Paper 2023/1511 Original: eprint.iacr.org/2023/1511 ↗

Lower bound of costs of formulas to compute image curves of $3$-isogenies in the framework of generalized Montgomery coordinates

Abstract

In 2022, Moriya, Onuki, Aikawa, and Takagi proposed a new framework named generalized Montgomery coordinates to treat one-coordinate type formulas to compute isogenies. This framework generalizes some already known one-coordinate type formulas of elliptic curves. Their result shows that a formula to compute image points under isogenies is unique in the framework of generalized Montogmery coordinates; however, a formula to compute image curves is not unique. Therefore, we have a question: What formula is the most efficient to compute image curves in the framework of generalized Montogmery coordinates? In this paper, we analyze the costs of formulas to compute image curves of $3$-isogenies in the framework of generalized Montgomery coordinates. From our result, the lower bound of the costs is $1\mathbf{M}+1\mathbf{S}$ as a formula whose output and input are in affine coordinates, $2\mathbf{S}$ as an affine formula whose output is projective, and $2\mathbf{M}+3\mathbf{S}$ as a projective formula.