diff --git a/ph23/kzg_hiding.md b/ph23/kzg_hiding.md index f218ce7..0181595 100644 --- a/ph23/kzg_hiding.md +++ b/ph23/kzg_hiding.md @@ -165,7 +165,7 @@ Therefore, we also need to add another different blinding factor to $[q(\tau)]_1 $$ \begin{aligned} -\mathsf{KZG.Commit}(q(X), {\color{green}s}) & = q_0 \cdot [1]_1 + q_1 \cdot [\tau]_1 + \cdots + q_d \cdot [\tau^d]_1 + {\color{green}s} \cdot {\color{red}[\gamma]_1} \\ +\mathsf{KZG.Commit}(q(X), {\color{green}s}) & = q_0 \cdot [1]_1 + q_1 \cdot [\tau]_1 + \cdots + q_d \cdot [\tau^{d-1}]_1 + {\color{green}s} \cdot {\color{red}[\gamma]_1} \\ & = [q(\tau) + {\color{green}s}\cdot{\color{red}\gamma}]_1 \end{aligned} $$ @@ -239,13 +239,13 @@ Where the red parts are provided by the Prover, and the blue parts are public va To prove the Degree Bound of $f(X)$, we also need to add a Blinding Factor to the polynomial $\hat{f}(X)$, then calculate its commitment as the Degree Bound proof of $f(X)$: $$ -\mathsf{cm}(\hat{f}) = [\tau^{D-d}\cdot \hat{f}(\tau)]_1 + {\color{red}\eta}\cdot[\gamma]_1 +\mathsf{cm}(\hat{f}) = [\tau^{D-d}\cdot f(\tau)]_1 + {\color{red}\eta}\cdot[\gamma]_1 $$ -At the same time, an additional element $C'\in\mathbb{G}_1$ is needed for balancing, +At the same time, an additional element $E \in\mathbb{G}_1$ is needed for balancing, $$ -C' = \rho\cdot[\tau^{D-d}]_1 - {\color{red}\eta}\cdot[1]_1 +E = \rho\cdot[\tau^{D-d}]_1 - {\color{red}\eta}\cdot[1]_1 $$ This way, the Verifier can verify the Degree Bound proof of $f(X)$ through the following equation: diff --git a/ph23/kzg_hiding.zh.md b/ph23/kzg_hiding.zh.md index 92a894b..22c43d4 100644 --- a/ph23/kzg_hiding.zh.md +++ b/ph23/kzg_hiding.zh.md @@ -165,7 +165,7 @@ $$ $$ \begin{aligned} -\mathsf{KZG.Commit}(q(X), {\color{green}s}) & = q_0 \cdot [1]_1 + q_1 \cdot [\tau]_1 + \cdots + q_d \cdot [\tau^d]_1 + {\color{green}s} \cdot {\color{red}[\gamma]_1} \\ +\mathsf{KZG.Commit}(q(X), {\color{green}s}) & = q_0 \cdot [1]_1 + q_1 \cdot [\tau]_1 + \cdots + q_d \cdot [\tau^{d-1}]_1 + {\color{green}s} \cdot {\color{red}[\gamma]_1} \\ & = [q(\tau) + {\color{green}s}\cdot{\color{red}\gamma}]_1 \end{aligned} $$ @@ -239,13 +239,13 @@ $$ 为了证明 $f(X)$ 的 Degree Bound,我们需要给多项式 $\hat{f}(X)$ 也加上 Blinding Factor,然后计算其承诺,作为 $f(X)$ 的 Degree Bound 证明: $$ -\mathsf{cm}(\hat{f}) = [\tau^{D-d}\cdot \hat{f}(\tau)]_1 + {\color{red}\eta}\cdot[\gamma]_1 +\mathsf{cm}(\hat{f}) = [\tau^{D-d}\cdot f(\tau)]_1 + {\color{red}\eta}\cdot[\gamma]_1 $$ -同时还要附加上一个用来配平的 元素 $C'\in\mathbb{G}_1$, +同时还要附加上一个用来配平的元素 $E\in\mathbb{G}_1$, $$ -C' = \rho\cdot[\tau^{D-d}]_1 - {\color{red}\eta}\cdot[1]_1 +E = \rho\cdot[\tau^{D-d}]_1 - {\color{red}\eta}\cdot[1]_1 $$ 这样 Verifier 可以用过下面的等式来验证 $f(X)$ 的 Degree Bound 证明: