Schnorr's signature scheme with addition of modulus secure? In Schnorr signature scheme,$s=(k-xe)$$ s_1=(k-xe_1)$ If the same key $k$ is used, then it is possible to reveal $k$ and $x$ value by using $\frac{s-s_1}{e_1-e}$. 
However, if we add $s=(k-xe) \bmod q$, $s_1=(k-xe_1) \bmod q$, $q$ is known by users. Will modulus usage prevent to reveal $k$ and $x$ value? If the modification scheme is secure, will there be more modification of Schnorr signature public key $g^x$ to something?
There's no chance to hide private key $x$ in case of re-using $k$ by calculating in the ring modulus $q$.
Formula $\frac{s - s_1}{e_1 - e}$ is well-defined over the ring modulo any prime $q$. In particular, extended Euclidean algorithm would do the division.