In the case of the puzzle transactions a publickey should be in the space 2^(bit-1) to (2^bit)-1,
we can just calculate 2^(bit-1) + 2^(bit-2) as a point and subtract it from our known publickey.
Our new generated public key will be within the space(ORDER - 2^(bit-2)) to 2 ^(bit-2).
Now we only have to check the x values from space 0 to 2^(bit-2).
When we find the solution to the calculated publickey, the solution to our orginal publickey will be either solution or (ORDER - solution) + subtraction factor.
Can you please clarify there is the benefit in your method? You are saying that "Now we only have to check the x values from space 0 to 2^(bit-2)", so we need to check 2^(bit-2) combinations. It is just 2 times less that the brutforce of the full range. Not effective.
However in Pollard method we should make only sqr (2^(bit-1)) operations, which is much much less than in your method.
Example: for 110 bit key, you suggest to check 2^108 combinations, but in Pollard method we need only 2^54.5 operations.
So why is your method valuable? What is the main idea and advantage?