book: Bring in the lookup argument description from HackMD

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)

book/src/design/lookup-argument.md

@@ -0,0 +1,75 @@
+# Generic Lookups with PLONK (DRAFT)
+
+_By Sean Bowe and Daira Hopwood_
+
+## Note on Language
+
+We use slightly different language than others, here's the overview.
+
+1. We like to think of PLONK-like arguments as tables, where each column corresponds to a "wire".
+2. We like to call "selector polynomials" and so on "fixed columns" instead.
+3. We call the others "advice columns" usually, when they're populated by the prover.
+4. We call the $Z(X)$ polynomial (the grand product argument polynomial for the permutation argument) the "permutation product" column.
+5. We use the term "rule" to refer to a "gate" like $A(X) \cdot q_A(X) + B(X) \cdot q_B(X) + A(X) \cdot B(X) \cdot q_M(X) + C(X) \cdot q_C(X) = 0.$
+
+## 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".
+
+* 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 $A$.
+
+$S$ 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.
+
+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:
+
+$$
+Z(X) (A(X) + \beta) (S(X) + \gamma) - Z(\omega^{-1} X) (A'(X) + \beta) (S'(X) + \gamma) = 0
+$$$$
+\ell_0(X) (Z(X) - 1) = 0
+$$
+
+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
+
+These generalizations are similar to those in sections 4  and 5 of the [Plookup paper](https://eprint.iacr.org/2020/315.pdf):
+
+* $A$ 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.
+
+That is, the differences from Plookup are in the subset argument. This argument can then be used in all the same ways.