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Suppose we have the following El Gamal digital signature scheme variant: We fix a prime number $p$ and a generator $g$ of the group $Z_p^*$. We choose $x \in Z_{p-1}^*$, which is going to be our secret key and we compute $y = g^x \pmod{p}$, which is going to be our public key. For the signing algorithm, we choose a random varue $k \in Z_p^*$, then compute $$r = g^k \pmod{p}$$ $$s = (m-kr)*x^{-1} \pmod{p-1}$$ and output $(r,s)$.

In order to verify the signature, it's easy to see that the following should hold: $$y^sr^r \equiv g^m \pmod{p}$$ Now, as far as i can tell, the signing algorithm is faster than the one in the original El Gamal scheme, as we compute the inverse of $x$ once and forever, while in the initial scheme we have to compute the inverse of the new random $k$ every time we want to sign a message.

However, what about security? Can we say that the new scheme provides the same security as the initial one or it can be beaten in some way?

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First, you should not use $p-1$ as modulus. Instead , use a big prime factor $q$ of $p-1$, as it is done in the DSA algorithm.

The DSA variant you describe exists as an elliptic curve version. It is called EC-GDSA ( Elliptic Curve German Digital Signature Algorithm) . It is described here: https://www.bsi.bund.de/SharedDocs/Downloads/EN/BSI/Publications/TechGuidelines/TR03111/BSI-TR-03111_pdf.pdf?__blob=publicationFile

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