Before we get deeper into the concept of Ethereum Blockchain, it is important to understand how Public Key, Private Key and Address are generated. To make this easier for beginners, I am trying to explain the technical concept with an example.

Ethereum key generation is based on elliptical curve ecp256k1. Eliptical curve (EC) is intense mathematics, and there are a lot of great articles on internet covering deep details of elliptical curve. There are standard libraries to generate Ethereum key pair in much safer way. In this article, just for demonstration, I will use elliptical curve Javascript libaray to perform elliptical curve operations.

Private Key

On EC ecp256k1 , any number between 1 to 2^256-1 is a valid private key. A good library generate a private key with taking sufficient randomness into account. For this exercise we will take "1" as private key which is an acceptable private key because it lies in specified range mentioned above. Ethereum requires private key to be 256 bit long. Here is the sample code for nodejs.

var privateKey=Buffer.alloc(32, 0);
privateKey[31]=1;

console.log("PK::"+privateKey.toString('hex'))

This will print private key in hex which is

PK::0000000000000000000000000000000000000000000000000000000000000001

Public key

Public key is described as follows in yellow paper.

Where pu is the public key, assumed to be a byte array of size 64 (formed from the concatenation of two positive integers each < 2256) and pr is the private key, a byte array of size 32 (or a single positive integer in the aforementioned range).

This is done using group operation of EC cryptography. To derive public key, private key is multiplied by G.Multiplication used to derive public is EC multiplication which is entirely different from normal multiplication for which I am going to use JS library . G is called generator point which is one of the domain parameters of EC cryptography. G has fixed value for ecp256k1, which is recommended by experts. you can read more here.

var EC = require('elliptic').ec;
var BN = require('bn.js');
var ec = new EC('secp256k1');

var G = ec.g; // Generator point
var pk = new BN('1'); // private key as big number
var pubPoint=G.mul(pk); // EC multiplication to determine public point 


var x = pubPoint.getX().toBuffer(); //32 bit x co-ordinate of public point 
var y = pubPoint.getY().toBuffer(); //32 bit y co-ordinate of public point 

var publicKey =Buffer.concat([x,y])

console.log("pub key::"+publicKey.toString('hex'))

Ethereum Address

Ethereum address is described as follows in yellow paper

For a given private key, pr, the Ethereum address A(pr) (a 160-bit value) to which it corresponds is defined as the right most 160-bits of the Keccak hash of the corresponding ECDSA public key.

To generate Ethereum address, take Keccak-256 hash of public key. Right most 20 bytes is your Ethereum address.

var EC = require('elliptic').ec;
var BN = require('bn.js');
var ec = new EC('secp256k1');
const keccak256 = require('js-sha3').keccak256;


var privateKey=Buffer.alloc(32, 0);
privateKey[31]=1;

console.log("PK::"+privateKey.toString('hex'))

var G = ec.g; // Generator point
var pk = new BN('1'); // private key as big number

var pubPoint=G.mul(pk); // EC multiplication to determine public point

var x = pubPoint.getX().toBuffer(); //32 bit x co-ordinate of public point
var y = pubPoint.getY().toBuffer(); //32 bit y co-ordinate of public point 

var publicKey =Buffer.concat([x,y])

console.log("public key::"+publicKey.toString('hex'))

const address = keccak256(publicKey) // keccak256 hash of  publicKey

const buf2 = Buffer.from(address, 'hex');
console.log("Ethereum Adress:::"+"0x"+buf2.slice(-20).toString('hex')) // take lat 20 bytes as ethereum adress

PK::0000000000000000000000000000000000000000000000000000000000000001

Ethereum Address:::0x7e5f4552091a69125d5dfcb7b8c2659029395bdf

This is just an example to understand the concept. Please do not use it to generate real wallet to hold ETH.

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