Discrete Logarithm Language

Definition

1. With $G_p$ as a Cyclic Group of prime order $p$, and Generator $g$
2. The Relation $dlo{g}_{G_p,g}:=\{(x,w)\in G_p\times \frac{\mathbb{Z}}{{\mathbb{Z}}_p}|g_w=x\}$ is Hard in the group of Quadratic Residues in $(\frac{\mathbb{Z}}{{\mathbb{Z}}_q}{)}^{\ast }$