This might be a solution, albeit a rather primitive one (I'm no expert on this).
1.Update Java and make sure you have the latest Minecraft launcher then try the login again.
2. If the update does not work try moving the file "trusted.certs" from the directory \Sun\Java\Deployment\Security (for me the directory would look like "C:\Users\Matt\AppData\LocalLow\Sun\Java\Deployment\security") to somewhere outside of that, like your desktop (Don't delete it just in case it doesn't work and you need to move it back). After that I would give the launcher another try.
3. And if that doesn't work you could try a complete uninstall/reinstall of Java and Minecraft.
And if I'm completely mistaken on this, as I said I'm no expert, feedback and alternatives would be great.
