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Schnorr Signatures#

There are several use-cases where you might need to verify a Schnorr signature on-chain.

This page describes how to do so in ErgoScript.

Initial Setup#

Ergo uses the same curve as Bitcoin (Secp256k1), which we call G.

The curve also defines a default generator g.

  1. Secret key is integer x
  2. Public key is Y = gx, an element of G

Signing#

Let the hash of the message to be signed be M. The signature is computed as follows:

  1. Generate a random integer r and compute U = gr.
  2. Compute the integer c = Hash(U || M)
  3. Compute s = r - cx.
  4. Send the value (c, s) to the verifier as the "signature"

Note that the signature is a pair of integers.

Verification#

Schnorr Identification#

To understand verification, first consider a variant called Schnorr identification.

In this, instead of (c, s), the value (U, s) (a group element and an integer) is sent.

The verifier computes c = Hash(U || M) and accepts if gs = U / Yc.

This works because LHS = gs = g(r - cx) = gr / (gx)c = RHS.

Schnorr Signature Verification#

Given the signature (c, s), we perform the "reverse" of the identification in some sense.

Recall that the verifier of the identification scheme computes c from U using Hash and then verifies some condition.

The verifier of the signature scheme instead computes U from c using the condition and then verifies Hash.

In other words, the verifier first computes U = gs Yc and accepts if c = Hash(U || M).

Verification in ErgoScript#

We use the following setup in our example:

  1. The public key Y is provided as a GroupElement in R4.
  2. The message M is provided as a Coll[Byte] in R5.
  3. The value c of the signature is provided as a Coll[Byte] (for convenience) in context variable 0.
  4. The value s of the signature is provided as a BigInt in context variable 1.
  5. The hash function is Sha256.

Which looks like this in ErgoScript

{ 
  // (Checking Schnorr signature in a script)

  // Getting the generator of the elliptic curve group 
  val g: GroupElement = groupGenerator

  // Getting the public key for a signature
  val Y = SELF.R4[GroupElement].get

  // Getting the message to be signed
  val M = SELF.R5[Coll[Byte]].get

  // Retrieving the c value of the signature (c, s)
  val cBytes = getVar[Coll[Byte]](0).get
  val c = byteArrayToBigInt(cBytes)

  // Retrieving the s value of the signature (c, s)
  val s = getVar[BigInt](1).get

  // Calculating U = g^s * Y^c
  val U = g.exp(s).multiply(Y.exp(c)).getEncoded // as a byte array

  // Checking the validity of the Schnorr signature
  sigmaProp(cBytes == sha256(U ++ M))
}

The complete process of signature generation off-chain and verification on-chain is explained in this test.

Off-chain code#

The problem with verifying signatures on-chain is that there is only 256-bits big integer data type.

Thus better to reduce number of bigints used by using simpler textbook version of Schnorr validation (message details missed):

{
    val message = ...
    // Computing challenge
    val e: Coll[Byte] = blake2b256(message) // weak Fiat-Shamir
    val eInt = byteArrayToBigInt(e) // challenge as big integer

     // a of signature in (a, z)
     val a = getVar[GroupElement](1).get
     val aBytes = a.getEncoded

     // z of signature in (a, z)
     val zBytes = getVar[Coll[Byte]](2).get
     val z = byteArrayToBigInt(zBytes)

     // Signature is valid if g^z = a * x^e
     val properSignature = g.exp(z) == a.multiply(holder.exp(eInt))

     sigmaProp(properSignature)
}

and then in offchain code we need to be sure that z big integer fits into 255 bits. The following code is simply iterating over signatures while one which can be provided used on the blockchain 

  def randBigInt: BigInt = {
    val random = new SecureRandom()
    val values = new Array[Byte](32)
    random.nextBytes(values)
    BigInt(values).mod(SecP256K1.q)
  }

  @tailrec
  def sign(msg: Array[Byte], secretKey: BigInt): (GroupElement, BigInt) = {
    val r = randBigInt
    val g: GroupElement = CryptoConstants.dlogGroup.generator
    val a: GroupElement = g.exp(r.bigInteger)
    val z = (r + secretKey * BigInt(scorex.crypto.hash.Blake2b256(msg))) % CryptoConstants.groupOrder

    if(z.bitLength <= 255) {
      (a, z)
    } else {
      sign(msg,secretKey)
    }
  }

Examples on building transactions can be found in ChainCash repository, e.g. this test https://github.com/kushti/chaincash/blob/master/src/test/scala/kiosk/ChainCashSpec.scala

Disclaimer

Please note that Schnorr here is using weak Fiat-Shamir transformation, but that should not be a problem as public key is fixed.