Supersingular isogeny graphs and endomorphism rings: reductions and solutions

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Authors

Colleges, School and Institutes

External organisations

  • The Pennsylvania State University, USA
  • Microsoft Research

Abstract

In this paper, we study several related computational problems for supersingular elliptic curves, their isogeny graphs, and their endomorphism rings. We prove reductions between the problem of path finding in the l-isogeny graph, computing maximal orders isomorphic to the endomorphism ring of a supersingular elliptic curve, and computing the endomorphism ring itself. We also give constructive versions of Deuring's correspondence, which associates to a maximal order in a certain quaternion algebra an isomorphism class of supersingular elliptic curves. The reductions are based on heuristics regarding the distribution of norms of elements in quaternion algebras. We show that conjugacy classes of maximal orders have a representative of polynomial size, and we define a way to represent endomorphism ring generators in a way that allows for efficient evaluation at points on the curve. We relate these problems to the security of the Charles-Goren-Lauter hash function. We provide a collision attack for special but natural parameters of the hash function and prove that for general parameters its preimage and collision resistance are also equivalent to the endomorphism ring computation problem.

Details

Original languageEnglish
Title of host publicationProceedings of 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT 2018)
Publication statusPublished - 31 Mar 2018
Event37th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT 2018) - Tel Aviv , Israel
Duration: 29 Apr 20183 May 2018

Publication series

NameLecture Notes in Computer Science
PublisherSpringer
Volume10822
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference37th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT 2018)
CountryIsrael
CityTel Aviv
Period29/04/183/05/18