# Algebraic approaches for solving isogeny problems of prime power degrees

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

## 抄録

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.

本文言語 英語 31-44 14 Journal of Mathematical Cryptology 15 1 https://doi.org/10.1515/jmc-2020-0072 出版済み - 1 1 2021

## All Science Journal Classification (ASJC) codes

• コンピュータ サイエンスの応用
• 計算数学
• 応用数学

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