mirror of https://github.com/zcash/orchard.git
Cosmetics and Markdown formatting.
Signed-off-by: Daira Hopwood <daira@jacaranda.org>
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Daira Hopwood
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@ -1,28 +1,28 @@
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# Commitments
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As in Sapling, we require two kinds of commitment schemes in Pollard:
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- $HomomorphicCommit$ is a linearly homomorphic commitment scheme with perfect hiding, and
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As in Sapling, we require two kinds of commitment schemes in Orchard:
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- $\mathit{HomomorphicCommit}$ is a linearly homomorphic commitment scheme with perfect hiding,
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and strong binding reducible to DL.
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- $\mathit{Commit}$ and $\mathit{ShortCommit}$ are commitment schemes with perfect hiding, and
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strong binding reducible to DL.
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- $Commit$ and $ShortCommit$ are commitment schemes with perfect hiding, and strong
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binding reducible to DL.
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By "strong binding" we mean that the scheme is collision resistant on the input and
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randomness.
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We instantiate $HomomorphicCommit$ with a Pedersen commitment, and use it for value
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commitments:
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We instantiate $\mathit{HomomorphicCommit}$ with a Pedersen commitment, and use it for
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value commitments:
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$$\mathsf{cv} = HomomorphicCommit^{\mathsf{cv}}_{\mathsf{rcv}}(v)$$
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$$\mathsf{cv} = \mathit{HomomorphicCommit}^{\mathsf{cv}}_{\mathsf{rcv}}(v)$$
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We instantiate $Commit$ and $ShortCommit$ with Sinsemilla, and use them for all other
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commitments:
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We instantiate $\mathit{Commit}$ and $\mathit{ShortCommit}$ with Sinsemilla, and use them
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for all other commitments:
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$$\mathsf{ivk} = \mathit{ShortCommit}^{\mathsf{ivk}}_{\mathsf{rivk}}(\mathsf{ak}, \mathsf{nk})$$
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$$\mathsf{cm} = Commit^{\mathsf{cm}}_{\mathsf{rcm}}(\text{rest of note})$$
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$$\mathsf{cm} = \mathit{Commit}^{\mathsf{cm}}_{\mathsf{rcm}}(\text{rest of note})$$
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This is the same split (and rationale) as in Sapling, but using the more PLONK-efficient
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Sinsemilla instead of Bowe-Hopwood Pedersen hashes.
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Sinsemilla instead of Bowe--Hopwood Pedersen hashes.
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Note that we also deviate from Sapling by using $ShortCommit$ to deriving $\mathsf{ivk}$
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Note that we also deviate from Sapling by using $\mathit{ShortCommit}$ to deriving $\mathsf{ivk}$
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instead of a full PRF. This removes an unnecessary (large) PRF primitive from the circuit,
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at the cost of requiring $\mathsf{rivk}$ to be part of the full viewing key.
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@ -5,6 +5,7 @@ The nullifier design we use for Orchard is
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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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- $F$ is a keyed circuit-efficient PRF (such as Rescue or Poseidon).
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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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@ -29,22 +30,26 @@ We care about several security properties for our nullifiers:
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- **Balance:** can I forge money?
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- **Note Privacy:** can I gain information about notes only from the public block chain?
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- This describes notes sent in-band.
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- **Note Privacy (OOB):** can I gain information about notes sent out-of-band, only from
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the public block chain?
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- In this case, we assume privacy of the channel over which the note is sent, and that
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the adversary does not have access to any notes sent to the same address which are
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then spent (so that the nullifier is on the block chain somewhere).
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- **Spend Unlinkability:** given the incoming viewing key for an address, and not the full
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viewing key, can I (possibly the sender) detect spends of any notes sent to that address?
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- We're giving $ivk$ to the attacker and allowing it to be the sender in order to make
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this property as strong as possible: they will have *all* the notes sent to that
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- We're giving $\mathsf{ivk}$ to the attacker and allowing it to be the sender in order
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to make this property as strong as possible: they will have *all* the notes sent to that
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address.
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- **Faerie Resistance:** can I perform a Faerie Gold attack (i.e. cause notes to be
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accepted that are unspendable)?
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- We're giving the full viewing key to the attacker and allowing it to be the sender in
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order to make this property as strong as possible: they will have *all* the notes sent
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to that address, and be able to derive *every* nullifier.
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@ -60,15 +65,17 @@ For our chosen design, our desired security properties rely on the following ass
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$$
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\begin{array}{|l|l|}
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\hline
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\text{Balance} & DL_E \\
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\text{Note Privacy} & HashDH^{KDF}_E \\
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\text{Note Privacy} & \mathit{HashDH}^{KDF}_E \\
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\text{Note Privacy (OOB)} & \text{Near perfect} \ddagger \\
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\text{Spend Unlinkability} & DDH_E^\dagger \vee PRF_F \\
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\text{Faerie Resistance} & DL_E \\
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\hline
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\end{array}
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$$
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$HashDH^{KDF}_E$ is computational Diffie-Hellman using $KDF$ for the key derivation, with
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$\mathit{HashDH}^{KDF}_E$ 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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@ -92,31 +99,33 @@ $$
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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{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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[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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[\mathsf{nk}] [\theta] H & (addr, v, H, \theta, \mathsf{rcm}) & DL_E & \mathit{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 & \mathit{HashDH}^{KDF}_E & \text{Perfect} & DDH_E & RO_{GH} \wedge DL_E & \text{No SU for DL-breaking} \\\hline
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\mathit{Hash}([\mathsf{nk}] [\theta] H) & (addr, v, H, \theta, \mathsf{rcm}) & DL_E & \mathit{HashDH}^{KDF}_E & \text{Perfect} & DDH_E \vee Pre_{\mathit{Hash}} & Coll_{\mathit{Hash}} \wedge RO_{GH} \wedge DL_E & Coll_{\mathit{Hash}} \text{ for FR} \\\hline
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\mathit{Hash}([\mathsf{nk}] H + [\mathsf{rnf}] \mathcal{I}) & (addr, v, H, \mathsf{rnf}, \mathsf{rcm}) & DL_E & \mathit{HashDH}^{KDF}_E & \text{Perfect} & DDH_E \vee Pre_{\mathit{Hash}} & Coll_{\mathit{Hash}} \wedge RO_{GH} \wedge DL_E & Coll_{\mathit{Hash}} \text{ for FR} \\\hline
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[F_{\mathsf{nk}}(\psi)] [\theta] H & (addr, v, H, \theta, \psi, \mathsf{rcm}) & DL_E & \mathit{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 & \mathit{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 & \mathit{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 & \mathit{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 & \mathit{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 & \mathit{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 & \mathit{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 & \mathit{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 & \mathit{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} + \mathit{Commit}^{\mathsf{nf}}_{\mathsf{rnf}}(v, \rho) & (addr, v, \rho, \mathsf{rnf}, \psi, \mathsf{rcm}) & DL_E & \mathit{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} + \mathit{Commit}^{\mathsf{nf}}_{\mathsf{rnf}}(v, \rho) & (addr, v, \rho, \mathsf{rnf}, \mathsf{rcm}) & DL_E & \mathit{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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- $Hash$ is a keyed circuit-efficient hash (such as Rescue).
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- $\mathit{Hash}$ 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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@ -153,9 +162,11 @@ requirements:
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- Publish a unique value $\rho$ at note creation time, and blind that value within the
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nullifier computation.
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- This is similar to the approach taken in Sprout and Sapling, which both implemented
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nullifiers as PRF outputs; Sprout uses the compression function from SHA-256, while
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Sapling uses BLAKE2s.
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- Derive a unique base $H$ from some unique value, publish that unique base at note
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creation time, and then blind the base (either additively or multiplicatively) during
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nullifier computation.
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@ -168,6 +179,7 @@ directly to the note (to avoid a DL-breaking adversary from immediately breaking
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We were considering using a design involving $H$ with the goal of eliminating all usages
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of a PRF inside the circuit, for two reasons:
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- Instantiating $PRF_F$ with a traditional hash function is expensive in the circuit.
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- We didn't want to solely rely on an algebraic hash function satisfying $PRF_F$ to
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achieve **Spend Unlinkability**.
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@ -196,6 +208,7 @@ is that it does not require an additional scalar multiplication, making it more
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inside the circuit.
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$\psi$'s derivation has two motivations:
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- Deriving from a random value $\mathsf{rseed}$ enables multiple derived values to be
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conveyed to the recipient within an action (such as the ephemeral secret $\mathsf{esk}$,
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per [ZIP 212](https://zips.z.cash/zip-0212)), while keeping the note plaintext short.
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@ -213,15 +226,16 @@ and reject it (as in Sapling).
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### Use of $\mathsf{cm}$
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The nullifier commits to the note value via $\mathsf{cm}$ for two reasons:
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- It domain-separates nullifiers for zero-valued notes from other notes. This is necessary
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because we do not require zero-valued notes to exist in the commitment tree.
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- Designs that bind the nullifier to $F_{\mathsf{nk}}(\rho)$ require $Coll_F$ to achieve
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**Faerie Resistance** (and similarly where $Hash$ is applied to a value derived from
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**Faerie Resistance** (and similarly where $\mathit{Hash}$ is applied to a value derived from
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$H$). Adding $\mathsf{cm}$ to the nullifier avoids this assumption: all of the bases
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used to derive $\mathsf{cm}$ are fixed and independent of $\mathcal{G}$, and so the
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nullifier can be viewed as a Pedersen hash where the input includes $\rho$ directly.
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The $Commit^{\mathsf{nf}}$ variants were considered to avoid directly depending on
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The $\mathit{Commit}^{\mathsf{nf}}$ variants were considered to avoid directly depending on
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$\mathsf{cm}$ (which in its native type is a base field element, not a group element). We
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decided instead to follow Sapling by defining an intermediate representation of
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$\mathsf{cm}$ as a group element, that is only used in nullifier computation. The circuit
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