In this paper, we define a novel function on elliptic curves called the generalized Montgomery coordinate that has the four coordinates described above as special cases. For a generalized Montgomery coordinate, we construct an explicit formula of scalar multiplication which includes the division polynomial, and both a formula of an image point under an isogeny and that of a coefficient of the codomain curve.
Finally, we expect numerous applications for the generalized Montgomery coefficient. As an experimental study, we present two applications of the theory of a generalized Montgomery coordinate. The first one is to construct a new efficient formula to compute isogenies on Montgomery curves. This formula is more efficient than the previous one for high degree isogenies as the $\sqrt{\vphantom{2}}$élu's formula in our implementation. The second one is to construct a new generalized Montgomery coordinate for Montgomery$^-$ curves used for CSURF.
Category / Keywords: public-key cryptography / isogeny-based cryptography, Velu's formulas, elliptic curves, generalized Montgomery coordinates Date: received 10 Feb 2022 Contact author: tomoki_moriya at mist i u-tokyo ac jp Available format(s): PDF | BibTeX Citation Version: 20220212:064653 (All versions of this report) Short URL: ia.cr/2022/150