Ethereum: Why is it impossible to obtain a private key from a public key?

Ethereum: Why is it not possible to derive a private key from a public key?

The Ethereum public key equation K = k * G may seem like a simple formula for deriving a private key from a public key. However, this assumption is fundamentally flawed in the context of cryptography and blockchain technology.

In cryptocurrency and smart contract platforms, including Ethereum, public and private keys are used to securely store and transfer sensitive information such as balances, transactions, and cryptographic secrets. The idea behind a public key infrastructure (PKI) like Ethereum is that a shared secret key, known as a private key (k), can be used to encrypt messages and decrypt them using the corresponding public key.

However, there are several reasons why it is not possible to derive a private key from an Ethereum public key:

• Key Exchange Protocol: In Ethereum, the public key equation K = k * G is used to exchange keys between parties (e.g. when two users want to agree on a shared secret key). However, this equation assumes that both parties have access to the same generator point (“G”). This means that even if the other party knows their private key (k), they cannot use it to encrypt and decrypt a message without access to the corresponding public key.

• Computational Complexity:

  The mathematical operations required to derive a private key from a public key are computationally expensive, making them impractical for large-scale applications. In particular, the multiplication k * G is an elliptic curve point doubling algorithm (ECDPA) with time complexity O(sqrt(n)), where n is the order of the generator (G). For most practical reasons, this means that even if the other party knows their private key, they cannot easily derive it from the public key.

• Mathematical limitations: The mathematical representation of a point on an elliptic curve (ECC) can be seen as a set of 2D coordinates, where each coordinate corresponds to the x and y components of the point. In the Ethereum implementation, the points are represented using 4 bytes (32 bits), which is relatively small compared to other cryptographic protocols that use more sophisticated elliptic curve algorithms, such as NIST-approved curves (e.g. secp256k1 or ed25519). This limited representation size makes it difficult to accurately represent the complexity of a point in ECC.

• Guarantee Protection: In Ethereum, the private key is often protected by a “nonce” value, which can be used to prevent replay attacks and ensure the integrity of transactions. Even if another party knows their private key, they cannot easily use it without knowing the corresponding nonce value.

In summary, while the public key equation “K = k * G” may seem like a simple formula for deriving a private key from a public key, it is fundamentally flawed due to computational complexity, mathematical limitations, and security considerations. Ethereum uses other protocols and mechanisms to securely store and exchange cryptographic secrets, such as the Elliptic Curve Digital Signature Algorithm (ECDSA) and HMAC-SHA256.

Recommendations:

• Use a more secure protocol such as ECDSA and HMAC-SHA256 for key storage and exchange.

• Consider using a security scheme such as zk-SNARK or zk-TREX to provide more efficient and secure cryptographic services.

• Always use secure practices such as password hashing and salting to protect user identities and sensitive data.

Hope this explanation helps!