How to derive the Private key when the two the nonces are k and k*2

Question

Is the same private key using k and k*2 respectively in two different signatures vulnerable? If yes How to calculate the key in this case

Answer 1

c0e2d0a89a348de88fda08211c70d1d7e52ccef2eb9459911bf977d587784c6e is the k1 nonce. a3c2dce777813dc3f0d7a105a5fd56d5894ba30ffa01ba79667cadba37a9d8df is the private key.

Answer 2

Assuming you're talking about ECDSA signing. For BIP340/Schnorr signatures, see this answer.

For valid signatures

Let:

• d: the private key
• P=d*G: the public key
• k1: the (first) nonce
• R1=k1*G: the public first nonce
• r1=R1.x mod n: the public first nonce as it will be encoded in the signature.
• k2=2*k1, R2=k2*G=2*R1, r2=R2.x mod n: the same for the second nonce
• z1 and z2: the respective message hash

The two signatures will then be the pairs (r,s) and (r',s') for which:

• s1 = (z1 + r1*d) / k1 mod n
• s2 = (z2 + r2*d) / k2 mod n = (m2 + r2*d) / (2*k1) mod n

Multiplying both sides of the equations by their denominator on the right hand side:

• s1*k1 = z1 + r1*d mod n
• 2*s2*k1 = z2 + r2*d mod n

Assuming 2*r1*s2 ≠ r2*s1, this is a set of two linear equations in two unknowns (k1 and d), with solution:

• d = (z2*s1 - 2*z1*s2) / (2*r1*s2 - r2*s1) mod n
• k1 = (z2*r1 - z1*r2) / (2*r1*s2 - r2*s1) mod n

Don't ever use related nonces for ECDSA (or Schnorr) signatures. Create a fresh, independent, nonce every time. The industry standard is to generate nonces using RFC6979.

For fake "signatures"

The values you've provided however do not correspond to real signatures, despite satisfying the equation. That's because for a signature to be valid, you have to give the messages which hash to the z values, not just the result. And you can't do that here, because z2 = (r2/r1)*z1. Such matching ratios will not (and cannot) occur for z values that are the result of hashing.

It turns out that given any ECDSA triplet (r,s,z), and integer a, another triplet can be found. Let R be an elliptic curve point whose X coordinate equals r (mod n), and R' = aR, and r' the X coordinate of R'. Then (r',r'*s/(a*r'),r'*z/r) is another ECDSA triplet, corresponding to multiplying k with a. However, again, this is not a valid signature because its z value will not be the hash of a computable message. And the formula above will also fail to retrieve the private key in this case, because the second "signature" wasn't created by using the private key.