I'm working on authoring Ethereum transactions using ECDSA with SECP256K1. On the tail end of an Ethereum transaction is the V, R, and S values of signing a hash of the message. V is defined as
chainID * 2 + 35 + RecoveryID where
chainID is some value unrelated to this question and
RecoveryID is somehow extracted from the signing process or signing keys.
I'm working with the mbedtls library to generate private keys, hash with keccak, and sign using the proper curve. However the output of the signature process is just the R and S values. I keep looking around and can't find a reputable source of what the RecoveryID value is and nothing in mbedtls's documentation talks about it.
In this random (poorly documented) library they check if the Y of the public key is odd. In that forum post they say it's the sign of the Y of the public key. I've tried both and for some private keys the V value works and others it fails to properly recover the public key.
Is there a definitive resource explaining where the Recovery ID comes from?