### The Generalized Montgomery Coordinate: A New Computational Tool for Isogeny-based Cryptography

##### Abstract

Recently, some studies have constructed one-coordinate arithmetics on elliptic curves. For example, formulas of the $x$-coordinate of Montgomery curves, $x$-coordinate of Montgomery$^-$ curves, $w$-coordinate of Edwards curves, $w$-coordinate of Huff's curves, $\omega$-coordinates of twisted Jacobi intersections have been proposed. These formulas are useful for isogeny-based cryptography because of their compactness and efficiency. In this paper, we define a novel function on elliptic curves called the generalized Montgomery coordinate that has the five coordinates described above as special cases. For a generalized Montgomery coordinate, we construct an explicit formula of scalar multiplication that 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 present two applications of the theory of a generalized Montgomery coordinate. The first one is the construction of 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 $\sqrt{\vphantom{2}}$\'{e}lu's formula in our implementation. The second one is the construction of a new generalized Montgomery coordinate for Montgomery$^-$ curves used for CSURF.

Available format(s)
Category
Public-key cryptography
Publication info
Preprint.
Keywords
isogeny-based cryptography Velu's formulas elliptic curves generalized Montgomery coordinates
Contact author(s)
tomoki_moriya @ mist i u-tokyo ac jp
History
2022-08-16: last of 2 revisions
See all versions
Short URL
https://ia.cr/2022/150

CC BY

BibTeX

@misc{cryptoeprint:2022/150,
author = {Tomoki Moriya and Hiroshi Onuki and Yusuke Aikawa and Tsuyoshi Takagi},
title = {The Generalized Montgomery Coordinate: A New Computational Tool for Isogeny-based Cryptography},
howpublished = {Cryptology ePrint Archive, Paper 2022/150},
year = {2022},
note = {\url{https://eprint.iacr.org/2022/150}},
url = {https://eprint.iacr.org/2022/150}
}

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