Backstory
The system I'm designing requires encrypted messages as well as message authentication. AES-GCM seems like the logical choice due to its authentication capabilities and high throughput.
The key used to encrypt a message is generally randomly generated per message and encrypted using the recipient's public key. The encrypted key is sent along with the message.
There are generally no persistent key between both parties.
Example
Let's take the following message as an example:
{
"timestamp": "2017-01-01",
"encryptedMessage": "...",
"sign": "..."
"key": "..."
}
The value "timestamp" is in clear text, "encryptedMessage" encrypted, "sign" holds the RSA signing of "timestamp" and "key" holds the encrypted key. In this example. Would it make sense to sign "encryptedMessage" as well?
Question clarification
Are there any benefits of using RSA signing over the authenticity provided by AES-GCM?
Would it make sense to sign a message already encrypted using AES-GCM?