Half-open connection
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A half-open connection refers to a TCP connection that is partially open.
The TCP protocol has a three state system for opening a connection. First, the originating site (A) sends a SYN packet to the destination (B). A is now half-open, and awaiting a response. B now updates its kernel information to indicate the incoming connection from A, and sends out a request to open a channel back (the SYN/ACK packet).
At this point, B is now "half-open" (it has sufficient information to receive packets, but not enough to send packets back). Note that B was put into this state by another machine, outside of B's control.
Under normal circumstances (see denial-of-service attack for deliberate failure cases), A will receive the SYN/ACK from B, update its tables (which now have enough information for A to both send and receive), and send a final ACK back to B.
Once B receives this final ACK, it also has sufficient information for two-way communication, and the connection is fully open.