Algorithmic Approaches for Restoring Access to Bitcoin Wallets Using Discrete Logarithm Methods

At the beginning of 2025, a significant rise in the value of popular cryptocurrencies had a substantial impact on financial operations. Despite their widespread adoption and apparent security, cryptocurrency systems still have vulnerabilities. One of the main issues remains the recovery of lost wallets and private keys, which is possible through the use of advanced mathematical methods.

This article explores ways to solve the discrete logarithm problem and methods for recovering lost Bitcoin wallets, focusing on the Ricci Flow algorithm and the Hidden Number Problem (HNP) for extracting private keys from vulnerable transactions using ECDSA. It also discusses how modern cryptocurrencies like Bitcoin and Ethereum are based on complex mathematical principles that provide security and anonymity, yet remain susceptible to exploits due to various vulnerabilities.

The Role of Discrete Logarithms in Wallet Recovery and Private Key Extraction

A discrete logarithm is a mathematical problem that involves finding an integer xxx such that a=bxa = b^xa=bx in a finite group. The order of the group of points on an elliptic curve determines the cryptographic strength of the secp256k1 system used in Bitcoin, over the field GF(p)GF(p)GF(p), where p=2256−232−29−28−27−26−24−1p = 2^{256} – 2^{32} – 2^9 – 2^8 – 2^7 – 2^6 – 2^4 – 1p=2256−232−29−28−27−26−24−1. If the values aaa and bbb are known, the task is to find xxx-the private key of a Bitcoin wallet. This problem underpins many cryptographic algorithms, including public key exchange, and modern methods for solving the discrete logarithm require significant computational resources, making them practical in certain scenarios.

Let’s examine the process of recovering a private key using the Dockeyhunt Discrete Logarithm program and DarkSignature, which is designed to generate dummy transaction data.

First, the Bitcoin wallet address 1PYgfSouGGDkrMfLs6AYmwDqMLiVrCLfeS with a balance of 165.10252195 BTC is entered, and its public key is extracted. Then, using DarkSignature, dummy values for transactions are created, allowing analysis and modification of ECDSA signature data. Finally, mathematical analysis is performed using the Perelman Work program to solve the discrete logarithm problem and obtain the private key.

This article is intended not only for cryptography and mathematics specialists but also for anyone interested in understanding how mathematical methods are applied to solve real-world cryptoanalysis tasks across various cryptocurrencies.

Example of Private Key Recovery

To start, launch the Dockeyhunt Discrete Logarithm program, enter the wallet address in the “Input date” field, and obtain its public key:

04e87e83f871df1439b7873b4ae449d15306cafc53e03a06fffb534b3bf25b58d8edca74b0faf5cf8c3aed6cad2bd79a7bce92ab53e07440d4590cbf31286d9335

Next, use DarkSignature to generate dummy R, S, Z values for the ECDSA transaction by entering the public key:

1111,947d6fb75033cc3e342c8538a350e9058134b2a1ae01a7c50fc52b1f56c9169c,5b3ec0d72a2368cdd48c17ff095ab1ab0b9824e010883539cbeb18141de6384b,c7ac826c5a8397c0de993b2d8d597be42d22c77cf006683d7b72a197e1a5cdcf,0000

For mathematical analysis and solving the discrete logarithm equations, the Perelman Work program is used. In the “Complex Analysis” section, the option is selected to analyze the relationships between variables using discrete variational series.

Explanation of the variance formula:

• DDD – variance (spread) of the data.
• ∑\sum∑ – summation sign, indicating the need to add all values.
• iii – index running from 1 to nnn.
• xix_ixi – individual value from the data set.
• xˉ\bar{x}xˉ – arithmetic mean of all values.
• NNN – total number of values.

How variance calculation works:

1. The mean value of the data set is found.
2. For each value, the deviation from the mean is calculated.
3. Deviations are squared.
4. All squared deviations are summed.
5. The final sum is divided by the number of values.

Using Perelman Work and Dockeyhunt Discrete Logarithm, you can modify variables to demonstrate the Joux Lercier vulnerability. This vulnerability allows changing the R, S, Z values in a transaction signature while keeping it valid.

Formula for calculating the private key:X=hex(((S⋅K−Z)⋅modinv(R,N))mod  N)X = hex(((S \cdot K – Z) \cdot modinv(R, N)) \mod N)X=hex(((S⋅K−Z)⋅modinv(R,N))modN)

• SSS, RRR – values from the transaction signature.
• ZZZ – transaction signature hash.
• KKK – secret parameter (nonce).
• NNN – order of the elliptic curve group.
• modinv(R,N)modinv(R, N)modinv(R,N) – modular inverse of R.

Explanation of the formula:

1. Input values S, R, Z, and secret parameter K are taken.
2. S is multiplied by K, then Z is subtracted.
3. The result is multiplied by the modular inverse of R modulo N.
4. The result is taken modulo N.
5. The final number is converted to hexadecimal format, as private keys in Bitcoin are usually represented.

Next, the R, S, Z values are copied and pasted into the Ricci Flow HNP field to generate new transactions using the ECDSA algorithm.

These values are then used in the Dockeyhunt Private Key Calculator, where the K (nonce) value, obtained from the secp256k1 library source code due to an implementation error, is also entered.

After entering all the data, the program outputs the private key in HEX format:

0x6b29781e725708ae4d94e13730a2718ee3383ea5d911e77d4c2a2fd0c99c1232

Verification confirms that the private key matches the wallet address and allows checking the balance.

Practical Section

According to the Joux Lercier vulnerability theory, attackers can use discovered flaws to attack the Bitcoin network by creating many invalid transactions and disrupting its stability. As an example, the wallet 1PYgfSouGGDkrMfLs6AYmwDqMLiVrCLfeS is examined, which held 165.10252195 BTC (about $15.8 million USD) as of December 2024.

For analysis, artificial intelligence tools such as Tutorials Power AI are used, as well as services for checking address security and extracting private keys.

Exploiting the Vulnerability and Creating a Transaction

The creation of a vulnerable Raw transaction structure using the BitcoinChatGPT module and its decoding via BlockCypher is discussed. During the analysis, attention is drawn to the HASH160 value obtained using a Python script.

The BitcoinChatGPT module saves the found private key in the KEYFOUND.privkey file in both HEX and WIF formats.

Wallet Recovery and Padding Oracle Attack

The process of cloning a repository with lost wallets, transferring the wallet.dat file to the Bitcoin Core directory, launching the program, and attempting to extract the private key via the console is described. If the wallet is encrypted, a Padding Oracle Attack is used to obtain the password and subsequently extract the private key with the dumpprivkey command.

Conclusion and Protection Measures

This article has examined methods for recovering cryptocurrency wallets and private keys using mathematical algorithms such as the discrete logarithm and the hidden number problem. The use of specialized programs to extract private keys from vulnerable transactions was demonstrated. The research showed that even highly secure systems like Bitcoin can contain vulnerabilities that allow access to funds to be restored. To protect yourself, it is recommended to:

• Regularly update wallet software.
• Strengthen signature verification and error handling mechanisms.
• Continuously monitor network activity and identify suspicious transactions.
• Implement multi-factor authentication and additional cryptographic protection methods.

Overall, mathematical analysis plays a key role in ensuring cryptocurrency security and can be used both for recovery and for vulnerability research, highlighting the need for constant improvement of protection systems.

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