Hi there again with more trouble and questions, I'd appreciate the time you'd spend to respond.
Is it possible to determine which X coordinate of our k is -N or is -k inverse without obviously looking at the k ?
No there is no knowed way to guess any information of private key (your k) with any information of public key (your X coordinate), even little.
Every actual attack (Lattice Attack etc...) oblige the attacker to know a part of the private part. and it is not an attack on the elliptic curve cryptography itself but on bad way to use signature.
ECC (and every asymmetric cryptography like RSA) is based on the assumption that the derivation of a private key in a public key "seems" perfectly randomly distributed.
Just for fun i tried a lot a cryptanalysis technical (statistics on huge amount of keys, pattern identification, deep learning ...) to find a bias in the distribution of key and believe me :
secp256k1 (curve used by bitcoin) seems really safeWell, I'm not interested in signatures and related stuff, the entire elliptic curve system revolves around public keys, so that is the only entry point for me to try all I got and find the best solution. "If there are no known method to correctly guess the position of any X coordinate of k, then finding a way should be a goal.
I have been studying the secp256k1 for the past 2 months, and tried at least 40-50 methods to figure out which one could be used to crack the target k by hand, not using automated existing tools.
What actually is bothering me is a lack of a safe environment to publish study results without worrying about other people exploiting them! Though I'm in the learning phase, no breakthroughs yet!😉