Following on from what several people have mentioned in here, i've had this same problem going on for a while now with this 'Cannot guarantee authenticity... certificate' dialog. I managed to fix the problem using the following method.
I'm running Win 7 SP1 with KTS 22.214.171.1248
Closed off Chrome then opened up TCPView
Started Chrome and waited for all the Chrome connections to show up in TCPView, then copied the Chrome entries in TCPView into a spreadsheet and grouped the IP addresses so I can see what the outbound connections were.
Looked at all the LAN IP addresses and found an address in there that was for my Chromecast device which Chrome was connecting to on ports 8008 and 8009.
With Chrome left open, I rebooted Chromecast from my phone and within 6 seconds of Chromecast rebooting, the security dialog appeared. Repeated this test 5 more times and each time same result so I knew this was the culprit.
Kaspersky > More Tools > My Network > Network Monitor
Find Google Chrome parent item in the list, then right click 'Application Network Rules'
Click on Exclusions tab
Tick: 'Do not scan all traffic' and 'Only for specified ports'
Enter port range: 1-8007,8010-65535 (this will exclude ports 8008 and 8009)
Save and close Kaspersky settings.
Closed Chrome off, opened Chrome and rebooted Chromecast several times and problem has disappered.
I have a Win 10 machine on the network as well running the same version of KTS but it doesn't seem to have the above security issue.
Hope this helps someone!