  CSIDH is a recent quantum-resistant primitive based on the difficulty of
finding isogeny paths between supersingular curves. Recently, two constant-time
versions of CSIDH have been proposed: first by Meyer, Campos and Reith, and
then by Onuki, Aikawa, Yamazaki and Takagi. While both offer protection against
timing attacks and simple power consumption analysis, they are vulnerable to
more powerful attacks such as fault injections. In this work, we identify and
repair two oversights in these algorithms that compromised their constant-time
character. By exploiting Edwards arithmetic and optimal addition chains, we
produce the fastest constant-time version of CSIDH to date. We then consider
the stronger attack scenario of fault injection, which is relevant for the
security of CSIDH static keys in embedded hardware. We propose and evaluate a
dummy-free CSIDH algorithm. While these CSIDH variants are slower, their
performance is still within a small constant factor of less-protected variants.
Finally, we discuss derandomized CSIDH algorithms.
