About the syntax and the identifiable abort functionality

Hi, I have two questions about the syntax utilized in this paper and how the identifiable abort functionality could be implemented:

  • First, in the Auxiliary Info. & Key Refresh in Three Rounds protocol (p.24, Figure 6, Output round, step 1), the parameter \mu is computed with exponent 1/N and reduced \mod N^2. However, is not clear to me if N is related to the Paillier modulus of the party (and if in that case should be written N_i instead). Otherwise, if N is not related to N_i, how is N obtained ?

  • Second, when obtaining the inverse of N, that is the exponent 1/N, in the computation of \mu, wouldn’t it be possible to obtain a value that has no inverse and therefore, culprit identification couldn’t be performed ?

Thanks in advance