Paper 2017/1198

Computing isogenies between Montgomery curves using the action of (0,0)

Joost Renes


A recent paper by Costello and Hisil at Asiacrypt'17 presents efficient formulas for computing isogenies with odd-degree cyclic kernels on Montgomery curves. We provide a constructive proof of a generalization of this theorem which shows the connection between the shape of the isogeny and the simple action of the point (0,0). This generalization removes the restriction of a cyclic kernel and allows for any separable isogeny whose kernel does not contain (0,0). As a particular case, we provide efficient formulas for 2-isogenies between Montgomery curves and show that these formulas can be used in isogeny-based cryptosystems without expensive square root computations and without knowledge of a special point of order 8. We also consider elliptic curves in triangular form containing an explicit point of order 3.

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Public-key cryptography
Publication info
Published elsewhere. MINOR revision.PQCrypto 2018
Velu's formulasMontgomery form2-isogeniesPost-quantum cryptoIsogeny-based crypto
Contact author(s)
j renes @ cs ru nl
2018-07-27: last of 3 revisions
2017-12-18: received
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      author = {Joost Renes},
      title = {Computing isogenies between Montgomery curves using the action of (0,0)},
      howpublished = {Cryptology ePrint Archive, Paper 2017/1198},
      year = {2017},
      note = {\url{}},
      url = {}
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