International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: Radical Isogenies on Montgomery Curves

Hiroshi Onuki , The University of Tokyo
Tomoki Moriya , The University of Tokyo
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Presentation: Slides
Conference: PKC 2022
Abstract: We work on some open problems in radical isogenies. Radical isogenies are formulas to compute chains of N-isogenies for small N and proposed by Castryck, Decru, and Vercauteren in Asiacrypt 2020. These formulas do not need to generate a point of order N generating the kernel and accelerate some isogeny-based cryptosystems like CSIDH. On the other hand, since these formulas use Tate normal forms, these need to transform Tate normal forms to curves with efficient arithmetic, e.g., Montgomery curves. In this paper, we propose radical-isogeny formulas of degrees 3 and 4 on Montgomery curves. Our formulas compute some values determining Montgomery curves, from which one can efficiently recover Montgomery coefficients. And our formulas are more efficient for some cryptosystems than the original radical isogenies. In addition, we prove a conjecture left open by Castryck et al. that relates to radical isogenies of degree 4.
Video from PKC 2022
  title={Radical Isogenies on Montgomery Curves},
  author={Hiroshi Onuki and Tomoki Moriya},