This problem is not related to Maverics. I got this problem since late 2011 and I was running Lion (10.7) at the time, with Safari 5.1. I had the problem too with Mountain Lion (10.8) and Safari 6. And now with Maverics.
May be I can give more technical details on what is happening. When Safari goes on a secure web site, the web site gives a certificate to be signed, and Safari has to sign that certificate using one from the keychain. There are plenty of them already in the keychain (added during the installation of the OS). In fact, all the ones we need should be there, and specifically the one needed for that web site. The web site certificate could be displayed by clicking on the lock on the left of the URL field on the top of the Safari window. For example, the one used in this web site here was delivered by Cybertrust Public SureServer. For some people, it seems that there is a confusion in the keychain between the right certificate and a wrong one, and then Safari tries to sign using the wrong certificate. I my case, after suppressing that unwelcomed certificate, the signing process starts to work as it should... at least for some time. Be careful not to suppress any system or root certificates, which will cause an access problem to their related web sites. The one shown on the top of this discussion is located in "my certificates" in the login or session keychain.
What I don't know so far is how that annoying certificate gets in the keychain and how can we prevent it to get there. I thought I came with some secured e-mails I received, but I did not succeed to match it. I am still looking for a solution.
I hope it helped.