So, the only way to get the private key of a public key, is simple guessing it/brute forcing it - Which can take forever to compute at some key ranges.
But What if we could reduce the key range? For example, if a public key is even, you can multiply it by 57896044618658097711785492504343953926418782139537452191302581570759080747169
And get a new public key, which is in the original range -1-bit range, which its private key is / 2 the value of the original private key.
It is like: guess a number between 1 and 10. If the correct key is 8, and you divide it by 2, you have to perform only 4 steps: 1,2,3,4 -> found!
But you don't know if the correct key is even or odd.
Knowing that is equal to have an additional information that let you to reshrink the search space.
But again you don't have this information, then you cannot reduce the key range.
if you divide each key by 2: [1 ... 10] -> [1/2, 2/2, 3/2, ... , 8/2, 9/2, 10/2] your range has the same size, it is not 'smaller'! Each of these 10 keys may be the correct one.