This is kind of a nitpick, but it's unlikely that the keyfob is doing public key cryptography. Those things have to be as energy-efficient as possible in order

This is kind of a nitpick, but it's unlikely that the keyfob is doing public key cryptography. Those things have to be as energy-efficient as possible in order to maximize battery life. An HMAC would accomplish effectively the same thing, and is much more efficient to compute.