Recovering an ECDSA Public Key from a Signature: A Step-by-Step Guide

Ethereum, being a decentralized platform built on top of the Bitcoin blockchain, utilizes 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 given ECDSA signature.

Understanding ECDSA Basics

Before diving into the solution, let’s quickly review some basic concepts:

• Public Key: A pair of keys used for authentication and non-repudiation: the public key (e.g., x) and the private key (e.g., y).

• Signature: A digital fingerprint that verifies a message has not been tampered with during transmission.

• Hash Function

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

Recovering ECDSA Public Key from the Signature

To recover the public key, we need to obtain the corresponding private key. This is typically done using the following steps:

• Get the raw signature: Obtain the raw signature by hashing the unsigned message with the signature.

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

• Extract the public key: Once the private key is obtained, use it to extract the corresponding public key.

Step-by-Step Solution

Here’s a detailed example of how to recover an ECDSA public key from a given signature:

import hashlib

import struct


def decrypt_signature(signature):

    """

    Decrypts the raw signature and returns the private key.

    

    Args:

        signature (bytes): The raw signature.

    

    Returns:

        bytes: The decrypted private key.

    """

    




Extract the signature length

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

    

    
Decrypt the signature using the ECDSA decryption function

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

    

    return public_key


def get_private_key(public_key):

    """

    Obtains the private key from a given public key.

    

    Args:

        public_key (bytes): The public key.

    

    Returns:

        bytes: The corresponding private key.

    """

    
Extract the 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



Example usage:

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

public_key = decrypt_signature(signature)


private_key = get_private_key(public_key)

print(private_key) 
Output: the corresponding private key

Note: In this example, we use a placeholder ecdsa_sign function (which is not implemented here). You should replace it with an actual implementation that provides a 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 the ECDSA public key from a given signature on the Ethereum blockchain or any other ECDSA-based platform.

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