I really want to know how this is possible with Bitcoin.
It is possible, because you can always add, subtract, multiply or divide any known public key, by any known number.
For example: only Satoshi knows the private key to the Genesis Block:
https://mempool.space/address/04678afdb0fe5548271967f1a67130b7105cd6a828e03909a67962e0ea1f61deb649f6bc3f4cef38c4f35504e51ec112de5c384df7ba0b8d578a4c702b6bf11d5fWe can compress it, to make it shorter, and preserve the same private key:
04 678afdb0fe5548271967f1a67130b7105cd6a828e03909a67962e0ea1f61deb6 49f6bc3f4cef38c4f35504e51ec112de5c384df7ba0b8d578a4c702b6bf11d5f
03 678afdb0fe5548271967f1a67130b7105cd6a828e03909a67962e0ea1f61deb6
And now, we can test some operations, where some known public key can be modified. Let's test addition:
SHA-256("addition")=77f930e4b1a527037bb9daf4dd0546e592166c78f229211635ec5387afe52d8c
privateKey*G=03678afdb0fe5548271967f1a67130b7105cd6a828e03909a67962e0ea1f61deb6
(privateKey*G)+77f930e4b1a527037bb9daf4dd0546e592166c78f229211635ec5387afe52d8c=03C920F3CD3775B74110C84CCD4C3EF49085B07B0908B6762F271A22F301583755Which means, that we can use 03C920F3CD3775B74110C84CCD4C3EF49085B07B0908B6762F271A22F301583755 as a new public key instead. And the recipient could simply add 77f930e4b1a527037bb9daf4dd0546e592166c78f229211635ec5387afe52d8c to the private key, and then get a matching key, allowing him to spend it. In the same way, we can test subtraction, by adding "n-number" to our public key:
n=fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
n-77f930e4b1a527037bb9daf4dd0546e592166c78f229211635ec5387afe52d8c=8806cf1b4e5ad8fc8446250b22fab9192898706dbd1f7f2589e60b05205113b5
03C920F3CD3775B74110C84CCD4C3EF49085B07B0908B6762F271A22F301583755+8806cf1b4e5ad8fc8446250b22fab9192898706dbd1f7f2589e60b05205113b5=03678AFDB0FE5548271967F1A67130B7105CD6A828E03909A67962E0EA1F61DEB6
And, as we can see, by starting from 03C920F3CD3775B74110C84CCD4C3EF49085B07B0908B6762F271A22F301583755 and adding 8806cf1b4e5ad8fc8446250b22fab9192898706dbd1f7f2589e60b05205113b5, we can go back to our 03678AFDB0FE5548271967F1A67130B7105CD6A828E03909A67962E0EA1F61DEB6 public key. Now, let's test multiplication in the same way:
SHA-256("multiplication")=6b8370563d27b789779ff9f2839f7575c0ad6918056d9b2b60609d6743bc0e76
03678afdb0fe5548271967f1a67130b7105cd6a828e03909a67962e0ea1f61deb6*6b8370563d27b789779ff9f2839f7575c0ad6918056d9b2b60609d6743bc0e76=027ECCA53971D9055E22148B38423D1B2F297BB027ACEE6B88200713BC47EAF2F0As previously, we can multiply known public key 03678afdb0fe5548271967f1a67130b7105cd6a828e03909a67962e0ea1f61deb6 by known number 6b8370563d27b789779ff9f2839f7575c0ad6918056d9b2b60609d6743bc0e76 and get 027ECCA53971D9055E22148B38423D1B2F297BB027ACEE6B88200713BC47EAF2F0 as a result. The recipient can multiply its private key, which would allow him to move the coins. And finally, we can test division:
n=fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
1/6b8370563d27b789779ff9f2839f7575c0ad6918056d9b2b60609d6743bc0e76=43f2a611a0a8b11bb9a627f7d69af10e3fe71e267bfa3e7475efc4f8412c5587
027ECCA53971D9055E22148B38423D1B2F297BB027ACEE6B88200713BC47EAF2F0*43f2a611a0a8b11bb9a627f7d69af10e3fe71e267bfa3e7475efc4f8412c5587=03678AFDB0FE5548271967F1A67130B7105CD6A828E03909A67962E0EA1F61DEB6
As we can see, by starting from 027ECCA53971D9055E22148B38423D1B2F297BB027ACEE6B88200713BC47EAF2F0 and multiplying it by 43f2a611a0a8b11bb9a627f7d69af10e3fe71e267bfa3e7475efc4f8412c5587, we can get back 03678AFDB0FE5548271967F1A67130B7105CD6A828E03909A67962E0EA1F61DEB6.
So, addition, subtraction, multiplication, and division works, as long as all numbers are known. Which means, that if you have a HD wallet, then you just start from some known public key, you use that kind of operations, to reach a different public key, and then, other people could compute your new addresses, send coins there, and you can apply the same operations on your private keys, to spend these coins later. This is how non-hardened HD wallets work. For hardened HD wallets, the same operations are just applied on private keys alone.
Because there are only fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364140 valid public keys, and a point at infinity. Every addition, subtraction, multiplication, and division, is always done in range from 1 to fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364140. If you start from some number, and you use these operations with known numbers, then private and public keys follow each other.
And now, if you want to understand, what is a signature, then it is just a relation between two public keys. You simply start from one public key Q, and you reveal two numbers, where "(Q+a)*b=R". Then, nobody knows the private key to Q public key, or R public key. But everyone knows "a" and "b", so all nodes can check, if it is valid. And because "a=z/r" and "b=r/s", and r-value is taken as x-value of the R public key, using randomly generated "a" and "b" won't break it. Also, z-value has to match SHA-256 of the simplified transaction, which means, that even if you would get random (r,s,z) values for some public key, which would pass secp256k1 validation alone, then still, you won't get a message, which will hash to this random z-value, without breaking SHA-256.
