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Abstract An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of ‘hard supersingular curves’ that is equations for supersingular curves for which computing the endomorphism ring is as difficult as it is for random supersingular curves. A related open problem is to produce a hash function to the vertices of the supersingular $$\ell $$-isogeny graph, which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring. Such a hash function would open up interesting cryptographic applications. In this paper, we document a number of (thus far) failed attempts to solve this problem, in the hope that we may spur further research, and shed light on the challenges and obstacles to this endeavour. The mathematical approaches contained in this article include: (i) iterative root-finding for the supersingular polynomial; (ii) gcd’s of specialized modular polynomials; (iii) using division polynomials to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces, and applying Kummer surfaces and (v) using quantum random walks. 

Abstract Qualitative spatial/temporal reasoning (QSR/QTR) plays a key role in research on human cognition, e.g., as it relates to navigation, as well as in work on robotics and artificial intelligence. Although previous work has mainly focused on various spatial and temporal calculi, more recently representation learning techniques such as embedding have been applied to reasoning and inference tasks such as query answering and knowledge base completion. These subsymbolic and learnable representations are well suited for handling noise and efficiency problems that plagued prior work. However, applying embedding techniques to spatial and temporal reasoning has received little attention to date. In this paper, we explore two research questions: (1) How do embedding-based methods perform empirically compared to traditional reasoning methods on QSR/QTR problems? (2) If the embedding-based methods are better, what causes this superiority? In order to answer these questions, we first propose a hyperbolic embedding model, called HyperQuaternionE, to capture varying properties of relations (such as symmetry and anti-symmetry), to learn inversion relations and relation compositions (i.e., composition tables), and to model hierarchical structures over entities induced by transitive relations. We conduct various experiments on two synthetic datasets to demonstrate the advantages of our proposed embedding-based method against existing embedding models as well as traditional reasoners with respect to entity inference and relation inference. Additionally, our qualitative analysis reveals that our method is able to learn conceptual neighborhoods implicitly. We conclude that the success of our method is attributed to its ability to model composition tables and learn conceptual neighbors, which are among the core building blocks of QSR/QTR.