Proving AIRs with multivariate sumchecks

Algebraic Intermediate Representation (AIR) is a powerful arithmetization that can be extended to Randomized Air with Preprocessing (RAP), which includes popular systems like PLONK. See this great note by Ariel Gabizon for descriptions of these arithmetizations.

Another popular arithmetization is given by R1CS. Two categories of proving systems based on R1CS are those using multivariate techniques (e.g. Spartan) and those using univariate techniques (e.g. Marlin).

To the best of my knowledge, every current AIR-based proving system uses univariate techniques. In this note, I’ll discuss what a multivariate AIR-based proving system could look like.

Differences between univariate and multivariate techniques

Before discussing AIR, let’s focus on the restricted arithmetization Trivial Air Representation (TAR) where the next row values are not used in the constraint.

Let $C \in \mathbb{F}[X_1,\dots,X_w]$ be a constraint polynomial, and assume we have $w$ columns of length $n$ (a power of 2) $f_1,\dots,f_w \in \mathbb{F}^n$. Every row satisfies $C\left(f_1[j],\dots,f_w[j]\right)=0$ and we want a succinct proof of this fact. Here are examples of how one could do this using univariate or multivariate techniques.

Univariate techniques

Let $H < \mathbb{F}^*$ be a subgroup of order $n$. Look at the columns as functions $f_i: H \to \mathbb{F}$. Using the FFT, interpolate these functions to get polynomials $\hat{f}_i: \mathbb{F}[X]^{<n}$ and compute the polynomial $C(\hat{f}_1,\dots,\hat{f}_w)\in \mathbb{F}[X]^{<deg(C)n}$. This latter polynomial evaluates to $0$ on $H$, so we can compute its quotient $Q(X)$ with the vanishing polynomial on $H$: $Z_H(X) = X^n - 1$.

Compute commitments of the polynomials $\hat{f}_1,\dots,\hat{f}_w,Q$ using your favorite univariate polynomial commitment scheme. Send these commitments to the verifier and receive a challenge $z \in \mathbb{F}$ in return. Send the evaluations $\hat{f}_1(z),\dots,\hat{f}_w(z),Q(z)$, along with their opening proofs, to the verifier. Finally, the verifier verifies the opening proofs and checks the equality $$ C\left(\hat{f}_1(z),\dots,\hat{f}_w(z)\right)\stackrel{?}{=} Q(z)Z_H(z). $$

Multivariate techniques

Let $v = \log n$, look at the columns as functions $f_i: \{0,1\}^v \to \mathbb{F}$ and compute the multilinear extensions $\tilde{f}_1,\dots,\tilde{f}_w \in \mathbb{F}[X_1,…,X_v]$. Let $F(x) = C\left(\tilde{f}_1(x),\dots,\tilde{f}_w(x)\right)$, by assumption the equality $F(x)=0$ holds for all $x \in \{0,1\}^v$. Define the polynomial

$$ P(t) = \sum_{x \in \{0,1\}^v} F(x) \beta(t, x), $$ where $\beta(t,x) = \prod _{i=1}^v (t_i x_i + (1-t_i)(1-x_i))$ is the multilinear polynomial such that $\beta(t,x)\ne 0$ iff $t=x$. $P(t)$ is a multilinear polynomial that evaluates to $0$ on $\{0,1\}^v$ and is therefore the zero polynomial. This is something we can prove to the verifier using the multivariate sumcheck protocol.

Commit to the polynomials $\tilde{f}_1,\dots, \tilde{f}_w$ using your favorite multivariate polynomial commitment scheme and send the commitments to the verifier. Then perform the multivariate sumcheck protocol to prove that $P(r)=0$ for a random $r\in \mathbb{F}^v$ chosen by the verifier (which implies that $P$ is the zero polynomial with high probability). In the last step of the sumcheck protocol, the verifier has to evaluate $F(z)\beta(r,z)$ for some random $z \in \mathbb{F}^v$. $\beta(r,z)$ can be evaluated in $O(v)$ and to evaluate $F(z)$, the prover opens $\tilde{f}_1(z),\dots, \tilde{f}_w(z)$ with which the verifier can compute $F(z) = C(\tilde{f}_1(z),\dots, \tilde{f}_w(z))$.

To summarize, the main differences between the univariate and multivariate techniques are

  1. The columns are seen as functions from $H \subset \mathbb{F}^*$ in the univariate case and as functions from $\{0,1\}^v \subset \mathbb{F}^v$ in the multivariate case.
  2. The columns are interpolated to polynomials of degree $<n$ in the univariate case and to multilinear polynomials in the multivariate case.
  3. To check that the constraint $C$ evaluates to $0$ on every row, a quotient polynomial argument is used in the univariate case, whereas the multivariate sumcheck protocol, along with the $\beta(t,x)$ trick, is used in the multivariate case.

Introducing next row values

To go from TAR to AIR, we just need to change the constraint $C$ to a $2w$-variate polynomial taking as inputs the $j$-th and $j+1$-th rows $$ C\left(f_1[j],\dots,f_w[j], f_1[j+1],\dots,f_w[j+1]\right)=0, \forall \ 0\leq j < n, $$ where $j+1$ is taken modulo $n$.

In the univariate setting, the proof system for TARs can be extended to AIRs as follows. Let $g\in H$ be a generator of $H$ and let the correspondence between rows and $H$ be given by $j$-th row $\leftrightarrow$ $g^j$, i.e., $f_i[j] = \hat{f}_i(g^j)$. Then, for every $h\in H$, the next row value of $\hat{f}_i(h)$ is $\hat{f}_i(gh)$.

With that, we can modify the univariate proof system for TAR to accommodate for the next row values. The polynomial $Q(X)$ is now computed as the quotient of $C\left(\hat{f}_1(X),\dots,\hat{f}_w(X),\hat{f}_1(gX),\dots,\hat{f}_w(gX)\right)$ and $Z_H(X)$, and the column polynomials are opened at $z$ and $gz$, so that the verifier can check $$ C\left(\hat{f}_1(z),\dots,\hat{f}_w(z),\hat{f}_1(gz),\dots,\hat{f}_w(gz)\right)\stackrel{?}{=} Q(z)Z_H(z). $$

Generalizing AIRs using endomorphisms

Let $R$ be the set of rows in a TAR and let $\sigma: R \to R$ be a function. We can extend TARs by using the constraint $$ C\left(f_1(r),\dots,f_w(r), f_1(\sigma r),\dots,f_w(\sigma r)\right)=0, \forall r \in R. $$ We’ll call this arithmetization an Endomorphism AIR Representation (EAR).

The endomorphism $\sigma$ on $R$ induces another endomorphism, that we also call $\sigma$, on functions $f: R \to \mathbb{F}$ by setting $(\sigma f)r = f(\sigma r)$ for all $r\in R$. This endomorphism can further be extended to polynomials using interpolation on $R$, i.e., $\sigma \hat{f} = \widehat{\sigma f}$.

AIRs can be recovered in this context by setting $R = H = \langle g \rangle$ and $\sigma = \mu_g$, the multiplication by $g$. Going through the previous section with this viewpoint, we observe that the only property of $\sigma$ used in the proving system for AIR is that there is an extension $\bar{\sigma} : \mathbb{F} \to \mathbb{F}$ of $\sigma$ that is efficient to compute and such that $(\sigma \hat{f})z = \hat{f}(\bar{\sigma} z)$ for all $z\in \mathbb{F}$. In the univariate context, $\bar{\sigma}$ is also $\mu_g$, the multiplication by $g$ with domain $\mathbb{F}$.

Note that, similarly to how AIRs can also be extended to also use the $j+2$-th row values in $C$ (or in general any shifted row values), EARs can be extended to use more than one endomorphism in the definition of $C$.

Furthermore, note that this construction can also be generalized to extend RAPs.

How to choose $\sigma$?

The choice of $\sigma$ can dramatically impact the expressiveness of the arithmetization. For example, setting $\sigma = \mathrm{id}_R$ gives a TAR, a rather useless arithmetization, and as seen previously setting $R = H, \sigma = \mu_g$ gives a regular AIR, a very powerful arithmetization.

Assuming for the moment that $\sigma$ is a permutation, a measure of expressiveness is the size of the largest cycle of $\sigma$, when seeing the permutation as a product of cycles. Interpreting an AIR as the repeated application of a function, the size of the largest cycle gives the maximum number of times the function can be applied.

Back to the multivariate setting

In the multivariate setting, the row domain is $R = \{0, 1\}^v$. So to get an EAR, we first need to set an endomorphism $\sigma : \{0, 1\}^v \to \{0, 1\}^v$. As noted previously, to make the proving system work, we need a $\sigma$ that can be extended to some $\bar{\sigma}: \mathbb{F}^v \to \mathbb{F}^v$ such that $(\sigma \tilde{f})z = \tilde{f}(\bar{\sigma} z)$ for all $z \in \mathbb{F}^v$ and functions $f: \{0, 1\}^v \to \mathbb{F}$.

A set of endomorphisms of $\mathbb{F}^v$ (which are actually permutations) satisfying this condition is given by compositions of bit-flips and coordinate permutations. Formally, they are given by $$ L_v = \{ \tau \theta \ \mid \ \tau \in S_v, \ \theta \in (\mathbb{Z}/2\mathbb{Z})^v \}, $$ where $S_v$ is the symmetric group on $v$ elements acting on $\mathbb{F}^v$ by permuting coordinates and $\theta = (\theta_1,\dots,\theta_v)$ acts on $x = (x_1, \dots, x_v)$ coordinate-wise by sending $x_i$ to $x_i$ if $\theta_i = 0$ and to $1-x_i$ if $\theta_i = 1$. It is clear that every element of $L_v$ restricts to a permutation on $\{0, 1\}^v$ and can be efficiently computed in $O(v)$ time.

The main “result” of this note can thus be stated as follows

For every $\sigma \in L_v$ and $C\in \mathbb{F}[X_1,\dots,X_{2w}]$, there is an efficient proving system based on the multivariate sumcheck protocol for the EAR on $n = 2^v$ rows, $w$ columns, endomorphism $\sigma$, and constraint $C$.

Here is a description of the full protocol, using the same notations as before:

  1. Commit to the multilinear polynomials $\tilde{f}_1,\dots,\tilde{f}_w$ and send the commitments to the verifier.
  2. Let $F(x) = C(\tilde{f}_1(x),\dots,\tilde{f}_w(x),\sigma\tilde{f}_1(x),\dots,\sigma\tilde{f}_w(x))$ and let $r\in \mathbb{F}^v$ be a random challenge sent by the verifier. Run the sumcheck argument for $P(r) = 0$, with $P(t) = \sum_{x \in \{0,1\}^v} F(x) \beta(t, x)$.
  3. In the last step of the sumcheck protocol, the verifier has to compute $F(z)$. The prover sends the openings $\tilde{f}_1(z),\dots, \tilde{f}_w(z), \tilde{f}_1(\sigma z),\dots,\tilde{f}_w(\sigma z)$ along with their opening proofs. The verifier then verifies the opening proofs and computes the value of $$ F(z) = C(\tilde{f}_1(z),\dots,\tilde{f}_w(z),\sigma\tilde{f}_1(z),\dots,\sigma\tilde{f}_w(z)) = C(\tilde{f}_1(z),\dots,\tilde{f}_w(z),\tilde{f}_1(\sigma z),\dots,\tilde{f}_w(\sigma z)) $$ to conclude the sumcheck protocol.
  4. The sumcheck protocol shows that (up to reasonable soundness) $P(t)$ is the zero polynomial, which implies that $F(x)=0$ for all $x\in \{0,1\}^v$, which is equivalent to the EAR statement.
How good are these permutations?

The sad thing is that the permutations in $L_v$ are not very good, measured in terms of their maximum cycle length. Denote by $M(v)$ the maximum cycle length over elements of $L_v$ and denote by $N(v)$ the number of length $M(v)$ cycles in a permutation achieving this maximum. Experiments give the following values of $M$ and $N$:

$v$ 10 11 12 13 14 15 16 17 18 19
$M(v)$ 60 60 120 120 168 210 280 420 420 840
$N(v)$ 12 30 24 48 72 108 216 216 540 432
$M(v)\cdot N(v)$ 720 1800 2880 5760 12096 22680 60480 90720 226800 362880
$n=2^v$ 1024 2048 4096 8192 16384 32768 65536 131072 263144 524288

So the bad news is that $M(v)$ is much smaller than $n$. The silver lining is that there are a lot of cycles achvieving $M(v)$ and so the number of elements in such cycles $M(v)\cdot N(v)$ is reasonably close to $n$.

EARs as “parallel” AIRs?

This suggests the following construction. Let $\sigma$ be a permutation with largest cycle of length $\mu = M(v)$ and $\eta = N(v)$ such cycles with supports $\Pi_1,\dots,\Pi_\eta \subset \{0,1\}^v$.

Then, the EAR with endomorphism $\sigma$ and constraint $C$ contains $\eta$ copies of an AIR with $\mu$ rows and constraint $C$. It can thus be used to compute $\eta$ AIR on the row sets $\Pi_1,\dots,\Pi_\eta$ in parallel.

There are a few issues with this construction:

  • The constraint $C$ must also hold outside $\bigcup \Pi_i$. The cycles outside this set do not have length $\mu$ which can be problematic. One solution is to add a preprocessed selector polynomial $\mathbb{1}_{\bigcup \Pi_i}$ and multiply $C$ by this polynomial to filter out smaller cycles.
  • One might want to use different constraints $C_1,\dots,C_\eta$ on the different row sets $\Pi_1,\dots,\Pi_\eta$. This can also be solved by adding preprocessed polynomials. Either $\eta$ selector polynomials taking binary values or one polynomial taking values $1$ to $\eta$.
  • Public inputs should be a bit annoying to work with in this model.
  • If implementing PLONK in this model, only rows belonging to the same cycle can be wired together.

Coming soon.