Digital signatures under plain RSA Show that digital signatures under plain RSA are insecure (Plain RSA means that signing is done by calculating $m^d\bmod n$, with $0\le m&lt;n$, and no padding or hashing of $m$).

Write an algorithm that, given someone’s public signature verification key ($e$, $n$), can easily generate a (message, signature) pair for some message that the private key holder never intended to sign. For full credit your algorithm should not require a signing oracle and your algorithm should generate “signed” messages where your message is different than its signature (Hints: The message need not have any meaning to count as a forgery. Think backwards).
I happened to see some similar question like this. The question mentioned about sending fake signature message. The method is like this:


Find some random string R.
Use the public key to encrypt the random string R, let the result be X.


(R,X) is your signature pair.(Think backwards)
When someone verifies the signature, he'll compare {R} with X which are equal.