We study the problem of generating the endomorphism ring of a supersingular elliptic curve by two cycles in ℓ-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 by Kohel in his thesis. We also give a criterion under which the ring 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 generalization of Schoof's algorithm can accomplish this computation efficiently. that led to a significant improvement in the running time analysis of the generalization of Schoof's algorithm, and for outlining the proofs of Proposition 2.4 and Lemma 6.9. Finally, we thank the Women in Numbers 4 conference and BIRS, for enabling us to start this project in a productive environment. K.E. and T.M. were partially supported by National Science Foundation awards DMS-1056703 and CNS-1617802. Isogeny graphs2.1. Definitions and properties. In this section, we recall several definitions and notation that are used throughout. We refer the reader to [Sil09] and [Koh96] for a detailed overview on some of the below. Let k be a field of characteristic p > 3.
In this paper, we study isogeny graphs of supersingular elliptic curves. Supersingular isogeny graphs were introduced as a hard problem into cryptography by Charles, Goren, and Lauter for the construction of cryptographic hash functions ([CGL06]). These are large expander graphs, and the hard problem is to find an efficient algorithm for routing, or pathfinding, between two vertices of the graph. We consider four aspects of supersingular isogeny graphs, study each thoroughly and, where appropriate, discuss how they relate to one another.First, we consider two related graphs that help us understand the structure: the 'spine' S, which is the subgraph of G (Fp) given by the j-invariants in Fp, and the graph G (Fp), in which both curves and isogenies must be defined over Fp. We show how to pass from the latter to the former. The graph S is relevant for cryptanalysis because routing between vertices in Fp is easier than in the full isogeny graph. The Fp-vertices are typically assumed to be randomly distributed in the graph, which is far from true. We provide an analysis of the distances of connected components of S.Next, we study the involution on G (Fp) that is given by the Frobenius of Fp and give heuristics on how often shortest paths between two conjugate j-invariants are preserved by this involution (mirror paths). We also study the related question of what proportion of conjugate j-invariants are -isogenous for = 2, 3. We conclude with experimental data on the diameters of supersingular isogeny graphs when = 2 and compare this with previous results on diameters of LPS graphs and random Ramanujan graphs.
We study genus 4 curves over finite fields and two invariants of the p-torsion part of their Jacobians: the a-number (a) and p-rank (f ). We collect and analyze statistical data of curves over Fp for p = 3, 5, 7, 11 and their invariants. Then, we study the existence of Cartier points, which are also related to the structure of J[p]. For curves with 0 ≤ a < g, the number of Cartier points is bounded, and it depends on a and f .
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