This research reveals a serious cryptographic vulnerability related to the incorrect specification of elliptic curve parameters in the Bitcoin system. The key issue is the incorrect calculation of the constant N, which determines the order of a group of points of the secp256k1 curve, which leads to the generation of invalid private keys and creates critical security risks for cryptocurrency users.

The essence of the identified vulnerability

The underlying problem is that the order of the secp256k1 elliptic curve group used in Bitcoin’s cryptographic infrastructure is incorrectly calculated. In the code under study, the constant N is defined as `(1 N[1]

These mathematical anomalies lead to practical cryptographic consequences, including signature incompatibility (about 43% of transactions are rejected by network nodes), the formation of side channels for information leakage, and the vulnerability of deterministic wallets to attacks.[1]

Vulnerable Components of the Bitcoin Ecosystem

The study identified a number of components of the Bitcoin ecosystem that are vulnerable to this vulnerability:

Software components

In the Bitcoin ecosystem, the most vulnerable to the described problem are:

1. Custom wallets generating keys that are not compatible with the network[1]
2. HSM modules with the ability to export keys through hardware vulnerabilities[1]
3. Web interfaces using legacy libraries like BitcoinJS[1]
4. Mobile apps with bugs in custom cryptographic implementations[1]

Analysis has shown that about 68% of home-made ECDSA implementations contain similar parametric errors[1]. This poses a systemic threat to the entire Bitcoin infrastructure, especially to solutions that use non-standard or home-made cryptographic libraries.

Threat classification

The threats to Bitcoin wallets associated with this vulnerability can be classified into the following types:

1. Parametric vulnerabilities (incorrect secp256k1 curve ordering) leading to invalid private keys[1]
2. Implementation vulnerabilities (weak RNG) that make brute-force attacks possible[1]
3. Protocol vulnerabilities (lack of signature verification) that allow double-spending[1]
4. Hardware vulnerabilities (HSM issues) leading to private key leakage[1]

Historical precedents for similar vulnerabilities

The vulnerability identified is not unique. The history of cryptocurrencies contains several significant precedents with similar problems:

1. BitcoinJS Randstorm vulnerability (2011-2016) due to a weak random number generator, affecting assets worth about $1 billion[1]
2. SafeNet HSM hardware vulnerability (2015) allowing private key extraction[1]
3. Private key collisions in Android Wallet (2013) due to bugs in SecureRandom() implementation[1]

These cases demonstrate that errors in cryptographic parameters and implementations regularly lead to serious consequences, including compromise of HD wallets and loss of funds by users[1].

Recommendations for eliminating the vulnerability

To eliminate the identified vulnerability, it is recommended to:

1. Correction of the constant N to the standard value:

   N = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141

2. Using proven cryptographic libraries instead of custom implementations:

   from ecdsa import SigningKey, SECP256k1

   def gen_private_key():
       return SigningKey.generate(curve=SECP256k1)

3. Implementation of additional validity checks for generated keys, including validation of the hex format of input data, correct exception handling, and boundary value testing[1]
4. Strict adherence to SECG SEC2 and NIST SP 800-186 standards when implementing cryptographic algorithms[1]

Conclusion

The identified problem of incorrect calculation of the elliptic curve order secp256k1 poses a serious threat to the security of Bitcoin and other cryptocurrencies that use similar cryptographic approaches. Incorrect specification of mathematical parameters leads to the generation of invalid private keys, which can have catastrophic consequences for the security and safety of cryptocurrency assets. Following standardized methods for implementing elliptic curve cryptography and using proven libraries is critical to ensuring the security of blockchain systems.