Fflonk

Presentation: https://www.youtube.com/watch?v=E5oLEqbx6F4

Introduction

When opening at a single point for $n$ polynomials, the common technique is to use a random linear combination, which requires $n - 1$ scalar multiplications. Eliminating these operations could lead to more efficient verification. Meanwhile, in SHPlonk, when opening KZG in batch, the verifier randomly samples a point $z \leftarrow \mathbb{F}$ and constructs a vanishing polynomial $L(X)$ for this point. This method allows an opening at $t$ points to be converted into an opening at a single point. However, the verifier still needs to perform $O(n)$ computations, and reducing this cost could improve verification efficiency.

A natural question arises: can "the problem of opening at a single point for multiple polynomials" be transformed into "the problem of opening at multiple points for a single polynomial"? If possible, then leveraging SHPlonk’s method could enable more efficient batch opening for the verifier.

Background

Please read SHPlonk beforehand!

Protocol Explanation

When performing Radix-2 FFT, $P(X)$ is decomposed into even and odd parts:

$$P(X) = P_E(X^2) + X \cdot P_O(X^2)$$

Now, consider a simple problem of opening at a single point $x$ for multiple polynomials $P_0(X), P_1(X)$. We define $P(X)$ as follows:

$$P(X) = P_0(X^2) + X \cdot P_1(X^2)$$

Let $x = z^2$, and open $P(X)$ at $\bm{S} = \{z, -z\}$:

$$P(z) = P_0(x) + z \cdot P_1(x) \\ P(-z) = P_0(x) - z \cdot P_1(x)$$

By performing the following computations, we can retrieve $P_0(x)$ and $P_1(x)$:

$$P_0(x) = \frac{P(z) + P(-z)}{2} \\ P_1(x) = \frac{P(z) - P(-z)}{2z}$$

Thus, the original problem is transformed into opening at multiple points $\bm{S} = \{z, -z\}$ for a single polynomial $P(X)$, allowing us to apply SHPlonk’s batch opening method.

To generalize this approach to $n$ polynomials, we need a way to combine $P_0(X), \dots, P_{n-1}(X)$ into a single polynomial and derive $\bm{S}$ from $x$.

Polynomial Combination and Decomposition

To merge multiple polynomials into one, we define the following functions:

• $\mathsf{combine}_n(\bm{P}): \mathbb{F}[X]^n \to \mathbb{F}[X]$: This constructs $P(X)$ from $P_0(X), \dots, P_{n-1}(X)$:

$$P(X) := \sum_{i < n} X^i \cdot P_i(X^n)$$

• $\mathsf{decompose}_n(P): \mathbb{F}[X] \rightarrow \mathbb{F}[X]^n$: This extracts $P_0(X), \dots, P_{n-1}(X)$ from $P(X)$.

By definition, we have:

$$\mathsf{decompose}_n(\mathsf{combine}_n(\bm{P})) = \bm{P}$$

By the way, $\mathsf{decompose}_n(P)$ is just introduced, but not actually used in this protocol!

Finding $\bm{S}$ from $x$

Next, we define a function to derive the set of opening points $\bm{S}$ from $x$:

• $\mathsf{roots}_n(x): \mathbb{F} \to \mathbb{F}^n$: This determines the point set $\bm{S}$:

$$\bm{S} := (z\omega^i)_{i < n}$$

where $z \in \mathbb{F}$ satisfies:

• $z^n = x$

• $z^i \neq x$ for $i < n$

$n$ must be a divisor of $p - 1$, and $\omega$ is the $n$-th root of unity.

For example, if $n = 4$, meaning $\bm{P} = \{P_0(X), P_1(X), P_2(X), P_3(X)\}$, and we want to open at $x$, we construct:

$$P(X) = P_0(X^4) + X \cdot P_1(X^4) + X^2 \cdot P_2(X^4) + X^3 \cdot P_3(X^4)$$

The corresponding root set is:

$$\bm{S} = \{z, z\omega, -z, -z\omega \}$$

This is because $\omega^2 = -1$ since $\omega^4 = 1$.

Opening at $\bm{S}$ gives:

$$P(z) = P_0(x) + z \cdot P_1(x) + z^2 \cdot P_2(x) + z^3 \cdot P_3(x) \\P(z\omega) = P_0(x) + z\omega \cdot P_1(x) - z^2 \cdot P_2(x) - z^3\omega \cdot P_3(x) \\P(-z) = P_0(x) - z \cdot P_1(x) + z^2 \cdot P_2(x) - z^3 \cdot P_3(x) \\P(-z\omega) = P_0(x) - z\omega \cdot P_1(x) - z^2 \cdot P_2(x) + z^3\omega \cdot P_3(x) \\$$

Remember that $x = z^4$ by definition!

By computing:

$$P_0(x) = \frac{P(z) + P(z\omega) + P(-z) + P(-z\omega)}{4} \\ P_1(x) = \frac{P(z) + P(z\omega) - P(-z) - P(-z\omega)}{4z} \\ P_2(x) = \frac{P(z) - P(z\omega) + P(-z) - P(-z\omega)}{4z^2} \\ P_3(x) = \frac{P(z) - P(z\omega) - P(-z) + P(-z\omega)}{4z^3} \\$$

we can recover $P_0(x), P_1(x), P_2(x), P_3(x)$.

Opening Multiple Polynomials at Multiple Points

Consider the same example from SHPlonk:

• $P_0(X)$ is opened at $x_0$ with value $y_0$.

• $P_1(X)$ is opened at $x_0, x_1$ with values $y_1, y_2$.

• $P_2(X)$ is opened at $x_0, x_1$ with values $y_3, y_4$.

Let $\bm{P} = \{P_0, P_1, P_2\}$, then we compute $P(X) = \mathsf{combine}_4(\bm{P})$, determine $\bm{S}_0 = \mathsf{roots}_4(x_0)$ and $\bm{S}_1 = \mathsf{roots}_4(x_1)$, and perform a batch opening with $S_0 \cup S_1$.

Let's set the maximum allowable number of polynomials as $A$. Then, if there are $n < A$ polynomials of degree $d$, and each polynomial is opened at $t$ points, the computational costs are as follows:

• The number of opening sets $s$: $1 \le s \le n$

• $\mathsf{srs}$ size: $A \cdot (d + 1) \mathbb{G}_1, 2\mathbb{G}_2$

• Prover work: $O(d)\mathbb{G}_1, O(s \cdot d \log d) \mathbb{F}$

• Proof length: $2\mathbb{G}_1$

• Verifier work: $O(1)\mathbb{G}_1, O(n)\mathbb{F}, 2\mathsf{P}$

Here, $\mathbb{G}_i$ represents scalar multiplication in $\mathbb{G}_i$​, $\mathbb{F}$ represents addition or multiplication in $\mathbb{F}$, and $\mathsf{P}$ denotes a pairing operation.

Conclusion

Fflonk further optimizes batch opening protocols using a trick inspired by FFT, (that's why the name starts with ff, which denotes "fast fourier") reducing verifier work while maintaining prover efficiency and proof size. This makes it particularly well-suited for large-scale applications where minimizing verifier overhead is critical. In https://ethresear.ch/t/on-the-gas-efficiency-of-the-whir-polynomial-commitment-scheme/21301, it says verifying groth16 and fflonk proofs costs are almost identical.

TODO(chokobole): compare the prover performance between groth16 and fflonk.

References

• https://eprint.iacr.org/2021/1167
• https://ethresear.ch/t/on-the-gas-efficiency-of-the-whir-polynomial-commitment-scheme/21301