In Schnorr signature scheme,
$ 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?

  • $\begingroup$ Please reformulate your question. As it is stated now, it does not make sense. $\endgroup$ – user27950 Oct 8 '16 at 15:08
  • $\begingroup$ 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. $\endgroup$ – user27950 Oct 8 '16 at 15:57
  • $\begingroup$ 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? $\endgroup$ – user27950 Oct 8 '16 at 17:03
  • $\begingroup$ @Cryptostasis..yes $\endgroup$ – peterpe Oct 8 '16 at 17:44
  • $\begingroup$ An then you want to calculate the first signature $s$ as a usual SChnorr signature and the second signature differently, say as $s_2 = s+s_1$ ? $\endgroup$ – 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.

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