How to determine first byte (recovery ID) for signatures (message signing)?

Question

I do realize that in bitcoin when we return the signature result for signing a message the first byte includes information for recovering the public key required for verification using a formula like this:

27 + (4 if comp. 0 if not) + (0<=num<=3)

What I am struggling with is how do we determine that num number? According to section 4.1.6 of SEC:1 for recovering public keys:

1. x = r + jn
2. Calculate Q based on R if invalid move to next
3. Calculate Q based on -R

Since j (cofactor of secp256k1) is 0 and 1 and there are two values for R. So there should be 4 possible public keys. Is num the number of public keys that we rejected before reaching the correct one? Because that is the only explanation that I can come up with based on comparisons I have made between signatures generated using Electrum and public key recovery I have done myself using SEC1.

Bonus question: Why was 27 chosen?

Answer 1

FWIW this is what I have found regarding this question. Basically there are two ways of finding what libraries call "recid" or recovery ID. Most of them use the first method but there is another way:

Method 1:

• Only requires r and s so it can be performed by anyone as long as you have the signature and the "message" that was signed.
• It is so much slower because it has 3 scalar multiplication and based on cofactor of the elliptic curve used the whole operation may be repeated 2*(h+1) times, which is up to 4 times for bitcoin with secp256k1 curve (99% of the time it is 1 or 2 times).

The steps are explained in section 4.1.6 of SEC:1, I won't repeat them here.

To find recid you check the calculated public key with the given public key or hash of it and report number of rejected public keys as recid. For example if x=r+(0*order) and Q was used then recid=0 and if -Q was used then recid=1 and so on.

method 2:

• It requires full r or R=(xR, yR) so it can only be done while signing (requires having the private key).
• It is much faster because there is no additional calculations. It just needs value checks which are fast.
• This is usually referred to as v in libraries

byte v = if(R.X > curve.N) then 2 else 0) | (if R.Y.IsEven then 0 else 1);

Basically it is check to see if xR is bigger than curve order (N) and whether yR is even or odd. Something that should not be forgotten is a "flip" that is performed based on s and whether s was used in the signature or -s

if s > curve.N/2 then v^=1 else do nothing

** Note that in the end you need to calculate the following:
recid = 27 + v + (if compressed 4 else 0)

Answer 2

This will be less of a theoretical answer and more of practical guidance; while developing Secp256k1 or Bitcoin libraries, I always found the following table handy:

y-parity	x-order	compression	recovery id	v
even	less than n	false	0	27
odd	less than n	false	1	28
even	more than n	false	2	29
odd	more than n	false	3	30
even	less than n	true	0	31
odd	less than n	true	1	32
even	more than n	true	2	33
odd	more than n	true	3	34

let me explain the columns:

• y-parity or often just parity describes whether the y-coordinate of the public key point R(x, y) is even. this is important because in most cases, the signature only contains r = R.x and not R.y. This speeds up signature recovery. (Secp256k1)
• x-order is a super rare edge-case where a point's x-coordinate is actually above the order of the curve n. In 99.99999999% of the cases, this is false. (Secp256k1)
• compression is just an indicator whether you want a compressed or an uncompressed public key. (Bitcoin)
• recovery_id just indicates which of the four potential point's y-coordinates is to be used. (Secp256k1)
• v is how we encode this value. For uncompressed keys we use v = recovery_id + 27 and for compressed v = recovery_id + 31. (Bitcoin)

If you want to understand how this works for Ethereum, you would have to read EIP-155 which also introduces an additional variable of chain_id for replay protection: v = 2 * chain_id + 35 + recovery_id. (Ethereum)

What I am struggling with is how do we determine that num number?

That's the recovery ID from the table above.

Answer 3

There may be another method to determine the Recovery ID(rec_id). Since the Recovery ID can only have 4 possible values(0/1/2/3), if you already have the message to sign, public key, and signature, you can try each Recovery ID value and determine which one yields the correct public key value.

There is already an implementation of this function called "ecRecover". For example, in Python, you can use ecdsa_raw_recover from py_ecc library(source code). This function allows you to derive possible public key from the given Recovery ID:

def ecdsa_raw_recover(msghash: bytes, vrs: Tuple[int, int, int]) -> "PlainPoint2D":
    v, r, s = vrs
    if not (27 <= v <= 34):
        raise ValueError("%d must in range 27-31" % v)
    x = r
    xcubedaxb = (x * x * x + A * x + B) % P
    beta = pow(xcubedaxb, (P + 1) // 4, P)
    y = beta if v % 2 ^ beta % 2 else (P - beta)
    # If xcubedaxb is not a quadratic residue, then r cannot be the x coord
    # for a point on the curve, and so the sig is invalid
    if (xcubedaxb - y * y) % P != 0 or not (r % N) or not (s % N):
        raise ValueError("sig is invalid, %d cannot be the x coord for point on curve" % r)
    z = bytes_to_int(msghash)
    Gz = jacobian_multiply(cast("PlainPoint3D", (Gx, Gy, 1)), (N - z) % N)
    XY = jacobian_multiply(cast("PlainPoint3D", (x, y, 1)), s)
    Qr = jacobian_add(Gz, XY)
    Q = jacobian_multiply(Qr, inv(r, N))
    Q_jacobian = from_jacobian(Q)

    return Q_jacobian