  We revisit theoretical background on OSIDH, that is an isogeny-based
key-exchange protocol proposed by Col\`o and Kohel at NutMiC 2019. We give a
proof of a fundamental theorem for OSIDH. The theorem was stated by Col\`o and
Kohel without proof. Furthermore, we consider parameters of OSIDH, give a
sufficient condition on the parameters that the protocol works, and estimate
the size of the parameters for a certain security level.

Number Theory