Bitcoin, as one of the most popular cryptocurrencies, uses elliptic curve cryptography secp256k1 to ensure transaction security. Private keys, which are fundamental elements in this process, can be vulnerable to recovery through various methods, including the use of modular arithmetic without checking the curve parameters. In this article, we will consider the problem of recovering private keys without checking the elliptic curve parameters and analyze the mathematical aspects of invalid private keys in Bitcoin wallets.
Theoretical basis
Elliptic Curve Cryptography
Elliptic curve cryptography is based on the difficulty of the discrete logarithm problem on elliptic curves. In the case of Bitcoin, the curve used is secp256k1, which is defined by the equation:
$$ y^2 = x^3 + 7 $$
in the field $$ \mathbb{F}_p $$, where $$ p = 2^{256} — 2^{32} — 2^9 — 2^8 — 2^7 — 2^6 — 2^4 — 1 $$.
Private keys and public keys
The private key in a Bitcoin wallet is an integer $$ k $$ in the range $$ [1, n-1] $$, where $$ n $$ is the order of the subgroup on the secp256k1 curve. The public key is obtained by multiplying the base point $$ G $$ by the private key:
$$ P = kG $$
Modular arithmetic
Modular arithmetic is used to perform operations to calculate private keys and signatures. However, if the curve parameters are not checked, mathematically incorrect private keys may arise.
Recovering private keys without checking parameters
Recovery method via short ECDSA signatures
One method to recover private keys is to use short ECDSA signatures. If a signature $$(R,S)$$ has a short component $$R$$, then one can try to recover the private key using the equation:
$$ k = (zS^{-1} – R) \mod n $$
where $$ z $$ is the message hash, $$ S $$ is the signature component, and $$ n $$ is the order of the subgroup on the curve.
The Problem of Incorrect Private Keys
If the curve parameters are not checked, the recovered key may not correspond to reality. For example, if $$ k $$ is out of the acceptable range or does not satisfy the curve equation, such a key will be incorrect.
Mathematical aspects of invalid private keys
Incorrect Private Keys and Security
Incorrect private keys can lead to security vulnerabilities in Bitcoin wallets. If a key is not restored correctly, it may not provide the necessary cryptographic protection, potentially allowing attackers to gain access to funds.
Impact on cryptographic strength
Bitcoin’s cryptographic security is based on the difficulty of the elliptic curve discrete logarithm problem. Using incorrect private keys can weaken this security, making the system more vulnerable to attack.
Conclusion
Recovering private keys without checking the elliptic curve secp256k1 parameters may result in mathematically incorrect private keys in Bitcoin wallets. This may have serious implications for the security of cryptocurrency transactions. Therefore, it is important to ensure proper verification of the curve parameters when recovering private keys to ensure the cryptographic strength and security of Bitcoin wallets.
Recommendations
- Checking the Curve Parameters : Always check the elliptic curve parameters when recovering private keys.
- Use proven methods : Use only proven and secure methods to recover private keys.
- Software Update : Update your software regularly to keep your cryptographic algorithms secure and up-to-date.
Citations:
[1] https://waymorr.ru/news/blog/chto-takoe-privatnyij-klyuch-bitkoin-koshelka
[2] https://habr.com/ru/articles/683802/
[3] https://yellow.com/ru/news/%D0%BA%D0%B0%D0%BA-%D0%B2%D0%BE%D1%81%D1%81%D1%82%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8%D1% 82%D1%8C-%D0%BF%D0%BE%D1%82%D0%B5%D1%80%D1%8F%D0%BD%D0%BD%D1%8B%D0%B9-%D0%B1%D0%B8%D1%82%D0%BA%D0%BE%D0%B9%D0% BD—%D0%BE%D1%82-%D1%81%D0%B8%D0%B4-%D1%84%D1%80%D0%B0%D0%B7%D1%8B-%D0%B4%D0%BE-%D0%B7%D0%B0%D0%BA%D1%80%D1%8B% D1%82%D1%8B%D1%85-%D0%BA%D0%BB%D1%8E%D1%87%D0%B5%D0%B9-%D0%BF%D0%BE%D0%BB%D0%BD%D1%8B%D0%B9-%D0%B3%D0%B8%D0%B4
[4] https://pikabu.ru/tag/%D0%91%D0%B8%D1%82%D0%BA%D0%BE%D0%B8%D0%BD%D1%8B,%D0%92%D0%B8%D0 %B4%D0%B5%D0%BE%20%D1%81%20%D1%82%D0%B5%D0%BA%D1%81%D1%82%D0%BE%D0%BC/best?page=67&mv=2
[5] https://vc.ru/crypto/1609511-kak-vosstanovit-blokchein-koshelek-6-osnovnyh-sposobov-s-instrukciyami
[6] https://www.ledger.com/ru/academy/%D1%82%D0%B5%D0%BC%D1%8B/crypto/how-to-find-and-recover-lost-bitcoin-wallets
[7] https://science-engineering.ru/ru/article/view?id=1247
[8] https://www.ledger.com/ru/academy/private-key-and-seed-phrase-whats-the-difference