Liftings of Reduction Maps for Quaternion Algebras

We construct liftings of reduction maps from CM points to supersingular points for general quaternion algebras and use these liftings to establish a precise correspondence between CM points on indefinite quaternion algebras with a given conductor and CM points on certain corresponding totally definite quaternion algebras.Comment: 17 page

Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem

Fix an ordinary abelian variety defined over a finite field. The ideal class group of its endomorphism ring acts freely on the set of isogenous varieties with same endomorphism ring, by complex multiplication. Any subgroup of the class group, and generating set thereof, induces an isogeny graph on the orbit of the variety for this subgroup. We compute (under the Generalized Riemann Hypothesis) some bounds on the norms of prime ideals generating it, such that the associated graph has good expansion properties. We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and Robert for computing explicit isogenies in genus 2, to prove random self-reducibility of the discrete logarithm problem within the subclasses of principally polarizable ordinary abelian surfaces with fixed endomorphism ring. In addition, we remove the heuristics in the complexity analysis of an algorithm of Galbraith for explicitly computing isogenies between two elliptic curves in the same isogeny class, and extend it to a more general setting including genus 2.Comment: 18 page

Global Divisibility of Heegner Points and Tamagawa Numbers

We improve Kolyvagin's upper bound on the order of the $p$-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and Swinnerton-Dyer conjectural formula.Comment: 20 page

Isogeny graphs of ordinary abelian varieties

Fix a prime number $\ell$. Graphs of isogenies of degree a power of $\ell$ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called $\mathfrak l$-isogenies, resolving that they are (almost) volcanoes in any dimension. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as $(\ell, \ell)$-isogenies: those whose kernels are maximal isotropic subgroups of the $\ell$-torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.Comment: 36 pages, 4 figure

XorSHAP: Privacy-Preserving Explainable AI for Decision Tree Models

Explainable AI (XAI) refers to the development of AI systems and machine learning models in a way that humans can understand, interpret and trust the predictions, decisions and outputs of these models. A common approach to explainability is feature importance, that is, determining which input features of the model have the most significant impact on the model prediction. Two major techniques for computing feature importance are LIME (Local Interpretable Model-agnostic Explanations) and SHAP (SHapley Additive exPlanations). While very generic, these methods are computationally expensive even in plaintext. Applying them in the privacy-preserving setting when part or all of the input data is private is therefore a major computational challenge. In this paper, we present $\texttt{XorSHAP}$ - the first practical privacy-preserving algorithm for computing Shapley values for decision tree ensemble models in the semi-honest Secure Multiparty Computation (SMPC) setting with full threshold. Our algorithm has complexity $O(T \widetilde{M} D 2^D)$, where $T$ is the number of decision trees in the ensemble, $D$ is the depth of the decision trees and $\widetilde{M}$ is the maximum of the number of features $M$ and $2^D$ (the number of leaf nodes of a tree), and scales to real-world datasets. Our implementation is based on Inpher\u27s $\texttt{Manticore}$ framework and simultaneously computes (in the SMPC setting) the Shapley values for 100 samples for an ensemble of $T = 60$ trees of depth $D = 4$ and $M = 100$ features in just 7.5 minutes, meaning that the Shapley values for a single prediction are computed in just 4.5 seconds for the same decision tree ensemble model. Additionally, it is parallelization-friendly, thus, enabling future work on massive hardware acceleration with GPUs