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@ -330,30 +330,30 @@ In the following protocol, we take it for granted that each polynomial $a_i(X, \
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1. $\prover$ and $\verifier$ proceed in the following $n_a$ rounds of interaction, where in round $j$ (starting at $0$)
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* $\prover$ sets $a'_j(X) = a_j(X, c_0, c_1, ..., c_{j - 1}, a_0(X, \cdots), ..., a_{j - 1}(X, \cdots, c_{j - 1}))$
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* $\prover$ sends a hiding commitment $A_j = \innerprod{\mathbf{a'}}{\mathbf{G}} + [\cdot] W$ where $\mathbf{a'}$ are the coefficients of the univariate polynomial $a'_j(X)$ and $\cdot$ is some random, independently sampled blinding factor elided for exposition. (This elision notation is used throughout this protocol description to simplify exposition.)
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* $\prover$ sends a hiding commitment $A_j = \innerprod{\mathbf{a'}}{\mathbf{G}} + [a^*_j] W$ where $\mathbf{a'}$ are the coefficients of the univariate polynomial $a'_j(X)$ and $a^*_j$ is some fresh (random and independently sampled) blinding factor. Similar notation is used throughout this protocol description.
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* $\verifier$ responds with a challenge $c_j$.
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2. $\prover$ sets $g'(X) = g(X, c_0, c_1, ..., c_{n_a - 1}, \cdots)$.
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3. $\prover$ sends a commitment $R = \innerprod{\mathbf{r}}{\mathbf{G}} + [\cdot] W$ where $\mathbf{r} \in \field^n$ are the coefficients of a randomly sampled univariate polynomial $r(X)$ of degree $n - 1$.
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3. $\prover$ sends a commitment $R = \innerprod{\mathbf{r}}{\mathbf{G}} + [r^*] W$ where $\mathbf{r} \in \field^n$ are the coefficients of a randomly sampled univariate polynomial $r(X)$ of degree $n - 1$.
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4. $\prover$ computes univariate polynomial $h(X) = \frac{g'(X)}{t(X)}$ of degree $n_g(n - 1) - n$.
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5. $\prover$ computes at most $n - 1$ degree polynomials $h_0(X), h_1(X), ..., h_{n_g - 2}(X)$ such that $h(X) = \sum\limits_{i=0}^{n_g - 2} X^{ni} h_i(X)$.
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6. $\prover$ sends commitments $H_i = \innerprod{\mathbf{h_i}}{\mathbf{G}} + [\cdot] W$ for all $i$ where $\mathbf{h_i}$ denotes the vector of coefficients for $h_i(X)$.
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6. $\prover$ sends commitments $H_i = \innerprod{\mathbf{h_i}}{\mathbf{G}} + [h^*_i] W$ for all $i$ where $\mathbf{h_i}$ denotes the vector of coefficients for $h_i(X)$, and $h^*_i$ are fresh blinding factors.
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7. $\verifier$ responds with challenge $x$ and computes $H' = \sum\limits_{i=0}^{n_g - 2} [x^{ni}] H_i$.
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8. $\prover$ sets $h'(X) = \sum\limits_{i=0}^{n_g - 2} x^{ni} h_i(X)$.
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8. $\prover$ sets $h'(X) = \sum\limits_{i=0}^{n_g - 2} x^{ni} h_i(X)$, and synthetic blinding factor $h'^* = \sum\limits_{i=0}^{n_g - 2} x^{ni} h^*_i$.
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9. $\prover$ sends $r = r(x)$ and for all $i \in [0, n_a)$ sends $\mathbf{a_i}$ such that $(\mathbf{a_i})_j = a'_i(\omega^{(\mathbf{p_i})_j} x)$ for all $j \in [0, n_e - 1]$.
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10. For all $i \in [0, n_a)$ $\prover$ and $\verifier$ set $s_i(X)$ to be the lowest degree univariate polynomial defined such that $s_i(\omega^{(\mathbf{p_i})_j} x) = (\mathbf{a_i})_j$ for all $j \in [0, n_e - 1)$.
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11. $\verifier$ responds with challenges $x_1, x_2$ and initializes $Q_0, Q_1, ..., Q_{n_q - 1} = \zero$.
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* Starting at $i=0$ and ending at $n_a - 1$ $\verifier$ sets $Q_{\sigma(i)} := [x_1] Q_{\sigma(i)} + A_i$.
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* $\verifier$ finally sets $Q_0 := [x_1^2] Q_0 + [x_1] H' + R$.
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12. $\prover$ initializes $q_0(X), q_1(X), ..., q_{n_q - 1}(X) = 0$.
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* Starting at $i=0$ and ending at $n_a - 1$ $\prover$ sets $q_{\sigma(i)} := x_1 q_{\sigma(i)} + a'(X)$.
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* $\prover$ finally sets $q_0(X) := x_1^2 q_0(X) + x_1 h'(X) + r(X)$.
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12. $\prover$ initializes $q_0(X), q_1(X), ..., q_{n_q - 1}(X) = 0$ and blinding factors $q^*_0, q^*_1, ..., q^*_{n_q-1} = 0$.
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* Starting at $i=0$ and ending at $n_a - 1$ $\prover$ sets $q_{\sigma(i)} := x_1 q_{\sigma(i)} + a'(X)$ and $q^*_{\sigma(i)} := x_1 q^*_{\sigma(i)} + a^*_i$.
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* $\prover$ finally sets $q_0(X) := x_1^2 q_0(X) + x_1 h'(X) + r(X)$, and its corresponding synthetic blinding factor $q^*_0 := x_1^2 q^*_0 + x_1 h'^* + r^*$.
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13. $\prover$ and $\verifier$ initialize $r_0(X), r_1(X), ..., r_{n_q - 1}(X) = 0$.
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* Starting at $i = 0$ and ending at $n_a - 1$ $\prover$ and $\verifier$ set $r_{\sigma(i)}(X) := x_1 r_{\sigma(i)}(X) + s_i(X)$.
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* Finally $\prover$ and $\verifier$ set $r_0 := x_1^2 r_0 + x_1 h + r$ and where $h$ is computed by $\verifier$ as $\frac{g'(x)}{t(x)}$ using the values $r, \mathbf{a}$ provided by $\prover$.
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14. $\prover$ sends $Q' = \innerprod{\mathbf{q'}}{\mathbf{G}} + [\cdot] W$ where $\mathbf{q'}$ defines the coefficients of the polynomial
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14. $\prover$ sends $Q' = \innerprod{\mathbf{q'}}{\mathbf{G}} + [q^{\prime *}] W$ where $q^{\prime *}$ is the synthetic blinding factor calculated in step 12, and $\mathbf{q'}$ defines the coefficients of the polynomial
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$$q'(X) = \sum\limits_{i=0}^{n_q - 1}
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x_2^i
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x_2^{n_q - 1 - i}
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\left(
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\frac
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{q_i(X) - r_i(X)}
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@ -369,11 +369,12 @@ $$
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15. $\verifier$ responds with challenge $x_3$.
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16. $\prover$ sends $\mathbf{u} \in \field^{n_q}$ such that $\mathbf{u}_i = q_i(x_3)$ for all $i \in [0, n_q)$.
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17. $\verifier$ responds with challenge $x_4$.
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18. $\verifier$ sets $P = Q' + x_4 \sum\limits_{i=0}^{n_q - 1} [x_4^i] Q_i$ and $v = $
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18. $\verifier$ sets $P = [x_4^{n_q}]Q' + \sum\limits_{i=0}^{n_q - 1} [x_4^{n_q - 1 - i}] Q_i$ and $v = $
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$$
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x_4^{n_q} \cdot
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\sum\limits_{i=0}^{n_q - 1}
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\left(
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x_2^i
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x_2^{n_q - 1 - i}
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\left(
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\frac
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{ \mathbf{u}_i - r_i(x_3) }
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@ -387,19 +388,19 @@ x_2^i
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\right)
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\right)
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+
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x_4 \sum\limits_{i=0}^{n_q - 1} x_4 \mathbf{u}_i
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\sum\limits_{i=0}^{n_q - 1} x_4^{n_q - 1 - i} \mathbf{u}_i
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$$
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19. $\prover$ sets $p(X) = q'(X) + [x_4] \sum\limits_{i=0}^{n_q - 1} x_4^i q_i(X)$.
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20. $\prover$ samples a random polynomial $s(X)$ of degree $n - 1$ with a root at $x_3$ and sends a commitment $S = \innerprod{\mathbf{s}}{\mathbf{G}} + [\cdot] W$ where $\mathbf{s}$ defines the coefficients of $s(X)$.
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19. $\prover$ sets $p(X) = x_4^{n_q} \cdot q'(X) + \sum\limits_{i=0}^{n_q - 1} x_4^{n_q - 1 - i} \cdot q_i(X)$ and the corresponding synthetic blinding factor $p^* = x_4^{n_q} \cdot q'^* + \sum\limits_{i=0}^{n_q - 1} x_4^{n_q - 1 - i} \cdot q^*_i$.
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20. $\prover$ samples a random polynomial $s(X)$ of degree $n - 1$ with a root at $x_3$ and sends a commitment $S = \innerprod{\mathbf{s}}{\mathbf{G}} + [s^{*}] W$ where $\mathbf{s}$ defines the coefficients of $s(X)$ and $s^{*}$ is a fresh blinding factor.
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21. $\verifier$ responds with challenges $\xi, z$.
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22. $\verifier$ sets $P' = P - [v] \mathbf{G}_0 + [\xi] S$.
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23. $\prover$ sets $p'(X) = p(X) - p(x_3) + \xi s(X)$ (where $p(x_3)$ should correspond with the verifier's computed value $v$).
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23. $\prover$ sets $p'(X) = p(X) - p(x_3) + \xi s(X)$ and $p^{\prime *} = s^* \cdot \xi + p^*$ (where $p(x_3)$ should correspond with the verifier's computed value $v$).
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24. Initialize $\mathbf{p'}$ as the coefficients of $p'(X)$ and $\mathbf{G'} = \mathbf{G}$ and $\mathbf{b} = (x_3^0, x_3^1, ..., x_3^{n - 1})$. $\prover$ and $\verifier$ will interact in the following $k$ rounds, where in the $j$th round starting in round $j=0$ and ending in round $j=k-1$:
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* $\prover$ sends $L_j = \innerprod{\mathbf{p'}_\hi}{\mathbf{G'}_\lo} + [z \innerprod{\mathbf{p'}_\hi}{\mathbf{b}_\lo}] U + [\cdot] W$ and $R_j = \innerprod{\mathbf{p'}_\lo}{\mathbf{G'}_\hi} + [z \innerprod{\mathbf{p'}_\lo}{\mathbf{b}_\hi}] U + [\cdot] W$.
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* $\prover$ sends $L_j = \innerprod{\mathbf{p'}_\hi}{\mathbf{G'}_\lo} + [z \innerprod{\mathbf{p'}_\hi}{\mathbf{b}_\lo}] U + [\ell_j^*] W$ and $R_j = \innerprod{\mathbf{p'}_\lo}{\mathbf{G'}_\hi} + [z \innerprod{\mathbf{p'}_\lo}{\mathbf{b}_\hi}] U + [r_j^*] W$ where $\ell_j^*$ and $r_j^*$ are fresh blinding factors.
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* $\verifier$ responds with challenge $u_j$ chosen such that $1 + u_{k-1-j} x_3^{2^j}$ is nonzero.
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* $\prover$ and $\verifier$ set $\mathbf{G'} := \mathbf{G'}_\lo + u_j \mathbf{G'}_\hi$ and $\mathbf{b} := \mathbf{b}_\lo + u_j \mathbf{b}_\hi$.
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* $\prover$ sets $\mathbf{p'} := \mathbf{p'}_\lo + u_j^{-1} \mathbf{p'}_\hi$.
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25. $\prover$ sends $c = \mathbf{p'}_0$ and synthetic blinding factor $f$ computed from the elided blinding factors.
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25. $\prover$ sends $c = \mathbf{p'}_0$ and synthetic blinding factor $f = p^{\prime *} + \sum_{j=0}^{k - 1}\left(\ell_j^* \cdot u_j^{-1} + r_j^* \cdot u_j\right)$.
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26. $\verifier$ accepts only if $\sum_{j=0}^{k - 1} [u_j^{-1}] L_j + P' + \sum_{j=0}^{k - 1} [u_j] R_j = [c] \mathbf{G'}_0 + [c \mathbf{b}_0 z] U + [f] W$.
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### Zero-knowledge and Completeness
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@ -825,4 +826,4 @@ Having established that these are each non-rational polynomials of degree at mos
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By construction of $h'(X)$ (from the representation $\repr{H'}{\mathbf{G}}$) in step 7 we know that $h'(x) = h(x)$ where by $h(X)$ we refer to the polynomial of degree at most $(n_g - 1) \cdot (n - 1)$ whose coefficients correspond to the concatenated representations of each $\repr{H_i}{\mathbf{G}}$. As before, suppose that $h(X)$ does _not_ take the form $g'(X) / t(X)$. Then because $h(X)$ is determined prior to the choice of $x$ then by the Schwartz-Zippel lemma we know that it would only agree with $g'(X) / t(X)$ at $(n_g - 1) \cdot (n - 1)$ points at most if the polynomials were not equal. By restricting again $|\badch(\trprefix{\tr'}{x})|/|\ch| \leq \frac{(n_g - 1) \cdot (n - 1)}{|\ch|} \leq \epsilon$ we obtain $h(X) = g'(X) / t(X)$ and because $h(X)$ is a non-rational polynomial by the factor theorem we obtain that $g'(X)$ vanishes over the domain $D$.
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We now have that $g'(X)$ vanishes over $D$ but wish to show that $g(X, C_0, C_1, \cdots)$ vanishes over $D$ at all points to complete the proof. This just involves a sequence of applying the same technique to each of the challenges; since the polynomial $g(\cdots)$ has degree at most $n_g \cdot (n - 1)$ in any indeterminate by definition, and because each polynomial $a_i(X, C_0, C_1, ..., C_{i - 1}, \cdots)$ is determined prior to the choice of concrete challenge $c_i$ by similarly bounding $|\badch(\trprefix{\tr'}{c_i})|/|\ch| \leq \frac{n_g \cdot (n - 1)}{|\ch|} \leq \epsilon$ we ensure that $g(X, C_0, C_1, \cdots)$ vanishes over $D$, completing the proof.
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We now have that $g'(X)$ vanishes over $D$ but wish to show that $g(X, C_0, C_1, \cdots)$ vanishes over $D$ at all points to complete the proof. This just involves a sequence of applying the same technique to each of the challenges; since the polynomial $g(\cdots)$ has degree at most $n_g \cdot (n - 1)$ in any indeterminate by definition, and because each polynomial $a_i(X, C_0, C_1, ..., C_{i - 1}, \cdots)$ is determined prior to the choice of concrete challenge $c_i$ by similarly bounding $|\badch(\trprefix{\tr'}{c_i})|/|\ch| \leq \frac{n_g \cdot (n - 1)}{|\ch|} \leq \epsilon$ we ensure that $g(X, C_0, C_1, \cdots)$ vanishes over $D$, completing the proof.
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ying tong
Daira-Emma Hopwood
adria0
Lasse Bramer Schmidt