A recap of lookup arguments

This is only a draft, and at the moment, it will be simply a list of papers to recap the different existing lookup arguments. I try to keep it in the order of publication.

• https://sci-hub.ru/10.1007/bf01185212
• https://eprint.iacr.org/2023/1555.pdf - Permem and gcq
• Nearly Linear-Time Zero-Knowledge Proofs for Correct Program Execution: https://eprint.iacr.org/2018/380
• Plookup: https://eprint.iacr.org/2020/315.pdf
• HyperPlonk: https://eprint.iacr.org/2022/1355
• MVLookup: https://github.com/Orbis-Tertius/MVlookups/blob/main/MVlookup.pdf
  • some interesting proofs about lookup derivative. Easy to read the lemmas, well-explained.
• cq: https://eprint.iacr.org/2022/1763

Introduction

A lookup argument is an argument to prove that the values of a commited polynomial $P(X) \in \mathbb{K}^{<n}[X]$ is contained in a table $t \in \mathbb{K}^N$, where $t$ is a precomputed table of values. It is used to avoid computations that would be expensive to constrain otherwise. In other terms, lookup arguments are used for relations of high algebraic complexities.

In terms of polynomials, a lookup argument aims to prove that the evaluations over a group $H$ of a polynomial $P(X)$ are equals to the evaluations of another polynomial $T$. If $H = <\omega^{i}>$, then the prover must convince that for each $\omega^{j}$, $P(\omega^{j}) = T(\omega^{k})$, for a certain $k$. $P(X)$ is part of the witness, where $T$ is public. You can note that with this notation, if $P$ and $T$ are of the same degree, a lookup argument is exactly proving that the evaluations of $P$ is a permutation of the evaluations of $T$, i.e. it is a permutation argument. That explains the statement that a permutation argument is a lookup argument.

An important note is that $P$ and $T$ can be the same polynomials. In the case of memory access, the same polynomials can be used to show which read/write operations, and with which values, the execution trace performs at each row. In the case of R/W operations, the order of the access must also be performed, it can be done by keeping track of the last time the values has been read/written and verify that the difference with the current instruction counter is in a predefined range.

Let’s take an example. Let’s the difference with the current instruction counters are of size $16 = 2^4$, i.e. we have an execution trace of 16 operations (very small program). The program fetches the memory at the instruction counter $IC$ 10. By checking that $16 - 10$ is in the range $0, …, 16$ and by keeping a column (i.e. a polynomial) that will contain the current instruction counter for the last fetch, we will keep track of the memory access at the instruction $10$, and that it happened during the execution of the program. If we later fetch the same address (e.g. instruction counter = $12$), we prove the property that it is after the next instruction by checking that the difference with it is still in the range $0, …, 16$, i.e. that $12 - 10$ is in the range. If the prover lies, and try to have the value before the instruction 10, the difference will be outside of the range if we suppoes the field is at least twice larger than the table size.

MVLookup

MVLookup are described in https://eprint.iacr.org/2022/1530.pdf. It is based on the sumcheck protocol. Note that the sumcheck protocol is simply another way to verify polynomial identities.

What to remember from the paper:

• there are conditions using the characteristic of $p$. In the case of SNARK, we use a large $p$, therefore we can ignore it.
• cached quotients -> we use some pre-computed quotients to speed up some computations
• batch column lookups -> on the same row, multiple columns query the same table.
• the main change in MVLookup is to use the mathetical objects called rational functions. Rational functions are well-defined mathematical objects, which are fractions of polynomials, i.e. $\frac{P(X)}{Q(X)}$.
• the notion of the logarithm derivative of a polynomial $P(X)$, defined as the rational function $\frac{P’(X)}{P(X)}$. We will use the term logup to talk about the logarithmic derivative of a polynomial $P(X)$, and we will use the notation $logup(P(X))$
• a property used in the lookup argument is the following. If a polynomial $P(X)$ can be decomposed in $\prod_{i = 1}^{n} (X + z_{i})$, then $logup(P(X)) = \sum_{i = 1}^{n} \frac{1}{X + z_{i}}$.
• $\prod_{i = 1}^{n} (X + a_{i}) = \prod_{i = 1}^{n} (X + b_{i})$ iff $\sum_{i = 1}^{n}\frac{1}{(X + a_{i})} = \sum_{i = 1}^{n}\frac{1}{(X + b_{i})}$
• The multiplicity of a root of a polynomial can be captured in its logup. More concretely, if $P(X) = \prod_{i = 1}^{n}(X + a_{i})^{m_{i}}$, where $m_{i}$ is the multiplicity of the root $a_{i}$, we have $logup(P(X)) = \sum_{i = 1}^{n}\frac{m_{i}}{(X + a_{i})}$
• If we represent $m_{i}$ as the evaluation of a polynomial over $\mathbb{K}$ (for $a \in \mathbb{K}$, $m(a) = 0$ if $a$ is a not root of $P(X)$ and $m_{i}$ if $a$ is a root), we can represent the logup using $logup(P(X)) = \sum_{a \in \mathbb{K}}\frac{m(a)}{(X + a)}$
• A corollary of the above gives us a good foundation for lookup arguments (as a reminder, we want to prove subset inclusion). More concretely, we have: if $f_{i}$ is a subset of $t_{j}$, then there exists a sequence $m_{i}$ such that $\sum_{i = 1}^{m}\frac{m_{i}}{(X + a_{i})} = \sum_{j = 1}^{n}\frac{1}{(X + b_{i})}$
• The concept of Lagrange Interactive Oracle Proofs (Lagrange IOP) is introduced. It can be used when the polynomial commitment scheme can be changed to not require the polynomial coefficients, neither when computing the commitment nor in an opening proof.
• Same for the concept of Lagrange commitment scheme. A Lagrange commitment scheme is a commitment scheme for functions over the boolean hypercube H that comes with an evaluation proof for Lagrange queries.

Homomorphic commitments

Useful to prove simultaneously (i.e with a single query) that $a_{i}$ and $b_{i}$ are in $T_{1}$ and $T_{2}$ by proving that $a_{i} + r b_{i}$ is in $T_{1} + r T_{2}$. We save one query. This is called a vector lookup. Example: baloo and Caulk

Vocabulary

• query: request to prove that a value is in the table.
• random folding procedure: when we have two elements $a$, $b$, a random folding procedure is computing $a + r * b$, with $r$ random.

Questions

Let $\mathbb{V} < G$ and $\mathbb{H} < G$, with $|V| = N$ and $|H| = n$, with $n < N$. Our table is the evaluations of a polynomial $T: V \rightarrow K$, and the value we look up are evaluations of a function $f: H \rightarrow G$. Can we have some nice properties if $f$ is a morphism?