fix: Poseidon spec's sparse-factorization w vector (#1278)

Co-authored-by: Jiaying Wang <[email protected]>
Co-authored-by: Ian Davis <[email protected]>

Author: 3 people

content/algorithms/crypto/poseidon.md

@@ -706,14 +706,10 @@ $\overline{\underline{\Function \textsf{sparse\_factorize}(m \typecolon \Zp^{[t
     0 & m_{t - 1, 1} & \dots & m_{t - 1, t - 1}
 \end{bmatrix} \\~
 \\
-\line{3} \wb \typecolon \Zp^{[t - 1 {\times} 1]} = \hat{m}_{\ast, 0} = \begin{bmatrix}
-    \hat{m}_{0, 0} \\
+\line{3} \wb \typecolon \Zp^{[t - 1 {\times} 1]} = m_{1 \dotdot, 0} = \begin{bmatrix}
+    m_{1, 0} \\
     \vdots \\
-    \hat{m}_{t - 2, 0}
-\end{bmatrix} = m_{1 \dotdot, 1} = \begin{bmatrix}
-    m_{1, 1} \\
-    \vdots \\
-    m_{t - 1, 1}
+    m_{t - 1, 0}
 \end{bmatrix} \\~
 \\
 \line{4} \hat\wb \typecolon \Zp^{[t - 1 {\times} 1]} = \hat{m}^{\neg 1} {\times} \hso \wb = \begin{bmatrix}
@@ -738,7 +734,7 @@ $\overline{\underline{\Function \textsf{sparse\_factorize}(m \typecolon \Zp^{[t
 **Algorithm Comments:**\
 **Line 1.** $\hat{m}$ is a submatrix of $m$ which excludes $m$'s first row and first column.\
 **Line 2.** $m'$ is a copy of $m$ where $m$'s first row and first column have been replaced with $[1, 0, \dots, 0]$.\
-**Line 3.** $\wb$ is a column vector which is the first column of $\hat{m}$ or the second column of $m$ excluding the first row's value.\
+**Line 3.** $\wb$ is a column vector whose values are the first column of $m$ excluding $m$'s first row.
 **Line 4.** $\hat\wb$ is the matrix-column vector product of $\hat{m}^{\neg 1}$ and $\wb$.\
 **Line 5.** $m''$ is a sparse matrix whose first row is the first row of $m$, remaining first column is $\hat\wb$, and remaining entries are the identity matrix.