Efficient Algorithm for Computing Odd-Degree Isogenies on Montgomery Curves

Abstract

Isogeny-based cryptography, such as commutative supersingular isogeny Diffie-Hellman (CSIDH), has been shown to be promising candidates for post-quantum cryptography. However, their speeds have remained unremarkable. For example, computing odd-degree isogenies between Montgomery curves is a dominant computation in CSIDH. To increase the speed of this isogeny computation, this study proposes a new technique called the “2-ADD-Skip method,” which reduces the required number of points to be computed. This technique is then used to develop a novel algorithm for isogeny computation. It is found that the proposed algorithm requires fewer field arithmetic operations for the degrees of \(\ell \ge 19\) compared with the algorithm of Meyer et al., which utilizes twisted Edwards curves. Further, a prototype CSIDH-512 implementation shows that the proposed algorithm can give a 6.7% speedup over the implementation by Meyer et al. Finally, individual experiments for each degree of isogeny show that the proposed algorithm requires the lowest number of clock cycles among existing algorithms for \(19 \le \ell \le 373\).