Algebraic approaches for solving isogeny problems of prime power degrees

Yasushi Takahashi, Momonari Kudo, Ryoya Fukasaku, Yasuhiko Ikematsu, Masaya Yasuda, Kazuhiro Yokoyama

Research output: Contribution to journalArticlepeer-review

Abstract

Recently, supersingular isogeny cryptosystems have received attention as a candidate of post-quantum cryptography (PQC). Their security relies on the hardness of solving isogeny problems over supersingular elliptic curves. The meet-in-the-middle approach seems the most practical to solve isogeny problems with classical computers. In this paper, we propose two algebraic approaches for isogeny problems of prime power degrees. Our strategy is to reduce isogeny problems to a system of algebraic equations, and to solve it by Gröbner basis computation. The first one uses modular polynomials, and the second one uses kernel polynomials of isogenies. We report running times for solving isogeny problems of 3-power degrees on supersingular elliptic curves over p2 with 503-bit prime p, extracted from the NIST PQC candidate SIKE. Our experiments show that our firstapproach is faster than the meet-in-the-middle approach for isogeny degrees up to 310.

Original languageEnglish
Pages (from-to)31-44
Number of pages14
JournalJournal of Mathematical Cryptology
Volume15
Issue number1
DOIs
Publication statusPublished - Jan 1 2021

All Science Journal Classification (ASJC) codes

  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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