hi albert0bsd, a small question. when the BSGS is searching for private key from public key, the key range is 2^256 unlike the other bruteforce methods that hash the private key to get ripemd 160 and check balance, in which case the key space is 2^160. if that is true this method is useful to solve only when the key space is know like the puzzle transaction otherwise regular bruteforce methods better because the key range is 2^160. correct me if I am wrong.
Interesting question, but to me, it's not as easy as one or the other. It's not black and white.
With ripemd160; what 2^160 range would you search? There are (roughly) 2^96, 2^160 ranges. Let's say you search the entire first 2^160 range, 0 to 2^160; you may not find the key you are looking for but instead you may find multiple collisions of the same keys within that range (not the key you are looking for). Now you search the next 2^160 range, 2^160-2^161, the key may not be in there. But if you stopped with the first range, let's say you found it in the first 2^160 range and only searched 50% of the keys before you found it, so you brute forced 2^159 keys to find it. With BSGS your search time would/could be reduced, depending on your hardware setup. I know with Kangaroo, to search for a key in the 2^256 range, you would roughly have to "search" through roughly 2^129 (group operations)....which is 2^30 times smaller than brute force approach.
