For assymetric encryption such as RSA, I understand that Alice can sign a document with her secret key and send it to Bob, which he can then verify.
Sign: $\sigma=m^d \bmod N$
Verify: $v=\sigma^e \bmod N$
This may be a silly question, but what if it was the other way around, Bob wants to send a document to Alice. How would he sign it to say it was from him. Can he do it without introducing "Bob's secret key" and just use the scheme provided, or does he have to do that? or is there better way?