As far as I know, SSL certificates do not contain contact information such as phone number. However, contact information is probably stored somewhere in order to contact me concerning the SSL cert if the need arises. Or am I wrong about this?
I bought an SSL certificate (from a reseller, if this matters), and have since acquired a new phone number. I have updated that new number at the reseller's website, so that's taken care of. The question is - is there need to update my contact info anywhere else?