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I want to calculate the v-value from the signature obtained using the OpenSSL ECDSA_do_sign function. I found that this value is calculated using the parity of y-coordinate of r. Here I also found that the parity of y is equal to the parity of r. I want to understand whether it depends on the implementation. Can anyone tell exactly how the coordinates x and y are stored inside r in OpenSSL. I can't find it in the code.

edited: I understand how v is calculated. My question is how to get the y-coordinate of r in the specific implementation of ECDSA in the OpenSSL library.

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The value r is just a number and doesn't explicitly store or encode any point coordinates. In a signature, r is set to the X coordinate of the point R, which is really k*G, where k is the secret nonce used during signing, then reduced mod the curve order. In secp256k1, this usually means that r is in fact the X coordinate (because r itself is usually very big), but it is not always so.

During verification, the verifier attempts to reconstruct the original R point by from the signature values (s, r), the message z and the signer's public key P by solving the equation s*R' = z*G + r*P, then reduce the X coordinate of R' mod the curve order and check that it is equal to r given in the signature.

I've never used openssl, but the process seems to be in ossl_ecdsa_verify_sig(). So you can see R' being computed here : https://github.com/openssl/openssl/blob/master/crypto/ec/ecdsa_ossl.c#L396-L404

After these lines, the reduction mod the order is done and finally the equality test. In simple signature verification, the Y coordinate of R' is not used, but it is available from the value point (which is R') in openssl's implementation.

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Also, I just realized I should add a something about pubkey recovery since you mention the v or recid value in your question. When doing pubkey recovery from message and signature (as in the case when the signer's pukey is not given), to get the Y coordinate from a possible R' point, you would try a few values based on r as the candidate X coordinates.

The possible X coordinates of R' will be r itself, or r + curve_order (and both can be valid X's). You can then solve the curve's equation y^2 = x^3 + 7 to get the possible Y values for each.

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