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[book] Fix p_poly to match implementation; specify synthetic blinding factor f construction #777

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[book] Fix p_poly to match implementation; specify synthetic blinding factor f construction #777

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67fb624
Update p_poly in protocol to match implementation
therealyingtong May 21, 2023
7d77466
Update book/src/design/protocol.md
therealyingtong May 22, 2023
62757bf
Fix step 14; fix v in step 18
therealyingtong May 23, 2023
ff9f593
added f calculation to protocol
May 25, 2023
2038845
changed challenge x to indeterminate X in step 19
May 25, 2023
dec8301
Formatting and copy fixes
therealyingtong May 27, 2023
ed2354c
[book] More detailed description of synthetic blinding factor
therealyingtong Jun 6, 2023
3039662
Apply suggestions from code review
therealyingtong Jun 27, 2023
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4 changes: 2 additions & 2 deletions book/src/design/protocol.md
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Expand Up @@ -369,7 +369,7 @@ $$
15. $\verifier$ responds with challenge $x_3$.
16. $\prover$ sends $\mathbf{u} \in \field^{n_q}$ such that $\mathbf{u}_i = q_i(x_3)$ for all $i \in [0, n_q)$.
17. $\verifier$ responds with challenge $x_4$.
18. $\verifier$ sets $P = Q' + x_4 \sum\limits_{i=0}^{n_q - 1} [x_4^i] Q_i$ and $v = $
18. $\verifier$ sets $P = [x_4^{n_q}]Q' + \sum\limits_{i=0}^{n_q - 1} [x_4^{n_q - 1 - i}] Q_i$ and $v = $
$$
\sum\limits_{i=0}^{n_q - 1}
\left(
Expand All @@ -389,7 +389,7 @@ x_2^i
+
x_4 \sum\limits_{i=0}^{n_q - 1} x_4 \mathbf{u}_i
$$
19. $\prover$ sets $p(X) = q'(X) + [x_4] \sum\limits_{i=0}^{n_q - 1} x_4^i q_i(X)$.
19. $\prover$ sets $p(X) = q'(X) \cdot x_4^{n_q} + \sum\limits_{i=0}^{n_q - 1} x_4^{n_q - 1 - i} q_i(X)$.
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 $s(X)$.
21. $\verifier$ responds with challenges $\xi, z$.
22. $\verifier$ sets $P' = P - [v] \mathbf{G}_0 + [\xi] S$.
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