From c2742c0d05391079a2814aad5d4bf63f0f77399b Mon Sep 17 00:00:00 2001 From: Jack Grigg Date: Tue, 22 Dec 2020 21:23:32 +0000 Subject: [PATCH] book: Edit the lookup argument page to fit the design section --- book/src/design/lookup-argument.md | 82 +++++++++++++++++++++--------- 1 file changed, 59 insertions(+), 23 deletions(-) diff --git a/book/src/design/lookup-argument.md b/book/src/design/lookup-argument.md index 5e72525..8107671 100644 --- a/book/src/design/lookup-argument.md +++ b/book/src/design/lookup-argument.md @@ -1,6 +1,7 @@ -# Generic Lookups with PLONK (DRAFT) +# Lookup argument -_By Sean Bowe and Daira Hopwood_ +halo2 uses the following lookup technique, which allows for lookups in arbitrary sets, and +is arguably simpler than Plookup. ## Note on Language @@ -11,18 +12,28 @@ In addition to the [general notes on language](../design.md#note-on-language): ## Technique Description -The following is a description of a lookup technique which allows for lookups in arbitrary sets, and is arguably simpler than Plookup. We will express lookups in terms of a "subset argument". +We express lookups in terms of a "subset argument" over a table with $2^k$ rows (numbered +from 0), and columns $A$ and $S$. -* There are $2^k$ rows (numbered from 0). -* There are columns $A$ and $S$. +The goal of the subset argument is to enforce that every cell in $A$ is equal to _some_ +cell in $S$. This means that more than one cell in $A$ can be equal to the _same_ cell in +$S$, and some cells in $S$ don't need to be equal to any of the cells in $A$. -The goal of the subset argument is to enforce that every cell in $A$ is equal to _some_ cell in $S$. This means that more than one cell in $A$ can be equal to the _same_ cell in $S$, and some cells in $S$ don't need to be equal to any of the cells in $A$. +- $S$ might be fixed, but it doesn't need to be. That is, we can support looking up values + in either fixed or variable tables (where the latter includes advice columns). +- $A$ and $S$ can contain duplicates. If the sets represented by $A$ and/or $S$ are not + naturally of size $2^k$, we extend $S$ with duplicates and $A$ with dummy values known + to be in $S$. + - Alternatively we could add a "lookup selector" that controls which elements of the $A$ + column participate in lookups. This would modify the occurrence of $A(X)$ in the + permutation rule below to replace $A$ with, say, $S_0$ if a lookup is not selected. -$S$ might be fixed, but it doesn't need to be. That is, we can support looking up values in either fixed or variable tables. $A$ and $S$ can contain duplicates. If the sets represented by $A$ and/or $S$ are not naturally of size $2^k$, we extend $S$ with duplicates and $A$ with dummy values known to be in $S$. Alternatively we can add a "lookup selector" that controls which elements of the $A$ column participate in lookups (this would modify the occurrence of $A(X)$ in the permutation rule below to replace $A$ with, say, $S_0$ if a lookup is not selected). +Let $\ell_i$ be the Lagrange basis polynomial that evaluates to $1$ at row $i$, and $0$ +otherwise. -Let $\ell_i$ be the Lagrange basis polynomial that evaluates to $1$ at row $i$ and $0$ otherwise. - -We'll start by allowing the prover to supply permutation columns of $A$ and $S$. Let's call these $A'$ and $S'$, respectively. We can enforce that they are permutations using a permutation argument with product column $Z$ with the rules: +We start by allowing the prover to supply permutation columns of $A$ and $S$. Let's call +these $A'$ and $S'$, respectively. We can enforce that they are permutations using a +permutation argument with product column $Z$ with the rules: $$ Z(X) (A(X) + \beta) (S(X) + \gamma) - Z(\omega^{-1} X) (A'(X) + \beta) (S'(X) + \gamma) = 0 @@ -30,12 +41,21 @@ $$$$ \ell_0(X) (Z(X) - 1) = 0 $$ -This is a version of the permutation argument which allows $A'$ and $S'$ to be permutations of $A$ and $S$, respectively, but doesn't specify the exact permutations. $\beta$ and $\gamma$ are separate challenges so that we can combine these two permutation arguments into one without worrying that they might interfere with each other. +This is a version of the permutation argument which allows $A'$ and $S'$ to be +permutations of $A$ and $S$, respectively, but doesn't specify the exact permutations. +$\beta$ and $\gamma$ are separate challenges so that we can combine these two permutation +arguments into one without worrying that they might interfere with each other. -The goal of these permutations is to allow $A'$ and $S'$ to be arranged by the prover in a particular way: +The goal of these permutations is to allow $A'$ and $S'$ to be arranged by the prover in a +particular way: -1. All the cells of column $A'$ are arranged so that like-valued cells are vertically adjacent to each other. This could be done by some kind of sorting algorithm but all that matters is that like-valued cells are on consecutive rows in column $A'$, and that $A'$ is a permutation of $A$. -2. The first row in a sequence of like values in $A'$ is the row that has the corresponding value in $S'$. Apart from this constraint, $S'$ is any arbitrary permutation of $S$. +1. All the cells of column $A'$ are arranged so that like-valued cells are vertically + adjacent to each other. This could be done by some kind of sorting algorithm, but all + that matters is that like-valued cells are on consecutive rows in column $A'$, and that + $A'$ is a permutation of $A$. +2. The first row in a sequence of like values in $A'$ is the row that has the + corresponding value in $S'.$ Apart from this constraint, $S'$ is any arbitrary + permutation of $S$. Now, we'll enforce that either $A'_i = S'_i$ or that $A'_i = A'_{i-1}$, using the rule @@ -49,7 +69,8 @@ $$ \ell_0(X) \cdot (A'(X) - S'(X)) = 0 $$ -Together these constraints effectively force every element in $A'$ (and thus $A$) to equal at least one element in $S'$ (and thus $S$). Proof: by induction on prefixes of the rows. +Together these constraints effectively force every element in $A'$ (and thus $A$) to equal +at least one element in $S'$ (and thus $S$). Proof: by induction on prefixes of the rows. ## Cost @@ -60,13 +81,28 @@ Together these constraints effectively force every element in $A'$ (and thus $A$ ## Generalizations -These generalizations are similar to those in sections 4 and 5 of the [Plookup paper](https://eprint.iacr.org/2020/315.pdf): +halo2's lookup argument implementation generalizes the above technique in the following +ways: -* $A$ and $S$ can be extended to multiple columns, combined using a random challenge. $A'$ and $S'$ stay as single columns. - * The commitments to the columns of $S$ can be precomputed, then combined cheaply once the challenge is known by taking advantage of the homomorphic property of Pedersen commitments. -* Then, a lookup argument for an arbitrary-width relation can be implemented in terms of a subset argument, i.e. to constrain $\mathcal{R}(x, y, ...)$ in each row, consider $\mathcal{R}$ as a set of tuples $S$ (using the method of the previous point) and check that $(x, y, ...) \in \mathcal{R}$. - * In the case where $\mathcal{R}$ represents a function, this implicitly also checks that the inputs are in the domain. This is typically what we want, and often saves an additional range check. -* We can support multiple tables in the same circuit, by combining them into a single table that includes a tag column to identify the original table. The tag column could be merged with the "lookup selector" mentioned earlier. -* The optimized range check technique in section 5 of the Plookup paper can also be used with this subset argument. +- $A$ and $S$ can be extended to multiple columns, combined using a random challenge. $A'$ + and $S'$ stay as single columns. + - The commitments to the columns of $S$ can be precomputed, then combined cheaply once + the challenge is known by taking advantage of the homomorphic property of Pedersen + commitments. +- Then, a lookup argument for an arbitrary-width relation can be implemented in terms of a + subset argument, i.e. to constrain $\mathcal{R}(x, y, ...)$ in each row, consider + $\mathcal{R}$ as a set of tuples $S$ (using the method of the previous point), and check + that $(x, y, ...) \in \mathcal{R}$. + - In the case where $\mathcal{R}$ represents a function, this implicitly also checks + that the inputs are in the domain. This is typically what we want, and often saves an + additional range check. +- We can support multiple tables in the same circuit, by combining them into a single + table that includes a tag column to identify the original table. + - The tag column could be merged with the "lookup selector" mentioned earlier, if this + were implemented. -That is, the differences from Plookup are in the subset argument. This argument can then be used in all the same ways. +These generalizations are similar to those in sections 4 and 5 of the +[Plookup paper](https://eprint.iacr.org/2020/315.pdf) That is, the differences from +Plookup are in the subset argument. This argument can then be used in all the same ways; +for instance, the optimized range check technique in section 5 of the Plookup paper can +also be used with this subset argument.