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So assuming Alice and Bob have agreed to use a particular symmetric cryptosystem and share an appropriate key, they want to achieve data origin authentication by computing the header checksum of an IP datagram and then encrypting it.

As far as I'm concerned this would be appropriate since

a) Bob will be able to determine that it was Alice that computed the checksum since it is assumed only Alice and Bob share the secret key.

b) When the checksum is then verified by Bob, he can be assured that the header was not modified in transit and therefore the source IP address is that of Alice's.

What am I missing?

Just to be clear, I am aware of IPSec but for simplicity I just want to consider IP.

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Right after the first packet has arrived, any intruder may listen then reproduce the same packet header. That's simply the reason.

That's known as the replay attack.

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