===================================================================
RCS file: /home/cvs/OpenXM/doc/sci-semi2001/factorb.tex,v
retrieving revision 1.3
retrieving revision 1.7
diff -u -p -r1.3 -r1.7
--- OpenXM/doc/sci-semi2001/factorb.tex	2001/07/24 09:35:54	1.3
+++ OpenXM/doc/sci-semi2001/factorb.tex	2001/07/28 06:37:40	1.7
@@ -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 1.6 2001/07/28 03:31:10 noro Exp $
 
 \Large
 \parskip 0pt
@@ -115,9 +115,6 @@ $\Rightarrow$ $B$3$l$G(B{\ec $B@0?t78?t$NB?9`<0$r?t
 
 \begin{slide}{}
 \fbox{\sc 3. $BB?9`<0$N0x?tJ,2r(B --- $BCf3X9b9;E*J}K!(B}
-{
-\Large\parskip 0pt
-
 \begin{enumerate}
 \item {\eec $B4cNOK!(B} ($B2r$H78?t$N4X78(B)
 
@@ -135,7 +132,6 @@ $x^2+ax+b=0$ $B$N:,(B ${-b \pm \sqrt{a^2-4b}} \over 
 
 $\Rightarrow$ $a^2-4b = t^2$ ($t$ : $B@0?t(B) $B$H$+$1$k$+$I$&$+D4$Y$k(B
 \end{enumerate}
-}
 \end{slide}
 
 \begin{slide}{}
@@ -161,12 +157,12 @@ $\Rightarrow$ {\bf \ec $x^2-t=0$ $B$N@0?t:,$rC5$9J}K!
 
 $\Rightarrow$ {\ec $B:,$rC5$9J}K!$,E,MQ$G$-$k(B}
 
-{\eec $B:,5r(B : $BCf4VCM$NDjM}(B}
-$B!V(B$f(a) < 0, f(b) > 0$ $B$J$i(B $a$, $b$ $B$N4V$K(B $f(c)=0$ $B$J$k(B $c$ $B$,$"$k(B.$B!W(B
-
 \begin{itemize}
 \item {\eec $BFsJ,K!(B}
 
+{\eec $B:,5r(B : $BCf4VCM$NDjM}(B}
+$B!V(B$f(a) < 0, f(b) > 0$ $B$J$i(B $a$, $b$ $B$N4V$K(B $f(c)=0$ $B$J$k(B $c$ $B$,$"$k(B.$B!W(B
+
 $B6h4V$rH>J,$:$D69$a$FDI$$9~$`(B
 
 \item {\eec Newton $BK!(B}
@@ -187,23 +183,28 @@ $\Rightarrow$ {\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}{}
 \fbox{\sc 4. $p$-$B?J6a;w$K$h$kB?9`<0$N0x?tJ,2r(B}
-{\Large\parskip 0pt
 
 \underline{\uc $B86M}(B} : {\eec $B@0?t(B $m$ $B$,(B 0} $\Leftrightarrow$
 
 {\eec $m$ $B$O$I$s$J@0?t$G$b3d$j@Z$l$k(B}
 
-({\eec $m$ $B$O==J,Bg$-$$@0?t$G3d$j@Z$l$k(B})
-
 $B$?$H$($P(B, 
 
 \begin{enumerate}
@@ -211,10 +212,10 @@ $\Rightarrow$ {\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}}
+\end{enumerate}
 \end{slide}
 
 \begin{slide}{}
@@ -238,21 +239,24 @@ $h_1$ $B$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}{}
@@ -278,6 +282,37 @@ $p = 3$ $B$H$9$k$H(B $a_0(x)=x^4+x^3+x+2$
 \end{slide}
 
 \begin{slide}{}
+\underline{\uc $f(x)$ $B$N(B $3$-$B?JE83+(B}
+
+$f(x)=(x^4+x^3+x+2)+3^1\cdot x+$
+
+$3^2(2x^3+x+2)+
+3^3(x^3+x^2+2x+2)+$
+
+$3^4(x^2+x+1)+
+3^5 \cdot x^3+
+3^6(2x^3+x+2)+$
+
+$3^7(x^3+x^2+x)+
+3^8(2x^3+x^2+2x)+$
+
+$3^9(x^2+2x+1)+
+3^{11}(2x^2+x+1)+$
+
+$3^{12}(x^2+2x+1)+
+3^{13}(x+1)+
+3^{14} \cdot 2+$
+
+$3^{15}(2x^2+x+2)+
+3^{16}(x^2+2)+
+3^{17} \cdot 2+$
+
+$3^{19} \cdot 2+
+3^{20}(x+2)+
+3^{21} \cdot 2$
+\end{slide}
+
+\begin{slide}{}
 \underline{\uc $B0l<!0x;R$,$"$k$+(B?}
 
 {\ec $b_0(x) = x+q$}, 
@@ -321,36 +356,36 @@ $q$, $r$, $s$, $t$ $B$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$b_1$}$))\bmod 3$
 
 $B:8JU$O(B $3$ $B$G2?2s$G$b3d$l$k(B $\Rightarrow$  $B1&JU$O(B $3$ $B$G3d$l$k(B
 
-$BJd@59`(B {\ec $b_1$}, {\ec $c_1$} : $x^2$ $B$N78?t$O(B 0 $B$H$7$F$h$$(B}
+$BJd@59`(B {\ec $b_1$}, {\ec $c_1$} : $x^2$ $B$N78?t$O(B 0 $B$H$7$F$h$$(B
 
 \end{slide}
 
 \begin{slide}{}
 \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9J}Dx<0(B}
 
-{\Large\parskip 0pt
 {\ec $b_1 = qx+r$},
 {\ec $c_1 = sx+t$} $B$H$*$/(B.
 
@@ -372,23 +407,40 @@ $2r+t \equiv 0 \bmod 3$}
 
 $B$3$s$I$OO"N)0l<!J}Dx<0(B($B9gF1<0(B). $B$3$l$r2r$/$H(B
 
-{\eec $(q,r,s,t) = (0,1,0,1)$} $B$9$J$o$A(B {\eec $b_1 = 1$}, {\eec $c_1 = 1$}}
+{\eec $(q,r,s,t) = (0,1,0,1)$} $B$9$J$o$A(B {\eec $b_1 = 1$}, {\eec $c_1 = 1$}
 
 \end{slide}
 
 \begin{slide}{}
-\underline{\uc $BFs<!0x;R$D$E$-(B --- $b_k$, $c_k$ $B$bF1MM(B}
+\underline{\uc $BFs<!0x;R$D$E$-(B --- $b_2$, $c_2$ $B$O(B $\bmod 3^3$ $B$G(B}
 
-{\Large\parskip 0pt
-$B$3$l$G(B, 
+$B$3$l$G(B, {\eec $f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$}
 
-\centerline{\eec $f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$}
+$B<!$O(B $a_2$, $b_2$, $c_2$ $B$^$G$H$C$F(B $\bmod 3^3$ $B$G8+$k(B
 
+\centerline{\eec $f \equiv a_0+3a_1+3^2a_2 \bmod 3^3$}
+
+\centerline{\ec $f \equiv (b_0+3b_1+3^2b_2)(c_0+3c_1+3^2c_2) \bmod 3^3$}
+
+$B$+$i(B {$((a_0+3a_1)-(b_0+3b_1)(c_0+3c_1))+$}
+
+\centerline{$3^2(a_2-(c_0b_2+b_0c_2)) \equiv 0 \bmod 3^3$}
+
+$BN>JU$r(B $3^2$ $B$G3d$C$F(B, {\ec $b_2=qx+r$}, {\ec $c_2=sx+t$}
+
+$\Rightarrow$ $BA0$HF1MM$K(B{\eec $BO"N)0l<!9gF1<0(B}$B$rF@$k(B
+
+\end{slide}
+
+\begin{slide}{}
+\underline{\uc $BFs<!0x;R$D$E$-(B --- $b_k$, $c_k$ $B$bF1MM(B}
+
 $B0J2<F1MM$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 $f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1}) \bmod 3^k$}
 
@@ -396,8 +448,7 @@ $2r+t \equiv 0 \bmod 3$}
 
 \centerline{\eec $f \equiv g_kh_k \bmod 3^k$}
 
-$B$H$J$k(B $g_k, h_k$ $B$,7h$^$k(B. }
-
+$B$H$J$k(B $g_k, h_k$ $B$,7h$^$k(B.
 \end{slide}
 
 \begin{slide}{}
@@ -427,12 +478,12 @@ $k$ & $g_k$ & $h_k$ \\ \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 +495,7 @@ $(x^2+9235x+342527)$}
 \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
 
@@ -573,7 +624,6 @@ $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
 \end{slide}
 
 \begin{slide}{}
-{\Large\parskip 0pt
 \underline{\uc A $B$5$s$H(B B $B$5$s$,80$r6&M-(B --- Diffie-Hellman}
 
 \begin{itemize}
@@ -586,7 +636,7 @@ $\Rightarrow$ $B7W;;5!$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}
@@ -594,10 +644,10 @@ $\Rightarrow$ $B7W;;5!$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}}
+\end{itemize}
 \end{slide}
 
 \begin{slide}{}
@@ -608,7 +658,7 @@ $\Rightarrow$ $B7W;;5!$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