M pinocchio-groth16.pdf → pinocchio-groth16.pdf

M pinocchio-groth16.tex → pinocchio-groth16.tex

@@ -587,10 +587,12 @@ $$e(g_a^{a_0(\tau)} \cdot g_a^{a_{io}(\tau)} \cdot g_a^{a_m},~ g_b^{b_0(\tau)} \cdot g_b^{b_{io}(\tau)} \cdot g_b^{b_m}) =
 e(g_c^{c_0(\tau)} \cdot g_c^{c_{io}(\tau)} \cdot g_c^{c_m}), g^1)$$
 
 ie.
-$$e([r_a \cdot (a_0(\tau) \cdot a_{io}(\tau) \cdot a_m)]_1,~ [r_b \cdot (b_0(\tau)} \cdot b_{io}(\tau) \cdot b_m)]_2) =
-e([r_c \cdot (c_0(\tau)} \cdot c_{io}(\tau) \cdot c_m)]_1, [1]_2)$$
+\begin{align*}
+    e([r_a \cdot (a_0(\tau) \cdot a_{io}(\tau) \cdot a_m)]_1,~ [r_b \cdot (b_0(\tau) &\cdot b_{io}(\tau) \cdot b_m)]_2) =\\
+&e([r_c \cdot (c_0(\tau) \cdot c_{io}(\tau) \cdot c_m)]_1, [1]_2)
+\end{align*}
 
-Next, it checks that the linear combinations computed over the QAP are in their appropiate spans:
+Next, it checks that the linear combinations computed over the QAP are in their appropriate spans:
 $$e([r_a \cdot a_m']_1, [1]_2) = e([r_a \cdot a_m]_1, [\alpha_a]_2)$$
 $$e([r_b \cdot b_m']_1, [1]_2) = e([r_b \cdot b_m]_1, [\alpha_b]_2)$$
 $$e([r_c \cdot c_m']_1, [1]_2) = e([r_c \cdot c_m]_1, [\alpha_c]_2)$$
@@ -600,7 +602,7 @@ And finally, it checks that the same coefficients were used in each of the linear combinations over the QAP
 
 $$e(g^z, g^{\gamma}) = e(g_a^{a_m} \cdot g_b^{b_m} \cdot g_c^{c_m}, g^{\beta \gamma})$$
 ie.
-$$e([z]_1, [\gamma}]_2 = e([r_a \cdot a_m]_1 \cdot [r_b \cdot b_m]_1 \cdot [r_c \cdot c_m]_1, [\beta \cdot \gamma]_2)$$
+$$e([z]_1, [\gamma]_2 = e([r_a \cdot a_m]_1 \cdot [r_b \cdot b_m]_1 \cdot [r_c \cdot c_m]_1, [\beta \cdot \gamma]_2)$$
 
 recall that $[z]_1$ comes from the proof.