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Diff for /OpenXM/doc/compalg/factor.tex between version 1.4 and 1.5

version 1.4, 2000/10/03 01:23:58 version 1.5, 2001/02/07 07:17:46
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 %$OpenXM: OpenXM/doc/compalg/factor.tex,v 1.3 2000/03/28 02:02:29 noro Exp $  %$OpenXM: OpenXM/doc/compalg/factor.tex,v 1.4 2000/10/03 01:23:58 noro Exp $
 \chapter{$BB?9`<0$N0x?tJ,2r(B}  \chapter{$BB?9`<0$N0x?tJ,2r(B}
   
 \section{$BM-8BBN(B}  \section{$BM-8BBN(B}
Line 423  $Q \leftarrow \pi$ $B$N9TNsI=8=(B\\
Line 423  $Q \leftarrow \pi$ $B$N9TNsI=8=(B\\
 $\{e_1 = 1, e_2, \cdots, e_r\} \leftarrow \Ker(Q-I)$ $B$N(B $K$-$B4pDl(B\\  $\{e_1 = 1, e_2, \cdots, e_r\} \leftarrow \Ker(Q-I)$ $B$N(B $K$-$B4pDl(B\\
 if ($r = 1$) then return $F$\\  if ($r = 1$) then return $F$\\
 while\= ($|F| < r$) do \{\\  while\= ($|F| < r$) do \{\\
 \> $g \leftarrow F$ $B$N85(B, \quad $F \leftarrow F \backslash \{g\}$\\  
 \> $(c_1,\cdots,c_r) \leftarrow$ $BMp?t%Y%/%H%k(B ($c_i \in GF(q)$)\\  \> $(c_1,\cdots,c_r) \leftarrow$ $BMp?t%Y%/%H%k(B ($c_i \in GF(q)$)\\
 \> $e \leftarrow \sum c_ie_i$ \\  \> $e \leftarrow \sum c_ie_i$ \\
 \> if \= $p=2$\\  \> if \= $p=2$\\
 \>\> $E \leftarrow \Tr(e)$\\  \>\> $E \leftarrow \Tr(e) \bmod f$\\
 \>else\\  \>else\\
 \>\> $E \leftarrow e^{(q-1)/2}-1$\\  \>\> $E \leftarrow e^{(q-1)/2}-1 \bmod f$\\
 \> $h \leftarrow \GCD(g,E)$\\  \> $F_1 \leftarrow \emptyset$\\
 \> if $h \neq 1,g$\\  \> while \= ($F \neq \emptyset$) do \{\\
 \>\> $F \leftarrow F \cup \{h,g/h\}$\\  \>        \> $g \leftarrow F$ $B$N85(B, \quad $F \leftarrow F \backslash \{g\}$\\
   \>        \> $h \leftarrow \GCD(g,E)$\\
   \>        \> if \= $h \neq 1,g$\\
   \>        \> \> $F_1 \leftarrow F_1 \cup \{h,g/h\}$\\
   \>        \> else \\
   \>        \> \> $F_1 \leftarrow F_1 \cup \{g\}$\\
   \> \}\\
   \>   $F \leftarrow F_1$ \\
 \}\\  \}\\
 return F  return F
 \end{tabbing}  \end{tabbing}
Line 554  Input : $f(x) \in GF(q)[x]$, $q=p^n$, $f$ $B$OL5J?J}$
Line 560  Input : $f(x) \in GF(q)[x]$, $q=p^n$, $f$ $B$OL5J?J}$
 Output : $f(x) = \prod f_i$, $f$ $B$N(B $B4{Ls0x;RJ,2r(B\\  Output : $f(x) = \prod f_i$, $f$ $B$N(B $B4{Ls0x;RJ,2r(B\\
 $r \leftarrow \deg(f)/d,\quad F \leftarrow \{f\}$\\  $r \leftarrow \deg(f)/d,\quad F \leftarrow \{f\}$\\
 while\= ($|F| < r$) do \{\\  while\= ($|F| < r$) do \{\\
 \> $h \leftarrow F$ $B$N(B $\deg(h)>d$ $B$J$k85(B,\quad $F \leftarrow F \backslash \{h\}$\\  
 \> $g \leftarrow 2d-1$ $B<!$N%i%s%@%`$JB?9`<0(B\\  \> $g \leftarrow 2d-1$ $B<!$N%i%s%@%`$JB?9`<0(B\\
 \> if \= $p=2$\\  \> if \= $p=2$\\
 \>\>$G \leftarrow \sum_{j=0}^{rd-1}g^{2^i}$\\  \>\>$G \leftarrow \sum_{j=0}^{rd-1}g^{2^i} \bmod f$\\
 \> else\\  \> else\\
 \>\> $G \leftarrow g^{(q^d-1)/2}-1$\\  \>\> $G \leftarrow g^{(q^d-1)/2}-1 \bmod f$\\
 \> $z \leftarrow \GCD(h,G)$\\  \> $F_1 \leftarrow \emptyset$\\
 \> if $z \neq 1,h$ \\  \> while \= ($F \neq \emptyset$) do \{\\
 \>\> $F \leftarrow F \cup \{z,h/z\}$\\  \>        \> $h \leftarrow F$ $B$N(B $\deg(h)>d$ $B$J$k85(B,\quad $F \leftarrow F \backslash \{h\}$\\
   \>        \> $z \leftarrow \GCD(h,G)$\\
   \>        \> if \= $z \neq 1,h$\\
   \>        \> \> $F_1 \leftarrow F_1 \cup \{z,h/z\}$\\
   \>        \> else \\
   \>        \> \> $F_1 \leftarrow F_1 \cup \{h\}$\\
   \> \}\\
   \>   $F \leftarrow F_1$ \\
 \}\\  \}\\
 return $F$\\  return $F$\\
 \end{tabbing}  \end{tabbing}
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