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ZeroJoin#

Mixers are basic tools that restore the fungibility of digital notes.

The first scheme on Ergo, ZeroJoin, is based on ring signatures and proof of knowledge for a Diffie-Hellman tuple (for publicly known g, h, u, v, there exists w, such as g&w == u && hw == v)

ZeroJoin is based on two-party interactions called Σ-protocols. Specifically, we need two types of Σ-protocols described below.

Let G be a multiplicative group of prime order q where the Decision Diffie-Hellman (DDH) problem is hard.

  1. The first protocol, denoted proveDlog(u), is a proof of knowledge of the Discrete Logarithm of some group element u with respect to some fixed generator g.
    1. That is, the prover proves knowledge of x such that u = gx by using Schnorr signatures.
  2. The second protocol primitive, denoted proveDHTuple(g, h, u, v), is a proof of knowledge of Diffie-Hellman Tuple, where the prover proves knowledge of x such that u = gx and v = hx for arbitrary generators g and h.

This is two instances of the first protocol running in parallel.

  1. The prover picks r ←R Zq, computes (t0, t1) = (gr , hr ) and sends (t0, t1) to the verifier.
  2. The verifier picks c R ← Zq and sends c to prover.
  3. The prover sends z = r + cx to the verifier, who accepts if gz = t0 · uc and hz = t1 · vc.

We use the non-interactive variant of the above protocol obtained via a Fiat-Shamir transformation, where c = H(t0‖t1‖m) for some message m to be signed.

Observe that proveDHTuple requires four exponentiations for verification, while proveDlog requires 2.

ErgoScript supports both protocols and thus has all the primitives needed to implement ZeroJoin.