I need to send confidential data to a server over a TCP connection. I have done a lot of researching and I understand the theoretical part. Based on what I have researched I
As you note, you are a beginner at crypto. If this is a fun toy project to learn about crypto, great. If this is real production code you are going to implement it insecurely. You should be using off-the-shelf tools like SSL/HTTPS/whatever to solve this problem rather than doing it wrong yourself.
I'll take this opportunity to point out areas where your sketch is fatally weak.
3) the client get's the server's public key.
OK. How? This is the most important step. The security of the entire system relies upon this step, and you have completely glossed over how it works. How does the client obtain the public key of the server? What stops an evil person from calling up the client and saying "hey client, I'm the server. Here's my public key!" And now the client is encrypting messages that can only be decrypted by the evildoer. The evildoer has the real server's public key, so the evildoer re-encrypts the message with the real public key and sends it on. Your whole system is thereby compromised. The public key cryptosystem is only secure if there is a secure key exchange mechanism. (And a reasonable question then is: if you have a secure key exchange mechanism, why not simply use it to exchange the message in the first place?)
4) since the server needs to be certain that the message comes from that specific client then the client will encrypt his name (signature) with his private key.
The client should encrypt a hash of the entire message as the signature, not just a part of the message. That way the server has evidence that the whole message was from the client.
6) the client will encrypt the message with the public key from the server. client will then send that message to the server.
This is extremely inefficient. Better is for the server and client to agree upon a key to a symmetric cryptosystem. The key can be transmitted between the server and the client using the public key cryptosystem. The server and client now have a shared secret key that they can use for this communication session.
9) lastly, the server will decrypt the client signature with the public key that was contained on the message to verify that the message is from that client.
How on earth does that help anything? I want to send you a message. You want to know who it comes from. So I send you a photocopy of my drivers license, so you can compare the signature on the license with the signature on the message. How do you know I sent you my drivers license and not a photocopy of someone else's? This doesn't solve the client authentication problem at all. Again, you need to solve the key distribution problem. The system depends on there being a secure key distribution infrastructure, which you have not specified.