I understand that digital signatures are usually implemented by encrypting a message digest with a private key, decrypting the resulting signature with a public key, and verifying that the message digest matches the message. This seems to be a normal application of an asymmetric encryption algorithm rather than a different algorithm, but is there anything about the scheme that makes it vulnerable to additional classes of attack over and above those that the underlying asymmetric encryption algorithm might suffer?
For example, does the use of a message digest algorithm, or the presence of the plaintext message, provide any additional information that an attacker can use to defeat the scheme?
To clarify what I'm asking: public-key encryption might be defeated by attacks such as chosen plaintext, or a breakthrough in factoring large numbers. Does using a public-key encryption algorithm in a digital signature scheme enable any additional types of attack?