Schnorr Signatures
There are several use-cases where we need to verify a Schnorr signature on-chain.
This post 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.
- Secret key is integer x
- Public key is Y = g^x, an element of G
Signing
Let the hash of the message to be signed be M. The signature is computed as follows:
- Generate a random integer r and compute U = g^r.
- Compute the integer c = Hash(U || M)
- Compute s = r - cx.
- 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 iff g^s = U / Y^c.
This works because LHS = g^s = g^(r - cx) = g^r / (g^x)^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 = g^s * Y^c and accepts if c = Hash(U || M).
Verification in ErgoScript
We use the following setup in our example:
- The public key Y is provided as a
GroupElement
in R4. - The message M is provided as a
Coll[Byte]
in R5. - The value c of the signature is provided as a
Coll[Byte]
(for convenience) in context variable 0. - The value s of the signature is provided as a
BigInt
in context variable 1. - The hash function is Sha256.
The following is the script.
{
// Checking Schnorr signature in a script
val g: GroupElement = groupGenerator
// Public key for a signature
val Y = SELF.R4[GroupElement].get
// Message to sign
val M = SELF.R5[Coll[Byte]].get
// c of signature in (c, s)
val cBytes = getVar[Coll[Byte]](0).get
val c = byteArrayToBigInt(cBytes)
// s of signature in (c, s)
val s = getVar[BigInt](1).get
val U = g.exp(s).multiply(Y.exp(c)).getEncoded // as a byte array
sigmaProp(cBytes == sha256(U ++ M))
}
The complete process of signature generation off-chain and verification on-chain is explained in this test.