Capturing stdout to zip and interrupting using CTRL-C gives a corrupted zip file

I am developing a C++ program that can run all day. It outputs to stdout and I want to compress this output. The uncompressed output can be many GB. A startup Bourne shell script compiles the C++ code and starts up the program like so:

./prog | gzip > output.gz

When I interrupt the script using CTRL-C, the .gz file is always corrupted. When I start the program from a terminal and interrupt it using CTRL-C, the .gz file is also always corrupted. When I start the program a terminal and terminate it using Linux killall, the .gz file is fine.

On the other hand, on a terminal cat <large_file> | gzip > cat.gz can be interrupted using CTRL-C and cat.gz is always fine. So I suspect cat has a signal handler of some sort that I have to implement as well in my program in C++... but looking at a cat implementation online, I found nothing like it. Never the less, I implemented this:

void SignalHandler(int aSignum)
{
  exit(0);
}

void Signals()
{
  signal(SIGINT,  SignalHandler);
  signal(SIGKILL, SignalHandler);
  signal(SIGTERM, SignalHandler);
}

...and even something in the bsh script, but nothing helps. After CTRL-C, the gz file is corrupted.

Questions:

• What does cat have what my program does not?
• How can I terminate my script/program using CTRL-C with the zip file in order?

Edit 1

Opening the resulting file using zcat gives some output, but then: gzip: file.gz: unexpected end of file. Opening it in Ubuntu's Archive Manager just gives a popup saying An error occurred while extracting files.

Edit 2

Tried flushing; no change in the problem was observed.

Edit 3

More info about the issue: Missing end (EOCDR) signature

Fix archive (-F) - assume mostly intact archive
    zip warning: bad archive - missing end signature
    zip warning: (If downloaded, was binary mode used?  If not, the
    zip warning:  archive may be scrambled and not recoverable)
    zip warning: Can't use -F to fix (try -FF)

zip error: Zip file structure invalid (file.gz)
maot@HP-Pavilion-dv7:~/temp$ zip -FF file.gz --out file2.gz
Fix archive (-FF) - salvage what can
    zip warning: Missing end (EOCDR) signature - either this archive
                     is not readable or the end is damaged
Is this a single-disk archive?  (y/n): y
  Assuming single-disk archive
Scanning for entries...
    zip warning: zip file empty
maot@HP-Pavilion-dv7:~/temp$ ls -lh file2.gz
-rw------- 1 maot maot 22 feb 15 15:18 file2.gz
maot@HP-Pavilion-dv7:~/temp$ 

Edit 4

Thanks @Maxim Egorushkin, but it does not work. The interruption of the script by CTRL-C kills prog before the signal handler of the script is executed. Hence, I can not send it a signal, it's already gone... and without output of SignalHandler. When prog is started from the command line, the output of SignalHandler is observed. Prog:

#include <iostream>
#include <unistd.h>
#include <csignal>

void SignalHandler(int aSignum)
{
  std::cout << "prog: Interrupt signal " << aSignum << " received.\n";
  fflush(nullptr);
  exit(0);
}

int main()
{
  for (int sig = 1; sig <=31; sig++)
  {
    std::cout << " sig " << sig;
    signal(sig,  SignalHandler);
  }

  while (true)
  {
    std::cout << "prog: Sleep ";
    fflush(nullptr);
    usleep(1e4);
  }
}

Script:

#!/bin/sh

onerror()
{
  echo "onerror(): Started."
  ps -jef | grep prog
  killall -s SIGINT prog
  exit
}

g++ -Wall prog.cpp -o prog

trap onerror 2

prog | gzip > file.gz

Result:

maot@HP-Pavilion-dv7:~/temp$ test.sh 
^Conerror(): Started.
maot     16733 16721 16721  5781  0 16:17 pts/1    00:00:00 grep prog
prog: no process found
maot@HP-Pavilion-dv7:~/temp$ 

Edit 5 minimal working solution

Implementation of the answer of Maxim Egorushkin. Script:

#!/bin/sh
g++ -Wall prog.cpp -o prog
prog | setsid gzip > file.gz & wait

Prog:

#include <iostream>
#include <unistd.h>
#include <csignal>

void SignalHandler(int aSignum)
{
  std::cout << "prog: Interrupt signal " << aSignum << " received.\n";
  exit(0);
}

int main()
{
  signal(SIGINT,  SignalHandler);

  while (true)
  {
    std::cout << "prog: Sleep ";
    usleep(1e4);
  }
}