For the non-math inclined developer is central to everything in the world of and cryptocurrencies. It’s used to create hash pointers that make blockchains possible and to create key pairs for wallets! Cryptography blockchain In this tutorial, we will be using the ‘ ’ and ‘ ’ javascript libraries in order to leverage principles of Elliptic Curve Cryptography (ECC) to derive a public key from a private key, and then convert the public key into it’s corresponding Ethereum address. elliptic js-sha3 Many online resources cover Elliptic Curve Cryptography from a mathematical perspective, but don’t offer applicable code examples. As a programmer, I find that seeing equations broken down into digestible code snippets can cement the principles into my mind. Before we dive into the code, let’s build a general foundation regarding what Elliptic Curve Cryptography is, why it is considered safe within the Ethereum ecosystem, and how it is used to generate key pairs. What is Elliptic Curve Cryptography? Elliptic Curve Example Elliptic Curve Cryptography is a form of asymmetric cryptography that leverages the discrete logarithm problem of elliptic curves (ECDSA) to create public and private key pairs. Essentially, it is easy to perform operations in one direction, but is very computationally expensive to go in the other direction. Is it possible to find a private key from a public key? Public Keys are generated as a result of using elliptic curve multiplication to multiply by in order to get private key (k) generator point (G) public key (K). K = k * G In , if we knew both and we could simply divide by in order to get . In other words, dividing the public key by the generator point would easily give us the private key. normal integer arithmetic K g, K g k k = K / G Elliptic Curve arithmetic and is therefore considered a one-way mathematical function. It is technically possible to find the private key by multiplying all of the possible factors that result in however, the larger the numbers involved, the longer it would take to compute. For more information, I recommend reading checking out this section from Andreas M. Antonopoulos and Gavin Wood’s book . does not have division K, Mastering Ethereum Overview As previously discussed, is actually another point on the elliptic curve. Therefore, where x and y are it’s coordinates**.** K = k * G K = (x, y) To convert the resulting public key ( into it’s corresponding Ethereum address, we must run it through the cryptographic hash function. The last 20 bytes of the resulting hash compose our Ethereum address. K) Keccak-256 Let’s jump into the code! First, let’s setup our project. (You can skip to the next part if you’re already comfortable setting up node projects.) Open up our terminal and create a directory for our code. mkdir deriveEthAddress Then, go into that directory using: cd deriveEthAddress To be able to npm install the modules we’ll be using, run npm init and fill in the prompts if you wish to. Next, let’s npm install the ‘elliptic’ and ‘js-sha3’ libraries. is a javascript library with various elliptic curve cryptography functions and provides us with a hash function we will be using. Elliptic js-sha3 npm install elliptic --save && npm install js-sha3 --save Lastly, we create our project file. touch deriveEthAddress.js Now that our project is set up, open up deriveEthAddress.js in a text editor. In deriveEthAddress.js we’ll require the modules we installed. Below, we create a new instance of an elliptic curve with the (a specific elliptic curve that both Ethereum and Bitcoin use) preset. We also require the hash function from secp256k1 keccak256 js-sha3. // Required Modules EC = require('elliptic').ec; ec = EC('secp256k1'); keccak256 = require('js-sha3').keccak256; const const new const We’ll then create variables for our private key and ethereum address. (This key pair is only for testing and shouldn’t be used to store actual funds. It was taken from the Ganache CLI - see the for more information.) github repo // Private Key and Public Ethereum address (from Ganache) givenPrivateKey = 'c87509a1c067bbde78beb793e6fa76530b6382a4c0241e5e4a9ec0a0f44dc0d3'; givenEthAddress = '0x627306090abab3a6e1400e9345bc60c78a8bef57'; const const Next, we’ll utilize the equation We get our generator point ( from the secp256k1 elliptic curve instance. is then multiplied by ( ). This results in ( ). As we discussed earlier, so we extract coordinates x and y and convert them to hexadecimal strings to be concatenated. K = k * G. G) generatorPoint privateKey k pubKeyCoordinates K K = (x,y), // Get secp256k1 generator point generatorPoint = ec.g; const // Public Key Coordinates calculated via Elliptic Curve Multiplication// PublicKeyCoordinates = privateKey * generatorPoint pubKeyCoordinates = generatorPoint.mul(givenPrivateKey); const x = pubKeyCoordinates.getX().toString('hex'); y = pubKeyCoordinates.getY().toString('hex'); const const // Public Key = X and Y concatenated publicKey = x + y; const console.log(publicKey);//af80b90d25145da28c583359beb47b21796b2fe1a23c1511e443e7a64dfdb27d7434c380f0aa4c500e220aa1a9d068514b1ff4d5019e624e7ba1efe82b340a59 We must then run our public key through the hash function. keccak256 // Use Keccak-256 hash function to get public key hash hashOfPublicKey = keccak256( Buffer(publicKey, 'hex')); const new console.log(hashOfPublicKey);// 3f704e8b0ee605967f55081f627306090abab3a6e1400e9345bc60c78a8bef57 If we look at our hash, the last 20 bytes (which would be the last 40 bits) compose our Ethereum address! Let’s convert this to a buffer so we can slice out the bytes we want to use and convert it to a hexadecimal string. We can then add the hexadecimal prefix to this. 0x // Convert hash to buffer ethAddressBuffer = Buffer(hashOfPublicKey, 'hex'); const new // Ethereum Address is '0x' concatenated with last 20 bytes// of the public key hash ethAddress = ethAddressBuffer.slice(-20).toString('hex'); ethAddressWithPrefix = `0x${ethAddress}`; const const Lastly, let’s compare our derived address to the address we were told our private key corresponded to. // Ethereum Address derived from private key// is equal to the given addressconsole.log(ethAddressWithPrefix === givenEthAddress); // true It’s a match! Here’s the full example - Closing We’ve only covered the basic principles of Elliptic Curve Cryptography. That being said, the mathematical concepts applied to the Ethereum network are pretty straightforward, and there are many libraries out there to help us! Thanks for reading! 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