There are other factors that can inhibit browsing of secure Web sites. One is if the PC's clock (time and date) is not set correctly.
Another possible reason why browsers would not be able to access secure sites, even if they use the latest encryption and their security settings and PC's time and date are correct, is that the maximum transfer unit (MTU) could be set incorrectly, as I discovered from an article at http://www.isaserver.org/tutorials/onlinebanking.html
(I had this problem a few years ago, so I am speaking from first-hand experience. The MTU is a feature of the IP networking protocol.).
If TCP/IP packets are too small, a secure server thinks the information in the packets is possibly malicious and drops them. The requesting computer sits there waiting until it finally times out the connection. Apparently this problem can also be caused by a router, a firewall or another computer in the path between your PC and the secure server that is blocking ICMP Code 3 Type 4 packets. This prevents the sending computer or firewall discovering the MTU path and/or size available. The solution is to increase the MTU value. There is a maximum MTU permissible value, which apparently is different for dial-up and ADSL connections. For ADSL the MTU should be no bigger than 1,492 bytes. (A few years ago I suddenly could not access my secure Amazon account after I had been tinkering with the MTU on my previous PC. As soon as I raised the MTU to 1,492 bytes I could access the secure site again.)
So perhaps your clock or your MTU setting are causing your problem. I have not changed the MTU for SL/Gentoo, but the following page shows how to do it for Debian using ifconfig:http://www.debianadmin.com/change-mtu-m ... rface.html
Perhaps with a bit of Googling you'll be able to come up with something for Gentoo.