Vol. 4, No. 1, 2020

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Supersingular curves with small noninteger endomorphisms

Jonathan Love and Dan Boneh

Vol. 4 (2020), No. 1, 7–22
Abstract

We introduce a special class of supersingular curves over 𝔽p2, characterized by the existence of noninteger endomorphisms of small degree. We prove a number of properties about this set. Most notably, we can partition this set into subsets such 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 distinct subsets can heuristically be computed efficiently, giving a technique for computing isogenies between certain prescribed curves that cannot be efficiently found by searching on -isogeny graphs.

Keywords
supersingular, elliptic curve, isogeny graph, M-small, endomorphism, quaternion, maximal order, Deuring correspondence, partition, archipelago, island, airport, orientation, Hilbert class polynomial
Mathematical Subject Classification 2010
Primary: 11G20, 11R52, 11T71
Milestones
Received: 20 February 2020
Accepted: 29 April 2020
Published: 29 December 2020
Authors
Jonathan Love
Department of Mathematics
Stanford University
Stanford, CA
United States
Dan Boneh
Computer Science Department
Stanford University
Stanford, CA
United States