In this article, we explore methods for solving the discrete logarithm problem and how to make money by recovering Bitcoin wallets, focusing on the Ricci Flow algorithm and the Hidden Number Problem for extracting private keys from vulnerable ECDSA transactions. We also discuss how modern cryptocurrencies like Bitcoin rely on complex mathematical foundations to provide security and anonymity, but are susceptible to exploitation due to various vulnerabilities.
We will go into detail about the process of recovering a private key using Dockeyhunt Discrete Logarithm software and DarkSignature tool to generate fake transaction data.
First, we will enter the Bitcoin wallet address: 1PYgfSouGGDkrMfLs6AYmwDqMLiVrCLfeS for the amount of: 165.10252195 BTC and get its public key. Then, using DarkSignature , we will create fake values for transactions, which will allow us to analyze and manipulate the signature data of the ECDSA algorithm. Finally, we will apply mathematical analysis through the Perelman Work software to solve the discrete logarithm and get the private key to the Bitcoin wallet.
This article is intended not only for cryptography and mathematics experts, but also for anyone who wants to understand how mathematical methods can be used to solve real-world problems in the world of cryptocurrencies.
First, we need to run the Dockeyhunt Discrete Logarithm software and "Input date"enter the Bitcoin Address 1PYgfSouGGDkrMfLs6AYmwDqMLiVrCLfeS in the field and get the public key of the wallet:
04e87e83f871df1439b7873b4ae449d15306cafc53e03a06fffb534b3bf25b58d8edca74b0faf5cf8c3aed6cad2bd79a7bce92ab53e07440d4590cbf31286d9335
Let’s use the DarkSignature tool to get fake R, S, Z values for the ECDSA algorithm transaction. In the field, "Input date"enter the public key of the Bitcoin Address and get the R, S, Z values
04e87e83f871df1439b7873b4ae449d15306cafc53e03a06fffb534b3bf25b58d8edca74b0faf5cf8c3aed6cad2bd79a7bce92ab53e07440d4590cbf31286d9335
Result:
1111,947d6fb75033cc3e342c8538a350e9058134b2a1ae01a7c50fc52b1f56c9169c,5b3ec0d72a2368cdd48c17ff095ab1ab0b9824e010883539cbeb18141de6384b,c7ac826c5a8397c0de993b2d8d597be42d22c77cf006683d7b72a197e1a5cdcf,0000
As a tool for mathematical analysis and solving discrete logarithm equations, we will use the Perelman Work software . We will select the option from the Complex Analysis section for a complete relationship between variables through the integration of Discrete variation series Variance: [ D = frac{sum_{i=1}^{n} (x_i – bar{x})^2}{N} ) ]
The formula for calculating discrete variation looks like this:
![Discrete variation series Variance formula: [ D = frac{sum_{i=1}^{n} (x_i - bar{x})^2}{N} ) ]](https://polynonce.ru/wp-content/uploads/2024/12/image-1.png)
Explanation of the formula components:
- D is the variance (variance) of your data set.
- ∑∑ is the sum symbol, which means that we will add the values.
- i is an index that runs through all values in the data set from 1 to n.
- xi is each individual value in your data set.
- ˉ x ˉ is the mean (or arithmetic average) of all the values in the data set.
- N is the total number of values in the data set.
How does this work?
- Calculating the Average: First, you find the average value of your data set.
- Difference from mean: Then for each value xi you calculate how much it differs from the mean ˉ x ˉ .
- Square the difference: You then take the square of that difference (to get rid of negative values and amplify the impact of large deviations).
- Summation: You add up all the squares of the differences.
- Dividing by the number of values: Finally, you divide the resulting sum by the total number of values N .
Using Perelman Work and Dockeyhunt Discrete Logarithm, we arbitrarily change variables to the Joux Lercier vulnerability , this is described in detail at the beginning of the article, this vulnerability in a Bitcoin transaction occurs due to the fact that it is possible to change the value of R, S , Z in the signature, while maintaining the validity of the signature, as well as in an arbitrary formula:
X=hex(((S⋅K−Z)⋅modinv(R,N))modN)
- S and R are the values from the transaction signature (RawTX).
- Z is the transaction signature hash.
- K is the secret key (nonce).
- N is the order of the elliptic curve group.
- modinv( R , N ) is the modular inverse function of R modulo N .
Explanation of the formula
- Input parameters :
- S and R : These values are obtained from the transaction signature. They are needed to recover the private key.
- Z : This is the signature hash, which is also used in the process.
- K : A secret key (nonce) that should only be known to the wallet owner.
- Calculations :
- First we multiply S by K.
- Then we subtract Z .
- The result is multiplied by the modular inverse of R modulo N . This allows us to “cancel” the influence of R to obtain a value that can be used to calculate the private key.
- Finally, the result is taken modulo N to ensure that it is within the acceptable range for private key values.
- Convert to hexadecimal format :
- After all the mathematical operations are performed, the result is converted to hexadecimal format using the function
hex(), which is the standard representation of private keys in Bitcoin.
- After all the mathematical operations are performed, the result is converted to hexadecimal format using the function
Ricci Flow Hidden Number Problem
Copy the values R, S, Z and paste them into the input field Ricci Flow HNP to build completely new transactions of the ECDSA algorithm .
Dockeyhunt Private Key Calculator
Copy the values and paste them into the Dockeyhunt Private Key Calculator R, S, Z software field
Now, according to the vulnerability of Joux Lercier, we copy from the code of ecdsa_impl.h the value of the secret key “K” called in cryptography NONCE – this is a secret, (pseudo) random parameter, which is usually denoted by “K”. Here NONCE, due to a bug in the code , fixed 0, 0, 0, 1, 0x45512319UL, 0x50B75FC4UL, 0x402DA172UL, 0x2FC9BAEEUL several HEX bits at the beginning (or at the end) of the record.
....
....
....
/** Difference between field and order, values 'p' and 'n' values defined in
* "Standards for Efficient Cryptography" (SEC2) 2.7.1.
* $ sage -c 'load("secp256k1_params.sage"); print(hex(P-N))'
* 0x14551231950b75fc4402da1722fc9baee
*/
static const secp256k1_fe secp256k1_ecdsa_const_p_minus_order = SECP256K1_FE_CONST(
0, 0, 0, 1, 0x45512319UL, 0x50B75FC4UL, 0x402DA172UL, 0x2FC9BAEEUL
);
....
....
....
Copy the K value and paste it into the Dockeyhunt Private Key Calculator software field , then click on the button: Calculate Private Key and get the private key to the Bitcoin Wallet 1PYgfSouGGDkrMfLs6AYmwDqMLiVrCLfeS
0x6b29781e725708ae4d94e13730a2718ee3383ea5d911e77d4c2a2fd0c99c1232
To start the algorithm for solving the discrete logarithm, click on the button: Private Key after that, we successfully receive a private key in HEX format
6b29781e725708ae4d94e13730a2718ee3383ea5d911e77d4c2a2fd0c99c1232
Next we need to make sure that we have received the required private key value in HEX format.
Click on the button
Bitcoin Addressand get the required value of the private key in HEX format
6b29781e725708ae4d94e13730a2718ee3383ea5d911e77d4c2a2fd0c99c1232: 1PYgfSouGGDkrMfLs6AYmwDqMLiVrCLfeS
We also click on the button Balance BTC and get the result of the balance amount:
6b29781e725708ae4d94e13730a2718ee3383ea5d911e77d4c2a2fd0c99c1232: 1PYgfSouGGDkrMfLs6AYmwDqMLiVrCLfeS: 165.10252195 BTC
_____________________________________________________________________________________________________
Private key received!
In conclusion , recovering a Bitcoin wallet using mathematically based methods such as the Ricci Flow Hidden Number Problem opens new horizons for understanding cryptographic vulnerabilities and opportunities. We demonstrated how Perelman Work, Dockeyhunt Discrete Logarithm , and DarkSignature software can be used to extract private keys and create fake transactions, highlighting the importance of mathematical analysis in the cryptocurrency space.
The results show that even in a complex system like Bitcoin , there are vulnerabilities that can be exploited to restore access to lost funds. This process requires deep knowledge of cryptography and mathematics, as well as skills in working with specialized software.
