Internet Checksum Example

Transmitting
1. Message: This is a message
2. Hex, ASCII codes, grouped into 16-bit units and padded with zeros
  5468 6973 2069 7320 6120 6d65 7373 6167 6500
3. Add them up
  5468 + 6973 + 2069 + 7320 + 6120 + 6d65 + 7373 + 6167 + 6500 = 359c3
4. Add in the carryout:
  59c3 + 0003 = 59c6
5. Invert the sum: a639
6. Not zeros, so don't need to replace with ones.
7. Transmit:
  5468 6973 2069 7320 6120 6d65 7373 6167 6500 a639
Checking
1. Receive:
  5468 6973 2069 7320 6120 6d65 7373 6167 6500 a639
2. Add them up
  5468 + 6973 + 2069 + 7320 + 6120 + 6d65 + 7373 + 6167 + 6500 + a639 = 3fffc
3. Add in the carryout:
  fffc + 0003 = ffff
4. Invert the sum: 0000
5. So the message checks.
So why does it work, even if you replace all-zeros with all-ones? (Numbers in this explanation are in hex.)
1. Call the initial checksum of the message data produced at step D on the transmission side S. (S=59c6 in the above example.)
2. We then invert S, call that Ŝ. (a639 in the example.)
3. If Ŝ=0, the checksum is C=ffff, otherwise C=Ŝ.
4. The verifier will then sum the received data, which is R=S+C, where addition is the odd 16-bit wrap-around-carryout kind.
5. If C=Ŝ, then R=S+Ŝ=ffff, which is inverted to 0000.
6. Suppose instead that we used the invert case to force C=ffff.
  1. This happens when Ŝ=0, so S=ffff.
  2. Now R=S+C=ffff+ffff=1fffe=fffe+0001=ffff.
  3. Which is inverted to 0000
7. So you get zero in either case.