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version 1.1, 2001/07/23 06:46:51 version 1.4, 2001/07/25 05:44:01
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 % $OpenXM$  % $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.3 2001/07/24 09:35:54 noro Exp $
   
 \Large  \Large
 \parskip 0pt  \parskip 0pt
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{\bf 1. $B$O$8$a$K(B}  \fbox{\sc 1. $B$O$8$a$K(B}
   
 computer = compute $B$9$k$?$a$N$b$N(B  computer = compute $B$9$k$?$a$N$b$N(B
   
 compute = $B7W;;$9$k(B  compute = {\ec $B7W;;(B}$B$9$k(B
   
 $B:G6a$G$O(B communication $B$N<jCJ$H$J$C$F$7$^$C$?(B  $B:G6a$G$O(B {\ec $B%G%8%?%k>pJsDL?.(B} $B$N<jCJ$H$J$C$F$7$^$C$?(B
   
 $B$5$^$6$^$J>pJs$r%G%8%?%k2=(B ($BId9f2=(B) $B$7$F%M%C%H%o!<%/$rDL$7$FAw<u?.(B  
   
 $\Rightarrow$ $B!V7W;;!W$K;H$C$F$$$k?M$O$4$/>/?t(B  $\Rightarrow$ $B!V7W;;!W$K;H$C$F$$$k?M$O$4$/>/?t(B
   
 $BNc(B : email, $B%&%'%V(B $\cdots$ $B!V%$%s%?!<%M%C%H$9$k!W(B  {\bf $BNc(B} : email, $B%&%'%V(B {\eec $B!V%$%s%?!<%M%C%H$9$k!W(B}
   
 $B$7$+$7(B, $B7W;;5!$NG=NO$O0[MM$K8~>e(B  $B$7$+$7(B, $B7W;;5!$NG=NO$O0[MM$K8~>e(B
   
 $B7W;;$K;H$o$J$$$N$O$b$C$?$$$J$$(B($B$H;W$&(B)  
   {\ec $B7W;;$K;H$o$J$$$N$O$b$C$?$$$J$$(B}($B$H;W$&(B)
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{\bf 2. $B%3%s%T%e!<%?$K$D$$$F$N%$%m%O(B}  \fbox{\sc 2. $B%3%s%T%e!<%?$K$D$$$F$N%$%m%O(B}
   
 \begin{itemize}  \begin{itemize}
 \item CPU  \item {\eec CPU}
   
 $B%W%m%0%i%`$K=>$C$FL?Na$r<B9T(B  $B%W%m%0%i%`$K=>$C$FL?Na$r<B9T(B
   
 \item $B%a%b%j(B  \item {\eec $B%a%b%j(B}
   
 $B%W%m%0%i%`(B, $B%G!<%?$rCV$/>l=j(B. $B>l=j(B ($BHVCO(B) $B$r;XDj$7$F9bB.$K=P$7F~$l$,$G$-$k(B.  $B%W%m%0%i%`(B, $B%G!<%?$rCV$/>l=j(B. $B>l=j(B ($BHVCO(B) $B$r;XDj$7$F9bB.$K=P$7F~$l$,$G$-$k(B.
   
 \item $B%l%8%9%?(B  \item {\eec $B%l%8%9%?(B}
   
 CPU $B$,;}$C$F$$$kFCJL$J%a%b%j$G(B, $B1i;;$NBP>]$K$J$k(B. $B?t$OB?$/$J$/(B, $BBg$-$5(B  CPU $B$,;}$C$F$$$kFCJL$J%a%b%j$G(B, $B1i;;$NBP>]$K$J$k(B. $B?t$OB?$/$J$/(B, $BBg$-$5(B
 ($BD9$5(B) $B$b>.$5$$(B.  ($BD9$5(B) $B$b>.$5$$(B.
Line 44  CPU $B$,;}$C$F$$$kFCJL$J%a%b%j$G(B, $B1i;;$NBP>]$K$
Line 43  CPU $B$,;}$C$F$$$kFCJL$J%a%b%j$G(B, $B1i;;$NBP>]$K$
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $BL?Na$NNc(B}  \underline{\uc $BL?Na$NNc(B}
 \begin{itemize}  \begin{itemize}
 \item 10 $BHVCO$+$i(B 1 $B%P%$%HFI$s$G(B A $B%l%8%9%?$KF~$l$h(B  \item 10 $BHVCO$+$i(B 1 $B%P%$%HFI$s$G(B A $B%l%8%9%?$KF~$l$h(B
 \item A, B $B%l%8%9%?$NCM$rB-$7$F(B C $B%l%8%9%?$KF~$l$h(B  \item A, B $B%l%8%9%?$NCM$rB-$7$F(B C $B%l%8%9%?$KF~$l$h(B
 \item A $B%l%8%9%?$NCM$,(B 0 $B$J$i(B 100 $BHVCO@h$KHt$Y(B  \item A $B%l%8%9%?$NCM$,(B 0 $B$J$i(B 100 $BHVCO@h$KHt$Y(B
 \end{itemize}  \end{itemize}
   
 \underline{\bf $B07$($k?t(B}  \underline{\uc $B07$($k?t(B}
   
 $B%l%8%9%?$NBg$-$5$G07$($k?t$NHO0O$,7h$^$k(B.  $B%l%8%9%?$NBg$-$5(B = $B07$($k?t$NHO0O(B
   
 32$B%S%C%H%l%8%9%?(B $\Rightarrow$ 0 $B$+$i(B $2^{32}-1$ $B$^$G$N@0?t$7$+07$($J$$(B  32$B%S%C%H%l%8%9%?(B $\Rightarrow$ 0 $B$+$i(B $2^{32}-1$ $B$^$G$N@0?t$7$+07$($J$$(B
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $B?t3X$K;H$&>l9g$r9M$($k$H(B...}  \underline{\uc $B?t3X$K;H$&>l9g$r9M$($k$H(B...}
   
 $11111111111 \times 11111111111$  $11111111111 \times 11111111111$
   
 $\Rightarrow 1332508849$ ??? (=$B7k2L$r(B $2^{32}$ $B$G3d$C$?M>$j(B)  $\Rightarrow$ {\ec 1332508849} ??? (=$B7k2L$r(B $2^{32}$ $B$G3d$C$?M>$j(B)
   
 $B$+$H$$$C$F(B  $B$+$H$$$C$F(B
   
Line 74  $\Rightarrow 1.234567 \times 10^{20}$ 
Line 73  $\Rightarrow 1.234567 \times 10^{20}$ 
   
 $B$b:$$k(B  $B$b:$$k(B
   
 $B8m:9$,F~$k$H?t3X$H$7$F$OL50UL#$J7W;;(B  {\ec $B8m:9$,F~$k$H?t3X$H$7$F$OL50UL#$J7W;;(B}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $B$H$j$"$($:Bg$-$J@0?t$O07$($J$$$H$$$1$J$$(B}  \underline{\uc $B$H$j$"$($:Bg$-$J@0?t$O07$($J$$$H$$$1$J$$(B}
   
 $\Rightarrow$ $B%W%m%0%i%`$r=q$1$P$h$$(B  $\Rightarrow$ {\eec $B%W%m%0%i%`(B}$B$r=q$1$P$h$$(B
   
 $B%a%b%j>e$K@0?t$rJB$Y$F(B, $B%3%s%T%e!<%?$K!VI.;;!W$r$5$;$l$P$h$$(B  $B%a%b%j>e$K@0?t$rJB$Y$F(B, $B%3%s%T%e!<%?$K(B {\eec $B!VI.;;!W(B}$B$r$5$;$l$P$h$$(B
   
 \begin{itemize}  \begin{itemize}
 \item $B?M4V(B  \item {\eec $B?M4V(B}
   
 $B$R$H$1$?(B : 0 $B0J>e(B 9 $B0J2<(B  $B$R$H$1$?(B : 0 $B0J>e(B 9 $B0J2<(B
   
 \item $B%3%s%T%e!<%?(B  \item {\eec $B%3%s%T%e!<%?(B}
   
 $B$R$H$1$?(B : 0 $B0J>e(B $2^{32}-1$ $B0J2<(B  $B$R$H$1$?(B : 0 $B0J>e(B $2^{32}-1$ $B0J2<(B
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $BNc(B : $B@0?t$NB-$7;;(B}  \underline{\uc $BNc(B : $B@0?t$NB-$7;;(B}
   
 \begin{tabular}{ccccc}  \begin{tabular}{ccccc} \\
 & 5 & 4001257187 & 1914644777 & (= $3^{42}$) \\  & 5 & 4001257187 & 1914644777 & (= $3^{42}$) \\
 + &  & 2830677074 & 689956897 & (= $3^{40}$) \\ \hline  + &  & 2830677074 & 689956897 & (= $3^{40}$) \\ \hline
 & 6 & 2536966965 & 2604601674 &  & 6 & 2536966965 & 2604601674 &
 \end{tabular}  \end{tabular}
   
 \underline{\bf $B0lJQ?tB?9`<0(B}  \vskip 1cm
   
 $B3F<!?t$N78?t$rJB$l$P$h$$(B  \underline{\uc $B0lJQ?tB?9`<0(B}
   
 $\Rightarrow$ $B$3$l$G(B, $B@0?t78?t$NB?9`<0$r?t3XE*$K07$($k(B  $B3F<!?t$N78?t$rJB$Y$l$P$h$$(B
   
   $\Rightarrow$ $B$3$l$G(B{\ec $B@0?t78?t$NB?9`<0$r?t3XE*$K07$($k(B}
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{\bf 3. $BB?9`<0$N0x?tJ,2r(B --- $BCf3X9b9;E*J}K!(B}  \fbox{\sc 3. $BB?9`<0$N0x?tJ,2r(B --- $BCf3X9b9;E*J}K!(B}
   {
   \Large\parskip 0pt
   
 \begin{enumerate}  \begin{enumerate}
 \item $B4cNOK!(B ($B2r$H78?t$N4X78(B)  \item {\eec $B4cNOK!(B} ($B2r$H78?t$N4X78(B)
   
 $x^2+ax+b \Rightarrow$ $BB-$7$F(B $a$, $B$+$1$F(B $b$ $B$K$J$k?t$NAH(B  $x^2+ax+b \Rightarrow$ $BB-$7$F(B $a$, $B$+$1$F(B $b$ $B$K$J$k?t$NAH(B
   
 $x^3+ax^2+bx+c$ $B$O$I$&$9$k(B?  $x^3+ax^2+bx+c$ $B$O$I$&$9$k(B?
   
 \item $B0x?tDjM}(B  \item {\eec $B0x?tDjM}(B}
   
 $BBeF~$7$F(B 0 $B$K$J$k?t$rC5$9(B ($B$I$&$d$C$FC5$9(B?)  $BBeF~$7$F(B 0 $B$K$J$k?t$rC5$9(B ($B$I$&$d$C$FC5$9(B?)
   
 \item $B2r$N8x<0(B  \item {\eec $B2r$N8x<0(B}
   
 $x^2+ax+b$ $B$N:,(B ${-b \pm \sqrt{a^2-4b}} \over 2$  $x^2+ax+b=0$ $B$N:,(B ${-b \pm \sqrt{a^2-4b}} \over 2$
   
 $\Rightarrow$ $a^2-4b = t^2$ ($t$ : $B@0?t(B) $B$H$+$1$k$+$I$&$+D4$Y$k(B  $\Rightarrow$ $a^2-4b = t^2$ ($t$ : $B@0?t(B) $B$H$+$1$k$+$I$&$+D4$Y$k(B
 \end{enumerate}  \end{enumerate}
   }
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $B4cNOK!$OLdBj$rFq$7$/$7$F$$$k(B}  \underline{\uc $B4cNOK!$OLdBj$rFq$7$/$7$F$$$k(B}
   
 $BNc(B : $x^2+11508x+28386587$  $BNc(B : $x^2+11508x+28386587$
   
 $28386587=3581\times 7927$ $B$,$a$N$3$GJ,$+$k?M$O>/$J$$(B($B$H;W$&(B)  $28386587=3581\times 7927$ $B$,4cNO$GJ,$+$k?M$O>/$J$$(B($B$H;W$&(B)
   
 \underline{\bf $B2r$N8x<0K!$OM-K>(B}  \vskip 1cm
   
   \underline{\uc $B2r$N8x<0K!$OM-K>(B}
   
 $(a^2-4b)/4 = 4717584 = 2172^2$ $B$J$i2?$H$+$J$k(B?  $(a^2-4b)/4 = 4717584 = 2172^2$ $B$J$i2?$H$+$J$k(B?
   
 $\Rightarrow$ $x^2-t$ $B$N@0?t:,$rC5$9J}K!$,$"$l$P$h$$(B  $\Rightarrow$ {\bf \ec $x^2-t=0$ $B$N@0?t:,$rC5$9J}K!$,$"$l$P$h$$(B}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf 3 $B<!0J2<$N>l9g(B}  \underline{\uc 3 $B<!0J2<$NB?9`<0(B}
   
 \underline{\bf $B@0?t>e$GJ,2r$G$-$k$J$i(B, $B0l<!0x;R$r;}$D(B}  {\eec $B@0?t>e$GJ,2r$G$-$k$J$i(B, $B0l<!0x;R$r;}$D(B}
   
 $B:,$rC5$9J}K!$,E,MQ$G$-$k(B.  $\Rightarrow$ {\ec $B:,$rC5$9J}K!$,E,MQ$G$-$k(B}
   
 $B:,5r(B : $BCf4VCM$NDjM}(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  $B!V(B$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}  \begin{itemize}
 \item $BFsJ,K!(B  \item {\eec $BFsJ,K!(B}
   
 $B6h4V$rH>J,$:$D69$a$FDI$$9~$`(B  $B6h4V$rH>J,$:$D69$a$FDI$$9~$`(B
   
 \item Newton $BK!(B  \item {\eec Newton $BK!(B}
   
 $BFsJ,K!$h$j$:$C$H9bB.(B  $BFsJ,K!$h$j$:$C$H9bB.(B
 \end{itemize}  \end{itemize}
Line 171  $\Rightarrow$ $x^2-t$ $B$N@0?t:,$rC5$9J}K!$,$"$l$P$h$
Line 177  $\Rightarrow$ $x^2-t$ $B$N@0?t:,$rC5$9J}K!$,$"$l$P$h$
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf 4 $B<!0J>e$N>l9g(B}  \underline{\uc 4 $B<!0J>e$N>l9g(B}
   
 $B$$$m$$$m$JJ,2r%Q%?!<%s$,$"$jF@$k(B  $B$$$m$$$m$JJ,2r%Q%?!<%s$,$"$jF@$k(B
   
 4 $B<!(B = 2 $B<!(B $\times$ 2 $B<!(B  4 $B<!(B = 2 $B<!(B $\times$ 2 $B<!(B
   
 $\Rightarrow$ $B:,$rC5$9J}K!$OE,MQ:$Fq(B  $\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}
   
 \underline{\bf $B%3%s%T%e!<%?MQ$K$O(B, $B$b$&>/$7E}0lE*$JJ}K!$,I,MW(B}  \vskip 1cm
   
 $BCf4VCM$NDjM}$O(B, $B<B?t$K$*$1$k(B {\bf $B6a;w(B} $B$NMxMQ(B  \underline{\uc $B%3%s%T%e!<%?$K9g$C$?J}K!$O(B?}
   
 $BJL$N6a;w(B $\Rightarrow$ {\bf $B3d$C$?M>$j(B}$B$KCmL\(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}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{\bf 4. $p$-$B?J6a;w$K$h$kB?9`<0$N0x?tJ,2r(B}  \fbox{\sc 4. $p$-$B?J6a;w$K$h$kB?9`<0$N0x?tJ,2r(B}
   {\Large\parskip 0pt
   
 \underline{\bf $B86M}(B} : {\bf $B@0?t(B $m$ $B$,(B 0} $\Leftrightarrow$  \underline{\uc $B86M}(B} : {\eec $B@0?t(B $m$ $B$,(B 0} $\Leftrightarrow$
   
 {\bf $m$ $B$O$I$s$J@0?t$G$b3d$j@Z$l$k(B}  {\eec $m$ $B$O$I$s$J@0?t$G$b3d$j@Z$l$k(B}
   
 ({\bf $m$ $B$O==J,Bg$-$$@0?t$G3d$j@Z$l$k(B})  ({\eec $m$ $B$O==J,Bg$-$$@0?t$G3d$j@Z$l$k(B})
   
 $B$?$H$($P(B,  $B$?$H$($P(B,
   
Line 202  $\Rightarrow$ $B:,$rC5$9J}K!$OE,MQ:$Fq(B
Line 219  $\Rightarrow$ $B:,$rC5$9J}K!$OE,MQ:$Fq(B
 $h_1$ $B$r8+$D$1$k(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  \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 $g_1$, $h_1$ $B$,Ev$?$j$J$i(B, $BE,Ev$J(B $k$ $B$N$H$3$m$G$[$s$H$K3d$j@Z$l$k$@$m$&(B.  \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}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $B8@$$$+$($l$P(B...}  \underline{\uc $B8@$$$+$($l$P(B...}
   
 $B0J2<(B, $B4JC1$N$?$a(B, $f(x)$ $B$*$h$S0x;R$N78?t$OA4$F@5$G$"$k$H$9$k(B.  $B0J2<(B, {\ec $B4JC1$N$?$a(B}, $f(x)$ $B$*$h$S0x;R$N78?t$OA4$F@5$G$"$k$H$9$k(B.
   
 $f(x) = a_0(x)+p\cdot a_1(x)+p^2\cdot a_2(x)+\cdots$  {\eec $f(x) = a_0(x)+p \cdot a_1(x)+p^2\cdot a_2(x)+\cdots$}
   
 $B$H!V$Y$-5i?tE83+!W$9$k(B. ( $a_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)  $B$H!V$Y$-5i?tE83+!W(B ( $a_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)
   
 $g(x) = b_0(x)+p\cdot b_1(x)+p^2\cdot b_2(x)+\cdots$  {\ec $g(x) = b_0(x)+p\cdot b_1(x)+p^2\cdot b_2(x)+\cdots$}
   
 $h(x) = c_0(x)+p\cdot c_1(x)+p^2\cdot c_2(x)+\cdots$  {\ec $h(x) = c_0(x)+p\cdot c_1(x)+p^2\cdot c_2(x)+\cdots$}
   
   ($b_i$, $c_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)
   
 $B$H$*$$$F(B $f(x)-g(x)h(x)=0$ $B$+$i(B $b_i$, $c_i$ $B$r7h$a$k(B.  $B$H$*$$$F(B $f(x)-g(x)h(x)=0$ $B$+$i(B $b_i$, $c_i$ $B$r7h$a$k(B.
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $B5-9f(B $a \equiv b \bmod M$}  \underline{\uc $B5-9f(B $a \equiv b \bmod M$}
   
 $M$ $B$r@0?t$H$9$k(B. $a \equiv b \bmod M$ $B$H$O(B  $M$ $B$r@0?t$H$9$k(B.
   
 \begin{itemize}  \begin{itemize}
 \item $a,b$ $B$,@0?t$N$H$-(B,  \item $a,b$ $B$,@0?t$N$H$-(B,
   
 $a-b$ $B$,(B $M$ $B$G3d$j@Z$l$k$3$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  \item $a,b$ $B$,@0?t78?tB?9`<0$N$H$-(B
   
 $a-b$ $B$N3F78?t$,(B $M$ $B$G3d$j@Z$l$k$3$H(B  {\eec $a \equiv b \bmod M$} $\Leftrightarrow$
   {\eec $a-b$ $B$N3F78?t$,(B $M$ $B$G3d$j@Z$l$k(B}
   
   \item {\eec $a$ $B$r(B $M$ $B$G3d$C$?M>$j(B} $B$b(B {\eec $a \bmod M$} $B$H=q$/(B
   
   \item $\equiv$ $B$G7k$P$l$?<0(B : {\eec $BEy<0$HF1MM$K07$($k(B}
 \end{itemize}  \end{itemize}
   
 $a$ $B$r(B $M$ $B$G3d$C$?M>$j$b(B $a \bmod M$ $B$H=q$/(B  
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf$b_0(x)$, $c_0(x)$ $B$+$i%9%?!<%H(B}  \underline{\uc $b_0(x)$, $c_0(x)$ $B$+$i%9%?!<%H(B}
   
 $f-gh = a_0-b_0c_0$ + ($p$$B$G3d$j@Z$l$kB?9`<0(B)  $f-gh$
   
 $B$@$+$i(B, $f=gh$ $B$J$i(B $a_0 \equiv b_0c_0 \bmod p$ $B$N$O$:(B  $\quad = a_0-${\ec $b_0c_0$} + ($p$$B$G3d$j@Z$l$kB?9`<0(B)
   
 \underline{$BNc(B}  $B$@$+$i(B, $f=gh$ $B$J$i(B
   
   $a_0 \equiv$ {\ec $b_0c_0$} $\bmod p$ $B$N$O$:(B
   
   \underline{\uc $BNc(B}
   
   {\eec
 \begin{tabbing}  \begin{tabbing}
 $f(x)=$ \= $x^4+17056x^3+72658809x^2$ \\  $f(x)=$ \= $x^4+17056x^3+72658809x^2$ \\
 \> $+3504023212x+30603759869$  \> $+3504023212x+30603759869$
 \end{tabbing}  \end{tabbing}}
   
 $p = 3$ $B$H$9$k$H(B $a_0(x)=x^4+x^3+x+2$  $p = 3$ $B$H$9$k$H(B $a_0(x)=x^4+x^3+x+2$
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $B0l<!0x;R$,$"$k$+(B?}  \underline{\uc $B0l<!0x;R$,$"$k$+(B?}
   
 $b_0(x) = x+p$, $h_0(x) = x^3+qx^2+rx+s$ $B$H$*$/(B.  {\ec $b_0(x) = x+q$},
   {\ec $c_0(x) = x^3+rx^2+sx+t$} $B$H$*$/(B.
   
 $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\  $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\
   
   {\ec
 $\left\{  $\left\{
 \parbox[c]{6in}{  \parbox[c]{6in}{
 $p+q \equiv 1 \bmod 3$ \\  $q+r \equiv 1 \bmod 3$ \\
 $pq+r \equiv 0 \bmod 3$ \\  $qr+s \equiv 0 \bmod 3$ \\
 $pr+s \equiv 1 \bmod 3$ \\  $qs+t \equiv 1 \bmod 3$ \\
 $ps \equiv 2 \bmod 3$}  $qt \equiv 2 \bmod 3$}
 \right.$\\  \right.$\\}
   
 $p$, $q$, $r$, $s$ $B$K(B 0, 1, 2 $B$r$I$&F~$l$F$b%@%a(B.  $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$r$I$&F~$l$F$b%@%a(B.
   
 $B$h$C$F(B, $B0l<!0x;R$O$J$$(B.  $B$h$C$F(B, {\eec $B0l<!0x;R$O$J$$(B}.
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $BFs<!0x;R$O$"$k$+(B? --- $B$^$:(B $b_0$, $c_0$ $B$rC5$9(B}  \underline{\uc $BFs<!0x;R$O$"$k$+(B? --- $B$^$:(B $b_0$, $c_0$ $B$rC5$9(B}
   
 $b_0(x) = x^2+px+q$, $h_0(x) = x^2+rx+s$ $B$H$*$/(B  {\ec $b_0(x) = x^2+qx+r$},
   {\ec $c_0(x) = x^2+sx+t$}
   
 $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\  $B$H$*$/$H(B, $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\
   
   {\ec
 $\left\{  $\left\{
 \parbox[c]{6in}{  \parbox[c]{6in}{
 $p+r \equiv 1 \bmod 3$ \\  $q+s \equiv 1 \bmod 3$ \\
 $pr+q+s \equiv 0 \bmod 3$ \\  $qs+r+t \equiv 0 \bmod 3$ \\
 $ps+qr \equiv 1 \bmod 3$ \\  $qt+rs \equiv 1 \bmod 3$ \\
 $sq \equiv 2 \bmod 3$}  $tr \equiv 2 \bmod 3$}
 \right.$\\  \right.$\\}
   
 $p$, $q$, $r$, $s$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$l$P(B  $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$l$P(B
   
 $(p,q,r,s) = (0,1,1,2), (1,2,0,1)$ $B$,8+$D$+$k(B.  {\eec $(q,r,s,t) = (0,1,1,2), (1,2,0,1)$}
   
 $B0lJ}$,(B $b_0$, $BB>J}$,(B $c_0$ $B$H$_$J$;$P$3$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}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $BFs<!0x;R$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}
   
 $b_0 = x^2+1$, $c_0 = x^2+x+2$ $B$H$9$k$H(B  {\Large\parskip 0pt
   {\eec $b_0 = x^2+1$},
   {\eec $c_0 = x^2+x+2$} $B$H$9$k$H(B
   
 \centerline{$f-b_0c_0 \equiv 0 \bmod 3$}  \centerline{\eec $f \equiv b_0c_0 \bmod 3$}
   
 $f-gh \equiv a_0-b_0c_0+p(a_1-(b_0c_1+b_1c_0)) \bmod 3^2$  $gh \equiv (b_0+3${\ec $b_1$}$)(c_0+3${\ec$c_1$}$) \bmod 3^2$ $B$h$j(B
   
 $B$h$j(B, $BN>JU$r(B 3 $B$G3d$C$F(B  $f-gh \equiv a_0-b_0c_0+3(a_1-$
   $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3^2$
   
 ${{f-gh} \over 3} \equiv {{a_0-b_0c_0}\over 3} + (a_1-(b_0c_1+b_1c_0)) \bmod 3$  $BN>JU$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  $B:8JU$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 $b_1$, $c_1$ : $x^2$ $B$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}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9J}Dx<0(B}  \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9J}Dx<0(B}
   
 $b_1 = px+q$, $c_1 = rx+s$ $B$H$*$/(B.  {\Large\parskip 0pt
   {\ec $b_1 = qx+r$},
   {\ec $c_1 = sx+t$} $B$H$*$/(B.
   
 \begin{tabbing}  \begin{tabbing}
 $B1&JU(B = \= $-(p+r)x^3-(p+q+s+1)x^2$\\  $B1&JU(B = \= {\ec $-(q+s)x^3-(q+r+t+1)x^2$}\\
 \> $-(2p+q+r-1)x-(2q+s)$  \> {\ec $-(2q+r+s-1)x-(2r+t)$}
 \end{tabbing}  \end{tabbing}
   
 $$B1&JU(B \equiv 0 \bmod 3$ $B$h$j(B\\  $$B1&JU(B \equiv 0 \bmod 3$ $B$h$j(B\\
   
   {\ec
 $\left\{  $\left\{
 \parbox[c]{6in}{  \parbox[c]{6in}{
 $p+r \equiv 0 \bmod 3$ \\  $q+s \equiv 0 \bmod 3$ \\
 $p+q+s+1 \equiv 0 \bmod 3$ \\  $q+r+t+1 \equiv 0 \bmod 3$ \\
 $2p+q+r-1 \equiv 0 \bmod 3$ \\  $2q+r+s-1 \equiv 0 \bmod 3$ \\
 $2q+s \equiv 0 \bmod 3$}  $2r+t \equiv 0 \bmod 3$}
 \right.$\\  \right.$\\}
   
 $B$3$s$I$OO"N)0l<!J}Dx<0(B($B9gF1<0(B). $B$3$l$r2r$/$H(B  $B$3$s$I$OO"N)0l<!J}Dx<0(B($B9gF1<0(B). $B$3$l$r2r$/$H(B
   
 $(p,q,r,s) = (0,1,0,1)$ $B$9$J$o$A(B $b_1 = 1$, $c_1 = 1$  {\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}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $BFs<!0x;R$D$E$-(B --- $b_k$, $c_k$ $B$bF1MM(B}  \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_k$, $c_k$ $B$bF1MM(B}
   
   {\Large\parskip 0pt
 $B$3$l$G(B,  $B$3$l$G(B,
   
 \centerline{$f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$}  \centerline{\eec $f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$}
   
 $B0J2<F1MM$K(B,  $B0J2<F1MM$K(B,
   
 \centerline{$b_k = px+q, c_k = rx+s$}  \centerline{\ec $b_i = qx+r, c_i = sx+t$}
   
 $B$H$*$$$F(B, $(p,q,r,s)$ $B$NO"N)0l<!J}Dx<0$r2r$1$P(B  ($i=2,3,\ldots$) $B$H$*$$$F(B, $(q,r,s,t)$ $B$NO"N)0l<!J}Dx<0$r=g<!(B
   $B2r$$$F$$$1$P(B
   
 \centerline{$f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1}) \bmod 3^k$}  \centerline{\eec $f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1}) \bmod 3^k$}
   
 $B$9$J$o$A(B  $B$9$J$o$A(B
   
 \centerline{$f \equiv g_kh_k \bmod 3^k$}  \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}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $(g_k, h_k)$ $B$NI=(B}  \underline{\uc $(g_k, h_k)$ $B$NI=(B}
   
 {\large  {\large
 \begin{tabular} { c | c c }  \begin{tabular} { c | c c }
Line 387  $k$ & $g_k$ & $h_k$ \\ \hline
Line 431  $k$ & $g_k$ & $h_k$ \\ \hline
 9&$x^2+7821x+10615$&$x^2+9235x+7916$\\ \hline  9&$x^2+7821x+10615$&$x^2+9235x+7916$\\ \hline
 10&$x^2+7821x+30298$&$x^2+9235x+47282$\\ \hline  10&$x^2+7821x+30298$&$x^2+9235x+47282$\\ \hline
 11&$x^2+7821x+89347$&$x^2+9235x+165380$\\ \hline  11&$x^2+7821x+89347$&$x^2+9235x+165380$\\ \hline
 12&$x^2+7821x+89347$&$x^2+9235x+342527$\\ \hline  12&{\ec $x^2+7821x+89347$}&{\ec $x^2+9235x+342527$}\\ \hline
 13&$x^2+7821x+89347$&$x^2+9235x+342527$\\ \hline  13&{\ec $x^2+7821x+89347$}&{\ec $x^2+9235x+342527$}\\ \hline
 \end{tabular}}  \end{tabular}}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $\bmod 3^k$ $B$G$N0x;R$+$i??$N0x;R$X(B}  \underline{\uc $\bmod 3^k$ $B$G$N0x;R$+$i??$N0x;R$X(B}
   
 $BI=$G8+$k$H(B, $k=12$ $B$+$i(B $k=13$ $B$GJQ2=$,$J$$(B  $BI=$G8+$k$H(B, {\eec $k=12 \rightarrow 13$ $B$GJQ2=$,$J$$(B}
   
 $\Rightarrow$ $f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0 $B$K$J$C$F$$$k(B!  $\Rightarrow$ {\ec $f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0}
   
 $B$9$J$o$A(B  {\eec
   $f(x) =$
   
 $f(x) = $  $ (x^2+7821x+89347)(x^2+9235x+342527)$}
   
 $(x^2+7821x+89347)(x^2+9235x+342527)$  \underline{\uc $B<B:]$K$O(B...}
   
 \underline{\bf $B<B:]$K$O(B...}  
   
 \begin{itemize}  \begin{itemize}
 \item $BIi$N78?t$N>l9g$r07$&$?$a$N9)IW$,I,MW(B  \item $BIi$N78?t$N>l9g$r07$&$?$a$N9)IW$,I,MW(B
   
Line 415  $(x^2+7821x+89347)(x^2+9235x+342527)$
Line 458  $(x^2+7821x+89347)(x^2+9235x+342527)$
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $\bmod p$ $B$G$NJ,2r$,0lHVBg@Z(B}  \underline{\uc $\bmod p$ $B$G$NJ,2r$,LdBj(B}
   
 $B=P$FMh$?(B, $B78?t$NJ}Dx<0(B  $B3F%9%F%C%W$G=P$FMh$k78?t$NJ}Dx<0(B
   
 \begin{itemize}  \begin{itemize}
 \item $k > 1$  \item {\eec $k > 1$}
   
 $BC1$J$kO"N)0l<!J}Dx<0(B  $BO"N)0l<!J}Dx<0(B ($B<B:]$K$O9gF1<0(B)
   
 \item $k = 1$  \item {\eec $k = 1$}
   
 $B0l<!J}Dx<0$G$J$$(B $\Rightarrow$ $B$7$i$_$D$V$7$G2r$/$N$O$"$^$j$K8zN((B  $B0l<!J}Dx<0$G$J$$(B
   
   $\Rightarrow$ $B$7$i$_$D$V$7$G2r$/$N$O$"$^$j$K8zN((B
 $B$,$o$k$$(B ($B$$$/$i%3%s%T%e!<%?$G$b(B)  $B$,$o$k$$(B ($B$$$/$i%3%s%T%e!<%?$G$b(B)
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{\bf 5. $BM-8BBN(B $GF(p) = \{0,1,\cdots,p-1\}$ }  \fbox{\sc 5. $BM-8BBN(B $GF(p) = \{0,1,\cdots,p-1\}$ }
   
 $p$ $B$,AG?t$N$H$-(B,  $p$ $B$,(B{\ec $BAG?t(B}$B$N$H$-(B,
 $GF(p) = \{0,1,\cdots,p-1\}$ $B$K(B, $+$, $-$, $\times$ $B$r(B  
 $B!V7k2L$r(B $p$ $B$G(B $B3d$C$?M>$j!W$GDj5A$9$k$H(B  
   
   {\eec $GF(p) = \{0,1,\cdots,p-1\}$} $B$K(B, $+$, $-$, $\times$ $B$r(B
   {\eec $B!V7k2L$r(B $p$ $B$G(B $B3d$C$?M>$j!W(B}$B$GDj5A$9$k$H(B
   
 \begin{enumerate}  \begin{enumerate}
 \item $B2C8:>h;;$GJD$8$F$$$k(B.  \item $B2C8:>h;;$GJD$8$F$$$k(B.
 \item 0 $B0J30$N85$G3d;;$,$G$-$k(B.  \item {\eec 0 $B0J30$N85$G3d;;$,$G$-$k(B. }
   
 $B!V(B$a {\not \equiv} 0 \bmod p$ $B$J$i(B $ab \equiv 1 \bmod p$ $B$H$J$k(B $b$ $B$,$H$l$k!W(B  $B!V(B$a {\not \equiv} 0 \bmod p$ $B$J$i(B $ab \equiv 1 \bmod p$ $B$J$k(B $b$ $B$,$"$k!W(B
 \end{enumerate}  \end{enumerate}
   
 $B$9$J$o$A(B, {\bf $GF(p)$ $B$OBN(B($B%?%$(B)}  $B$9$J$o$A(B, {\eec $GF(p)$ $B$OBN(B($B%?%$(B)}
   
 $B85$N8D?t$,M-8B8D(B ($p$ $B8D(B)$B$J$N$G(B {\bf $BM-8BBN(B} $B$H$h$V(B.  $B85$N8D?t$,M-8B8D(B ($p$ $B8D(B)$B$J$N$G(B {\ec $BM-8BBN(B} $B$H$h$V(B.
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $k=1$ $B$G$N7W;;$O(B, $BM-8BBN>e$G$N0x?tJ,2r(B}  \underline{\uc $k=1$ $\Rightarrow$  $BM-8BBN>e$G$N0x?tJ,2r(B}
   
 $a_0 \equiv f \bmod p$ $B$r(B $GF(p)$ $B78?tB?9`<0$H$_$k(B.  $a_0 \equiv f \bmod p$ $B$r(B $GF(p)$ $B78?tB?9`<0$H$_$k(B.
   
 $\Rightarrow$ $a_0 \equiv b_0c_0 \bmod p$ $B$H$J$k(B $b_0$, $c_0$ $B$r(B  $\Rightarrow$ $a_0 \equiv b_0c_0 \bmod p$ $B$H$J$k(B $b_0$, $c_0$ $B$r(B
 $B5a$a$k$3$H$O(B, $GF(p)$ $B>e$G$N0x?tJ,2r$KAjEv(B  $B5a$a$k$3$H$O(B, $GF(p)$ $B>e$G$N0x?tJ,2r$KAjEv(B
   
 $\Rightarrow$ {\bf $B$h$$%"%k%4%j%:%`$,$?$/$5$s$"$k(B}  $\Rightarrow$ {\eec $B<B$O$h$$%"%k%4%j%:%`$,$"$k(B}
   
 \underline{\bf $k > 1$ $B$G$N7W;;$O(B, $BM-8BBN>e$G$NO"N)0l<!J}Dx<05a2r(B}  \vskip 1cm
   
   \underline{\uc $k > 1$ $\Rightarrow$ $BM-8BBN>e$G$NO"N)0l<!J}Dx<05a2r(B}
   
 $B<B:]$K$O(B, $k=1$ $B$N7k2L$+$i5!3#E*$K7W;;$G$-$k(B.  $B<B:]$K$O(B, $k=1$ $B$N7k2L$+$i5!3#E*$K7W;;$G$-$k(B.
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $B0x?tJ,2r$N$^$H$a(B (Zassenhaus $B%"%k%4%j%:%`(B)}  \underline{\uc $B0x?tJ,2r$^$H$a(B (Zassenhaus $B%"%k%4%j%:%`(B)}
   
 \begin{enumerate}  \begin{enumerate}
 \item $B$h$$AG?t(B $p$ $B$rA*$s$G(B $f \bmod p$ $B$r0x?tJ,2r(B  \item {\eec $B$h$$AG?t(B $p$ $B$rA*$s$G(B $f \bmod p$ $B$r0x?tJ,2r(B}
   
 $f$ $B$N:G9b<!78?t$r3d$i$J$$(B  {\eec $B!V$h$$!W(B} $B$H$O(B
   
 $GF(p)$ $B$G$N0x;R$,A4$F0[$J$k(B etc.  \begin{itemize}
   \item $f$ $B$N:G9b<!78?t$r3d$i$J$$(B
   
 \item $B<!$r7+$jJV$7(B  \item $GF(p)$ $B$G$N0x;R$,A4$F0[$J$k(B etc.
   \end{itemize}
   
   \item {\eec $B<!$r7+$jJV$7(B}
   
 \begin{enumerate}  \begin{enumerate}
 \item $GF(p)$ $B>e$N0x;R$rFsAH$KJ,$1$k(B  \item $GF(p)$ $B>e$N0x;R$rFsAH$KJ,$1$k(B
   
Line 492  $GF(p)$ $B$G$N0x;R$,A4$F0[$J$k(B etc.
Line 544  $GF(p)$ $B$G$N0x;R$,A4$F0[$J$k(B etc.
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $BN"$K$O$$$m$$$m?t3X$,1#$l$F$$$k(B}  \underline{\uc $BN"$K$O$$$m$$$m?t3X$,1#$l$F$$$k(B}
   
 \begin{itemize}  \begin{itemize}
 \item $BBN$N>e$G$N0x?tJ,2r$N0l0U@-(B  \item {\eec $BBN$N>e$G$N0x?tJ,2r$N0l0U@-(B}
   
 $BBN>e$NB?9`<04D$N@-<A(B  $BBN>e$NB?9`<04D$N@-<A(B
   
 \item $BM-8BBN>e$G$N0x?tJ,2r%"%k%4%j%:%`(B  \item {\eec $BM-8BBN>e$G$N0x?tJ,2r%"%k%4%j%:%`(B}
   
 Berlekamp $B%"%k%4%j%:%`(B  Berlekamp $B%"%k%4%j%:%`(B
   
 \item $\bmod p$ $B$+$i(B $\bmod p^k$ $B$X$N;}$A>e$2(B  \item {\eec $\bmod p$ $B$+$i(B $\bmod p^k$ $B$X$N;}$A>e$2(B}
   
 Euclid $B$N8_=|K!(B, Hensel $B$NJdBj(B  Euclid $B$N8_=|K!(B, Hensel $B$NJdBj(B
 \end{itemize}  \end{itemize}
   
 $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B. $B?t3X$r$&$^$/MxMQ$7$?(B  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
 $B%"%k%4%j%:%`@_7W$,I,MW$H$$$&$3$H(B.  
   
   {\ec $B?t3X$r$&$^$/;H$C$?%"%k%4%j%:%`@_7W$,I,MW(B}
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{\bf 6. $BM-8BBN$N1~MQNc(B : $B0E9f(B}  \fbox{\sc 6. $BM-8BBN$N1~MQNc(B : $B0E9f(B}
   
 \underline{\bf $B%M%C%H%o!<%/>e$G$NDL?.$O4pK\E*$KE{H4$1(B}  \underline{\uc $B%M%C%H%o!<%/>e$G$NDL?.$O4pK\E*$KE{H4$1(B}
   
 $B<+J,$N?H$O<+J,$G<i$k(B $\Rightarrow$ $BDL?.FbMF$r(B{\bf $B0E9f(B}$B2=(B  $B<+J,$N?H$O<+J,$G<i$k(B $\Rightarrow$ $BDL?.FbMF$r(B{\ec $B0E9f(B}$B2=(B
   
 \underline{\bf $B0E9f2=DL?.$N0lNc(B}  \underline{\uc $B0E9f2=DL?.$N0lNc(B}
   
 \begin{enumerate}  \begin{enumerate}
 \item $B6&DL$N0E9f2=(B/$BI|9f2=80$r6&M-$9$k(B.  \item $B0E9f2=(B/$BI|9f2=(B{\ec $B80(B}$B$r(B{\ec $B6&M-(B}$B$9$k(B.
   
 \item $BAw?.B&(B : $B80$G0E9f2=(B $\Rightarrow$ $B<u?.B&(B : $B80$GI|9f2=(B  \item $BAw?.B&(B : $B80$G0E9f2=(B $\Rightarrow$ $B<u?.B&(B : $B80$GI|9f2=(B
 \end{enumerate}  \end{enumerate}
   
 \underline{\bf $BLdBj(B : $BDL?.O)$,E{H4$1$N$H$-$K(B, $B$I$&$d$C$F80$r6&M-(B?}  \underline{\uc $BLdBj(B : $BDL?.O)$,E{H4$1$N$H$-$K(B,}
   
   \underline{\uc $B$I$&$d$C$F80$r6&M-(B?}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf A $B$5$s$H(B B $B$5$s$,80$r6&M-(B --- Diffie-Hellman}  {\Large\parskip 0pt
   \underline{\uc A $B$5$s$H(B B $B$5$s$,80$r6&M-(B --- Diffie-Hellman}
   
 \begin{itemize}  \begin{itemize}
 \item $B8x3+>pJs(B  \item {\eec $B8x3+>pJs(B}
   
 $BBg$-$$AG?t(B $p$, $0 < g < p$ $B$J$k@0?t(B $g$  $BBg$-$$AG?t(B $p$, $0 < g < p$ $B$J$k@0?t(B $g$
   
 \item A $B$5$s$N;E;v(B  \item {\eec A $B$5$s$N;E;v(B}
   
 \begin{enumerate}  \begin{enumerate}
 \item $0 < s_A < p$ $B$J$k@0?t(B $s_A$ ($BHkL)(B) $B$r:n$k(B.  \item $0 < s_A < p$ $B$J$k@0?t(B {\eec $s_A$} ($BHkL)(B) $B$r:n$k(B.
 \item $w_A = g^{s_A} \bmod p$ $B$r(B B $B$5$s$KAw$k(B.  \item $w_A =$ {\eec $g^{s_A} \bmod p$} $B$r(B B $B$5$s$KAw$k(B.
 \item $s = w_B^{s_A} \bmod p$ $B$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}  \end{enumerate}
   
 \item B $B$5$s$N;E;v(B  \item {\eec B $B$5$s$N;E;v(B}
   
 \begin{enumerate}  \begin{enumerate}
 \item $0 < s_B < p$ $B$J$k@0?t(B $s_B$ ($BHkL)(B) $B$r:n$k(B.  \item $0 < s_B < p$ $B$J$k@0?t(B {\eec $s_B$} ($BHkL)(B) $B$r:n$k(B.
 \item $w_B = g^{s_B} \bmod p$ $B$r(B A $B$5$s$KAw$k(B.  \item $w_B =$ {\eec $g^{s_B} \bmod p$} $B$r(B A $B$5$s$KAw$k(B.
 \item $s = w_A^{s_B} \bmod p$ $B$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{enumerate}
   
 \end{itemize}  \end{itemize}}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $BBg;v$JE@(B}  \underline{\uc $BBg;v$JE@(B}
   
 \begin{itemize}  \begin{itemize}
 \item $w_B^{s_A} = w_A^{s_B} \bmod p$  \item {\eec $w_B^{s_A} = w_A^{s_B} \bmod p$}
   
 $B$3$l$G80$,6&M-$G$-$?(B  $B$3$l$G80$,6&M-$G$-$?(B
   
 \item $w_A$, $w_B$ $B$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$$(B  $g^{s_A} \bmod p$ $B$+$i(B $s_A$ $B$r5a$a$k$N$OFq$7$$(B
   
 ($BM-8BBN$N>hK!72$K$*$1$kN%;6BP?tLdBj(B)  {\ec ($BM-8BBN$N>hK!72$K$*$1$kN%;6BP?tLdBj(B)}
   
 \item $\overline{a^b} = a^b \bmod p$ $B$O(B $p$ $BDxEY$N?t$N$+$1;;3d;;$K5"Ce(B  \item $\overline{a^b} = a^b \bmod p$ $B$O(B {\eec $p$ $BDxEY$N?t$N$+$1;;3d;;$K5"Ce(B}
   
 $\overline{a^{100}} = \overline{(\overline{a^{50}})^2}$,  $\overline{a^{100}} = \overline{(\overline{a^{50}})^2}$,
 $\overline{a^{50}} = \overline{(\overline{a^{25}})^2}$,  $\overline{a^{50}} = \overline{(\overline{a^{25}})^2}$,
Line 585  $\overline{a^{3}} = \overline{\overline{(\overline{a})
Line 641  $\overline{a^{3}} = \overline{\overline{(\overline{a})
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\bf $BB>$K$b$$$m$$$m$"$k(B}  \underline{\uc $BB>$K$b$$$m$$$m$"$k(B}
   
 \begin{itemize}  \begin{itemize}
 \item RSA $B0E9f(B  \item {\eec RSA $B0E9f(B}
   
 $BBg$-$J@0?t$NAG0x?tJ,2r$NFq$7$5$rMxMQ(B  {\eec $BBg$-$J@0?t$NAG0x?tJ,2r$NFq$7$5(B}$B$rMxMQ(B
   
 \item $BBJ1_6J@~0E9f(B  \item {\eec $BBJ1_6J@~0E9f(B}
   
 $BM-8BBN>e$G(B $y^2=x^3+ax+b$ $B$N2r(B $P=(x,y)$ $B$r9M$($k$H(B,  $BM-8BBN>e$G(B $y^2=x^3+ax+b$ $B$N2r(B $P=(x,y)$ $B$r9M$($k$H(B,
 $k$ $BG\;;(B $kP$ $B$,Dj5A$G$-$k(B.  $k$ $BG\;;(B $kP$ $B$,Dj5A$G$-$k(B.
   
 $kP$ $B$+$i(B $k$ $B$r5a$a$k7W;;$NFq$7$5$rMxMQ(B  {\eec $kP$ $B$+$i(B $k$ $B$r5a$a$k7W;;$NFq$7$5(B}$B$rMxMQ(B
 \end{itemize}  \end{itemize}
   
 $\Rightarrow$ {\bf $B$$$:$l$b(B, $BD>@\(B, $B4V@\$K@0?t$N>jM>1i;;$,4XM?(B}  $\Rightarrow$ {\ec $BD>@\(B, $B4V@\$K@0?t$N>jM>1i;;$,4XM?(B}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{\bf 7. $B$^$H$a(B}  \fbox{\sc 7. $B$^$H$a(B}
   
 \begin{enumerate}  \begin{enumerate}
 \item {\bf $BB?9`<00x?tJ,2rDxEY$G$b(B, $B8zN($h$$<B8=$OBgJQ(B}  \item {\eec $BB?9`<00x?tJ,2rDxEY$G$b(B, $B8zN($h$$<B8=$OBgJQ(B}
   
 $B?t3X$,0U30$KLr$KN)$D(B $\cdots$ $BFC$KM-8BBN(B  $B?t3X$,0U30$KLr$KN)$D(B $\cdots$ $BFC$K(B{\ec $BM-8BBN(B}
   
 \item {\bf $B$G$bM-8BBN$J$s$FB>$K2?$NLr$KN)$D$N(B?}  \item {\eec $B$G$bM-8BBN$J$s$FB>$K2?$NLr$KN)$D$N(B?}
   
 $B<B$O(B IT $B<R2q$rN"$G;Y$($F$$$k(B.  $B<B$O(B IT $B<R2q$rN"$G;Y$($F$$$?$j$9$k(B.
   
 \item {\bf $B?t3X$N1|?<$5(B}  \item {\eec $B?t3X$N2{$N?<$5(B}
   
 $B8e$K$J$C$F$H$s$G$b$J$$$H$3$m$K1~MQ$5$l$k2DG=@-$,$"$k(B  $B8e$K$J$C$F$H$s$G$b$J$$$H$3$m$K1~MQ$5$l$k2DG=@-(B
 $B$H$$$&3Z$7$5(B, $B1|?<$5(B  
   $B7W;;$NFq$7$5$,Lr$KN)$D$3$H$b$"$k$H$$$&IT;W5D(B
   
   
 \end{enumerate}  \end{enumerate}
   
 \end{slide}  \end{slide}
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