The Mathematical Connection: How Different Addresses Are Generated from the Same Private Key
In the world of cryptocurrencies, especially Ethereum, understanding how different addresses are generated from a single private key is crucial to understanding the underlying mathematics. One concept that comes to mind is the relationship between private keys and their corresponding public keys, which ultimately become unique addresses on the blockchain.
For those unfamiliar, here’s the lowdown: when you create an Ethereum wallet, it generates two types of addresses:
- Private Key: A secret code used to sign transactions (or “sign” them).
- Public Key (or
Address): A unique string that identifies and represents your Ethereum wallet.
The connection between private keys and public addresses lies in the mathematical concept of elliptic curves, which are used to encode and verify cryptographic operations on the blockchain.
From Private Key to Public Address
Consider the following steps:
- Generate a Pair: You create a pair of numbers (p, q) that define an elliptic curve using your private key.
- Select e: This is a large integer (e) that determines the difficulty level of performing certain mathematical operations on the curve.
- Calculate k and n
: These values are chosen so that n is the product of (p-1), q, p, and e.
From Public Key to Address
The public key consists of four hexadecimal digits: 0x[base58-encoded private key]. To translate this into a unique address on the Ethereum network:
- Subtract 2: You need to “subtract” your base58-encoded private key from 2.
- Modify with e: The resulting number, after applying the modulus operation using e, becomes your public key.
Why Multiple Addresses from One Private Key?
Now let’s see why you can get multiple addresses when generating an Ethereum wallet:
- Base58 Encoding: This step converts the private key into a compact format that can be easily stored and transmitted (base58 encoded).
- Hash Function: The base58 encoded public key is then passed through a hash function that generates a fixed-length string.
This mathematical manipulation ensures that different addresses from a single private key are generated randomly and unpredictably, providing an additional layer of security against potential attacks or unauthorized transactions.
To summarize:
- The private key generates two types of Ethereum addresses: a public key (or address) and another pair of numbers (p, q) in an elliptic curve.
- The public key is further modified by subtracting 2 from the base58 encoded private key and applying modulo operations using a large integer e.
- This process provides multiple unique addresses for each wallet, ensuring security and preventing attacks.
Understanding how Ethereum addresses are generated from a single private key can help you navigate the world of cryptocurrencies with more confidence.
