mirror of https://github.com/zcash/orchard.git
book: Revert to the previous nullifier design
We examined the nullifier designs more closely, and determined that the previously-selected design was actually fine, but for a somewhat-subtle reason: even though an adversary with knowledge of a victim's full viewing key could choose psi to cancel out Hash_nk(rho), the nullifier still directly depends on rho via the note commitment.
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Jack Grigg
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@ -2,34 +2,25 @@
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The nullifier design we use for Orchard is
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$$\mathsf{nf} = [Hash_{\mathsf{nk}}(\psi)] H + [\mathsf{rnf}] \mathcal{I},$$
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$$\mathsf{nf} = [F_{\mathsf{nk}}(\rho) + \psi \pmod{p}] \mathcal{G} + \mathsf{cm},$$
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where:
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- $Hash$ is a keyed circuit-efficient hash (such as Rescue).
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- $GH$ is a cryptographic hash into the group (such as BLAKE2s with simplified SWU).
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- $\mathcal{I}$ is a fixed base, independent of any others returned by $GH$.
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- $H$ is a base unique to this output.
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- For non-zero-valued notes, $H = GH(\rho)$. As with $\mathsf{h_{Sig}}$ in Sprout,
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$\rho$ includes the nullifiers of any Orchard notes being spent in the same action.
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Given that an action consists of a single spend and a single output, we set $\rho$ to
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be the nullifier of the spent note.
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- For zero-valued notes, $H$ is constrained by the circuit to a fixed base independent
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of $\mathcal{I}$ and any others returned by $GH$.
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- $F$ is a keyed circuit-efficient PRF (such as Rescue).
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- $\rho$ is unique to this output. As with $\mathsf{h_{Sig}}$ in Sprout, $\rho$ includes
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the nullifiers of any Orchard notes being spent in the same action. Given that an action
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consists of a single spend and a single output, we set $\rho$ to be the nullifier of the
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spent note.
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- $\psi$ is sender-controlled randomness. It is not required to be unique, and in practice
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is derived from both $\rho$ and a sender-selected random value $\mathsf{rseed}$:
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$\psi = KDF^\psi(\rho, \mathsf{rseed})$.
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- $\mathsf{rnf}$ is a blinding scalar, similarly generated as
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$\mathsf{rnf} = KDF^\mathsf{rnf}(\rho, \mathsf{rseed})$.
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$$\psi = KDF^\psi(\rho, \mathsf{rseed}).$$
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- $\mathcal{G}$ is a fixed independent base.
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This gives a note structure of
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$$(addr, v, H, \psi, \mathsf{rnf}, \mathsf{rcm}).$$
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$$(addr, v, \rho, \psi, \mathsf{rcm}).$$
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The note plaintext includes $\mathsf{rseed}$ in place of $\psi$, $\mathsf{rnf}$, and
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$\mathsf{rcm}$. $H$ is omitted entirely from the action:
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- Consensus nodes directly derive $GH(\rho)$ and provide it as a public input to the
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circuit (which ignores it for zero-valued notes, as with the commitment tree anchor).
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- The recipient can recompute the correct $H$ given their additional knowledge of $v$.
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The note plaintext includes $\mathsf{rseed}$ in place of $\psi$ and $\mathsf{rcm}$, and
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omits $\rho$ (which is a public part of the action).
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## Security properties
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@ -60,6 +51,8 @@ We care about several security properties for our nullifiers:
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We assume (and instantiate elsewhere) the following primitives:
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- $GH$ is a cryptographic hash into the group (such as BLAKE2s with simplified SWU), used
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to derive all fixed independent bases.
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- $E$ is an elliptic curve (such as Pallas).
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- $KDF$ is the note encryption key derivation function.
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@ -69,19 +62,25 @@ $$
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\begin{array}{|l|l|}
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\text{Balance} & DL_E \\
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\text{Note Privacy} & HashDH^{KDF}_E \\
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\text{Note Privacy (OOB)} & \text{Perfect} \\
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\text{Spend Unlinkability} & DDH_E^\dagger \vee PRF_{Hash} \\
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\text{Faerie Resistance} & (RO_{GH} \vee (Coll_{GH} \wedge RO_{Hash})) \wedge DL_E \\
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\text{Note Privacy (OOB)} & \text{Near perfect} \ddagger \\
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\text{Spend Unlinkability} & DDH_E^\dagger \vee PRF_F \\
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\text{Faerie Resistance} & DL_E \\
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\end{array}
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$$
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$HashDH^{F}_E$ is computational Diffie-Hellman using $F$ for the key derivation, with
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$HashDH^{KDF}_E$ is computational Diffie-Hellman using $KDF$ for the key derivation, with
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one-time ephemeral keys. This assumption is heuristically weaker than $DDH_E$ but stronger
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than $DL_E$.
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We omit $RO_{GH}$ as a security assumption because we only rely on the random oracle
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applied to fixed inputs defined by the protocol, i.e. to generate the fixed base
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$\mathcal{G}$, not to attacker-specified inputs.
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> $\dagger$ We additionally assume that for any input $x$,
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> $\{Hash_{\mathsf{nk}}(x) : \mathsf{nk} \in E\}$ gives a scalar in an adequate range for
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> $DDH_E$. (Otherwise, $Hash$ could be trivial, e.g. independent of $\mathsf{nk}$.)
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> $\{F_{\mathsf{nk}}(x) : \mathsf{nk} \in E\}$ gives a scalar in an adequate range for
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> $DDH_E$. (Otherwise, $F$ could be trivial, e.g. independent of $\mathsf{nk}$.)
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>
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> $\ddagger$ Statistical distance $< 2^{-167.8}$ from perfect.
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## Considered alternatives
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@ -92,26 +91,36 @@ not fully rigorous.
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$$
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\begin{array}{|c|l|c|c|c|c|c|}
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\hline
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\mathsf{nf} & Note & \text{Balance} & \text{Note Privacy} & \text{Note Privacy (OOB)} & \text{Spend Unlinkability} & \text{Faerie Resistance} & \text{Rejected because} \\\hline
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\mathsf{nf} & Note & \text{Balance} & \text{Note Privacy} & \text{Note Privacy (OOB)} & \text{Spend Unlinkability} & \text{Faerie Resistance} & \text{Reason not to use} \\\hline
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[\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
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[\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
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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
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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
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[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
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[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
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[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
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[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
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[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
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[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
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[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
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[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
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[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
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[F_{\mathsf{nk}}(\psi)] [\theta] H & (addr, v, H, \theta, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E^\dagger \vee PRF_F & RO_{GH} \wedge DL_E & \text{Performance (2 variable-base)} \\\hline
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[F_{\mathsf{nk}}(\psi)] H + [\mathsf{rnf}] \mathcal{I} & (addr, v, H, \mathsf{rnf}, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E^\dagger \vee PRF_F & RO_{GH} \wedge DL_E & \text{Performance (1 variable- + 1 fixed-base)} \\\hline
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[F_{\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_F & RO_{GH} \wedge DL_E & \text{Performance (1 variable- + 1 fixed-base)} \\\hline
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[F_{\mathsf{nk}}(\psi)] H + \mathsf{cm} & (addr, v, H, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & DDH_E^\dagger & DDH_E^\dagger \vee PRF_F & RO_{GH} \wedge DL_E & \text{NP(OOB) not perfect} \\\hline
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[F_{\mathsf{nk}}(\rho, \psi)] \mathcal{G} + \mathsf{cm} & (addr, v, \rho, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & DDH_E^\dagger & DDH_E^\dagger \vee PRF_F & DL_E & \text{NP(OOB) not perfect} \\\hline
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[F_{\mathsf{nk}}(\rho)] \mathcal{G} + \mathsf{cm} & (addr, v, \rho, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & DDH_E^\dagger & DDH_E^\dagger \vee PRF_F & DL_E & \text{NP(OOB) not perfect} \\\hline
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[F_{\mathsf{nk}}(\rho, \psi)] \mathcal{G_v} + [\mathsf{rnf}] \mathcal{I} & (addr, v, \rho, \mathsf{rnf}, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E^\dagger \vee PRF_F & Coll_F \wedge DL_E & Coll_F \text{ for FR} \\\hline
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[F_{\mathsf{nk}}(\rho)] \mathcal{G_v} + [\mathsf{rnf}] \mathcal{I} & (addr, v, \rho, \mathsf{rnf}, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Perfect} & DDH_E^\dagger \vee PRF_F & Coll_F \wedge DL_E & Coll_F \text{ for FR} \\\hline
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[F_{\mathsf{nk}}(\rho) + \psi \pmod{p}] \mathcal{G_v} & (addr, v, \rho, \psi, \mathsf{rcm}) & DL_E & HashDH^{KDF}_E & \text{Near perfect} \ddagger & DDH_E^\dagger \vee PRF_F & \color{red}{\text{broken}} & \text{broken for FR} \\\hline
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[F_{\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_F & DL_E & \text{Performance (2 fixed-base)} \\\hline
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[F_{\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_F & DL_E & \text{Performance (2 fixed-base)} \\\hline
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\end{array}
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$$
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In the above alternatives:
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- $\mathcal{G}$ is an fixed independent base, independent of $\mathcal{I}$ and any others
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- $Hash$ is a keyed circuit-efficient hash (such as Rescue).
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- $\mathcal{I}$ is an fixed independent base, independent of $\mathcal{G}$ and any others
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returned by $GH$.
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- $\mathcal{G_v}$ is a pair of fixed independent bases (independent of all others), where
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the specific choice of base depends on whether the note has zero value.
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- $H$ is a base unique to this output.
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- For non-zero-valued notes, $H = GH(\rho)$. As with $\mathsf{h_{Sig}}$ in Sprout,
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$\rho$ includes the nullifiers of any Orchard notes being spent in the same action.
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- For zero-valued notes, $H$ is constrained by the circuit to a fixed base independent
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of $\mathcal{I}$ and any others returned by $GH$.
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The $Commit^{\mathsf{nf}}$ variants enabled nullifier domain separation based on note
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value, without directly depending on $\mathsf{cm}$ (which in its native type is a base
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