% $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.6 2001/07/28 03:31:10 noro Exp $

\Large
\parskip 0pt

\begin{slide}{}
\fbox{\sc 1. $B$O$8$a$K(B}

computer = compute $B$9$k$?$a$N$b$N(B

compute = {\ec $B7W;;(B}$B$9$k(B

$B:G6a$G$O(B {\ec $B%G%8%?%k>pJsDL?.(B} $B$N<jCJ$H$J$C$F$7$^$C$?(B

$\Rightarrow$ $B!V7W;;!W$K;H$C$F$$$k?M$O$4$/>/?t(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


{\ec $B7W;;$K;H$o$J$$$N$O$b$C$?$$$J$$(B}($B$H;W$&(B)
\end{slide}

\begin{slide}{}
\fbox{\sc 2. $B%3%s%T%e!<%?$K$D$$$F$N%$%m%O(B}

\begin{itemize}
\item {\eec CPU}

$B%W%m%0%i%`$K=>$C$FL?Na$r<B9T(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. 

\item {\eec $B%l%8%9%?(B}

CPU $B$,;}$C$F$$$kFCJL$J%a%b%j$G(B, $B1i;;$NBP>]$K$J$k(B. $B?t$OB?$/$J$/(B, $BBg$-$5(B
($BD9$5(B) $B$b>.$5$$(B.
\end{itemize}

\end{slide}

\begin{slide}{}
\underline{\uc $BL?Na$NNc(B}
\begin{itemize}
\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 $B%l%8%9%?$NCM$,(B 0 $B$J$i(B 100 $BHVCO@h$KHt$Y(B
\end{itemize}

\underline{\uc $B07$($k?t(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

\end{slide}

\begin{slide}{}
\underline{\uc $B?t3X$K;H$&>l9g$r9M$($k$H(B...}

$11111111111 \times 11111111111$

$\Rightarrow$ {\ec 1332508849} ??? (=$B7k2L$r(B $2^{32}$ $B$G3d$C$?M>$j(B)

$B$+$H$$$C$F(B

$11111111111 \times 11111111111$

$\Rightarrow 1.234567 \times 10^{20}$ 

$B$b:$$k(B

{\ec $B8m:9$,F~$k$H?t3X$H$7$F$OL50UL#$J7W;;(B}
\end{slide}

\begin{slide}{}
\underline{\uc $B$H$j$"$($:Bg$-$J@0?t$O07$($J$$$H$$$1$J$$(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(B {\eec $B!VI.;;!W(B}$B$r$5$;$l$P$h$$(B

\begin{itemize}
\item {\eec $B?M4V(B}

$B$R$H$1$?(B : 0 $B0J>e(B 9 $B0J2<(B

\item {\eec $B%3%s%T%e!<%?(B}

$B$R$H$1$?(B : 0 $B0J>e(B $2^{32}-1$ $B0J2<(B
\end{itemize}
\end{slide}

\begin{slide}{}
\underline{\uc $BNc(B : $B@0?t$NB-$7;;(B}

\begin{tabular}{ccccc} \\
& 5 & 4001257187 & 1914644777 & (= $3^{42}$) \\
+ &  & 2830677074 & 689956897 & (= $3^{40}$) \\ \hline
& 6 & 2536966965 & 2604601674 & 
\end{tabular}

\vskip 1cm

\underline{\uc $B0lJQ?tB?9`<0(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}

\begin{slide}{}
\fbox{\sc 3. $BB?9`<0$N0x?tJ,2r(B --- $BCf3X9b9;E*J}K!(B}
\begin{enumerate}
\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

$x^3+ax^2+bx+c$ $B$O$I$&$9$k(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?)

\item {\eec $B2r$N8x<0(B}

$x^2+ax+b=0$ $B$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
\end{enumerate}
\end{slide}

\begin{slide}{}
\underline{\uc $B4cNOK!$OLdBj$rFq$7$/$7$F$$$k(B}

$BNc(B : $x^2+11508x+28386587$

$28386587=3581\times 7927$ $B$,4cNO$GJ,$+$k?M$O>/$J$$(B($B$H;W$&(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?

$\Rightarrow$ {\bf \ec $x^2-t=0$ $B$N@0?t:,$rC5$9J}K!$,$"$l$P$h$$(B}
\end{slide}

\begin{slide}{}
\underline{\uc 3 $B<!0J2<$NB?9`<0(B}

{\eec $B@0?t>e$GJ,2r$G$-$k$J$i(B, $B0l<!0x;R$r;}$D(B}

$\Rightarrow$ {\ec $B:,$rC5$9J}K!$,E,MQ$G$-$k(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}

$BFsJ,K!$h$j$:$C$H9bB.(B
\end{itemize}

\end{slide}

\begin{slide}{}
\underline{\uc 4 $B<!0J>e$N>l9g(B}

$B$$$m$$$m$JJ,2r%Q%?!<%s$,$"$jF@$k(B

4 $B<!(B = 2 $B<!(B $\times$ 2 $B<!(B

$\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}

\vskip 1cm

\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}

\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}

$B$?$H$($P(B, 

\begin{enumerate}
\item $B:G=i(B, $f(x)-g_1(x)h_1(x)$ $B$N78?t$,@0?t(B $p$ $B$G3d$j@Z$l$k$h$&$J(B $g_1$,
$h_1$ $B$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
$B=g<!:n$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{slide}

\begin{slide}{}
\underline{\uc $B8@$$$+$($l$P(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. 

{\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(B ( $a_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)

{\ec $g(x) = b_0(x)+p\cdot b_1(x)+p^2\cdot b_2(x)+\cdots$}

{\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. 
\end{slide}

\begin{slide}{}
\underline{\uc $B5-9f(B $a \equiv b \bmod M$}

$M$ $B$r@0?t$H$9$k(B.

\begin{itemize}
\item $a,b$ $B$,@0?t$N$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 $a \equiv b \bmod M$} $\Leftrightarrow$
{\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{slide}

\begin{slide}{}
\underline{\uc $b_0(x)$, $c_0(x)$ $B$+$i%9%?!<%H(B}

$f-gh$

$\quad = a_0-${\ec $b_0c_0$} + ($p$$B$G3d$j@Z$l$kB?9`<0(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}
$f(x)=$ \= $x^4+17056x^3+72658809x^2$ \\
\> $+3504023212x+30603759869$
\end{tabbing}}

$p = 3$ $B$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 $b_0(x) = x+q$}, 
{\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\\

{\ec
$\left\{
\parbox[c]{6in}{
$q+r \equiv 1 \bmod 3$ \\
$qr+s \equiv 0 \bmod 3$ \\
$qs+t \equiv 1 \bmod 3$ \\
$qt \equiv 2 \bmod 3$}
\right.$\\}

$q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$r$I$&F~$l$F$b%@%a(B. 

$B$h$C$F(B, {\eec $B0l<!0x;R$O$J$$(B}.

\end{slide}

\begin{slide}{}
\underline{\uc $BFs<!0x;R$O$"$k$+(B? --- $B$^$:(B $b_0$, $c_0$ $B$rC5$9(B}

{\ec $b_0(x) = x^2+qx+r$},
{\ec $c_0(x) = x^2+sx+t$}

$B$H$*$/$H(B, $a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\

{\ec
$\left\{
\parbox[c]{6in}{
$q+s \equiv 1 \bmod 3$ \\
$qs+r+t \equiv 0 \bmod 3$ \\
$qt+rs \equiv 1 \bmod 3$ \\
$tr \equiv 2 \bmod 3$}
\right.$\\}

$q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$l$P(B

{\eec $(q,r,s,t) = (0,1,1,2), (1,2,0,1)$}

($b_0$,$c_0$) $B$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>r7o(B}

{\eec $b_0 = x^2+1$}, 
{\eec $c_0 = x^2+x+2$} $B$H$9$k$H(B

\centerline{\eec $f \equiv b_0c_0 \bmod 3$}

$gh \equiv (b_0+3${\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$b_1$}$))\bmod 3^2$

$BN>JU$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

\end{slide}

\begin{slide}{}
\underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9J}Dx<0(B}

{\ec $b_1 = qx+r$},
{\ec $c_1 = sx+t$} $B$H$*$/(B.

\begin{tabbing}
$B1&JU(B = \= {\ec $-(q+s)x^3-(q+r+t+1)x^2$}\\
\> {\ec $-(2q+r+s-1)x-(2r+t)$}
\end{tabbing}

$$B1&JU(B \equiv 0 \bmod 3$ $B$h$j(B\\

{\ec 
$\left\{
\parbox[c]{6in}{
$q+s \equiv 0 \bmod 3$ \\
$q+r+t+1 \equiv 0 \bmod 3$ \\
$2q+r+s-1 \equiv 0 \bmod 3$ \\
$2r+t \equiv 0 \bmod 3$}
\right.$\\}

$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 $c_1 = 1$}

\end{slide}

\begin{slide}{}
\underline{\uc $BFs<!0x;R$D$E$-(B --- $b_2$, $c_2$ $B$O(B $\bmod 3^3$ $B$G(B}

$B$3$l$G(B, {\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 $c_2=sx+t$}

$\Rightarrow$ $k=1$ $B$HF1MM$N(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 $b_i = qx+r, c_i = sx+t$}

($i=2,3,\ldots$) $B$H$*$$$F(B, $(q,r,s,t)$ $B$NO"N)0l<!J}Dx<0$r=g<!(B
$B2r$$$F$$$1$P(B

\centerline{\eec $f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1}) \bmod 3^k$}

$B$9$J$o$A(B 

\centerline{\eec $f \equiv g_kh_k \bmod 3^k$}

$B$H$J$k(B $g_k, h_k$ $B$,7h$^$k(B.
\end{slide}

\begin{slide}{}
\underline{\uc $(g_k, h_k)$ $B$NI=(B}

{\large
\begin{tabular} { c | c c }
$k$ & $g_k$ & $h_k$ \\ \hline
1&$x^2+1$&$x^2+x+2$\\ \hline
2&$x^2+4$&$x^2+x+5$\\ \hline
3&$x^2+18x+4$&$x^2+x+5$\\ \hline
4&$x^2+45x+4$&$x^2+x+59$\\ \hline
5&$x^2+45x+166$&$x^2+x+140$\\ \hline
6&$x^2+531x+409$&$x^2+487x+626$\\ \hline
7&$x^2+1260x+1867$&$x^2+487x+1355$\\ \hline
8&$x^2+1260x+4054$&$x^2+2674x+1355$\\ \hline
9&$x^2+7821x+10615$&$x^2+9235x+7916$\\ \hline
10&$x^2+7821x+30298$&$x^2+9235x+47282$\\ \hline
11&$x^2+7821x+89347$&$x^2+9235x+165380$\\ \hline
12&{\ec $x^2+7821x+89347$}&{\ec $x^2+9235x+342527$}\\ \hline
13&{\ec $x^2+7821x+89347$}&{\ec $x^2+9235x+342527$}\\ \hline
\end{tabular}}
\end{slide}

\begin{slide}{}
\underline{\uc $\bmod 3^k$ $B$G$N0x;R$+$i??$N0x;R$X(B}

$BI=$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}

{\eec
$f(x) =$

$ (x^2+7821x+89347)(x^2+9235x+342527)$}

\underline{\uc $B<B:]$K$O(B...}

\begin{itemize}
\item $BIi$N78?t$N>l9g$r07$&$?$a$N9)IW$,I,MW(B

\item $B<:GT$N2DG=@-$b$"$k$N$G(B, $k$ $B$r$I$3$^$G>e$2$l$P$$$$$+$N>e8B$,I,MW(B
\end{itemize}
\end{slide}

\begin{slide}{}
\underline{\uc $\bmod p$ $B$G$NJ,2r$,LdBj(B}

$B3F%9%F%C%W$G=P$FMh$k78?t$NJ}Dx<0(B

\begin{itemize}
\item {\eec $k > 1$}

$BO"N)0l<!J}Dx<0(B ($B<B:]$K$O9gF1<0(B)

\item {\eec $k = 1$}

$B0l<!J}Dx<0$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)
\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{\sc 5. $BM-8BBN(B $GF(p) = \{0,1,\cdots,p-1\}$ }

$p$ $B$,(B{\ec $BAG?t(B}$B$N$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}
\item $B2C8:>h;;$GJD$8$F$$$k(B. 
\item {\eec 0 $B0J30$N85$G3d;;$,$G$-$k(B. }

$B!V(B$a {\not \equiv} 0 \bmod p$ $B$J$i(B $ab \equiv 1 \bmod p$ $B$J$k(B $b$ $B$,$"$k!W(B
\end{enumerate}

$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 {\ec $BM-8BBN(B} $B$H$h$V(B. 

\end{slide}

\begin{slide}{}
\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. 

$\Rightarrow$ $a_0 \equiv b_0c_0 \bmod p$ $B$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

$\Rightarrow$ {\eec $B<B$O$h$$%"%k%4%j%:%`$,$"$k(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.
\end{slide}

\begin{slide}{}
\underline{\uc $B0x?tJ,2r$^$H$a(B (Zassenhaus $B%"%k%4%j%:%`(B)}

\begin{enumerate}
\item {\eec $B$h$$AG?t(B $p$ $B$rA*$s$G(B $f \bmod p$ $B$r0x?tJ,2r(B}

{\eec $B!V$h$$!W(B} $B$H$O(B

\begin{itemize}
\item $f$ $B$N:G9b<!78?t$r3d$i$J$$(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}
\item $GF(p)$ $B>e$N0x;R$rFsAH$KJ,$1$k(B

\item $B3FAH$N@Q$r(B $g_1$, $h_1$ $B$H$9$k(B. 

\item $f \equiv g_kh_k \bmod p^k$ $B$J$k(B $g_k$, $h_k$ $B$r:n$k(B

\item $B78?t$N@5Ii$rD4@a$7$F;n$73d$j(B
\end{enumerate}

\end{enumerate}
\end{slide}

\begin{slide}{}
\underline{\uc $BN"$K$O$$$m$$$m?t3X$,1#$l$F$$$k(B}

\begin{itemize}
\item {\eec $BBN$N>e$G$N0x?tJ,2r$N0l0U@-(B}

$BBN>e$NB?9`<04D$N@-<A(B

\item {\eec $BM-8BBN>e$G$N0x?tJ,2r%"%k%4%j%:%`(B}

Berlekamp $B%"%k%4%j%:%`(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
\end{itemize}

$\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.

{\ec $B?t3X$r$&$^$/;H$C$?%"%k%4%j%:%`@_7W$,I,MW(B}

\end{slide}

\begin{slide}{}
\fbox{\sc 6. $BM-8BBN$N1~MQNc(B : $B0E9f(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{\ec $B0E9f(B}$B2=(B

\underline{\uc $B0E9f2=DL?.$N0lNc(B}

\begin{enumerate}
\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
\end{enumerate}

\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}

\begin{slide}{}
\underline{\uc A $B$5$s$H(B B $B$5$s$,80$r6&M-(B --- Diffie-Hellman}

\begin{itemize}
\item {\eec $B8x3+>pJs(B}

$BBg$-$$AG?t(B $p$, $0 < g < p$ $B$J$k@0?t(B $g$

\item {\eec A $B$5$s$N;E;v(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 $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}

\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 $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{slide}

\begin{slide}{}
\underline{\uc $BBg;v$JE@(B}

\begin{itemize}
\item {\eec $w_B^{s_A} = w_A^{s_B} \bmod p$}

$B$3$l$G80$,6&M-$G$-$?(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

{\ec ($BM-8BBN$N>hK!72$K$*$1$kN%;6BP?tLdBj(B)}

\item $\overline{a^b} = a^b \bmod p$ $B$O(B {\eec $p$ $BDxEY$N?t$N$+$1;;3d;;$K5"Ce(B}

$\overline{a^{100}} = \overline{(\overline{a^{50}})^2}$,
$\overline{a^{50}} = \overline{(\overline{a^{25}})^2}$,
$\overline{a^{25}} = \overline{\overline{(\overline{a^{12}})^2} \times \overline{a}}$,
$\overline{a^{12}} = \overline{(\overline{a^{6}})^2}$,
$\overline{a^{6}} = \overline{(\overline{a^{3}})^2}$,
$\overline{a^{3}} = \overline{\overline{(\overline{a})^2} \times \overline{a}}$

\end{itemize}
\end{slide}

\begin{slide}{}
\underline{\uc $BB>$K$b$$$m$$$m$"$k(B}

\begin{itemize}
\item {\eec RSA $B0E9f(B}

{\eec $BBg$-$J@0?t$NAG0x?tJ,2r$NFq$7$5(B}$B$rMxMQ(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,
$k$ $BG\;;(B $kP$ $B$,Dj5A$G$-$k(B.

{\eec $kP$ $B$+$i(B $k$ $B$r5a$a$k7W;;$NFq$7$5(B}$B$rMxMQ(B
\end{itemize}

$\Rightarrow$ {\ec $BD>@\(B, $B4V@\$K@0?t$N>jM>1i;;$,4XM?(B}
\end{slide}

\begin{slide}{}
\fbox{\sc 7. $B$^$H$a(B}

\begin{enumerate}
\item {\eec $BB?9`<00x?tJ,2rDxEY$G$b(B, $B8zN($h$$<B8=$OBgJQ(B}

$B?t3X$,0U30$KLr$KN)$D(B $\cdots$ $BFC$K(B{\ec $BM-8BBN(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$$$?$j$9$k(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=@-(B

$B7W;;$NFq$7$5$,Lr$KN)$D$3$H$b$"$k$H$$$&IT;W5D(B


\end{enumerate}

\end{slide}

%\begin{slide}{}
%\fbox{\bf}
%\end{slide}
