Private key not used in a signature scheme
Compute $v^e \equiv a^m b^s c \pmod n$. Output $(e,s,v)$.
  


This should perhaps read:


Solve $v^e \equiv a^m b^s c \pmod n$ for $v$.  Output $(e,s,v)$.


If you can solve $v^e \equiv u \pmod n$ for $v$ without using $p$ in general, you have found a major breakthrough in cryptanalysis!