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Computing Isomorphisms between Products of Supersingular Elliptic Curves

Pierrick Gaudry, Julien Soumier, Pierre-Jean Spaenlehauer

TL;DR

This paper presents methods for explicitly computing isomorphisms between products of supersingular elliptic curves over finite fields in polynomial time, and leverages the Deuring correspondence to reformulate computational isogeny problems into algebraic problems in quaternions.

Abstract

The Deligne-Ogus-Shioda theorem guarantees the existence of isomorphisms between products of supersingular elliptic curves over finite fields. In this paper, we present methods for explicitly computing these isomorphisms in polynomial time, given the endomorphism rings of the curves involved. Our approach leverages the Deuring correspondence, enabling us to reformulate computational isogeny problems into algebraic problems in quaternions. Specifically, we reduce the computation of isomorphisms to solving systems of quadratic and linear equations over the integers derived from norm equations. We develop $\ell$-adic techniques for solving these equations when we have access to a low discriminant subring. Combining these results leads to the description of an efficient probabilistic Las Vegas algorithm for computing the desired isomorphisms. Under GRH, it is proved to run in expected polynomial time.

Computing Isomorphisms between Products of Supersingular Elliptic Curves

TL;DR

This paper presents methods for explicitly computing isomorphisms between products of supersingular elliptic curves over finite fields in polynomial time, and leverages the Deuring correspondence to reformulate computational isogeny problems into algebraic problems in quaternions.

Abstract

The Deligne-Ogus-Shioda theorem guarantees the existence of isomorphisms between products of supersingular elliptic curves over finite fields. In this paper, we present methods for explicitly computing these isomorphisms in polynomial time, given the endomorphism rings of the curves involved. Our approach leverages the Deuring correspondence, enabling us to reformulate computational isogeny problems into algebraic problems in quaternions. Specifically, we reduce the computation of isomorphisms to solving systems of quadratic and linear equations over the integers derived from norm equations. We develop -adic techniques for solving these equations when we have access to a low discriminant subring. Combining these results leads to the description of an efficient probabilistic Las Vegas algorithm for computing the desired isomorphisms. Under GRH, it is proved to run in expected polynomial time.

Paper Structure

This paper contains 26 sections, 28 theorems, 32 equations, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related works
  4. Organization of the paper
  5. Acknowledgements
  6. Background
  7. Deuring correspondence
  8. Quaternion algebras.
  9. The correspondence.
  10. Efficient representations of isogenies
  11. Ideal representation.
  12. Knowing the endomorphism ring of a curve.
  13. Superspecial Abelian varieties
  14. Main Results
  15. Tools
  16. ...and 11 more sections

Key Result

Proposition 2.8

QuatAlg Let $\mathcal{O}_1, \mathcal{O}_2, \mathcal{O}_3\subset\mathcal{B}_{p,\infty}$ be three maximal orders. If $I\in\mathop{\mathrm{Conn}}\nolimits(\mathcal{O}_1, \mathcal{O}_2)$ and $J\in\mathop{\mathrm{Conn}}\nolimits(\mathcal{O}_2,\mathcal{O}_3)$, then $I\cdot J\in\mathop{\mathrm{Conn}}\nolim

Theorems & Definitions (69)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: Quaternion order
  • Example 2.4
  • Definition 2.5: Left/Right Order
  • Remark 2.6
  • Definition 2.7: Ideal norm
  • Proposition 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 59 more