The Supersingular Isogeny Problem in Genus 2 and Beyond

Abstract

Let $A/\overline{\mathbb {F}}_p$ and $A'/\overline{\mathbb {F}}_p$ be superspecial principally polarized abelian varieties of dimension $g>1$. For any prime $\ell \ne p$, we give an algorithm that finds a path $\phi :A \rightarrow A'$ 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.

A A Proof-of-Concept Implementation

We include a naive Magma implementation of the product finding stage (i.e. Steps 1–3) of Algorithm 1 in dimension $g=2$ with $\ell =2$. First, it generates a challenge by walking from the known superspecial node corresponding to the curve $\mathcal {C}:y^2=x^5+x$ over a given $\mathbb {F}_{p^2}$ to a random abelian surface in $\varGamma _{2}({2};p)$, which becomes the target $\mathcal {A}$. Then it starts computing random walks of length slightly larger than $\log _2(p)$, whose steps correspond to (2, 2)-isogenies. As each step is taken, it checks whether we have landed on a product of two elliptic curves (at which point it will terminate) before continuing.

Magma’s built-in functionality for (2, 2)-isogenies makes this rather straightforward. At a given node, the function RichelotIsogenousSurfaces computes all 15 of its neighbours, so our random walks are simply a matter of generating enough entropy to choose one of these neighbours at each of the $O(\log (p))$ steps. For the sake of replicability, we have used Magma’s inbuilt implementation of SHA-1 to produce pseudo-random walks that are deterministically generated by an input seed. SHA-1 produces 160-bit strings, which correspond to 40 integers in $[0,1,\dots , 15]$; this gives a straightforward way to take 40 pseudo-random steps in $\varGamma _{2}({2};p)$, where no step is taken if the integer is 0, and otherwise the index is used to choose one of the 15 neighbours.

The seed processor can be used to generate independent walks across multiple processors. We always used the seed “0” to generate the target surface, and set processor to be the string “1” to kickstart a single process for very small primes. For the second and third largest primes, we used the strings “1”, “2”, ..., “16” as seeds to 16 different deterministic processes. For the largest prime, we seeded 128 different processes.

For the prime $p=\mathbf{127}=2^7-1$, the seed “0” walks us to the starting node corresponding to $C_0/\mathbb {F}_{p^2} :y^2=(41i + 63)x^6 +\dots + (6i +12)x + 70$. The single processor seeded with “1” found a product variety $E_1 \times E_2$ on its second walk after taking 53 steps in total, with $E_1/\mathbb {F}_{p^2} :y^2 = x^3 + (93i + 43)x^2 + (23i + 93)x + (2i + 31)$ and $E_2/\mathbb {F}_{p^2} :y^2 = x^3 + (98i + 73)x^2 + (30i + 61)x + (41i + 8)$.

For the prime $p=\mathbf{8191}=2^{13}-1$, the single processor seeded with “1” found a product variety on its 175-th walk after taking 6554 steps in total.

For the prime $p=\mathbf{524287}=2^{19}-1$, all 16 processors were used. The processor seeded with “2” was the first to find a product variety on its 311-th walk after taking 11680 steps in total. Given that all processors walk at roughly the same pace, at this stage we would have walked close to $16 \cdot 11680 = \mathbf{186880}$ steps.

For the 25-bit prime $p=\mathbf{17915903}=2^{13}3^7-1$, the processor seeded with “13” found a product variety after taking 341 walks and a total of 12698 steps. At this stage the 16 processors would have collectively taken around 203168 steps.

The largest experiment that we have conducted to date is with the prime $p=\mathbf{2147483647}=2^{31}-1$, where 128 processors walked in parallel. Here the processor seeded with “95” found a product variety after taking 10025 walks and a total of 375703 steps. At this stage the processors would have collectively taken around 48089984 steps.

In all of the above cases we see that product varieties are found with around p steps. The Magma script that follows can be used to verify the experiments³, or to experiment with other primes.