diff --git a/book/book.toml b/book/book.toml index c6278076..005ef54b 100644 --- a/book/book.toml +++ b/book/book.toml @@ -1,5 +1,10 @@ [book] -authors = ["Jack Grigg"] +authors = [ + "Jack Grigg", + "Sean Bowe", + "Daira Hopwood", + "Ying Tong Lai", +] language = "en" multilingual = false src = "src" diff --git a/book/src/SUMMARY.md b/book/src/SUMMARY.md index 33639964..1bee6b2b 100644 --- a/book/src/SUMMARY.md +++ b/book/src/SUMMARY.md @@ -10,3 +10,4 @@ - [Tips and tricks](user/tips-and-tricks.md) - [Design](design.md) - [Permutation argument](design/permutation.md) + - [Lookup argument](design/lookup-argument.md) diff --git a/book/src/design.md b/book/src/design.md index 3d14cb7c..db8a6d09 100644 --- a/book/src/design.md +++ b/book/src/design.md @@ -1 +1,17 @@ # Design + +## Note on Language + +We use slightly different language than others to describe PLONK concepts. Here's the +overview: + +1. We like to think of PLONK-like arguments as tables, where each column corresponds to a + "wire". We refer to entries in this table as "cells". +2. We like to call "selector polynomials" and so on "fixed columns" instead. We then refer + specifically to a "selector constraint" when a cell in a fixed column is being used to + control whether a particular constraint is enabled in that row. +3. We call the other polynomials "advice columns" usually, when they're populated by the + prover. +4. We use the term "rule" to refer to a "gate" like + $$A(X) \cdot q_A(X) + B(X) \cdot q_B(X) + A(X) \cdot B(X) \cdot q_M(X) + C(X) \cdot q_C(X) = 0.$$ + - TODO: Check how consistent we are with this, and update the code and docs to match. diff --git a/book/src/design/lookup-argument.md b/book/src/design/lookup-argument.md new file mode 100644 index 00000000..81076711 --- /dev/null +++ b/book/src/design/lookup-argument.md @@ -0,0 +1,108 @@ +# Lookup argument + +halo2 uses the following lookup technique, which allows for lookups in arbitrary sets, and +is arguably simpler than Plookup. + +## Note on Language + +In addition to the [general notes on language](../design.md#note-on-language): + +- We call the $Z(X)$ polynomial (the grand product argument polynomial for the permutation + argument) the "permutation product" column. + +## Technique Description + +We express lookups in terms of a "subset argument" over a table with $2^k$ rows (numbered +from 0), and 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 $A$. + +- $S$ 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. + +Let $\ell_i$ be the Lagrange basis polynomial that evaluates to $1$ at row $i$, and $0$ +otherwise. + +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: + +$$ +Z(X) (A(X) + \beta) (S(X) + \gamma) - Z(\omega^{-1} X) (A'(X) + \beta) (S'(X) + \gamma) = 0 +$$$$ +\ell_0(X) (Z(X) - 1) = 0 +$$ + +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. + +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 $S$. + +Now, we'll enforce that either $A'_i = S'_i$ or that $A'_i = A'_{i-1}$, using the rule + +$$ +(A'(X) - S'(X)) \cdot (A'(X) - A'(\omega^{-1} X)) = 0 +$$ + +In addition, we enforce $A'_0 = S'_0$ using the rule + +$$ +\ell_0(X) \cdot (A'(X) - S'(X)) = 0 +$$ + +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. + +## Cost + +* There is the original column $A$ and the fixed column $S$. +* There is a permutation product column $Z$. +* There are the two permutations $A'$ and $S'$. +* The gates are all of low degree. + +## Generalizations + +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, if this + were implemented. + +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.