Definition of Schemes with Prekey

Instead of being a separate component of the signature schemes, (AllFam, alljtest) could be a component of the zero-knowledge proof schemes, in particular because parts of V and Obs may be equal to all test. However, I did not want to change the definition of zero-knowledge proof schemes more than necessary. 

The outputs of gertg are written (prek, aux). The first output, prefe, is called a prekey the second output, aux, is only needed to convince the signer s entity of the correctness of prek in the zero-knowledge proof. Note that the inputs to geng are only the two security parameters par = ( 1 , 1 ), and not the message bound N. 

A functional version, gen/i, is defined as usual It outputs sk, mk), where sk consists of skjemp and a random string 

These components must have the following additional properties  

The domain of the first parameter consists of values pk = (prek, mk) where parameters par = ( 1 , V, 1 ) exist such that prek e and 

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