Enric Nart, Christophe Ritzenthaler
We exhibit the isogeny classes of supersingular abelian threefolds over F_{2^n} containing the Jacobian of a genus 3 curve. In particular, we prove that for even n>6 there always exist a maximal and a minimal curve over F_{2^n}. All the curves can be obtained explicitly.
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Supersingular Isogeny Graphs from Algebraic Modular Curves
本文描述并比较了计算超奇异同源图的算法，推导出一般超奇异同源图邻接矩阵迹的公式，并证明了Nakaya最近提出的猜想。
We introduce a category of $\mathcal{O}$-orientedsupersingularellipticcurves and derive properties of the associated oriented and nonoriented $\ell$-isogeny supersingular isogeny graphs. As an application we introduce an oriented super-singular isogeny Diffie-Hellman protocol (OSIDH), analogous to the supersingular isogeny Diffie-Hellman (SIDH) protocol and generalizing the commutative supersingular isogeny Diffie-Hellman (CSIDH) protocol.
Supersingular isogeny graphs and Hecke modules with level structure
本文研究带有层次结构的奇异同态图及其相关的伽罗瓦表示，探讨其数学结构与性质。
Double-orientations on supersingular isogeny graphs
本文定义了超奇异同源图与四元数理想图的精确同构，引入双定向概念并研究其图结构。
We describe the neighborhood of the vertex $[E_0]$ (resp. $[E_{1728}]$) in the $\ell$-isogeny graph $\mathcal{G}_\ell(\mathbb{F}_{p^2}, -2p)$ of supersingular elliptic curves over the finite field $\mathbb{F}_{p^2}$ when $p>3\ell^2$ (resp. $p>4\ell^2$) with $E_0: y^2=x^3+1$ (resp. $E_{1728}: y^2=x^3+x$) supersingular.
Supersingular isogeny graphs are known to have very few loops and multi-edges. We formalize this idea by studying and finding bounds for the number of loops and multi-edges in such graphs. We also find conditions under which the supersingular isogeny graph $Λ_p(\ell)$ is simple. The methods presented in this paper can be used to study many kinds of collisions in supersingular isogeny graphs. As an application, we introduce the notion of bi-route number for two graphs $Λ_p(\ell_1),Λ_p(\ell_2)$ and compute bounds for it. We also study the number of edges in common between the graphs $Λ_p(\ell_1)…
Adding Level Structure to Supersingular Elliptic Curve Isogeny Graphs
面向基于同源的密码学，本文将层结构引入超奇异椭圆曲线，建立其与四元数代数中 Eichler 序的对应，并研究相关同源图。
Quasi-isogeny groups of supersingular abelian surfaces via pro-étale fundamental groups
本文通过 pro-étale 基本群分类非正规概型的几何覆盖并计算曲线基本群，证明了 Siegel 三重 fold 上的 torsor 诱导出超奇异阿贝尔曲面自拟同源群与有限生成自由群的同构。
Automorphisms of the Supersingular Isogeny Graph
本文研究特征 p 的超奇异 l-同源图的自同构，给出了使其仅具有 Frobenius 作用这一非平凡自同构的条件，并证明该条件对密度为 1 的素数 l 成立。
Let p>3 be a prime and let E, E' be supersingular elliptic curves over F_p. We want to construct an isogeny phi: E --> E'. The currently fastest algorithm for finding isogenies between supersingular elliptic curves solves this problem by performing a "meet-in-the-middle" breadth-first search in the full supersingular 2-isogeny graph over F_{p^2}. In this paper we consider the structure of the isogeny graph of supersingular elliptic curves over F_p. We give an algorithm to construct isogenies between such supersingular elliptic curves that works faster than the usual algorithm. We then discuss …
Orientations and cycles in supersingular isogeny graphs
本文研究了定向超奇异同源火山与同源图中闭游走的关系，建立了定向火山边缘与同源环之间的双射，并据此精确计数了给定长度的同源环且给出显式上界。
We describe an algorithm that we used to compute the q-expansions of all weight 2 cusp forms of prime level at most 2,000,000 and dimension at most 6. We also present an algorithm that we used to verify that there was only one cusp form of dimension 7 or more per Atkin-Lehner eigenspace for prime levels between 10,000 and 1,000,000. Our algorithm is based on Mestre's Méthode des Graphes, and involves supersingular isogeny graphs and Wiedemann's algorithm for finding the minimal polynomial of sparse matrices over finite fields.
Isogeny graphs associated to Moret-Bailly families of supersingular abelian surfaces
研究特征p下阿贝尔曲面的Moret-Bailly族等价类与相对(ℓ,ℓ)-同源构成的有限有向图，将其与GSp₄/ℚ的内形式上的代数模形式空间关联，并利用Jacquet-Langlands对应估计邻接矩阵的特征值。
The Spine of a Supersingular $\ell$-Isogeny graph
本文研究超奇异ℓ-同源图的结构性质，重点分析由F_p-顶点诱导的脊的直径和中心顶点数量，数值结果显示其中心行为与随机(ℓ+1)-正则图一致，为相关密码协议的安全性假设提供了证据。
Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems. In this paper we give a new algorithm for computing the endomorphism ring of a supersingular elliptic curve $E$ that runs, under certain heuristics, in time $O((\log p)^2p^{1/2})$. The algorithm works by first finding two cycles of a certain form in the supersingular $\ell$-isogeny graph $G(p,\ell)$, generating an order $Λ\subseteq \operatorname{End}(E)$. Then all maxi…
We prove that the path-finding problem in $\ell$-isogeny graphs and the endomorphism ring problem for supersingular elliptic curves are equivalent under reductions of polynomial expected time, assuming the generalised Riemann hypothesis. The presumed hardness of these problems is foundational for isogeny-based cryptography. As an essential tool, we develop a rigorous algorithm for the quaternion analog of the path-finding problem, building upon the heuristic method of Kohel, Lauter, Petit and Tignol. This problem, and its (previously heuristic) resolution, are both a powerful cryptanalytic too…
Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field $k$ of characteristic $p$. We combine these results with computations of the automorphism groups to study Oort's conjecture; we prove that every generic three-dimensional principally polarised supersingular abelian variety over $k$ of characteristic $\neq 2$ has automorphism group $\{ \pm 1 \}$.
We study the problem of generating the endomorphism ring of a supersingular elliptic curve by two cycles in $\ell$-isogeny graphs. We prove a necessary and sufficient condition for the two endomorphisms corresponding to two cycles to be linearly independent, expanding on the work in Kohel's thesis. We also give a criterion under which the order generated by two cycles is not a maximal order. We give some examples in which we compute cycles which generate the full endomorphism ring. The most difficult part of these computations is the calculation of the trace of these cycles. We show that a gen…
Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves
本文利用可嵌入自同态环的虚二次序及素数分裂条件，在超奇异同源图中构造自环和环，分析其长度并给出意外2-环对应的素数上界。
Cycles and Cuts in Supersingular L-Isogeny Graphs
本文研究了超奇异L同余图中的循环与割问题，提出基于Brandt矩阵迹的循环计数方法和贪心邻居算法优化图割，为同余密码学提供了理论支持和算法改进。
We investigate the isogeny graphs of supersingular elliptic curves over $\mathbb{F}_{p^2}$ equipped with a $d$-isogeny to their Galois conjugate. These curves are interesting because they are, in a sense, a generalization of curves defined over $\mathbb{F}_p$, and there is an action of the ideal class group of $\mathbb{Q}(\sqrt{-dp})$ on the isogeny graphs. We investigate constructive and destructive aspects of these graphs in isogeny-based cryptography, including generalizations of the CSIDH cryptosystem and the Delfs-Galbraith algorithm.
We explain why the first Galbraith-Petit-Shani-Ti attack on the Supersingular Isogeny Diffie-Hellman and the Supersingular Isogeny Key Encapsulation fails in some cases.
Refined Humbert Invariants in Supersingular Isogeny Degree Analysis
研究主极化超特殊阿贝尔曲面的精细洪堡不变量，通过计算该不变量，在基于同构的密码学中实现检查极化同构、确定曲面几何类型、证明同构度上界及提供同构频率证据等应用，并从新视角分析固定同构度问题。
Failing to hash into supersingular isogeny graphs
本文针对无可信权威下构造不泄露自同态环的超奇异 $\ell$-同源图哈希函数这一开放问题，整理并分析了五类失败的数学构造尝试。
Distribution of cycles in supersingular $\ell$-isogeny graphs
本文扩展了超奇异 $\ell$-同源图中长度为 $r$ 的环的计数工作，推导了沿脊分布的环数量及其在 $p \to \infty$ 时的平均值公式，并证明此类环不成比例地倾向于沿脊出现。
Let $p$ and $q$ be distinct primes, and let $X_{p,q}$ be the $(q+1)$-regular graph whose nodes are supersingular elliptic curves over $\overline{\mathbb{F}}_p$ and whose edges are $q$-isogenies. For fixed $p$, we compute the distribution of the $\ell$-Sylow subgroup of the sandpile group (i.e.\ Jacobian) of $X_{p,q}$ as $q \to \infty$. We find that the distribution disagrees with the Cohen-Lenstra heuristic in this context. Our proof is via Galois representations attached to modular curves. As a corollary of our result, we give an upper bound on the probability that the Jacobian is cyclic, whi…
Minimal degree of an isogeny between a supersingular elliptic curve and its conjugate
研究超奇异椭圆曲线与其共轭曲线间同构最小次数，通过新技术计算次数并给出界，此界渐近最优且在多情况下精确，还基于数据给出多个猜想。
We determine what isogeny classes of supersingular abelian surfaces over a finite field k of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and to count the number of curves, up to k-isomorphism, leading to the same zeta function.
On the Zeta functions of supersingular isogeny graphs and modular curves
本文证明了模曲线的 Hasse--Weil zeta 函数与配备级结构的超奇异椭圆曲线同源图的 Ihara zeta 函数之间的关系公式，推广了 Sugiyama 的结果。
Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields
本文针对有限域上三维阿贝尔簇同源类是否包含亏格3超椭圆曲线雅可比簇的问题，基于韦伊多项式分析提出新的阻碍判据，并猜想其中两个阻碍可在q趋于无穷时渐近分类所有同源类。
We investigate to what extent an abelian variety over a finite field can be lifted to one in characteristic zero with small Mumford-Tate group. We prove that supersingular abelian surfaces, respectively threefolds, can be lifted to ones isogenous to a square, respectively product, of elliptic curves. On the other hand, we show that supersingular abelian threefolds cannot be lifted to one isogenous to the cube of an elliptic curve over the Witt vectors.
We show that supersingular K3 surfaces in characteristic $p\geq5$ are related sequences of very special correspondences. This is not enough to conclude that they are unirational. As a byproduct, we exhibit a fibration structure on the moduli space of rigidified K3 crystals. We also establish Shioda-Inose type isogeny theorems for K3 surfaces with Picard rank $ρ\geq19$ in positive characteristic.
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 $ϕ\colon A \rightarrow A'$ 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…
Ordinary 3-Isogeny Graphs and Improvement of Supersingularity Testing for Twisted Hessian Curves over Prime Fields
本文利用扭曲Hessian曲线的3-同源性质，证明当p≡2 mod 3时普通曲线位于3-火山表面，并据此改进Sutherland超奇异检测算法，同时推广了超奇异j-不变量为立方的结论。
On the Spectral theory of Isogeny Graphs and Quantum Sampling of Secure Supersingular Elliptic curves
本文提出首个证明的量子多项式时间算法，用于高概率采样无内射环的奇异椭圆曲线，基于Booher等人的算法变种，在量子门复杂度上达到O(log^4 p)并基于广义黎曼假设达到O(log^13 p)，并提供安全的CGL哈希函数实例。
Distributions of Iwasawa $λ$-invariants of $\mathbf{Z}_p$-towers over supersingular isogeny graphs
本文固定素数 $r$ 和 $p$，让 $\ell$ 遍历无穷多个素数，研究超奇异 $\ell$-同源图上常 $\mathbf{Z}_p$ 塔的 Iwasawa $\lambda$ 不变量的分布，揭示图论、Iwasawa 理论、椭圆曲线和新形式 Galois 表示之间的联系。
On supersingular isogeny graphs of Drinfeld modules
本文研究A=𝔽_q[T]上秩2的Drinfeld模的超奇异同构图，证明其连通性，引入完备性数并推导其上、下界，证实相关猜想并反驳旧界，还给出计算该数的算法准则。
Neighborhood of vertices in the isogeny graph of principally polarized superspecial abelian surfaces
本研究确定了当其中一条超奇异椭圆曲线定义在$\u2119_p$上时，主极化超特殊阿贝尔曲面的$(\ell, \ell)$-同源图中对应乘积曲面顶点的局部结构，并给出了已有文献主定理的简单新证明。
The supersingular Endomorphism Ring and One Endomorphism problems are equivalent
本文证明超奇异自同态环问题与单自同态问题在概率多项式时间归约下等价，并推导出哈希函数抗碰撞、协议可靠、任意同源计算等价及无条件O~(sqrt(p))算法等推论。
Adventures in Supersingularland
本文系统研究超奇异同源图的结构与路由问题，重点分析脊图S、Frobenius对合下的镜像路径及共轭j-不变量的同源比例，并给出l=2时的直径实验数据，为密码分析提供理论依据。
We revisit theoretical background on OSIDH, that is an isogeny-based key-exchange protocol proposed by Colò and Kohel at NutMiC 2019. We give a proof of a fundamental theorem for OSIDH. The theorem was stated by Colò and Kohel without proof. Furthermore, we consider parameters of OSIDH, give a sufficient condition on the parameters that the protocol works, and estimate the size of the parameters for a certain security level.
Computation of Hilbert class polynomials and modular polynomials from supersingular elliptic curves
该研究针对素数模下类多项式与模多项式的计算问题，借助同源密码学推动下发展的超奇异曲线Deuring对应研究成果，提出多种基于超奇异椭圆曲线的新启发式算法，打破了此前超奇异情形效率低的固有认知。
Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p odd and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves. We prove the analogous results for p=2 under the additional assumption that E is not supersingular at primes above 2.
Given an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits structural differences between ordinary and supersingular isogeny graphs. This yields a simple algorithm that, given E and a suitable non-residue in F_p^2, determines the supersingularity of E in O(n^3 log^2 n) time and O(n) space, where n=O(log p). Both these complexity bounds are significant improvements over existing methods, as we demonstrat…
Supersingular elliptic curves, quaternion algebras and applications to cryptography
本文综述了超奇异同源图及其密码学应用，介绍三个近期结果，并将讨论推广到超特殊同源图，同时证明了一个关于全实域上格生成的一般定理。
An elliptic curve $E$ defined over a $p$-adic field $K$ with a $p$-isogeny $ϕ:E\rightarrow E^\prime$ comes equipped with an invariant $α_{ϕ/K}$ that measures the valuation of the leading term of the formal group homomorphism $Φ:\hat E \rightarrow \hat E^\prime$. We prove that if $K/\mathbb{Q}_p$ is unramified and $E$ has additive, potentially supersingular reduction, then $α_{ϕ/K}$ is determined by the number of distinct geometric components on the special fibers of the minimal proper regular models of $E$ and $E^\prime$.
Stabilization of isogeny spaces between supersingular Drinfeld modules
本文通过Brandt矩阵理论和自动形式的L函数性质，证明了奇点Drinfeld模的形态空间维度稳定公式，并由此构造了半田秩度码。
Abelian surfaces in Hesse form and explicit isogeny formulas
本文提出了一种基于Hesse形式计算主极化阿贝尔面(3,3)-同源的新方法，并探讨了其在更高维和不同同源度的应用。
Let $F$ be a totally real field with ring of integers $O_F$, and $D$ be a totally definite quaternion algebra over $F$. A well-known formula established by Eichler and then extended by Körner computes the class number of any $O_F$-order in $D$. In this paper we generalize the Eichler class number formula so that it works for arbitrary $\mathbb{Z}$-orders in $D$. The motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a prime finite field $\mathbb{F}_p$. We give explicit formulas for the number of these isomorphism classes for all prim…
On towers of Isogeny graphs with full level structure
本文定义带 $p^nN$ 层结构、以 $l$-同源为边的椭圆曲线图，证明其连通分支仅在超奇异情形形成 Galois 覆盖塔，并推广到定向超奇异曲线图。
Weber modular curves and modular isogenies
本文研究韦伯函数定义的模曲线及其关联的模多项式、SL₂(ℤ)作用及椭圆曲线参数化，旨在探讨其确定的同源图，特别是奇异椭圆曲线的性质，并应用于密码学中的高效同源计算及显式伽罗瓦表示。
We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension $g$ only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga-Satake construction, we also show that only finitely many supersingular $K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory an…
Supersingularity and Superspeciality Verification of Abelian Surfaces
本文提出高效算法验证阿贝尔曲面的超奇异性和超特殊性，基于结构分析实现对数时间复杂度，并推广到任意维数。
In this paper we provide an algorithm to classify groups of points on abelian threefolds over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given $\mathbb{F}_q$-isogeny class. This work completes partial classification given by S.Rybakov.
Small degree isogenies between conjugate principally polarized superspecial abelian surfaces
本文证明在有限域上每个主极化超特殊阿贝尔曲面到其Frobenius共轭存在小乘子同源，推广了超奇异椭圆曲线的已知结果。
Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of $E^g$. The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null…
We introduce a special class of supersingular curves over $\mathbb{F}_{p^2}$, characterized by the existence of non-integer endomorphisms of small degree. A number of properties of this set is proved. Most notably, we show that when this set partitions into subsets in such a way that curves within each subset have small-degree isogenies between them, but curves in distinct subsets have no small-degree isogenies between them. Despite this, we show that isogenies between these curves can be computed efficiently, giving a technique for computing isogenies between certain prescribed curves that ca…
Endomorphism Rings of Supersingular Elliptic Curves and Ternary Quadratic Forms
本文研究了超奇异椭圆曲线的自同态环，提出一种通过求解有限域平方根，从 Eichler order 高效计算 $B_{p,\infty}$ 中极大阶的算法，并给出了核理想右阶的基。
Computing Isomorphisms between Products of Supersingular Elliptic Curves
研究有限域上超奇异椭圆曲线乘积间同构的计算方法，利用德林对应将问题转化为四元数代数问题，通过求解相关方程组，结合ℓ - 进技术，给出高效概率算法，在广义黎曼假设下有期望多项式时间复杂度。
Isogeny graphs of superspecial abelian varieties and Brandt matrices
本文研究了 superspecial abelian varieties 的 isogeny 图和 Brandt 矩阵，证明了三种 isogeny 图的连通性，并探讨了它们的性质和 Ramanujan 性质。
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