Ok let me explain, imagine #125 as number 750, if we subtract 500 (2^124, half of 2^125 or in our example 2^125 = 1000) if we subtract 500 from 750 we'd have 250, now all we have to do is to figure out what number is closest to our 250 (which we don't know it's exact value) and try to subtract that guessed number from our 1/3 of #125 puzzle, and the rest would be easy.
Guessing the right number is very hard. And you can always guess it wrong, subtract 800 from 750, and you will get -50. Then, assuming that your N is 1000000, you will try to break 999950, instead of trying to break 750. Also, as far as I know, people already start from some offset, so they are not trying to scan keys outside of that 500 to 1000 range, instead they just pick for example 550 to 560 range, and try to scan only that, and that alone takes a lot of time.
Though if you could explain how to figure out which key is greater or smaller than the other, that would be awesome!
You cannot figure it out, based only on a single public key. It is always relative to the base point. For example, if you have (basePoint,yourPoint) pair, you could have "yourPointPrivKey" as some even number, for example 250. But it is only even in a relation between this particular base point! So, only "basePoint*250=yourPoint" gives you "250" as an even number. However, you could use (basePoint*2,yourPoint) pair instead, and then your private key would be odd, and equal to 125. See? Your private key is not universally assigned to your public key alone, it is always relative to your base point. And for that reason, if there would be any algorithm that could give you a private key, based on some public key, it would take at least a pair of public keys as arguments.
For the same reason, there is no such thing as "which key is greater or smaller than the other". ECDSA is working like a clock, you can give me some two public keys, and then I could calculate "(firstPubKey+secondPubKey)/2", and then say that the distance between my point, and any of your two points, is identical.
Can you tell me why is this happening? Ok, if I subtract this
02A1940074961CDF60CB2A0E7BC7157A7970B05469C58EEB5AD1C0462CE0C17811
from #125 puzzle, the result is :
02ed01ff219ed5c1afc12d991a82e3063ddcee1fd53b46f7cad52a0d87a7112aed
And if I subtract this
0290ad85b389d6b936463f9d0512678de208cc330b11307fffab7ac63e3fb04ed4
which is actually 2^124 public key, if I subtract it from #125 puzzle the result is :
03ed01ff219ed5c1afc12d991a82e3063ddcee1fd53b46f7cad52a0d87a7112aed
I know why this is happening though, because the first subtraction is actually inverse addition, the distance between the first public key and the second one is exactly the private key of #125.
This is why I'm stuck and can't solve this puzzle, and this is one of the easiest of puzzles since we know exactly it starts with a 1.
But I am close God willing, it should be solved soon.
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