Private Key Recovery in Case of Nonce Reuse
- Extracts
PubKey(integerOrder(G)),Signature(r, s)from inputs of transactions - Derives private key and K value that was used to sign two inputs in case pubkey and R values have been reused.
pk = Private Key (unknown at first)
K = K value that was used (unknown at first)
N = integer order of G (part of public key, known)
# From Signing Defintion
s1 = (L1 + pk * R) / K Mod N and s2 = (L2 + pk * R) / K Mod N
# Rearrange
K = (L1 + pk * R) / s1 Mod N and K = (L2 + pk * R) / s2 Mod N
# Set Equal
(L1 + pk * R) / s1 = (L2 + pk * R) / s2 Mod N
# Solve for pk (private key)
pk Mod N = (s2 * L1 - s1 * L2) / R * (s1 - s2)
pk Mod N = (s2 * L1 - s1 * L2) * (R * (s1 - s2)) ** -1
Make sure zeromq is installed for realtime streaming features.
This can be done on OSX with brew install zeromq.