Paper 2021/699

Radical Isogenies on Montgomery Curves

Hiroshi Onuki and Tomoki Moriya


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.

Available format(s)
Public-key cryptography
Publication info
A minor revision of an IACR publication in PKC 2022
Post-quantum cryptographyradical isogeniesMontgomery curvesCSIDH
Contact author(s)
onuki @ mist i u-tokyo ac jp
2021-12-25: last of 2 revisions
2021-05-28: received
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Creative Commons Attribution


      author = {Hiroshi Onuki and Tomoki Moriya},
      title = {Radical Isogenies on Montgomery Curves},
      howpublished = {Cryptology ePrint Archive, Paper 2021/699},
      year = {2021},
      note = {\url{}},
      url = {}
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