Efficient Isogeny Computations on Twisted Edwards Curves

The isogeny-based cryptosystem is the most recent category in the field of postquantum cryptography. However, it is widely studied due to short key sizes and compatibility with the current elliptic curve primitives. The main building blocks when implementing the isogeny-based cryptosystem are isogeny computations and point operations. From isogeny construction perspective, since the cryptosystem moves along the isogeny graph, isogeny formula cannot be optimized for specific coefficients of elliptic curves. Therefore, Montgomery curves are used in the literature, due to the efficient point operation on an arbitrary elliptic curve. In this paper, we propose formulas for computing 3 and 4 isogenies on twisted Edwards curves. Additionally, we further optimize our isogeny formulas on Edwards curves and compare the computational cost of Montgomery curves. We also present the implementation results of our isogeny computations and demonstrate that isogenies on Edwards curves are as efficient as those on Montgomery curves.