In a cryptographic digital signature or MAC system, digital signature forgery is the ability to create a pair consisting of a message, , and a signature (or MAC), , that is valid for , but has not been created in the past by the legitimate signer. There are different types of forgery. To each of these types, security definitions can be associated. A signature scheme is secure by a specific definition if no forgery of the associated type is possible. Types The following definitions are ordered from lowest to highest achieved security, in other words, from most powerful to the weakest attack. The definitions form a hierarchy, meaning that an attacker able to do mount a specific attack can execute all the attacks further down the list. Likewise, a scheme that reaches a certain security goal also reaches all prior ones. Total break More general than the following attacks, there is also a total break: when adversary can recover the private information and keys used by the signer, they can create any possible signature on any message. Universal forgery (universal unforgeability, UUF) Universal forgery is the creation (by an adversary) of a valid signature, , for any given message, . An adversary capable of universal forgery is able to sign messages they chose themselves (as in selective forgery), messages chosen at random, or even specific messages provided by an opponent. Selective forgery (selective unforgeability, SUF) Selective forgery is the creation of a message/signature pair by an adversary, where has been chosen by the attacker prior to the attack. may be chosen to have interesting mathematical properties with respect to the signature algorithm; however, in selective forgery, must be fixed before the start of the attack. The ability to successfully conduct a selective forgery attack implies the ability to successfully conduct an existential forgery attack. Existential forgery Existential forgery (existential unforgeability, EUF) is the creation (by an adversary) of at least one message/signature pair, , where has never been signed by the legitimate signer. The adversary can choose freely; need not have any particular meaning; the message content is irrelevant — as long as the pair, , is valid, the adversary has succeeded in constructing an existential forgery. Thus, creating an existential forgery is easier than a selective forgery, because the attacker may select a message for which a forgery can easily be created, whereas in the case of a selective