I admit I don't know how you'd make browser goto ONLY https sites, but the suggestion
Quote:
It might be worth while to install the HTTPS Everywhere add-on to Firefox. This program forces secure connections to web sites wherever possible.
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is recommended and definitely a good start.
I would also recommend "noscript" addon (firefox), but it does need tuning for each site. I'm guessing there would be a limited num of sites?
Possibly also "uBlock Origin".
In fact there are a number of security addons available, but you would need to read up on them.
Possibly you could describe a bit more exactly what this system is for and where you are getting the (security) requirements from - that could help us advise you better.
You may even want to look at the Security forum here at LQ