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ECDSA specs require private key ($d$) to be a random integer ($r$) in the range $[1, N-1]$, where $N$ is the order of the base point ($G$). What if the private key instead is set to the x-coordinate of the EC point $r \cdot G$ (i.e., $G$ added $r$ times)?

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    $\begingroup$ That’s effectively the formula for the public key. You’d need to pretty much change everything. $\endgroup$ – bartonjs May 11 '18 at 17:18
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What if the private key instead is set to the x-coordinate of the EC point $r\cdot G$

Then your "private" key is the x-coordinate of your public key and thus public, unless you also want to redefine how the public key is constructed.

Additionally, the idea of ECDSA / Schnorr signatures is to prove knowledge of a solution to the discrete logarithm problem at the example of a message hash and if your private key is the x-coordinate of a point, you don't know said discrete logarithm and thus also need to come up with a new signature scheme.

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