2

Private key is a number. Is the question positive or negative? Is it possible to know a positive private key or negative if you only know the public key?

For instance:

Positive private key: 2

public key:

04C6047F9441ED7D6D3045406E95C07CD85C778E4B8CEF3CA7ABAC09B95C709EE51AE168FEA63DC339A3C58419466CEAEEF7F632653266D0E1236431A950CFE52A

Negative Private Key: FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD036413F

public key:

04C6047F9441ED7D6D3045406E95C07CD85C778E4B8CEF3CA7ABAC09B95C709EE5E51E970159C23CC65C3A7BE6B99315110809CD9ACD992F1EDC9BCE55AF301705

Now I will write two public keys to "3" and "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD036413E":

04F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F9388F7B0F632DE8140FE337E62A37F3566500A99934C2231B6CB9FD7584B8E672

04F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F9C77084F09CD217EBF01CC819D5C80CA99AFF5666CB3DDCE4934602897B4715BD

How to determine which is a positive and which is a negative private key to these two public keys?

3

There really isn't a "positive" or "negative" for private keys. Private keys are unsigned integers, so they're basically all "positive". Your "negative" numbers are not actually negative, they're the modular additive inverse of the "positive" private key.

Since private keys are really unsigned integers, you can't glean any information about signed-ness from the public key because there are no signs to even look at. Every private key is the "negative" of another private key, so a private key is both "positive" and "negative". Every private key has an additive inverse.

However, given two public keys, you can determine whether their private keys are additive inverses of each other by checking whether the X coordinates are the same. If so, one should have a Y coordinate that is odd, and the other have a Y coordinate that is even. This only tells you that their private keys are additive inverses of each other. But you won't know which is the lower number or what one of them is (unless you have the private key for one of them).

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