How to determine S using priv_publickey_XY, nonce_publickey_XY, and Message(z)

Hey everyone, I've got some example data to share: priv_publickey_XY: px: 0xbb6c1de01f36618ae05f7c183c22dfa8797e779f39537752c27e2dc045b0e694 py: 0x2f8af53270bf045f2258834b6dad7481ad6fca009d80f5b54697b08d104fc7b3 nonce_publickey_XY: px: 0xcabc3692f1f7ba75a8572dc5d270b35bcc00650534f6e5ecd6338e55355454d5 py: 0xafa7746c07a124cb59e190f00955952a7329591b805c4d9d04d34abe8a803a74 Message(z): 0x9b076ad2fe6b2ce63acf4edf7fc82d5152d3c8bffb36b944da7a1cce038f544a # So, I need to figure out the S value using priv_publickey_XY, nonce_publickey_XY, and Message(z). S = 0xf65bfc44435a91814c142a3b8ee288a9183e6a3f012b84545d1fe334ccfac25e My question is how do I calculate S with just the d publickey_XY, k publickey_XY, and z? I get the S calculation formula: s = (z + x*r) / k, but I'm looking for an alternative way to get S, like s = (z + priv_publickey_XY * r) / nonce_publickey_XY. I think this could work. Sorry if my English isn’t perfect. Edit: just wanted to share this possible idea.

I assume this is the Schnorr Signature. Im not too familiar with it but I think the equation for generating s is:   s = k+m*x   m = message hash (i think this is SHA256)   x  = private key   k  = a random generated private key   P = public key   K = public key to the random private key listed above k to check if s is signed with the correct private key you would have to multiply s by G. G is the starting point of the elliptic curve, bitcoin uses secp256k1.   s*G = (k+m*x)*G = k*G + m*(x*G) = K + m*P ECDSA is distributive and associative so its possible to rearrange it like this. Now you can compute K + m*P to determine if it matches the output of s*G. If it does then it is a good signature. This video goes through all of this in better detail than i can: https://www.youtube.com/watch?v=wjACBRJDfxc&ab_channel=Bitcoinology Edit: actually this seems to be more of what your looking for: https://learnmeabitcoin.com/technical/cryptography/elliptic-curve/ecdsa/