The question is somewhat complex and directed to clearing thing out.
n is the order of the cyclic group. It
n - 1 is the number of all private keys possible
n = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
We also know that every private and public key has its modular inverse.
To get a modular inverse of a private key, we need to subtract the private key from
n - privKey
To get a modular inverse of a public key, we'll have to multiply its
y coordinate by
-1 and modulo by the
p - order of the finite field.
x,y = x, -y % p
A modular inversed public key has the same
x coordinate as original public key, but different
y coordinate, and the
y coordinate is always different in its polarity.
If the original
y was odd, in a modular inversed key it will be even, and vice versa.
If a compressed public key has
"02" index at the biggining then it has even
y. If it is
"03" then it is odd.
The question is, if the
y coordinate of a public key is even, does it mean that the corresponding private key is less than
n/2 by its value? If the
y is odd, the private key is more than
Is there any relationship between the eveness/oddness of the
x) coordinate and the value of the corresponding private key?
Is there any way to know that the private key is more or less than
n/2 while not knowing the private key itself?
Is there a way to find out the public key of an address that never sent Bitcoin but only received it?