[Ethereum] Lost password on parity! I do have the recovery 12-word phrases thankfully

There is no way for the Ethereum Foundation to reset your password, but some enterprising community members have developed tools to assist with the password recovery process.

If you are familiar with Python (or willing to learn some basics), the best tool so far is a particular version of pyethrecover. Let us suppose that you know part of your password, but can't remember all of it (or perhaps you put in a typo during the presale or when you created the wallet). You can easily input the parts of the password you remember, and then add possible characters to check for those parts of the password you've forgotten. The program will then check all permutations of that possible password space to see if any combination unlocks your wallet. This is a lot easier than manually typing every possibility.

I've also saved a screen shot of the original presale page in case that helps jog your memory.

I never found any proper documentation about the Recovery ID but I did talk with somebody on Reddit and they gave me my answer:

id = y1 & 1; // Where (x1,y1) = k x G;
if (s > curve.n / 2) id = id ^ 1; // Invert id if s of signature is over half the n

I had to modify the mbedtls library to pass back the Recovery ID but when I did I could generate transactions that Geth accepted 100% of the time.

The long explanation:

During signing, a point is generated (X, Y) called R and a number called S. R's X goes on to become r and S becomes s. In order to generate the Recovery ID you take the one's bit from Y. If S is bigger than half the curve's N parameter you invert that bit. That bit is the Recovery ID. Ethereum goes on to manipulate it to indicate compressed or uncompressed addresses as well as indicate what chain the transaction was signed for (so the transaction can't be replayed on another Ethereum chain that the private key might be present on). These modifications to the Recovery ID become v.

There's also a super rare chance that you need to set the second bit of the recovery id meaning the recovery id could in theory be 0, 1, 2, or 3. But there's a 0.000000000000000000000000000000000000373% of needing to set the second bit according to a question on Bitcoin.SE.