From 6f86b8d365fbbf23ebd295b8cdbe600c719edc23 Mon Sep 17 00:00:00 2001 From: eittah Date: Sun, 16 Jun 2024 17:50:00 +0800 Subject: [PATCH 1/5] co-learn-notes --- Plonk/co-learn notes/Eta/01_R1CS_Plonklish.md | 154 ++++++++++++++++++ Plonk/co-learn notes/Eta/02_Lagrange.md | 0 Plonk/co-learn notes/Eta/03_Permutation.md | 0 .../Eta/04_Copy_Constraints_Optimization.md | 0 .../Eta/05_Polynomial_Commitments.md | 0 Plonk/co-learn notes/Eta/images/28_ring.png | Bin 0 -> 417290 bytes .../co-learn notes/Eta/images/29_plonkish.png | Bin 0 -> 108736 bytes 7 files changed, 154 insertions(+) create mode 100644 Plonk/co-learn notes/Eta/01_R1CS_Plonklish.md create mode 100644 Plonk/co-learn notes/Eta/02_Lagrange.md create mode 100644 Plonk/co-learn notes/Eta/03_Permutation.md create mode 100644 Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md create mode 100644 Plonk/co-learn notes/Eta/05_Polynomial_Commitments.md create mode 100644 Plonk/co-learn notes/Eta/images/28_ring.png create mode 100644 Plonk/co-learn notes/Eta/images/29_plonkish.png diff --git a/Plonk/co-learn notes/Eta/01_R1CS_Plonklish.md b/Plonk/co-learn notes/Eta/01_R1CS_Plonklish.md new file mode 100644 index 0000000..d4e3bc7 --- /dev/null +++ b/Plonk/co-learn notes/Eta/01_R1CS_Plonklish.md @@ -0,0 +1,154 @@ +# ETAAcademy-ZKMeme: 01. R1CS V.S. Plonkish + + + + + + + + + + +
titletags
01. R1CS_Plonkish + + + + + + + +
zk-memebasicquick_readR1CS_Plonkish
+
+ +[Github](https://github.com/ETAAcademy)|[Twitter](https://twitter.com/ETAAcademy)|[ETA-ZK-Meme](https://github.com/ETAAcademy/ETAAcademy-ZK-Meme) + +Authors: [Eta](https://twitter.com/pwhattie), looking forward to your joining + +## Plonkish Arithmetization + +Plonkish Arithmetization represents a significant advancement in Zero-knowledge proofs (ZKPs) that allow one party, the Prover, to prove to another party, the Verifier, that a statement is true without revealing any additional information. Let's delve into the intricacies of Plonkish Arithmetization and its comparison with the well-established R1CS framework. + +### 1. R1CS: A Foundation in Multiplication Gates + +R1CS, or Rank-1 Constraint Systems, **revolves around multiplication gates**, where three "selector" matrices - U, V, and W - designate the "left input", "right input", and "output" of each multiplication gate, respectively, 1 of which indicates involvement in computations and 0 non-involvement. The selector matrix usually has a number of columns equivalent to the number of variables, with the number of rows matching the count of multiplication gates. Additionally, R1CS accommodates constant multiplication gates, a special case treated as addition gates. + +
+ +
+ +Consider a scenario where we have four input variables labeled $(x_1, x_2, x_3, x_4).$ In this scenario, $x_5$ acts as the output of the second multiplication gate and also functions as the right input of the first multiplication gate, while \(out\) represents the output of the first multiplication gate. + +$$\vec{a} = (x_1, x_2, x_3,x_4,x_5,out)$$ + +Imagine a circuit with multiplication gates. Three matrices \(U\), \(V\), and \(W \in \mathbb{F}^{n\times m}\) describe how the gates are connected (circuit's setup). W is a table with rows matching the number of gates (n) and columns hinting at the number of wires (m). Each row in W tells how the corresponding gate connects its inputs and output. + +Left input matrix \(U\): + +$$ +\begin{array}{|c|c|c|c|c|} +\hline +x_1 & x_2 & x_3 & x_4 & x_5 & out & \texttt{i} \\ +\hline +1 & 1 & 0 & 0 & 0 & 0 & \texttt{1}\\ +\hline +0 & 0 & 1 & 0 & 0 & 0 & \texttt{2}\\ +\hline +\end{array} +$$ + +Right input matrix \(V\): + +$$ +\begin{array}{|c|c|c|c|c|} +\hline +x_1 & x_2 & x_3 & x_4 & x_5 & out &\texttt{i}\\ +\hline +0 & 0 & 0 & 0 & 1 & 0 & \texttt{1}\\ +\hline +0 & 0 & 0 & 1 & 0 & 0 & \texttt{2}\\ +\hline +\end{array} +$$ + +Output matrix \(W\): + +$$ +\begin{array}{|c|c|c|c|c|} +\hline +x_1 & x_2 & x_3 & x_4 & x_5 & out & \texttt{i}\\ +\hline +0 & 0 & 0 & 0 & 0 & 1 & \texttt{1}\\ +\hline +0 & 0 & 0 & 0 & 1 & 0 & \texttt{2}\\ +\hline +\end{array} +$$ + +We can easily verify the following equation: +$$(U \cdot \vec{a}) \circ (V \cdot \vec{a}) = (W \cdot\vec{a})$$ + +## 2. Plonkish Arithmetization: A Leap Forward + +Plonkish arithmetization is a more efficient way to express circuits for zero-knowledge proofs: + +**Circuit Setup:** + +Instead of three tables (U,V,W) with potentially many columns, Plonk uses a single W table with just 3 columns to reduces complexity. + +Plonk can handle more than just addition and multiplication gates, such as comparisons, lookups, and even memory access (RAM). + +**Key Tables: $\mathsf{Plonkish}_0 \triangleq (Q, \sigma;$ W)** + +- W table: Stores assignments (secret). +- Q table (Selector): Defines operation for each row (addition or multiplication). +- Permutation table: Ensures consistency when values are moved around in W. + +We define a matrix $W\in\mathbb{F}^{n\times 3}$ to represent constraints where n is the number of arithmetic gates): + +$$ +\begin{array}{c|c|c|c|} +i & w_a & w_b & w_c \\ +\hline +1 & \boxed{x_6} & \underline{x_5} & [out] \\ +2 & x_1 & x_2 & \boxed{x_6} \\ +3 & x_3 & x_4 & \underline{x_5} \\ +4 & 0 & 0 & [out] \\ +\end{array} +$$ + +To distinguish addition and multiplication, we define a vector $Q\in\mathbb{F}^{n\times 5}$ to represent operators: + +$$ +\begin{array}{c|c|c|c|} +i & q_L & q_R & q_M & q_C & q_O \\ +\hline +1 & 0 & 0 & 1 & 0& 1 \\ +2 & 1 & 1 & 0 & 0& 1 \\ +3 & 0 & 0 & 1 & 0& 1 \\ +4 & 0 & 0 & 0 & out& 1 \\ +\end{array} +$$ + +Permutation proofs confirm this consistency post rearrangement, verified by Prover. + +$$ +\begin{array}{c|c|c|c|} +i & w_a & w_b & w_c \\ +\hline +1 & \boxed{x_6} & \underline{x_5} & out \\ +2 & x_1 & x_2 & \boxed{x_6} \\ +3 & x_3 & x_4 & \underline{x_5} \\ +\end{array} +$$ + +If three or more vector positions must share a value, Prover cyclically shifts them, ensuring equality with the original. + +$$ + A = (b_1, b_2, \underline{a_1}, b_3, \underline{a_2}, b_4, \underline{a_3}) = (b_1, b_2, \underline{a_3}, b_3, \underline{a_1}, b_2, \underline{a_2}) = A' +$$ + +To ensure that $w_c$ in the first row equal to the out variable, an additional copy constraint is added to the $\sigma$ matrix. In short, if the Prover is honest, then for $i \in (1,2,3,4),$ the following arithmetic constraint equation holds: + +$$ +q_{L,i} \circ w_{a,i} + q_{R,i} \circ w_{b,i} + q_{M,i}\circ(w_{a,i}\cdot w_{b,i}) + q_{C,i} - q_{O,i}\circ w_{c,i} = 0 +$$ diff --git a/Plonk/co-learn notes/Eta/02_Lagrange.md b/Plonk/co-learn notes/Eta/02_Lagrange.md new file mode 100644 index 0000000..e69de29 diff --git a/Plonk/co-learn notes/Eta/03_Permutation.md b/Plonk/co-learn notes/Eta/03_Permutation.md new file mode 100644 index 0000000..e69de29 diff --git a/Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md b/Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md new file mode 100644 index 0000000..e69de29 diff --git a/Plonk/co-learn notes/Eta/05_Polynomial_Commitments.md b/Plonk/co-learn notes/Eta/05_Polynomial_Commitments.md new file mode 100644 index 0000000..e69de29 diff --git a/Plonk/co-learn notes/Eta/images/28_ring.png b/Plonk/co-learn notes/Eta/images/28_ring.png new file mode 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8c579f1422dda543202949bbeda5aae42ff67c89 Mon Sep 17 00:00:00 2001 From: eittah Date: Sun, 16 Jun 2024 17:51:13 +0800 Subject: [PATCH 2/5] co-learn-notes --- Plonk/co-learn notes/Eta/02_Lagrange.md | 88 +++++++++++++++++++++++++ 1 file changed, 88 insertions(+) diff --git a/Plonk/co-learn notes/Eta/02_Lagrange.md b/Plonk/co-learn notes/Eta/02_Lagrange.md index e69de29..bc2e911 100644 --- a/Plonk/co-learn notes/Eta/02_Lagrange.md +++ b/Plonk/co-learn notes/Eta/02_Lagrange.md @@ -0,0 +1,88 @@ +# ETAAcademy-ZKMeme: 02. Lagrange Interpolation + + + + + + + + + + +
titletags
02. Lagrange Interpolation + + + + + + + +
zk-memebasicquick_readLagrange_Interpolation
+
+ +[Github](https://github.com/ETAAcademy)|[Twitter](https://twitter.com/ETAAcademy)|[ETA-ZK-Meme](https://github.com/ETAAcademy/ETAAcademy-ZK-Meme) + +Authors: [Eta](https://twitter.com/pwhattie), looking forward to your joining + +## Lagrange Interpolation + +Provers can combine multiple constraints into a single polynomial, which reduces communication and simplifies verification for Verifiers. + +### Probabilistic Check of Polynomials + +1. **Schwartz-Zippel Theorem**: The Verifier provides a random challenge $\zeta \in \mathbb{F}$ for two polynomials $f(\zeta) = g(\zeta)$. If the degree of the polynomials is small compared to the size of the field they're working in (like the number of elements in a set), the chance of $f(X) = g(X)$ getting a wrong answer is less than $\leq \frac{d}{|\mathbb{F}|}$(since a polynomial of degree d is determined by \(d+1\) points). + +2. **Vector Verification**: Similar to above, vectors (lists of numbers) can be encoded as polynomials. To verify that three vectors $\vec{a} + \vec{b} = \vec{c}$, the direct method is to encode the vectors into polynomials (using the vector as polynomial coefficients), such as $a(X) = a_0 + a_1X + a_2X^2 + \cdots + a_{n-1}X^{n-1}$. We can create polynomials a(X), b(X), and c(X) from the lists a, b, and c, respectively. If the property $a_i + b_i = c_i$ holds true, then interestingly, adding the polynomials a(X) + b(X) will result in the polynomial c(X), a(X) + b(X) = c(X). By challenging with a random number $\zeta$, If we evaluate both sides of the equation a(ζ) + b(ζ) = c(ζ), then $\vec{a} + \vec{b} = \vec{c}$. + +### Lagrange Interpolation and Evaluation Form + +Multiplying polynomials can get messy. So, for checking if the element-wise product (think multiplication of corresponding entries) of two vectors equals a third vector, a special kind of polynomial encoding is used called Lagrange interpolation. + +- Using Lagrange Interpolation to validate $\vec{a} \circ \vec{b} \overset{?}{=} \vec{c}$, where terms $a_i \cdot b_i$ and $c_i$ do not correspond to the coefficients of $X^i$. +- Instead, we use Lagrange interpolation polynomials Interpolation: $\{L_i(X)\}_{i\in[0,N-1]}$, where $L_i(w_i)=1$, and $L_i(w_j)=0 (j\neq i)$. Then $\vec{a}$ can be encoded as follows: + +$$ +a(X)=a_0\cdot L_0(X) + a_1\cdot L_1(X)+ a_2\cdot L_2(X) + \cdots + a_{N-1}\cdot L_{N-1}(X) +$$ + +- The element-wise product of vectors $a_i \cdot b_i = c_i$ resulting in $a(w_i) \cdot b(w_i) = c(w_i)$. Similarly, $\vec{a} \circ \vec{b} = \vec{c}$ translates to functions $a(X) \cdot b(X) = c(X)$ for all $X \in H$. + +### Single Challenge Verification + +To detect Prover’s cheating with a single challenge, transform the above equality by removing the specific X values (since X should cover a large range like $\mathbb{F}:$ + +$$ +a(X) \cdot b(X) - c(X) = q(X) \cdot z_H(X), \quad \forall X \in \mathbb{F} +$$ + +Since $f(X) = 0$ for all $X \in H$ and $f(X) = a(X) \cdot b(X) - c(X)$,H is the root set of f(X): + +$$ +f(X)=(X-w_0)(X-w_1)(X-w_2)\cdots(X-w_{N-1})\cdot q(X) +$$ + +Thus, f(X) is divisible by the vanishing polynomial $z_H(X) = (X - w_0)(X - w_1) \cdots (X - w_{N-1})$. The Prover computes q(X) and sends it to the Verifier. Since H is a known system parameter, the Verifier can compute $z_H(X)$ and check: + +$$a(\zeta) \cdot b(\zeta) - c(\zeta) \overset{?}{=} q(\zeta) \cdot z_H(\zeta)$$ + +### Optimization: Roots of Unity + +There's a clever way to choose the roots of unity to simplify calculations. The subgroup H is formed by the powers of $\omega$: + +$$ +H = (1, \omega, \omega^2, \ldots, \omega^{N-1}) +$$ + +These elements satisfy certain symmetries, e.g., $\omega = -\omega^{\frac{N}{2} + 1}$ and $\omega^i = -\omega^{\frac{N}{2} + i}$. Summing all roots of unity yields zero: + +$$ +\sum\_{i=0}^{N-1} \omega^i = 0 +$$ + +In practical, we choose a large finite field with a large powers-of-2 multiplicative subgroup. Due to the symmetry of $\omega^i$, on the subgroup H, we have: + +$$ +z_H(X) = \prod*{i=0}^{N-1} (X - \omega^i) = X^N - 1 +$$ + +In essence, polynomial encoding and Lagrange interpolation provide a powerful way to compress and verify complex relationships between data, simplifies many calculations and enhances efficiency in verification. From 5b560c100d3033e9a4116d37a1bbc58ab82924d0 Mon Sep 17 00:00:00 2001 From: eittah Date: Sun, 16 Jun 2024 17:52:11 +0800 Subject: [PATCH 3/5] co-learn-notes --- Plonk/co-learn notes/Eta/03_Permutation.md | 121 +++++++++++++++++++++ 1 file changed, 121 insertions(+) diff --git a/Plonk/co-learn notes/Eta/03_Permutation.md b/Plonk/co-learn notes/Eta/03_Permutation.md index e69de29..b38a06e 100644 --- a/Plonk/co-learn notes/Eta/03_Permutation.md +++ b/Plonk/co-learn notes/Eta/03_Permutation.md @@ -0,0 +1,121 @@ +# ETAAcademy-ZKMeme: 03. Permutation + + + + + + + + + + +
titletags
03. Permutation + + + + + + + +
zk-memebasicquick_readPermutation
+
+ +[Github](https://github.com/ETAAcademy)|[Twitter](https://twitter.com/ETAAcademy)|[ETA-ZK-Meme](https://github.com/ETAAcademy/ETAAcademy-ZK-Meme) + +Authors: [Eta](https://twitter.com/pwhattie), looking forward to your joining + +## Permutation + +A verifier can verify multiple copy constraints within a table $\omega$ using just one random challenge, without even seeing the table itself. For $\omega$, cyclic permutations are performed for the positions that need to be equal, and then the prover demonstrates that the original $\omega$ table is equal to the cyclically permuted $\omega'$ table. **Verifying table equality can be achieved through polynomial encoding and probabilistic checking** + +### Cold Start: Grand Product + +1. **Product Relation:** + The product $p = q_0 \cdot q_1 \cdot q_2 \cdot \cdots \cdot q_{n-2}$ is proved by converting a cumulative product calculation into multiple single multiplication. An auxiliary vector $\vec{r}$, acting as an accumulator, is introduced to represent the "intermediate values" after each multiplication, essentially capturing the entire calculation trace of the product process. The leftmost column represents the original vector to be multiplied { $q_i$ }, the middle column { $r_i$ } is the auxiliary variable that stores the intermediate value "before each single multiplication," and the rightmost column shows the value after each multiplication. + + Interestingly, when we shift the "middle column" vector, r , up one row, it becomes almost identical to the "rightmost column," except for the last element. + +2. **Polynomial Constraints:** + When we encode the three columns of the table as polynomials, they satisfy three specific constraints based on Lagrange Basis polynomials: + + - Initial value: $r_0 = 1, r_{n-1} = p$: + $L_0(X) \cdot (r(X) - 1) = 0,$ + $L_{n-1}(X) \cdot (r(X) - p) = 0, \quad \forall X \in H.$ + - Recursive multiplication relation (removing $\omega^{-1}$ from the set): + $q(X) \cdot r(X) = r(\omega \cdot X), \quad \forall X \in H \setminus \{\omega^{-1}\}.$ + + Simplifying the Constraints: + + - We add a row to the product table, setting $q_{n-1} = 1/p$ (note: p is the product of vector $\vec{q}$. Thus, $r_n = r_0 = 1$. The rightmost column is exactly a cyclic shift of $vec{r}$. + - The verifier can challenge the following polynomial equation: + $L_0(X) \cdot (r(X) - 1) + \alpha \cdot (q(X) \cdot r(X) - r(\omega \cdot X)) = h(X) \cdot z_H(X).$ + Here, $\alpha$ is a random challenge number used to combine multiple polynomial constraints, and $h(X)$ represents the quotient polynomial, with $z_H(X) = (X-1)(X-\omega) \cdots (X-\omega^{n-1})$. + +$$ +\begin{array}{c|c|c} +q_i & r_i & q_i\cdot r_i \\ +\hline +q_0 & 1 & r_0\\ +q_1 & r_0 & r_1\\ +q_2 & r_1 & r_2\\ +\vdots & \vdots & \vdots\\ +q_{n-2} & r_{n-2} & r_{n-1}\\ +q_{n-1}=\frac{1}{p} & r_{n-1} & 1 \\ +\end{array} +$$ + +### Using Grand Product to Prove Multiset Equality: + +1. If two polynomials p(X) and q(X) are equal, they must share the same set of roots { $q_i$ }. + For example: $\prod_{i}(X - q_i) = q(X) = p(X) = \prod_{i}(X - p_i),$ $\\{q_i\\}=_{multiset}\\{p_i\\}$. + + By requesting a random number $\gamma$ from the verifier, the prover can prove that the vectors { $p_i$ } and { $q_i$} are equal in the multiset sense through the following equation: + $\prod_{i \in [n]}(\gamma - p_i) = \prod_{i \in [n]}(\gamma - q_i)$ + +2. **Product Proof:** + As mentioned earlier, the multiplication process is converted into a series of single multiplications using auxiliary vectors. Interestingly, two multiplications can be merged into one, as shown here: + $\prod_{i \in [n]} \frac{(\gamma - p_i)}{(\gamma - q_i)} = 1.$ + +### From Multiset Equality to Permutation Proof: + +If two vectors $\vec{a}$ and $\vec{b}$ satisfy a permutation $\sigma$, the combined vectors $\vec{a}'$ and $\vec{b}'$ will satisfy the multiset equality relationship. By combining vectors and position values, treating $(a_i, i)$ as one element and $(b_i, \sigma(i))$ as another element, a "permutation proof" can be converted into a "multiset equality." By requesting a random number $\beta$ from the verifier, the tuples can be "folded" into a single value. + +$$ +\begin{array}{|c|c | c|c|} +a_i & {i} & b_i & \sigma({i}) \\ +\hline +a_0 & 0 & b_0=a_1 & 1 \\ +a_1 & 1 & b_1=a_0 & 0 \\ +a_2 & 2 & b_2=a_3 & 3 \\ +a_3 & 3 & b_3=a_2 & 2 \\ +\vdots & \vdots & \vdots & \vdots \\ +a_{n-1} & n-1 & b_{n-1}=a_n & n \\ +a_n & n & b_n=a_{n-1} & n-1 \\ +\end{array} +$$ + +$$ +\begin{array}{|c|c|} +a'_i=(a_i, i) & b'_i=({b}_i, \sigma(i)) \\ +\hline +(a_0, 0) & (b_0=a_1, 1) \\ +(a_1, 1) & (b_1=a_0, 0) \\ +\vdots & \vdots \\ +(a\_{n-1}, n-1) & (b\_{n-1}=a\_{n}, n) \\ +(a\_n, n) & (b\_n=a\_{n-1}, n-1) \\ +\end{array} +$$ + +$$ +\begin{array}{|c|c|} +a'_i=(a_i+\beta\cdot i) & b_i'=(b + \beta\cdot \sigma(i)) \\ +\hline +(a_0 + \beta\cdot 0) & (b_0 + \beta\cdot 1) \\ +(a_1 + \beta\cdot 1) & (b_1 + \beta\cdot 0) \\ +\vdots & \vdots \\ +(a\_{n-1} + \beta\cdot n-1) & (b\_{n-1} + \beta\cdot n) \\ +(a\_n + \beta\cdot n) & (b\_n + \beta\cdot (n-1))\\ +\end{array} +$$ + +The verification process leverages multiset equivalence as a stepping stone to prove the existence of a permutation between two vectors, which relies on random challenges and polynomial encoding to ensure the validity of the proof. From fa9bbda4bde715c389996659f73281e2c15bd918 Mon Sep 17 00:00:00 2001 From: eittah Date: Sun, 16 Jun 2024 17:53:11 +0800 Subject: [PATCH 4/5] co-learn-notes --- .../Eta/04_Copy_Constraints_Optimization.md | 169 ++++++++++++++++++ 1 file changed, 169 insertions(+) diff --git a/Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md b/Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md index e69de29..f1c4e1e 100644 --- a/Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md +++ b/Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md @@ -0,0 +1,169 @@ +# ETAAcademy-ZKMeme: 04. Copy Constraints and Optimization + + + + + + + + + + +
titletags
04. Copy Constraints and Optimization + + + + + + + +
zk-memebasicquick_readCopy_Constraints_Optimization
+
+ +[Github](https://github.com/ETAAcademy)|[Twitter](https://twitter.com/ETAAcademy)|[ETA-ZK-Meme](https://github.com/ETAAcademy/ETAAcademy-ZK-Meme) + +Authors: [Eta](https://twitter.com/pwhattie), looking forward to your joining + +## Plonk Protocol: Copy Constraints and Position Vector Optimization + +### Copy Constraints + +The copy constraints in the Plonk protocol ensure that the values of certain variables remain consistent across different columns of the W table. This is necessary for the protocol to correctly verify the computation of a circuit. + +**Table W** + +$$ +\begin{array}{c|c|c|c|} +i & w_a & w_b & w_c \\ +\hline +0 & 0 & 0 & {\color{green}out} \\ +1 & {\color{red}x_6} & {\color{blue}x_5} & {\color{green}out} \\ +2 & x_1 & x_2 & {\color{red}x_6} \\ +3 & x_3 & x_4 & {\color{blue}x_5} \\ +\end{array} +$$ + +**Table of IDs** + +$$ +\begin{array}{c|c|c|c|} +i & id_{a,i} & id_{b,i} & id_{c,i} \\ +\hline +0 & 0 & 4 & {\color{green}8} \\ +1 & {\color{red}1} & {\color{blue}5} & {\color{green}9} \\ +2 & 2 & 6 & {\color{red}10} \\ +3 & 3 & 7 & {\color{blue}11} \\ +\end{array} +$$ + +**Permutation Vectors** + +Introduce permutation vectors $\sigma_a, \sigma_b, \sigma_c:$ + +$$ +\begin{array}{c|c|c|c|} +i & \sigma_{a,i} & \sigma_{b,i} & \sigma_{c,i} \\ +\hline +0 & 0 & 4 & {\color{green}9} \\ +1 & {\color{red}10} & {\color{blue}11} & {\color{green}8} \\ +2 & 2 & 6 & {\color{red}1} \\ +3 & 3 & 7 & {\color{blue}5} \\ +\end{array} +$$ + +**Prover's Process** + +The prover uses random numbers $\beta$ and $\gamma$ provided by the verifier to merge and permute vectors, obtaining the multisets $f_i$ and $g_i$, which could be encoded into polynomials: + +$$ +\begin{split} +f(X)&=\Big(w_a(X)+\beta\cdot {id_a}(X)+\gamma\Big)\Big(w_b(X)+\beta\cdot {id_b}(X)+\gamma\Big)\Big(w_c(X)+\beta\cdot {id_c}(X)+\gamma\Big)\\ +g(X)&=\Big(w_a(X)+\beta\cdot {\sigma_a}(X)+\gamma\Big)\Big(w_b(X)+\beta\cdot {\sigma_b}(X)+\gamma\Big)\Big(w_c(X)+\beta\cdot {\sigma_c}(X)+\gamma\Big)\\ +\end{split} +$$ + +**Auxiliary Vector $\vec{z}$** + +Construct $\vec{z}$ to represent intermediate steps: + +$$ +z_0 = 1, \qquad z_{i+1}=z_i\cdot \frac{f_i}{g_i}\\ +$$ + +$$ +\begin{array}{|c|c|c|} +i & H_i & z_i\\ +\hline +0 & \omega^0=1 & 1\\ +1 & \omega^1 & 1\cdot \frac{f_0}{g_0}\\ +2 & \omega^2 & \frac{f_0}{g_0}\cdot \frac{f_1}{g_1}\\ +3 & \omega^3 & \frac{f_0f_1}{g_0g_1}\cdot \frac{f_2}{g_2}\\ +\vdots & & \vdots\\ +N-1 & \omega^{N-1} & \frac{f_0f_1\cdots f_{N-3}}{g_0g_1\cdots g_{N-3}}\cdot \frac{f_{N-2}}{g_{N-2}} \\ +N & \omega^{N}=1 & \frac{f_0f_1\cdots f_{N-1}}{g_0g_1\cdots g_{N-1}} = 1 +\end{array} +$$ + +If $\vec{f}$ and $\vec{g}$ are equivalent, $z_N = 1$. This is verified by ensuring: + +$$ +z(\omega^0) = 1 +$$ + +$$ +z(\omega\cdot X)g(X) = z(X)f(X) +$$ + +where: + +- $z_0$ is an initial value of 1 +- $z_i$ is the value of the copy constraint at step $i$ +- $f_i$ is a polynomial expression representing the value of a variable at step $i$ +- $g_i$ is a polynomial expression representing the value of a variable at step $i$ +- $X$ is a variable +- $\omega$ is a generator of the multiplicative subgroup $H$ + +### Position Vector Optimization + +The σ vectors only need to use distinct values to mark the permutations, not necessarily increasing natural numbers. If we use H = (1, ω, $ω^2,$ ...), then the polynomial ${id_a}(X)$ will be greatly simplified: + +$$ +\begin{split} +\vec{id}_a &= (1,\omega,\omega^2,\omega^3)\\ +\vec{id}_b &= (k_1,k_1\omega,k_1\omega^2,k_1\omega^3)\\ +\vec{id}_c &= (k_2,k_2\omega,k_2\omega^2,k_2\omega^3)\\ +\end{split} +$$ + +where $k_i$ are distinct quadratic non-residues. + +$$ +id_a(X) = X, \quad id_b(X) = k_1 \cdot X, \quad id_a(X) = k_2 \cdot X +$$ + +If there are more columns in σ, then we need to choose multiple $k_1, k_2, k_3, ... = g^1, g^2, g^3, ...$ where g is a generator of the multiplicative subgroup T, and $|T|\*2^λ = p-1.$ + +### Public Inputs + +$$ +\begin{array}{c|c|c|c|} +i & q_L & q_R & q_M & q_C & q_O \\ +\hline +0 & 0 & 0 & 0 & 99 & 1 \\ +1 & 0 & 0 & 1 & 0 & 1 \\ +2 & 1 & 1 & 0 & 0 & 1 \\ +3 & 0 & 0 & 1 & 0 & 1 \\ +\end{array} +$$ + +To prove that a circuit output equals a specific public value (e.g., out=99), the simplest method is to use the $q_C$ column with constraints $q_L = q_R = q_M = 0:$ + +$q_C(X) - q_O(X)w_c(X) = 0$ + +This method fixes the public values as constants, requiring recomputation of the $q_C(X)$ polynomial if values change, which is inefficient. + +Introduce a new column $\phi$ for public parameters, which allow public input changes without affecting other parts of the circuit. The new arithmetic constraint is: + +$q_L(X)w_a(X) + q_R(X)w_b(X) + q_M(X)w_a(X)w_b(X) - q_O(X)w_c(X) + q_C(X) + \phi(X) = 0$ + +In summary, handling public inputs efficiently in the Plonk protocol involves introducing a dedicated column for public parameters to avoid frequent recalculations of constraint polynomials. Additionally, optimizing position vectors and managing copy constraints with auxiliary vectors and random merging helps ensure the integrity and efficiency of the proof system. From 208d93e98d6bc82308f782d5a5e848faa9146f06 Mon Sep 17 00:00:00 2001 From: eittah Date: Sun, 16 Jun 2024 17:54:26 +0800 Subject: [PATCH 5/5] co-learn-notes --- Plonk/co-learn notes/Eta/01_R1CS_Plonklish.md | 2 +- Plonk/co-learn notes/Eta/02_Lagrange.md | 2 +- Plonk/co-learn notes/Eta/03_Permutation.md | 2 +- .../Eta/04_Copy_Constraints_Optimization.md | 2 +- .../Eta/05_Polynomial_Commitments.md | 175 ++++++++++++++++++ 5 files changed, 179 insertions(+), 4 deletions(-) diff --git a/Plonk/co-learn notes/Eta/01_R1CS_Plonklish.md b/Plonk/co-learn notes/Eta/01_R1CS_Plonklish.md index d4e3bc7..f1ab765 100644 --- a/Plonk/co-learn notes/Eta/01_R1CS_Plonklish.md +++ b/Plonk/co-learn notes/Eta/01_R1CS_Plonklish.md @@ -1,4 +1,4 @@ -# ETAAcademy-ZKMeme: 01. R1CS V.S. Plonkish +# 01. R1CS V.S. Plonkish diff --git a/Plonk/co-learn notes/Eta/02_Lagrange.md b/Plonk/co-learn notes/Eta/02_Lagrange.md index bc2e911..93c7d6f 100644 --- a/Plonk/co-learn notes/Eta/02_Lagrange.md +++ b/Plonk/co-learn notes/Eta/02_Lagrange.md @@ -1,4 +1,4 @@ -# ETAAcademy-ZKMeme: 02. Lagrange Interpolation +# 02. Lagrange Interpolation
diff --git a/Plonk/co-learn notes/Eta/03_Permutation.md b/Plonk/co-learn notes/Eta/03_Permutation.md index b38a06e..e8a4fd1 100644 --- a/Plonk/co-learn notes/Eta/03_Permutation.md +++ b/Plonk/co-learn notes/Eta/03_Permutation.md @@ -1,4 +1,4 @@ -# ETAAcademy-ZKMeme: 03. Permutation +# 03. Permutation
diff --git a/Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md b/Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md index f1c4e1e..e4d318a 100644 --- a/Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md +++ b/Plonk/co-learn notes/Eta/04_Copy_Constraints_Optimization.md @@ -1,4 +1,4 @@ -# ETAAcademy-ZKMeme: 04. Copy Constraints and Optimization +# 04. Copy Constraints and Optimization
diff --git a/Plonk/co-learn notes/Eta/05_Polynomial_Commitments.md b/Plonk/co-learn notes/Eta/05_Polynomial_Commitments.md index e69de29..0b25ab8 100644 --- a/Plonk/co-learn notes/Eta/05_Polynomial_Commitments.md +++ b/Plonk/co-learn notes/Eta/05_Polynomial_Commitments.md @@ -0,0 +1,175 @@ +# 05. Polynomial Commitments + +
+ + + + + + + + +
titletags
05. Polynomial Commitments + + + + + + + +
zk-memebasicquick_readPolynomial_Commitments
+
+ +[Github](https://github.com/ETAAcademy)|[Twitter](https://twitter.com/ETAAcademy)|[ETA-ZK-Meme](https://github.com/ETAAcademy/ETAAcademy-ZK-Meme) + +Authors: [Eta](https://twitter.com/pwhattie), looking forward to your joining + +## Polynomial Commitments + +Polynomial commitment, `c = commit(x)`, is a cryptographic scheme that allows a prover to commit to a polynomial $f(X)$ in such a way that they can later prove that they know the value of $f(X)$ at a given point $x$ without revealing the entire polynomial. + +**Hash** + +The most basic construction of a polynomial commitment is to simply hash the coefficients of the polynomial. For example, if $f(X) = a_0 + a_1X + a_2X^2 + \cdots + a_nX^n$, then the commitment $C$ is: + +$$ +C = SHA256(a_0 || a_1 || a_2 || ... || a_n) +$$ + +**Petersen Commitment** + +Another construction is the Petersen commitment, using a set of randomly chosen bases to compute an ECC point: + +$$ +C = a_0 G_0 + a_1 G_1 + \cdots + a_n G_n +$$ + +**KZG Commitment** + +One of the most efficient construction is the KZG commitment, using a set of bases with internal algebraic structure $(G, \chi G, \chi^2G, \ldots, \chi^{d-1}G, H, \chi H)$. The KZG commitment to a polynomial $f(X)$ is: + +$$ +C_{f(X)} = a_0 G_0 + a_1 G_1 + \cdots + a_{n-1} G_{n-1} += a_0 G + a_1 \chi G + \cdots + a_{n-1}\chi^{n-1} G += f(\chi) G +$$ + +where $\chi$ is a random provided by a trusted third party, known as the trapdoor, which is destroyed after the setup phase. There is a bilinear pairing $e\in \mathbb{G}_1\times\mathbb{G}_2\to \mathbb{G}_T$, with $G\in \mathbb{G}_1$ and $H\in \mathbb{G}_2.$ + +## Optimization Round One🍄: Polynomial Remainder Theorem + +KZG is a powerful tool for proving knowledge of a polynomial without revealing the polynomial itself. However, they can be computationally expensive, especially when proving the value of the polynomial at a point. The polynomial remainder theorem can be used to optimize KZG polynomial commitments and reduce the computational cost of proving polynomial values. + +**Polynomial Remainder Theorem** + +To make KZG polynomial commitments more succinct, we can use the Polynomial Remainder Theorem: + +$$ +f(X) = q(X) \cdot (X - \zeta) + y +$$ + +The prover provides the commitment to the polynomial q(X), $C_q,$ to prove $f(\zeta) = y$. The verifier checks whether $[q(\chi)]$ satisfies the divisibility: + +$$ +(f(X)-y) \cdot 1 \overset{?}{=} q(X) \cdot (X - \zeta) +$$ + +Using Groth's notation, where $[1]_1 \triangleq G$ and $[1]_2 \triangleq H$ are generators of two groups, we can express the verification equation as: + +$$ +e(C\_{f(X)} - y[1]_1, [1]_2) \overset{?}{=} e(C\_{q(X)}, [\chi]_2 - \zeta [1]_2) +$$ + +To reduce the expensive operations on \(\mathbb{G}2\), we can simplify to: + +$$ +f(X) + \zeta \cdot q(X) - y = q(X) \cdot X +$$ + +Hence: + +$$ +e(C\_{f(X)} + \zeta\cdot C\_{q(X)} -y\cdot[1]_1,\ [1]_2)\overset{?}{=} e(C\_{q(X)},\ [\chi]_2) +$$ + +This equation is equivalent to the original equation, but it reduces the number of operations in $\mathbb{G}\_2$. + +## Batch Verification of Multiple Polynomial Commitments + +One approach to batch verification is to use random linearization. Multiple polynomials can be combined into a larger polynomial using random numbers, allowing batch verification by opening at a single point: + +**Combining Two Polynomials**: + +- Given two polynomials $f_1(X)$ and $f_2(X)$, we can form a new polynomial $g(X) = f_1(X) + \nu \cdot f_2(X),$ where $\nu$ is a random number. +- Similarly, $q(X) = q_1(X) + \nu \cdot q_2(X)$ leads to $[q(\chi)]\_1 = \pi = \pi_1 + \nu \cdot \pi_2.$ +- The commitment for the new polynomial g is $C_g = C_1 + \nu \cdot C_2.$ +- The value of the polynomial at a single point $X = \zeta$ can be computed as $y_g = y_1 + \nu \cdot y_2.$ +- The verifier checks the relation: + +$$ + e(C - y \cdot G_0, H_0) \overset{?}{=} e(\pi, H_1 - x \cdot H_0) +$$ + +**Combining Multiple Polynomials**: + +- For a more complex case, consider the polynomial constraint $f_1(X) f_2(X) + h_1(X) h_2(X) h_3(X) + g(X) = 0.$ +- The straightforward approach involves the prover opening the commitments at $X = \zeta$ and sending the evaluations and corresponding proofs for each polynomial: + +$$ +(f_1(\zeta),\pi_{f_1}),(f_2(\zeta),\pi_{f_2}),(h_1(\zeta),\pi_{h_1}),(h_2(\zeta),\pi_{h_2}),(h_3(\zeta),\pi_{h_3}),(g(\zeta),\pi_{g}) +$$ + +- The verifier then checks all six evaluation proofs and verifies the polynomial constraint: + +$$ +f_1(\zeta)f_2(\zeta) + h_1(\zeta)h_2(\zeta)h_3(\zeta) + g(\zeta) \overset{?}{=} 0 +$$ + +## Optimization Round Two 🍄🏰: Reducing the Number of Openings + +we will further optimize the process by reducing the number of openings required. + +Consider a simple polynomial constraint of the form: + +- The prover first opens $\bar{f} = f(\zeta)$ and sends the proof $\pi_{f(\zeta)}.$ +- Introduce an auxiliary polynomial $L(X) = \bar{f} \cdot g(X) - h(X),$ then open $L(X)$ at $X = \zeta.$ + +In this optimized scheme, the prover needs to open only twice. The first opening is $\bar{f},$ and the second opening is for the constant 0, which does not need to be sent to the verifier. Thus, the prover only needs to send two evaluation proofs. Using the aggregation method from the previous section, with a challenge number 𝜈, the prover can aggregate the two polynomial commitments and only needs to send one evaluation proof. + +**Complex Polynomial Constraint** + +Similarly, consider a more complex polynomial constraint involving six polynomials: + +$$ +f_1(X) f_2(X) + h_1(X) h_2(X) h_3(X) + g(X) = 0 +$$ + +This constraint can be optimized by introducing an auxiliary polynomial: + +$$ +L(X) = \bar{f}_1 ⋅ f_2(X) + \bar{h}_1 \bar{h}_2 ⋅ h_3(X) + g(X) +$$ + +The verifier sends a random number $\nu$, and the prover computes the commitment of the folded polynomial: + +$$ +[F]_1 = [L]_1 + \nu ⋅ [f_1(\chi)]_1 + \nu^2 [h_1(\chi)]_1 + \nu^3 [h_2(\chi)]_1 +$$ + +and the value of the folded polynomial at $X = \zeta$: + +$$ +E = \nu ⋅ \bar{f}_1 + \nu^2 ⋅ \bar{h}_1 + \nu^3 ⋅ \bar{h}_2 +$$ + +The verifier checks the following bilinear mapping equation: + +$$ +e([F]_1 - [E]_1 + \zeta [q(\chi)]_1, [1]_2) \overset{?}{=} e([q(\chi)]_1, [\chi]_2) +$$ + +This optimized protocol reduces the prover's workload to sending only three openings and one evaluation proof, compared to the original six openings and six evaluation proofs, thus significantly reducing the communication overhead (i.e., the proof size). + +### Conclusion + +Polynomial commitments are a powerful tool for proving knowledge of polynomials in a cryptographically secure way. They have a wide range of applications in cryptography and are likely to become even more important in the future as applications such as zk-SNARKs and zk-rollups become more widely adopted.