It's not only the browser
I'm on a Mandriva 2011 machine but have a working opera.
One needs however to upgrade several things.
1. upgrade openssl to OpenSSL 1.0.2e 3 Dec 2015 where
the patch openssl-1.0.2-disable-sslv2v3.patch should be revoked.
So do NOT disable SSLv3.
2. upgrade rootcerts to rootcerts-20151029.00
3. upgrade wget to wget-1.14-4.2
4. upgrade curl to curl-7.28.1-8
5. upgrade nss3 to nss-3.21.0-1
for step 5 one needs to correct and patch a very serious
error inside libstdc++ which on Mandriva 2011 is a part
of gcc 4.6.1. The details of the bug are here :
ftp://ftp.crashrecovery.org/pub/linux/gcc/RPMS/mdv2011/3.1a/README.txt
the details of the upgrade of nss3 to nss-3.21.0 are here :
ftp://ftp.crashrecovery.org/pub/linux/nss3/RPMS/mdv2011/README.txt
After that opera will have no issues whatsoever connecting
to high profile websites using https.