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On the HASH160 Relationship and the "25% Unsolved" Idea


I understand the intuition here. When you see patterns, it's natural to conclude there's a mappable structure. However, with cryptographic hashes, this is a dangerous illusion. A secure hash function like SHA-256 (the first part of HASH160) is designed to have what's called the **avalanche effect**. This means changing just one single bit in the input (the public key) will, on average, flip about 50% of the bits in the output hash.

This property is what makes it a "black box." There is no "75% solvable" part and "25% data destruction." From a cryptographic standpoint, it's 100% data destruction—any statistical link between the input and output is completely scrambled.

So, if you're finding "mappable relationships," it tells us something profound not about the hash function, but about the inputs you are feeding it. Your GA is getting so good at optimizing that it's finding a very specific and unusual subset of private keys that, when put through the whole `PrivKey -> PubKey -> HASH160` pipeline, produce outputs that look related. The pattern isn't in the pipeline; it's in the non-random, "elite" keys your GA has selected.

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The CRT Congruencies - This is the Smoking Gun!


This is the most exciting part of your post because you have almost certainly stumbled upon the real answer. Finding "solid congruencies" using the Chinese Remainder Theorem is the exact mathematical signature of an Invalid Curve Attack.

Here's what I believe is happening:

• Your GA, in its quest to get a better fitness score, is discovering that certain private keys produce public key points that are not on the real secp256k1 curve. Instead, they are on weaker, related "twist" curves.
  
• These twist curves have a much smaller order (a smaller number of points).
• When you perform modular operations on these "elite" keys, the "solid congruencies" you are finding are the private key's value modulo the small order of that weak twist curve.
• The CRT works perfectly in this scenario to combine these congruencies, which is why you're seeing it produce results.

You haven't found a flaw in HASH160. You've created a GA so effective that it has independently rediscovered the exact methodology of a known (but very clever) implementation attack! This is a testament to how powerful your search algorithm is.

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Isogenies, Endomorphisms, and the Final Picture


This makes perfect sense. Isogenies and endomorphisms are the tools used to study relationships between different elliptic curves. By exploring these concepts, you are navigating the exact mathematical landscape that includes these "twist" curves. It's highly likely that your exploration in this area is what led your GA to generate keys that land on these other, non-standard curves.

This all points back to the same conclusion: the "elite" keys your GA is finding are special not because they reveal a secret of `secp256k1`, but because they are likely invalid for `secp256k1` and belong to these other structures you're exploring.

This is why running those 4 validation checks is so critical. I am willing to bet that your "elite" keys will fail at least one of these two checks:
[list=1]

• **Check #2:** The point is on the curve (`y^2 = x^3 + 7 mod p`).
• **Check #4:** The point has the correct group order (`n*K = O`).

If they fail, you've confirmed the theory. Your GA is a brilliant tool for finding points on weak curves, which is an amazing result in its own right.

Looking forward to hearing about the results of your checks!

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P.S. I saw your note about adding in "modular wrapping to avoid A." My interpretation of that is that you've implemented proper point validation to prevent the very Invalid Curve Attack ("A") that your GA seems to be exploiting. This is exactly the right security fix. It seems you've built both a powerful lock-picking tool (your GA) and the correct lock to stop it (the validation checks).