What's the probability of getting the same signature using ECDSA?

Question

If I hash the message "HELLO" - keccak256("Hello") then run it to an ECDSA, what's the probability that I will get the same r, s and v value everytime?

Do I need to add some unique identifier other than the actual message then do keccak256(abi.encode("Hello", uniqueIdentifier)) to make sure that it's unique?

Answer 1

The ECDSA implementation creates and uses a random number internally. So, the probability of getting the same signature given the same message is really low, it's not feasible.

You can try a couple of ECDSA implementations and play around with the results. For example, using nodejs, you can install starkbank-ecdsa and try it out:

npm install starkbank-ecdsa npm install keccak256

const keccak256 = require("keccak256");
const ellipticcurve = require("starkbank-ecdsa");
const Ecdsa = ellipticcurve.Ecdsa;
const PrivateKey = ellipticcurve.PrivateKey;

// Generate privateKey from PEM string
const privateKey = PrivateKey.fromPem(
  "-----BEGIN EC PARAMETERS-----\nBgUrgQQACg==\n-----END EC PARAMETERS-----\n-----BEGIN EC PRIVATE KEY-----\nMHQCAQEEIODvZuS34wFbt0X53+P5EnSj6tMjfVK01dD1dgDH02RzoAcGBSuBBAAK\noUQDQgAE/nvHu/SQQaos9TUljQsUuKI15Zr5SabPrbwtbfT/408rkVVzq8vAisbB\nRmpeRREXj5aog/Mq8RrdYy75W9q/Ig==\n-----END EC PRIVATE KEY-----\n"
);

const messageHash = keccak256("Hello").toString("hex");

console.log("Message keccak256 hash: ", messageHash);

signature = Ecdsa.sign(messageHash, privateKey);
signature2 = Ecdsa.sign(messageHash, privateKey);
signature3 = Ecdsa.sign(messageHash, privateKey);
signature4 = Ecdsa.sign(messageHash, privateKey);

// Even though these signatures where created for the same message, they are different.
console.log("signature: ", signature.toBase64());
console.log("signature2: ", signature2.toBase64());
console.log("signature3: ", signature3.toBase64());
console.log("signature4: ", signature4.toBase64());

// To double check if message matches the signature
let publicKey = privateKey.publicKey();

console.log(Ecdsa.verify(messageHash, signature, publicKey));
console.log(Ecdsa.verify(messageHash, signature2, publicKey));
console.log(Ecdsa.verify(messageHash, signature3, publicKey));
console.log(Ecdsa.verify(messageHash, signature4, publicKey));

I modified their example from npm: https://www.npmjs.com/package/starkbank-ecdsa

It would yield something like this:

Message keccak256 hash:  06b3dfaec148fb1bb2b066f10ec285e7c9bf402ab32aa78a5d38e34566810cd2
signature:  MEYCIQD7odNPByFXKAoaOzni4BI1hRQ8qcolcHUQcPMITf7j+QIhALoIW5vOSCGGdsfeD1zWwc/VrXN+fukKE213HAiRIlgq
signature2:  MEUCIQC7X4HZpSu1kjNB2Dz7IEtnhtut1nKJ/zJrzcL04SxJxAIgDtccNTtn0tu7j+7oMPovez8NgiULALS+k2B1veRAp5Y=
signature3:  MEUCIBLk94Uyx8EaVSZqEbi85zP9B5qtlLgUezjb0ctXz8MWAiEA3ubrR5CgaDiOD/NZ1r9moGa0F8xLOtjWEHfTKupzjcQ=
signature4:  MEQCIEAiBKQXk2M+DmEjxONKs0N6qmUuhrjFhVuj+ZzBtpmdAiAHVtdy/e3s4yfL77wzYiR399h0JKFPdH8tVPtMLiiyzg==
true
true
true
true

You will notice that given the same message all the signatures are different, and all of them yield true when verifying the message.