(Short Paper) A Faster Constant-Time Algorithm of CSIDH Keeping Two Points

At ASIACRYPT 2018, Castryck, Lange, Martindale, Panny and Renes proposed CSIDH, which is a key-exchange protocol based on isogenies between elliptic curves, and a candidate for post-quantum cryptography. However, the implementation by Castryck et al. is not constant-time. Specifically, a part of the secret key could be recovered by the side-channel attacks. Recently, Meyer, Campos, and Reith proposed a constant-time implementation of CSIDH by introducing dummy isogenies and taking secret exponents only from intervals of non-negative integers. Their non-negative intervals make the calculation cost of their implementation of CSIDH twice that of the worst case of the standard (variable-time) implementation of CSIDH. In this paper, we propose a more efficient constant-time algorithm that takes secret exponents from intervals symmetric with respect to the zero. For using these intervals, we need to keep two torsion points on an elliptic curve and calculation for these points. We implemented our algorithm by extending the implementation in C of Meyer et al. (originally from Castryck et al.). Then our implementation achieved 152.8 million clock cycles, which is about 29.03% faster than that of Meyer et al.