===================================================================
RCS file: /home/cvs/OpenXM/doc/sci-semi2001/factorb.tex,v
retrieving revision 1.3
retrieving revision 1.4
diff -u -p -r1.3 -r1.4
--- OpenXM/doc/sci-semi2001/factorb.tex	2001/07/24 09:35:54	1.3
+++ OpenXM/doc/sci-semi2001/factorb.tex	2001/07/25 05:44:01	1.4
@@ -1,4 +1,4 @@
-% $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.2 2001/07/24 08:02:47 noro Exp $
+% $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.3 2001/07/24 09:35:54 noro Exp $
 
 \Large
 \parskip 0pt
@@ -187,11 +187,19 @@ $\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}
 
 \vskip 1cm
 
-\underline{\uc $B%3%s%T%e!<%?$K$ONO5;(B($B7+$jJV$7(B)$B$,;w9g$&(B}
+\underline{\uc $B%3%s%T%e!<%?$K9g$C$?J}K!$O(B?}
 
+\begin{itemize}
+\item {\eec $B!V6a;w!W(B}$B$r$&$^$/;H$&(B
+
 {\eec $BCf4VCM$NDjM}(B} = {\eec $B<B?t$K$*$1$k6a;w(B} $B$NMxMQ(B
 
 $BJL$N6a;w(B $\Rightarrow$ {\ec $B3d$C$?M>$j(B}$B$KCmL\(B
+
+\item $B%3%s%T%e!<%?$O(B{\eec $B7+$jJV$7(B}$B$,F@0U(B
+
+$B6a;w$r7+$jJV$7$F@:EY$r>e$2$k(B 
+\end{itemize}
 \end{slide}
 
 \begin{slide}{}
@@ -211,7 +219,7 @@ $\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}
 $h_1$ $B$r8+$D$1$k(B. 
 
 \item $f(x)-g_k(x)h_k(x)$ $B$N78?t$,(B $p^k$ $B$G3d$j@Z$l$k$h$&$K(B $g_k$, $h_k$ $B$r(B
-$B:n$C$F$$$/(B ($k=1,2,\ldots$)
+$B=g<!:n$C$F$$$/(B ($k=2,3,\ldots$)
 
 \item $g_1$, $h_1$ $B$,@52r$KBP1~$7$F$$$l$P(B, $BE,Ev$J(B $k$ $B$N$H$3$m$G$[$s$H$K3d$j@Z$l$k$@$m$&(B. 
 \end{enumerate}}
@@ -238,21 +246,24 @@ $h_1$ $B$r8+$D$1$k(B. 
 \begin{slide}{}
 \underline{\uc $B5-9f(B $a \equiv b \bmod M$}
 
-$M$ $B$r@0?t$H$9$k(B. {\eec $a \equiv b \bmod M$} $B$H$O(B
+$M$ $B$r@0?t$H$9$k(B.
 
 \begin{itemize}
 \item $a,b$ $B$,@0?t$N$H$-(B, 
 
-{\eec $a-b$ $B$,(B $M$ $B$G3d$j@Z$l$k(B}$B$3$H(B
+{\eec $a \equiv b \bmod M$} $\Leftrightarrow$
+{\eec $a-b$ $B$,(B $M$ $B$G3d$j@Z$l$k(B}
 
 \item $a,b$ $B$,@0?t78?tB?9`<0$N$H$-(B
 
-{\eec $a-b$ $B$N3F78?t$,(B $M$ $B$G3d$j@Z$l$k(B}$B$3$H(B
-\end{itemize}
+{\eec $a \equiv b \bmod M$} $\Leftrightarrow$
+{\eec $a-b$ $B$N3F78?t$,(B $M$ $B$G3d$j@Z$l$k(B}
 
-\vskip 1cm
+\item {\eec $a$ $B$r(B $M$ $B$G3d$C$?M>$j(B} $B$b(B {\eec $a \bmod M$} $B$H=q$/(B
 
-\underline{\uc $a$ $B$r(B $M$ $B$G3d$C$?M>$j$b(B $a \bmod M$ $B$H=q$/(B}
+\item $\equiv$ $B$G7k$P$l$?<0(B : {\eec $BEy<0$HF1MM$K07$($k(B}
+\end{itemize}
+
 \end{slide}
 
 \begin{slide}{}
@@ -321,22 +332,24 @@ $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$
 
 {\eec $(q,r,s,t) = (0,1,1,2), (1,2,0,1)$}
 
-$B0lJ}$,(B $b_0$, $BB>J}$,(B $c_0$ $\Rightarrow$ $B$3$l$i$OF1$8$b$N(B
+($b_0$,$c_0$) $B$N%Z%"$H$7$F$O$3$l$i$OF1$8$b$N(B
 \end{slide}
 
 \begin{slide}{}
-\underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9@-<A(B}
+\underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9>r7o(B}
 
 {\Large\parskip 0pt
 {\eec $b_0 = x^2+1$}, 
 {\eec $c_0 = x^2+x+2$} $B$H$9$k$H(B
 
-\centerline{\eec $f-b_0c_0 \equiv 0 \bmod 3$}
+\centerline{\eec $f \equiv b_0c_0 \bmod 3$}
 
-$f-gh \equiv a_0-b_0c_0+p(a_1-$
+$gh \equiv (b_0+3${\ec $b_1$}$)(c_0+3${\ec$c_1$}$) \bmod 3^2$ $B$h$j(B
+
+$f-gh \equiv a_0-b_0c_0+3(a_1-$
 $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3^2$
 
-$B$h$j(B, $BN>JU$r(B 3 $B$G3d$C$F(B
+$BN>JU$r(B 3 $B$G3d$C$F(B
 
 ${{f-gh}\over 3} \equiv {{a_0-b_0c_0}\over 3}+(a_1-$
 $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3$
@@ -386,9 +399,10 @@ $2r+t \equiv 0 \bmod 3$}
 
 $B0J2<F1MM$K(B, 
 
-\centerline{\ec $b_k = qx+r, c_k = sx+t$}
+\centerline{\ec $b_i = qx+r, c_i = sx+t$}
 
-$B$H$*$$$F(B, $(q,r,s,t)$ $B$NO"N)0l<!J}Dx<0$r2r$1$P(B
+($i=2,3,\ldots$) $B$H$*$$$F(B, $(q,r,s,t)$ $B$NO"N)0l<!J}Dx<0$r=g<!(B
+$B2r$$$F$$$1$P(B
 
 \centerline{\eec $f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1}) \bmod 3^k$}
 
@@ -427,12 +441,12 @@ $k$ & $g_k$ & $h_k$ \\ \hline
 
 $BI=$G8+$k$H(B, {\eec $k=12 \rightarrow 13$ $B$GJQ2=$,$J$$(B}
 
-$\Rightarrow$ {\ec $f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0!}
+$\Rightarrow$ {\ec $f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0}
 
 {\eec
-$f(x) = (x^2+7821x+89347) \times$
+$f(x) =$
 
-$(x^2+9235x+342527)$}
+$ (x^2+7821x+89347)(x^2+9235x+342527)$}
 
 \underline{\uc $B<B:]$K$O(B...}
 
@@ -444,7 +458,7 @@ $(x^2+9235x+342527)$}
 \end{slide}
 
 \begin{slide}{}
-\underline{\uc $\bmod p$ $B$G$NJ,2r$,0lHVBg@Z(B}
+\underline{\uc $\bmod p$ $B$G$NJ,2r$,LdBj(B}
 
 $B3F%9%F%C%W$G=P$FMh$k78?t$NJ}Dx<0(B
 
@@ -586,7 +600,7 @@ $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
 \begin{enumerate}
 \item $0 < s_A < p$ $B$J$k@0?t(B {\eec $s_A$} ($BHkL)(B) $B$r:n$k(B. 
 \item $w_A =$ {\eec $g^{s_A} \bmod p$} $B$r(B B $B$5$s$KAw$k(B.
-\item $s =$ {\eec $w_B^{s_A} \bmod p$} $B$r:n$k(B. 
+\item $B<u$1<h$C$?(B $w_B$ $B$+$i(B $s =$ {\eec $w_B^{s_A} \bmod p$} $B$r:n$k(B. 
 \end{enumerate}
 
 \item {\eec B $B$5$s$N;E;v(B}
@@ -594,7 +608,7 @@ $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
 \begin{enumerate}
 \item $0 < s_B < p$ $B$J$k@0?t(B {\eec $s_B$} ($BHkL)(B) $B$r:n$k(B.
 \item $w_B =$ {\eec $g^{s_B} \bmod p$} $B$r(B A $B$5$s$KAw$k(B.
-\item $s =$ {\eec $w_A^{s_B} \bmod p$} $B$r:n$k(B. 
+\item $B<u$1<h$C$?(B $w_A$ $B$+$i(B $s =$ {\eec $w_A^{s_B} \bmod p$} $B$r:n$k(B. 
 \end{enumerate}
 
 \end{itemize}}
@@ -608,7 +622,7 @@ $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
 
 $B$3$l$G80$,6&M-$G$-$?(B
 
-\item {\eec $w_A$, $w_B$ $B$O0E9f2=$5$l$J$$(B}
+\item {\eec $w_A$, $w_B$ $B$O0E9f2=$NI,MW$J$7(B}
 
 $g^{s_A} \bmod p$ $B$+$i(B $s_A$ $B$r5a$a$k$N$OFq$7$$(B