orchard/book/src/design/nullifiers.md

Nullifiers

The nullifier design we use for Orchard is

\mathsf{nf} = [Hash_{\mathsf{nk}}(\psi)] H + [\mathsf{rnf}] \mathcal{I},

where:

• Hash is a keyed circuit-efficient hash (such as Rescue).
• GH is a cryptographic hash into the group (such as BLAKE2s with simplified SWU).
• \mathcal{I} is a fixed base, independent of any others returned by GH.
• H is a base unique to this output.
  • For non-zero-valued notes, H = GH(\rho). As with \mathsf{h_{Sig}} in Sprout, \rho includes the nullifiers of any Orchard notes being spent in the same action. Given that an action consists of a single spend and a single output, we set \rho to be the nullifier of the spent note.
  • For zero-valued notes, H is constrained by the circuit to a fixed base independent of \mathcal{I} and any others returned by GH.
• \psi is sender-controlled randomness. It is not required to be unique, and in practice is derived from both \rho and a sender-selected random value \mathsf{rseed}: \psi = KDF^\psi(\rho, \mathsf{rseed}).
• \mathsf{rnf} is a blinding scalar, similarly generated as \mathsf{rnf} = KDF^\mathsf{rnf}(\rho, \mathsf{rseed}).

This gives a note structure of

(addr, v, H, \psi, \mathsf{rnf}, \mathsf{rcm}).

The note plaintext includes \mathsf{rseed} in place of \psi, \mathsf{rnf}, and \mathsf{rcm}. H is omitted entirely from the action:

• Consensus nodes directly derive GH(\rho) and provide it as a public input to the circuit (which ignores it for zero-valued notes, as with the commitment tree anchor).
• The recipient can recompute the correct H given their additional knowledge of v.

Security properties

We care about several security properties for our nullifiers:

• Balance: can I forge money?

• Note Privacy: can I gain information about notes only from the public block chain?

  • This describes notes sent in-band.

• Note Privacy (OOB): can I gain information about notes sent out-of-band, only from the public block chain?

  • In this case, we assume privacy of the channel over which the note is sent, and that the adversary does not have access to any notes sent to the same address which are then spent (so that the nullifier is on the block chain somewhere).

• Spend Unlinkability: given the incoming viewing key for an address, and not the full viewing key, can I (possibly the sender) detect spends of any notes sent to that address?

  • We're giving ivk to the attacker and allowing it to be the sender in order to make this property as strong as possible: they will have all the notes sent to that address.

• Faerie Resistance: can I perform a Faerie Gold attack (i.e. cause notes to be accepted that are unspendable)?

  • We're giving the full viewing key to the attacker and allowing it to be the sender in order to make this property as strong as possible: they will have all the notes sent to that address, and be able to derive every nullifier.

We assume (and instantiate elsewhere) the following primitives:

• E is an elliptic curve (such as Pallas).
• KDF is the note encryption key derivation function.

For our chosen design, our desired security properties rely on the following assumptions:

\begin{array}{|l|l|}
\text{Balance} & DL_E \\
\text{Note Privacy} & HashDH^{KDF}_E \\
\text{Note Privacy (OOB)} & \text{Perfect} \\
\text{Spend Unlinkability} & DDH_E^\dagger \vee PRF_{Hash} \\
\text{Faerie Resistance} & (RO_{GH} \vee (Coll_{GH} \wedge RO_{Hash})) \wedge DL_E \\
\end{array}

HashDH^{F}_E is computational Diffie-Hellman using F for the key derivation, with one-time ephemeral keys. This assumption is heuristically weaker than DDH_E but stronger than DL_E.

\dagger We additionally assume that for any input x, \{Hash_{\mathsf{nk}}(x) : \mathsf{nk} \in E\} gives a scalar in an adequate range for DDH_E. (Otherwise, Hash could be trivial, e.g. independent of \mathsf{nk}.)

Considered alternatives

\color{red}{\textsf{⚠ Caution}}: be skeptical of the claims in this table about what problem(s) each security property depends on. They may not be accurate and are definitely not fully rigorous.

\begin{array}{|c|l|c|c|c|c|c|}
\hline
\mathsf{nf} & Note & \text{Balance} & \text{Note Privacy} & \text{Note Privacy (OOB)} & \text{Spend Unlinkability} & \text{Faerie Resistance} & \text{Rejected because} \\\hline
[\mathsf{nk}] [\theta] H & (addr, v, H, \theta, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E & RO_{GH} \wedge DL_E & \text{No SU for DL-breaking} \\\hline
[\mathsf{nk}] H + [\mathsf{rnf}] \mathcal{I} & (addr, v, H, \mathsf{rnf}, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E & RO_{GH} \wedge DL_E & \text{No SU for DL-breaking} \\\hline
Hash([\mathsf{nk}] [\theta] H) & (addr, v, H, \theta, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E \vee Pre_{Hash} & Coll_{Hash} \wedge RO_{GH} \wedge DL_E & Coll_{Hash} \text{ for FR} \\\hline
Hash([\mathsf{nk}] H + [\mathsf{rnf}] \mathcal{I}) & (addr, v, H, \mathsf{rnf}, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E \vee Pre_{Hash} & Coll_{Hash} \wedge RO_{GH} \wedge DL_E & Coll_{Hash} \text{ for FR} \\\hline
[Hash_{\mathsf{nk}}(\psi)] [\theta] H & (addr, v, H, \theta, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E^\dagger \vee PRF_{Hash} & RO_{GH} \wedge DL_E & 2 \text{ variable-base scalar mults} \\\hline
[Hash_{\mathsf{nk}}(\psi)] \mathcal{G} + [\theta] H & (addr, v, H, \theta, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E^\dagger \vee PRF_{Hash} & RO_{GH} \wedge DL_E \\\hline
[Hash_{\mathsf{nk}}(\psi)] H + \mathsf{cm} & (addr, v, H, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & PRF_{Hash} & DDH_E^\dagger \vee PRF_{Hash} & (RO_{GH} \vee (Coll_{GH} \wedge RO_{Hash})) \wedge DL_E & PRF_{Hash} \text{ for NP(OOB)} \\\hline
[Hash_{\mathsf{nk}}(\rho, \psi)] \mathcal{G} + \mathsf{cm} & (addr, v, \rho, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & PRF_{Hash} & DDH_E^\dagger \vee PRF_{Hash} & Coll_{Hash} \wedge DL_E & PRF_{Hash} \text{ for NP(OOB)} \\\hline
[Hash_{\mathsf{nk}}(\rho)] \mathcal{G} + \mathsf{cm} & (addr, v, \rho, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & PRF_{Hash} & DDH_E^\dagger \vee PRF_{Hash} & Coll_{Hash} \wedge DL_E & PRF_{Hash} \text{ for NP(OOB)} \\\hline
[Hash_{\mathsf{nk}}(\rho, \psi)] \mathcal{G} + Commit^{\mathsf{nf}}_{\mathsf{rnf}}(v, \rho) & (addr, v, \rho, \mathsf{rnf}, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E^\dagger \vee PRF_{Hash} & Coll_{Hash} \wedge DL_E & Coll_{Hash} \text{ for FR} \\\hline
[Hash_{\mathsf{nk}}(\rho)] \mathcal{G} + Commit^{\mathsf{nf}}_{\mathsf{rnf}}(v, \rho) & (addr, v, \rho, \mathsf{rnf}, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E^\dagger \vee PRF_{Hash} & Coll_{Hash} \wedge DL_E & Coll_{Hash} \text{ for FR} \\\hline
[Hash_{\mathsf{nk}}(\rho, \psi)] \mathcal{G} + [\mathsf{rnf}] \mathcal{I} + \mathsf{cm} & (addr, v, \rho, \mathsf{rnf}, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E^\dagger \vee PRF_{Hash} & Coll_{Hash} \wedge DL_E & Coll_{Hash} \text{ for FR} \\\hline
[Hash_{\mathsf{nk}}(\rho)] \mathcal{G} + [\mathsf{rnf}] \mathcal{I} + \mathsf{cm} & (addr, v, \rho, \mathsf{rnf}, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E^\dagger \vee PRF_{Hash} & Coll_{Hash} \wedge DL_E & Coll_{Hash} \text{ for FR} \\\hline
\end{array}

In the above alternatives:

• \mathcal{G} is an fixed independent base, independent of \mathcal{I} and any others returned by GH.

The Commit^{\mathsf{nf}} variants enabled nullifier domain separation based on note value, without directly depending on \mathsf{cm} (which in its native type is a base field element, not a group element). We decided instead to follow Sapling by defining an intermediate representation of \mathsf{cm} as a group element, that is only used in nullifier computation.