HTTPS connections over proxy servers

TLS/SSL (The S in HTTPS) guarantees that there are no eavesdroppers between you and the server you are contacting, i.e. no proxies. Normally, you use CONNECT to open up a TCP connection through the proxy. In this case, the proxy will not be able to cache, read, or modify any requests/responses, and therefore be rather useless.

If you want the proxy to be able to read information, you can take the following approach:

1. Client starts HTTPS session
2. Proxy transparently intercepts the connection and returns an ad-hoc generated(possibly weak) certificate K_a, signed by a certificate authority that is unconditionally trusted by the client.
3. Proxy starts HTTPS session to target
4. Proxy verifies integrity of SSL certificate; displays error if the cert is not valid.
5. Proxy streams content, decrypts it and re-encrypts it with K_a
6. Client displays stuff

An example is Squid's SSL bump. Similarly, burp can be configured to do this. This has also been used in a less-benign context by an Egyptian ISP.

Note that modern websites and browsers can employ HPKP or built-in certificate pins which defeat this approach.

as far as i can remember, you need to use a HTTP CONNECT query on the proxy. this will convert the request connection to a transparent TCP/IP tunnel.

so you need to know if the proxy server you use support this protocol.

I don't think "have HTTPS connections over proxy servers" means the Man-in-the-Middle attack type of proxy server. I think it's asking whether one can connect to a http proxy server over TLS. And the answer is yes.

Is it possible to have HTTPS connections over proxy servers?

Yes, see my question and answer here. HTTPs proxy server only works in SwitchOmega

If yes, what kind of proxy server allows this?

The kind of proxy server deploys SSL certificates, like how ordinary websites do. But you need a pac file for the brower to configure proxy connection over SSL.