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[book] Fix Sarkar square root explanation

This commit is contained in:
therealyingtong 2021-06-30 19:31:23 +08:00
parent 93aabf26c6
commit 69e7fb8221
1 changed files with 5 additions and 5 deletions
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10
book/src/design/implementation/fields.md
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@ -29,7 +29,7 @@ $$
$$
invtab = \begin{bmatrix}
(g^{2^{-24}})^0 & (g^{2^{-24}})^1 & ... & (g^{2^{-24}})^{2^8 - 1}
(g^{-2^{24}})^0 & (g^{-2^{24}})^1 & ... & (g^{-2^{24}})^{2^8 - 1}
\end{bmatrix}
$$
@ -41,7 +41,7 @@ Let $x_3 = x, x_2 = x_3^{2^8}, x_1 = x_2^{2^8}, x_0 = x_1^{2^8}.$
### i = 0, 1
Using $invtab$, we lookup $t_0$ such that
$$
x_0 = (g^{2^{-24}})^{t_0} \implies x_0 \cdot g^{t_0 \cdot 2^{24}} = 1.
x_0 = (g^{-2^{24}})^{t_0} \implies x_0 \cdot g^{t_0 \cdot 2^{24}} = 1.
$$
Define $\alpha_1 = x_1 \cdot (g^{2^{16}})^{t_0}.$
@ -50,7 +50,7 @@ Define $\alpha_1 = x_1 \cdot (g^{2^{16}})^{t_0}.$
Lookup $t_1$ s.t.
$$
\begin{array}{ll}
\alpha_1 = (g^{2^{-24}})^{t_1} &\implies x_1 \cdot (g^{2^{16}})^{t_0} = (g^{2^{-24}})^{t_1} \\
\alpha_1 = (g^{-2^{24}})^{t_1} &\implies x_1 \cdot (g^{2^{16}})^{t_0} = (g^{-2^{24}})^{t_1} \\
&\implies
x_1 \cdot g^{(t_0 + 2^8 \cdot t_1) \cdot 2^{16}} = 1.
\end{array}
@ -63,7 +63,7 @@ Lookup $t_2$ s.t.
$$
\begin{array}{ll}
\alpha_2 = (g^{2^{-24}})^{t_2} &\implies x_2 \cdot (g^{2^8})^{t_0 + 2^8\cdot {t_1}} = (g^{2^{-24}})^{t_2} \\
\alpha_2 = (g^{-2^{24}})^{t_2} &\implies x_2 \cdot (g^{2^8})^{t_0 + 2^8\cdot {t_1}} = (g^{-2^{24}})^{t_2} \\
&\implies x_2 \cdot g^{(t_0 + 2^8 \cdot t_1 + 2^{16} \cdot t_2) \cdot 2^8} = 1.
\end{array}
$$
@ -75,7 +75,7 @@ Lookup $t_3$ such that
$$
\begin{array}{ll}
\alpha_3 = (g^{2^{-24}})^{t_3} &\implies x_3 \cdot g^{t_0 + 2^8\cdot {t_1} + 2^{16} \cdot t_2} = (g^{2^{-24}})^{t_3} \\
\alpha_3 = (g^{-2^{24}})^{t_3} &\implies x_3 \cdot g^{t_0 + 2^8\cdot {t_1} + 2^{16} \cdot t_2} = (g^{-2^{24}})^{t_3} \\
&\implies x_3 \cdot g^{t_0 + 2^8 \cdot t_1 + 2^{16} \cdot t_2 + 2^{24} \cdot t_3} = 1.
\end{array}
$$