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108 lines
4.9 KiB
Markdown
108 lines
4.9 KiB
Markdown
# Lookup argument
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halo2 uses the following lookup technique, which allows for lookups in arbitrary sets, and
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is arguably simpler than Plookup.
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## Note on Language
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In addition to the [general notes on language](../design.md#note-on-language):
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- We call the $Z(X)$ polynomial (the grand product argument polynomial for the permutation
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argument) the "permutation product" column.
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## Technique Description
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We express lookups in terms of a "subset argument" over a table with $2^k$ rows (numbered
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from 0), and columns $A$ and $S$.
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The goal of the subset argument is to enforce that every cell in $A$ is equal to _some_
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cell in $S$. This means that more than one cell in $A$ can be equal to the _same_ cell in
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$S$, and some cells in $S$ don't need to be equal to any of the cells in $A$.
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- $S$ might be fixed, but it doesn't need to be. That is, we can support looking up values
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in either fixed or variable tables (where the latter includes advice columns).
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- $A$ and $S$ can contain duplicates. If the sets represented by $A$ and/or $S$ are not
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naturally of size $2^k$, we extend $S$ with duplicates and $A$ with dummy values known
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to be in $S$.
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- Alternatively we could add a "lookup selector" that controls which elements of the $A$
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column participate in lookups. This would modify the occurrence of $A(X)$ in the
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permutation rule below to replace $A$ with, say, $S_0$ if a lookup is not selected.
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Let $\ell_i$ be the Lagrange basis polynomial that evaluates to $1$ at row $i$, and $0$
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otherwise.
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We start by allowing the prover to supply permutation columns of $A$ and $S$. Let's call
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these $A'$ and $S'$, respectively. We can enforce that they are permutations using a
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permutation argument with product column $Z$ with the rules:
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$$
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Z(X) (A(X) + \beta) (S(X) + \gamma) - Z(\omega^{-1} X) (A'(X) + \beta) (S'(X) + \gamma) = 0
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$$$$
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\ell_0(X) (Z(X) - 1) = 0
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$$
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This is a version of the permutation argument which allows $A'$ and $S'$ to be
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permutations of $A$ and $S$, respectively, but doesn't specify the exact permutations.
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$\beta$ and $\gamma$ are separate challenges so that we can combine these two permutation
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arguments into one without worrying that they might interfere with each other.
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The goal of these permutations is to allow $A'$ and $S'$ to be arranged by the prover in a
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particular way:
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1. All the cells of column $A'$ are arranged so that like-valued cells are vertically
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adjacent to each other. This could be done by some kind of sorting algorithm, but all
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that matters is that like-valued cells are on consecutive rows in column $A'$, and that
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$A'$ is a permutation of $A$.
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2. The first row in a sequence of like values in $A'$ is the row that has the
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corresponding value in $S'.$ Apart from this constraint, $S'$ is any arbitrary
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permutation of $S$.
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Now, we'll enforce that either $A'_i = S'_i$ or that $A'_i = A'_{i-1}$, using the rule
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$$
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(A'(X) - S'(X)) \cdot (A'(X) - A'(\omega^{-1} X)) = 0
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$$
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In addition, we enforce $A'_0 = S'_0$ using the rule
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$$
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\ell_0(X) \cdot (A'(X) - S'(X)) = 0
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$$
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Together these constraints effectively force every element in $A'$ (and thus $A$) to equal
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at least one element in $S'$ (and thus $S$). Proof: by induction on prefixes of the rows.
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## Cost
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* There is the original column $A$ and the fixed column $S$.
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* There is a permutation product column $Z$.
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* There are the two permutations $A'$ and $S'$.
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* The gates are all of low degree.
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## Generalizations
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halo2's lookup argument implementation generalizes the above technique in the following
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ways:
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- $A$ and $S$ can be extended to multiple columns, combined using a random challenge. $A'$
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and $S'$ stay as single columns.
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- The commitments to the columns of $S$ can be precomputed, then combined cheaply once
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the challenge is known by taking advantage of the homomorphic property of Pedersen
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commitments.
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- Then, a lookup argument for an arbitrary-width relation can be implemented in terms of a
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subset argument, i.e. to constrain $\mathcal{R}(x, y, ...)$ in each row, consider
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$\mathcal{R}$ as a set of tuples $S$ (using the method of the previous point), and check
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that $(x, y, ...) \in \mathcal{R}$.
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- In the case where $\mathcal{R}$ represents a function, this implicitly also checks
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that the inputs are in the domain. This is typically what we want, and often saves an
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additional range check.
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- We can support multiple tables in the same circuit, by combining them into a single
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table that includes a tag column to identify the original table.
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- The tag column could be merged with the "lookup selector" mentioned earlier, if this
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were implemented.
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These generalizations are similar to those in sections 4 and 5 of the
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[Plookup paper](https://eprint.iacr.org/2020/315.pdf) That is, the differences from
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Plookup are in the subset argument. This argument can then be used in all the same ways;
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for instance, the optimized range check technique in section 5 of the Plookup paper can
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also be used with this subset argument.
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