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

The Ethereum public key equation K = k * G may seem like a simple formula for deriving the private key from the 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 transmit 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 the private key (k), can be used to encrypt messages and decrypt them with a corresponding public key.

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

• Key Exchange Protocol:

  In Ethereum, the public key equation K = k * G is used for key exchange between parties (for example, 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 one of the parties knows your private key (k), they cannot use it to encrypt a message and decrypt it without having access to the corresponding public key.

• Computational Complexity: The mathematical operations required to derive the private key from the 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), which has a time complexity of O(sqrt(n)), where n is the order of the generator (G). For most practical purposes, this means that even if a party knows your private key, they cannot easily derive it from your public key.

• Mathematical Limitations:

  

  The mathematical representation of a point on an elliptic curve (ECC) can be viewed as a set of 2D coordinates, where each coordinate corresponds to the x and y components of the point. In the Ethereum implementation, points are represented using 4 bytes (32 bits), which is relatively small compared to other cryptographic protocols that use more advanced 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 an ECC.

• Guaranteed Security: In Ethereum, the private key is often guaranteed by a “nonce” value, which can be used to prevent replay attacks and ensure transaction integrity. Even if a party knows your private key, they cannot easily use it without knowing the corresponding nonce value.

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

Recommendations:

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

• Consider using a zero-knowledge verification system such as zk-SNARKs or zk-TREX to provide more efficient and secure cryptographic services.

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

I hope this explanation helps!

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