New paper dropped! We explore the expansion properties of the superspecial isogeny graph in dimension g > 1 (with level structure), and give several applications of this result.
tl;dr: they are not optimal expander graphs but still provide enough expansion for cryptographic use!
ia.cr/2026/2429
ia.cr
Rapid mixing in superspecial isogeny graphs
We prove expansion properties of superspecial isogeny graphs in dimension $g \geq 2$, including arbitrary level structure. In particular, we provide explicit bounds for the spectral gap, and a fine ch...
![Abstract. We introduce WayFinder, a framework for generalizing the Delfs-Galbraith and SuperSolver algorithms for the supersingular isogeny problem. Our framework extends the search for elliptic curves with an orientation by an order containing $\mathbb{Z}[\ell \sqrt{-p}]$ to more general orders, and we derive a cost model for such generalisations. Our cost model not only works in a more general context, but also provides more accurate predictions when applied to SuperSolver. We instantiate WayFinder for orders containing $\mathbb{Z}[\ell_1\sqrt{-\ell_2p}]$ where ℓ_(i) are 1 or primes such that the modular curve X₀(ℓ₂) has genus 0. We then introduce a low-storage algorithm for computing an isogeny between two oriented supersingular elliptic curves, even when the curves are oriented by distinct orders. Together, these provide an algorithm that improves on the state of the art for solving the isogeny problem, and a cost model with potential applications to parameter selection in isogeny-based cryptography.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:fwa55bujvdrwlwlwgqmmxmuf/bafkreidps3jwwz33rxdmp6w7ok6w462ioosoazxxabir2thhplfnfytz6m)
