For the BTC Puzzle
https://bitcointalk.org/index.php?topic=5218972.0 to try the possibility of simplification using the probabilistic approach.
If we start analyzing all the Found Keys we see that most of them are NOT PRIME.
![](https://ip.bitcointalk.org/?u=https%3A%2F%2Fwww.talkimg.com%2Fimages%2F2024%2F07%2F14%2Foqq9l.jpeg&t=665&c=t8UgkRZFf1LmyQ)
So instead of blind bruteforcing we could try to find the maximum likelihood range. This off course may not work for each of the remaining puzzle but if we see them together as a whole there seems to be a way of approach for many of them.
Privatekey being a composite number will have several factors. We are not interested in the Prime Factors but instead all the possible factors. Because once we have (somehow) a factor of the privatekey then the difficulty of that Key is reduced. Before thinking about how we can get a factor of the unknown Key, Lets analyze this concept in the already Found Keys first. There are some interesting observation. Some Keys are hopeless and have very less number of factors but some Keys are juicy and have a factor in almost all bit.
![](https://ip.bitcointalk.org/?u=https%3A%2F%2Fwww.talkimg.com%2Fimages%2F2024%2F07%2F14%2Foqxzm.png&t=665&c=OFxuBonkKQTYLg)
If we looks closely in every 10 Keys there is definitely 1 or more juicy Key present. Another observation is we are looking for Factors which are not very small and not very big. Small factor will leave rest of Key bruteforcing almost same level and Big Factor finding itself is a huge challenge. The Best is if the factor is somewhere in closer to the mid level of the Key. So again lets try for this search in existing Keys.
![](https://ip.bitcointalk.org/?u=https%3A%2F%2Fwww.talkimg.com%2Fimages%2F2024%2F07%2F14%2Fo0AZH.png&t=665&c=2AhJIF6SmPQz4A)
Here if we look for Factors closer to Q1 then we have much narrower range to search for. The same thing can be also observed in Q2 or Q3. Now if we use this range of -10 to 25% of distance for the closest factor and check the same in existing Keys, we can validate that we would have found these puzzles as shown in transparent lines. The beauty of big mid factor is that second factor will also be closer to similar strength and will help in search range limit.
Applying the same idea to the remaining Puzzles becomes interesting as the reduction in difficulty level would be significant.
![](https://ip.bitcointalk.org/?u=https%3A%2F%2Fwww.talkimg.com%2Fimages%2F2024%2F07%2F14%2Fo0Gl2.png&t=665&c=jaoP74BfLyvv0g)
Note:
1. This whole analysis is to focus on the most relevant range, if you do not want to search for everything (hardware or time shortage)
2. If we still have to scan through all the factors and for each Factor all the Keys, then there is no gain in the search
2. For the Pubkey Kangaroo search the overhead of switching factors range would soon overpower the advantage of stride search of Keys