Nils Schneider

Recovering Bitcoin private keys using weak signatures from the blockchain

On December 25th of last year I discovered a potential weakness in some Bitcoin implementations. Have a look at this transaction:

transaction: 9ec4bc49e828d924af1d1029cacf709431abbde46d59554b62bc270e3b29c4b1

input script 1:

input script 2:

This transactions has two inputs and one output. If you look closely at the two input scripts you will notice there are quite a few equal bytes at the start and at the end. Those bytes at the end is the hex-encoded public key of the address spending the coins so there’s nothing wrong with that. However, the first half of the script is the actual signature (r, s):

r1: d47ce4c025c35ec440bc81d99834a624875161a26bf56ef7fdc0f5d52f843ad1
r2: d47ce4c025c35ec440bc81d99834a624875161a26bf56ef7fdc0f5d52f843ad1

s1: 44e1ff2dfd8102cf7a47c21d5c9fd5701610d04953c6836596b4fe9dd2f53e3e
s2: 9a5f1c75e461d7ceb1cf3cab9013eb2dc85b6d0da8c3c6e27e3a5a5b3faa5bab

As you can see, r1 equals r2. This is a huge problem. We’ll be able to recover the private key to this public key:

private key = (z1*s2 - z2*s1)/(r*(s1-s2))

We just need to find z1 and z2! These are the hashes of the outputs to be signed. Let’s fetch the output transations and calculate them (it is calculated by OP_CHECKSIG):

z1: c0e2d0a89a348de88fda08211c70d1d7e52ccef2eb9459911bf977d587784c6e
z2: 17b0f41c8c337ac1e18c98759e83a8cccbc368dd9d89e5f03cb633c265fd0ddc

That’s it. Let’s setup our sage notebook like this:

r  = 0xd47ce4c025c35ec440bc81d99834a624875161a26bf56ef7fdc0f5d52f843ad1
s1 = 0x44e1ff2dfd8102cf7a47c21d5c9fd5701610d04953c6836596b4fe9dd2f53e3e
s2 = 0x9a5f1c75e461d7ceb1cf3cab9013eb2dc85b6d0da8c3c6e27e3a5a5b3faa5bab
z1 = 0xc0e2d0a89a348de88fda08211c70d1d7e52ccef2eb9459911bf977d587784c6e
z2 = 0x17b0f41c8c337ac1e18c98759e83a8cccbc368dd9d89e5f03cb633c265fd0ddc

p is just the order of G, a parameter of the secp256k1 curve used by Bitcoin. Let’s create a field for our calculations:

K = GF(p)

And calculate the private key within this field:

K((z1*s2 - z2*s1)/(r*(s1-s2)))

Convert it to a more suitable format:

hex: c477f9f65c22cce20657faa5b2d1d8122336f851a508a1ed04e479c34985bf96

And import it to your favourite Bitcoin wallet. It’ll calculate the correct bitcoin address and you’ll be able to spend coins send to this address.

There are a few vulnerable bitcoin addresses in the blockchain. After some research I was able to contact the owner of this address. He allowed me to spend the funds.

Why did this work? ECDSA requires a random number for each signature. If this random number is ever used twice with the same private key it can be recovered. This transaction was generated by a hardware bitcoin wallet using a pseudo-random number generator that was returning the same “random” number every time.