Verification in Threshold RSA or Threshold Paillier
This is to ensure $v$ is in the cyclic sub-group $G$ of $Z_N^*$ that has a large enough order $m=p'\cdot q'$. Moreover, with a large probability $v$ is a generator of $G$, so that $v_i=v^{s_i}$ is a one-to-one mapping from $s_i$ to $v_i$, which is important when proving correctness of the shares.