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Double-orientations on supersingular isogeny graphs

Do Eon Cha, Imin Chen

TL;DR

The paper introduces double-orientations on supersingular elliptic curves by embedding two imaginary quadratic fields into the quaternion endomorphism algebra B_0 and studies the resulting double-oriented isogeny graphs and quaternion ideal graphs. It develops a contravariant equivalence among curves with double-orientations, primitive/connecting ideals, and maximal orders, extending Deuring’s correspondence to the double-orientation setting and deriving detailed edge correspondences. It provides explicit graph isomorphisms between double-oriented graphs and quaternion-ideal graphs, characterizes local/global roots, and links these graphs to Bass orders and Bruhat–Tits trees, thereby enriching the algebraic-combinatorial toolbox for isogeny-based cryptography. A key computational result is a probabilistic polynomial-time reduction from the Isogeny Path Finding problem to the Maximal Order problem under plausible heuristics or GRH, tying cryptanalytic hardness to quaternion algebra structure and informing potential protocol design. Altogether, the work deepens structural understanding of isogeny graphs through double-orientation data and reveals new connections to lattice-theoretic objects with potential cryptographic implications.

Abstract

We recall and define various kinds of supersingular $\ell$-isogeny graphs and precise graph isomorphism with a corresponding quaternion $\ell$-ideal graph. In particular, we introduce the notion of double-orientations on supersingular elliptic curves and study the structure of double-oriented supersingular $\ell$-isogeny graphs.

Double-orientations on supersingular isogeny graphs

TL;DR

The paper introduces double-orientations on supersingular elliptic curves by embedding two imaginary quadratic fields into the quaternion endomorphism algebra B_0 and studies the resulting double-oriented isogeny graphs and quaternion ideal graphs. It develops a contravariant equivalence among curves with double-orientations, primitive/connecting ideals, and maximal orders, extending Deuring’s correspondence to the double-orientation setting and deriving detailed edge correspondences. It provides explicit graph isomorphisms between double-oriented graphs and quaternion-ideal graphs, characterizes local/global roots, and links these graphs to Bass orders and Bruhat–Tits trees, thereby enriching the algebraic-combinatorial toolbox for isogeny-based cryptography. A key computational result is a probabilistic polynomial-time reduction from the Isogeny Path Finding problem to the Maximal Order problem under plausible heuristics or GRH, tying cryptanalytic hardness to quaternion algebra structure and informing potential protocol design. Altogether, the work deepens structural understanding of isogeny graphs through double-orientation data and reveals new connections to lattice-theoretic objects with potential cryptographic implications.

Abstract

We recall and define various kinds of supersingular -isogeny graphs and precise graph isomorphism with a corresponding quaternion -ideal graph. In particular, we introduce the notion of double-orientations on supersingular elliptic curves and study the structure of double-oriented supersingular -isogeny graphs.
Paper Structure (31 sections, 73 theorems, 322 equations, 3 algorithms)

This paper contains 31 sections, 73 theorems, 322 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Supersingular isogeny graphs
  5. Quadratic orders
  6. Orientations
  7. Quaternion Algebras
  8. Equivalence of categories in terms of quaternion algebras
  9. Correspondence between curves and ideals
  10. Cyclic isogenies and primitive ideals
  11. Correspondences between curves, primitive ideals, and maximal orders
  12. Connecting ideals
  13. Orientations and maximal orders
  14. Single-orientations with maximal orders
  15. Double-orientations with maximal orders
  16. ...and 16 more sections

Key Result

Theorem 1.1

Given a set $S$ of an even number of places of $\mathbb{Q}$, there is one isomorphism class of quaternion algebras over $\mathbb{Q}$ ramified exactly at $S$.

Theorems & Definitions (198)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • proof
  • proof
  • Definition 3.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.2
  • proof
  • ...and 188 more