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When ownership of a digital signature secret key is bound to a specific user, a valid signature shows that the message was sent by that user.
A trapdoor permutation family is a family of permutations, specified by a parameter, that is easy to compute in the forward direction, but is difficult to compute in the reverse direction without already knowing the private key ("trapdoor").
Without the hash function, the text "to be signed" may have to be split (separated) in blocks small enough for the signature scheme to act on them directly.
However, the receiver of the signed blocks is not able to recognize if all the blocks are present and in the appropriate order.
Digital signatures are a standard element of most cryptographic protocol suites, and are commonly used for software distribution, financial transactions, contract management software, and in other cases where it is important to detect forgery or tampering.
Digital signatures are often used to implement electronic signatures, which includes any electronic data that carries the intent of a signature, electronic signatures have legal significance. In many instances they provide a layer of validation and security to messages sent through a non-secure channel: Properly implemented, a digital signature gives the receiver reason to believe the message was sent by the claimed sender.