You need to both sign and encrypt data using RSA.
Sign the concatenation of the public key of the message recipient and the data you actually wish to sign. Then concatenate the signature to the plaintext, and encrypt everything, in multiple messages if necessary.
Naïve implementations where a message is both signed and encrypted with public key cryptography tend to be insecure. Simply signing data with a private key and then encrypting the data with a public key isn’t secure, even if the signature is part of the data you encrypt. Such a scheme is susceptible to an attack called surreptitious forwarding . For example, suppose that there are two servers, S1 and S2. The client C signs a message and encrypts it with S1’s public key. Once S1 decrypts the message, it can reencrypt it with S2’s public key and make it look as if the message came from C.
In a connection-oriented protocol, it could allow a compromised S1 to replay a key transport between C and S1 to a second server S2. That is, if an attacker compromises S1, he may be able to imitate C to S2. In a document-based environment such as an electronic mail system, if Alice sends email to Bob, Bob can forward it to Charlie, making it look as if it came from Alice instead of Bob. For example, if Alice sends important corporate secrets to Bob, who also works for the company, Bob can send the secrets to the competition and make it look as if it came from Alice. ...