Problems, Countermeasures, and Emerging Security Notions

In Section 2.4, just as in the early literature, the notion of breaking had not been made precise. Implicitly, one assumed the following problem Given the public key and a message, find a corresponding signature. However, one noticed that some schemes that seem infeasible to break in this sense are vulnerable if one allows stronger attacks and weaker forms of success. 

the terminology that has later been developed for different types of attacks and success, e.g., in [G0MR88], is introduced. 

The most important types of attacks, in the order of increasing strength, are  

In the most general type of active attack, the attacker can choose those messages at any time. This is called an adaptive chosen-message attack. For technical reasons, weaker forms were considered where the attacker must choose all the messages before the signer publishes her public key or at least before the signer issues the first signature. 

The most important types of success, in decreasing order, are  

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