Improved Supersingularity Testing of Elliptic Curves Using Legendre Form

Yuji Hashimoto, Koji Nuida

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

2 Citations (Scopus)


There are two types of elliptic curves, ordinary elliptic curves and supersingular elliptic curves. In 2012, Sutherland proposed an efficient and almost deterministic algorithm for determining whether a given curve is ordinary or supersingular. Sutherland’s algorithm is based on sequences of isogenies started from the input curve, and computation of each isogeny requires square root computations, which is the dominant cost of the algorithm. In this paper, we reduce this dominant cost of Sutherland’s algorithm to approximately a half of the original. In contrast to Sutherland’s algorithm using j-invariants and modular polynomials, our proposed algorithm is based on Legendre form of elliptic curves, which simplifies the expression of each isogeny. Moreover, by carefully selecting the type of isogenies to be computed, we succeeded in gathering square root computations at two consecutive steps of Sutherland’s algorithm into just a single fourth root computation (with experimentally almost the same cost as a single square root computation). The results of our experiments using Magma are supporting our argument; for cases of characteristic p of 768-bit to 1024-bit lengths, our algorithm runs 43.6% to 55.7% faster than Sutherland’s algorithm.

Original languageEnglish
Title of host publicationComputer Algebra in Scientific Computing - 23rd International Workshop, CASC 2021, Proceedings
EditorsFrançois Boulier, Matthew England, Timur M. Sadykov, Evgenii V. Vorozhtsov
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages15
ISBN (Print)9783030851644
Publication statusPublished - 2021
Event23rd International Workshop on Computer Algebra in Scientific Computing, CASC 2021 - Sochi, Russian Federation
Duration: Sept 13 2021Sept 17 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12865 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference23rd International Workshop on Computer Algebra in Scientific Computing, CASC 2021
Country/TerritoryRussian Federation

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science


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