# 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?

• Please reformulate your question. As it is stated now, it does not make sense. – user27950 Oct 8 '16 at 15:08
• What shall "addition of modulus" mean? Also, the last sentence "Any changes to public parameters, for example, g^x mod q" cannot be understood, because the verb is missing. – user27950 Oct 8 '16 at 15:57
• I am still not clear what is your intention. Do you want to construct two (or more) Schnorr signatures and always use the same secret k value? – user27950 Oct 8 '16 at 17:03
• @Cryptostasis..yes – peterpe Oct 8 '16 at 17:44
• An then you want to calculate the first signature $s$ as a usual SChnorr signature and the second signature differently, say as $s_2 = s+s_1$ ? – user27950 Oct 8 '16 at 17:48

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.