I've recently looked into fundamentals of RSA and think I understand how it works. However, there's one basic problem I can't seem to figure out. If someone received ciphertext for a given message, what is to stop them from sending the same message/ciphertext to other nodes in a network and pretending to be the private key holder?
For example, suppose Alice is a general and commands soldiers Bob, Sue, and Jerry. Alice generates private key (n, d) and public key (n, e) and sends (n, e) to all soldiers. The soldiers wait for Alice's instructions every day, which is either "attack" or "don't attack". Alice sends encrypted message "attack" to the soldiers via RSA as ciphertext to all soldiers. Now suppose one of the soldiers is malicious and has hacked Alice's delivery method (e.g. email), but not her private key. And on a separate day sends the same "attack" encrypted cypher text to all other soldiers. What's to prevent this from happening (even if you use padding which will always result in decrypted message "attack")?
I suppose one way is to add a timestamp to all messages and soldiers must make sure the date matches with current day. But it still doesn't seem right that someone can send the same exact message to others by just having the cyphertext... Or am I missing something?