Error opening https sites from home?

[Maverick]

Quite often I face a problem of accessing https sites like https://www.bankofamerica.com and many others. Apparently all these websites are standard website with valid certificates and work fine from office internet access and my friend's place. However from my home where I have a Linksys router running on default configuration at times (after certain hours of usage) I start getting certificate security errors (ERR_CERT_INVALID) for all the most commonly access websites and the examples are mentioned above. The same happens on my spouse laptop as well. I am not sure if it is linksys issue or same malware/somecrap sniffer creating problem.
Something more to add - this problem occurs intermittent, only known pattern that fix it for me is router restart - does not always work though. I use the same laptop at work and at home and am facing this problem for over a year now. I do have all windows update automatically synced up. I'll try and see how to sync/update root certificates. and yes I won't be able to gives votes- my current level does not allow.
Actual message from firefox screen - it happened just now
The certificate is only valid for the following names:
*.s3.amazonaws.com , s3.amazonaws.com

(Error code: ssl_error_bad_cert_domain)

[Noah23453]

You have a virus, oh no!!!!!

[DunbarPappy®ϟϟ]

Most common problem:
Right click tray time> "Adjust..."> Update.
While here, tick 'Automatic synchronize...'

I don't believe it's related to the function or settings of the router; that is a straight forward device that does not examine 'Certificates' or packet contents for that matter: only headers.
*****
Taking a different tack, you might be encountering "3rd party assets" (remote sources, usually advertising) that have a Certificate issued, but it does not come from the 1st party (the site in the address bar) and so Firefox is flagging it, for a reason I'm not sure of just now.

Try making these tweaks and see if it wok relieve some of the problems:
FIREFOX: Tools> Options> Privacy> top drop slot set to "Firefox will use custom settings..."> Check 'Accept Cookies from sites'
â–ºUn-Check 'Accept 3rd Party cookies'.

A good collection of effective security & privacy add-on's (Official Mozilla) is here:
https://addons.mozilla.org/collections/d...nbarpappy/
"AdBlock Plus" (with 'subscription' added) helps defray ad rubbish.

(From that collection, also get "Perspectives" (a Certificate Notary service) and "Certificate Patrol" (monitors and notifies of Certificate changes and essentially interrupts the 'Trust on first use' behavior of browsers until you can look it over and accept or deny it).