Efficient Algorithm to Compute Odd-Degree Isogenies Between Montgomery Curves for CSIDH

Isogeny-based cryptography, such as commutative supersingular isogeny Diffie-Hellman (CSIDH), have been shown to be promising candidates for post-quantum cryptography. However, their speeds have remained unremarkable. This study focuses on computing odd-degree isogeny between Montgomery curves, which is a dominant computation in CSIDH. Our proposed “2-ADD-Skip method” technique reduces the required number of points to be computed during isogeny computation. A novel algorithm for isogeny computation is also proposed to efficiently utilize the 2-ADD-Skip method. Our proposed algorithm with the optimized parameter reduces computational cost by approximately 12% compared with the algorithm proposed by Meyer and Reith. Further, individual experiments for each degree of isogeny ℓ show that the proposed algorithm is the fastest for 19≤ℓ≤373 among previous studies focusing on isogeny computation including the Õ(√ℓ) algorithm proposed by Bernstein et al. The experimental results also show that the proposed algorithm achieves the fastest on CSIDH-512. For CSIDH-1024, the proposed algorithm is faster than the algorithm by Meyer and Reith although it is slower than the algorithm by Bernstein et al.