What is the Nonce value K in the Bitcoin blockchain?

In the Bitcoin blockchain, the private key K, also known as Nonce, plays a crucial role in ensuring the security of transactions. In this article, we will look at the meaning of K and its role in Bitcoin cryptography, especially in the context of ECDSA signatures.

ECDSA Cryptography and the Secret Key K

Cryptographic Basics

ECDSA (Elliptic Curve Digital Signature with Schnorr) is a digital signature algorithm used in Bitcoin to authenticate transactions. It is based on the mathematical properties of elliptic curves and provides a high level of security through the use of a secret key.

The role of the secret key K (Nonce)

The secret key K in ECDSA is a one-time code, chosen randomly or pseudo-randomly. It is used to generate a signature along with the message hash and the private key. K must be unique for each transaction to prevent replay attacks.

Vulnerabilities and attacks

Frey-Rück Attack

One of the known vulnerabilities in Bitcoin is the possibility of a Frey-Rück attack. This attack allows attackers to recover the secret key K if it has been reused or if there is a vulnerability in the ECDSA implementation. Knowing K, it is possible to recover the wallet’s private key, which poses a serious security threat 1 .

Short ECDSA signatures

Short ECDSA signatures also pose a security risk. If the RRR signature is too short, it can leak information about the KKK private key. Knowing RRR, SSS, and ZZZ, the private key can be calculated using a certain formula, which allows you to recover the wallet 3 .

Conclusion

The KKK private key in the Bitcoin blockchain is a critical component of transaction security. Its uniqueness and randomness provide protection against replay attacks. However, if a KKK is reused or if there is a vulnerability in the ECDSA implementation, this can lead to serious consequences, including recovery of the private key. Therefore, it is important to use reliable methods for generating KKK and to keep the software up to date to protect against potential attacks.

References

Implementation of Frey-Rück Attack for KKK Secret Key Recovery . Habr.
Nonce in cryptography . Wikipedia.
Bitcoin Wallet Recovery via ECDSA Short Signatures . Pikabu.
Bitcoin in 10 minutes . European Bitcoiners.
Blockchain – a new market for information security . IKS Media.

Citations:

What other attacks can be used to reveal the private key in the Bitcoin blockchain & Implementation of Frey-Rück Attack to recover the private key 𝐾

Implementation of Frey-Rück Attack to Recover KKK Secret Key

Frey-Rück Attack is a method used to recover the KKK (Nonce) private key in ECDSA signatures, which can lead to the private key of a Bitcoin wallet. This attack is possible due to a critical vulnerability in Bitcoin blockchain transactions.

Theoretical foundations

ECDSA Vulnerabilities : Frey-Rück Attack is based on a solution to the discrete logarithm problem, which occurs when the KKK secret key is reused or there are vulnerabilities in the ECDSA implementation.
Discrete Logarithm : The discrete logarithm problem is to find xxx such that gx≡ymod pg^x \equiv y \mod pgx≡ymodp, where ggg is the generator, yyy is a known value, and ppp is a prime number. In the context of ECDSA, this means recovering the KKK from the signature.

Practical implementation

To implement Frey-Rück Attack, the following steps can be used:

Vulnerable Transaction Collection : Finding transactions in the Bitcoin blockchain where the KKK private key has been reused or there is a vulnerability in the ECDSA implementation.
Extracting ECDSA Signatures : Extracting signatures from vulnerable transactions.
Solving the Discrete Logarithm Problem : Using algorithms such as Babbage’s algorithm or Pollard’s rho algorithm to solve the discrete logarithm problem and recover the KKK secret key.
Getting the private key : Knowing the KKK, you can recover the private key of your Bitcoin wallet.

Example of implementation in Python

Python and libraries such as cryptographyand can be used to implement the Frey-Rück Attack ecdsa. However, direct implementation requires a deep understanding of cryptography and access to vulnerable transactions.

pythonimport ecdsa
from ecdsa.curves import SECP256k1

# Пример восстановления секретного ключа K
def recover_k(signature, message_hash):
    # Восстановление K из подписи и хэша сообщения
    # Это требует решения задачи дискретного логарифмирования
    # и наличия уязвимой реализации ECDSA
    pass

# Пример использования
signature = b'...'  # Подпись ECDSA
message_hash = b'...'  # Хэш сообщения

k = recover_k(signature, message_hash)
if k:
    print("Секретный ключ K восстановлен:", k)
else:
    print("Не удалось восстановить K")

Important Notes

The Frey-Rück Attack is a complex task and requires a deep understanding of cryptography and access to vulnerable transactions.
Carrying out such an attack may be illegal and ethically questionable.
To protect against such attacks, it is important to use secure methods for generating and storing secret keys, and to keep your software up to date.

Sources

Citations:

Bitcoin Wallet Recovery via ECDSA Short Signatures

Short ECDSA signatures are a potential security threat to Bitcoin wallets, as they can lead to a full recovery of the private key and, as a result, to the loss of coins. Let’s look at how this happens and how you can recover a wallet through such signatures.

What are ECDSA Short Signatures?

Short ECDSA signatures are signatures that, due to bugs or vulnerabilities in the implementation of the ECDSA algorithm, are smaller than expected. This can lead to information leakage about the KKK secret key used to generate the signature.

How to Recover Bitcoin Wallet via ECDSA Short Signatures?

To restore your wallet, you can use the following steps:

Harvesting vulnerable transactions : Finding transactions with short ECDSA signatures in the Bitcoin blockchain.
Extract Signatures : Extract signatures from these transactions.
Calculating the KKK Secret Key : Using the RRR, SSS, and ZZZ values from the signature, the KKK secret key can be calculated using a specific formula.
Getting a private key : Knowing KKK, you can restore the private key of the wallet using the formula: Privkey=((((S∗K)−Z)∗modinv(R,N))Privkey = ((((S * K) – Z) * modinv(R,N)) % N)Privkey=((((S∗K)−Z)∗modinv(R,N))

Example of implementation in Python

To restore your wallet, you can use Python scripts like calculate.py, which work with the repository CryptoDeepTools. Here’s an example of how to do this:

import ecdsa
from ecdsa.curves import SECP256k1

def recover_private_key(R, S, Z, K):
    # Определение констант для эллиптической кривой secp256k1
    N = SECP256k1.order
    
    # Вычисление приватного ключа
    priv_key = ((((S * K) - Z) * modinv(R, N)) % N)
    
    return priv_key

# Функция для вычисления модульного обратного
def modinv(a, m):
    def extended_gcd(a, b):
        if a == 0:
            return b, 0, 1
        else:
            gcd, x, y = extended_gcd(b % a, a)
            return gcd, y - (b // a) * x, x

    gcd, x, _ = extended_gcd(a, m)
    if gcd != 1:
        raise ValueError('Modular inverse does not exist')
    else:
        return x % m

# Пример использования

K = 0x39588951cd20e38a6dc86d6b436da7abd2bcad84af3dd16b6f8a83c946c1d3c6 # Value of K (secret key)
R = 0xaafe80d17b0d30de09cbe39a85514aaae0a388135987ab80207e1eed3c915280 # Value of R from signature
S = 0x0d46fb28a4b30599d33325aa8b7633dd0f584f8125bb2e136c88a3e91a6f4238 # Value of S from signature
Z = 0xbbfd05c3355957cbdf44d283b9199eb9741f775a16081288187a82f544fac11f # Value of Z (message hash)


priv_key = recover_private_key(R, S, Z, K)
print("Приватный ключ:", priv_key)

Пример: https://github.com/demining/CryptoDeepTools/blob/main/38QuantumAttacks/calculate.py

Important Notes

Security : Short ECDSA signatures pose a serious security threat to Bitcoin wallets. To avoid problems, it is important to use up-to-date software and avoid using vulnerable ECDSA implementations.
Ethical Considerations : Recovering someone else’s wallet without permission is illegal and unethical. These methods should only be used to recover your own lost wallets or for educational purposes.