ECDSA Public Key Recovery from Signature: Step by Step Guide

Ethereum, as a decentralized platform built on top of the Bitcoin blockchain, uses the Elliptic Curve Digital Signature Algorithm (ECDSA) for secure communication and transaction processing. In this article, we will explore how to recover the public key associated with a specific ECDSA signature.

Understanding ECDSA Basics

Before we dive into the solution, let’s briefly review some basic concepts:

• Public Key: A pair of keys used for authentication and non-repudiation: a public key (eg x) and a private key (eg y).

• Signature: A digital fingerprint that verifies that the message has not been altered during transmission.

• Hash Function: A one-way function used to create a fixed-size string of characters (known as a hash) from arbitrary input.

Recovery of ECDSA public key from signature

To recover the public key, we must obtain the corresponding private key. This is usually done using the following steps:

• Obtain a raw signature: Obtain a raw signature by hashing an unsigned signed message.

• Decrypt signature: Use the decrypt function (eg ecdsa_sign or ecdsa_recover) to decrypt the signature, which reveals the private key.

• Extract Public Key: Once you have the private key, use it to extract the corresponding public key.

Step by step solution

Here is a detailed example of recovering an ECDSA public key from a default signature:

import hashlib

import struct


def decrypt_signature(signature):

    """

    Decrypts the raw signature and returns the private key.

    

    Arguments:

        signature (bytes): raw signature.

    

    Return:

        bytes: Decrypted private key.

    """

    
Extract the signature length

    signature_len = int.from_bytes(signature[:32], byteorder='big')

    

    
Decrypt the signature using the ECDSA decrypt function

    public_key, _ = struct.unpack('!BBHH', signature[32:1024])

    

    return public_key


def get_private_key(public_key):

    """

    Gets the private key from the given public key.

    

    Arguments:

        public_key (bytes): Public key.

    

    Return:

        bytes: The corresponding private key.

    """

    
Extract ECDSA parameters

    n, x, _ = struct.unpack('!BBH', public_key[:32])

    

    
Compute the private key using the RSA decryption function

    private_key = pow(x, (n - 1) % 2, n)

    

    return private_key



Usage example:

signature = b'\x01\x02\x03\x04\x05' 
Replace with your actual signature

public_key = signature_decrypt(signature)


private_key = get_private_key(public_key)

print(private_key) 
Output: the corresponding private key

Note: In this example we use the ecdsa_sign placeholder function (which is not implemented here). You should replace it with an actual implementation that provides similar functionality. Additionally, you should ensure that you have the necessary cryptographic libraries and tools installed on your system.

By following these steps, you can recover an ECDSA public key from a given signature on the Ethereum blockchain or any other ECDSA-based platform.

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