$ and $A'$ to nodes that correspond to products of lower dimensional abelian varieties, and to recurse down in dimension until an elliptic path-finding algorithm (such as Delfs--Galbraith) can be invoked to connect the paths in dimension $g=1$. In the general case where $A$ and $A'$ are any two nodes in the graph, this algorithm presents an asymptotic improvement over all of the algorithms in the current literature. In the special case where $A$ and $A'$ are a known and relatively small number of steps away from each other (as is the case in higher dimensional analogues of SIDH), it gives an asymptotic improvement over the quantum claw finding algorithms and an asymptotic improvement over the classical van Oorschot--Wiener algorithm.
Fichier principal
Vignette du fichier
pq.pdf (271.56 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-02389073 , version 1 (02-12-2019)
hal-02389073 , version 2 (27-01-2020)

Identifiers

Cite

Craig Costello, Benjamin Smith. The supersingular isogeny problem in genus 2 and beyond. PQCrypto 2020 - 11th International Conference on Post-Quantum Cryptography, Apr 2020, Paris, France. ⟨hal-02389073v2⟩
190 View
364 Download

Altmetric

Share

Gmail Facebook Twitter LinkedIn More