In this paper, we present an efficient method to compute arbitrary odd-degree isogenies on Edwards curves. By using the $w$-coordinate, we optimized the isogeny formula on Edwards curves by Moody \textit{et al}.. The state-of-the-art implementation of isogeny-based cryptosystems works entirely with Montgomery curves since they provide efficient isogeny computation and elliptic curve arithmetic. However, we demonstrated that the same computational costs of elliptic curve arithmetic and isogeny evaluation could be achieved by using the $w$-coordinate on Edwards curves, with additional benefit when computing isogenous curves. For $\ell$-degree isogeny where $\ell=2s+1$, our isogeny formula on Edwards curves outperforms Montgomery curves when $s \geq 2$. The result of our work opens the door for the usage of Edwards curves in isogeny-based cryptography, especially in CSIDH which requires higher degree isogenies.

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