% $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.1 2001/07/23 06:46:51 noro Exp $

\Large
\parskip 0pt

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

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

compute = $B7W;;$9$k(B

$B:G6a$G$O(B communication $B$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

$BNc(B : email, $B%&%'%V(B $\cdots$ $B!V%$%s%?!<%M%C%H$9$k!W(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)
\end{slide}

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

\begin{itemize}
\item CPU

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

\item $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 $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{\bf $BL?Na$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{\bf $B07$($k?t(B}

$B%l%8%9%?$NBg$-$5$G07$($k?t$NHO0O$,7h$^$k(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{\bf $B?t3X$K;H$&>l9g$r9M$($k$H(B...}

$11111111111 \times 11111111111$

$\Rightarrow 1332508849$ ??? (=$B7k2L$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

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

\begin{slide}{}
\underline{\bf $B$H$j$"$($:Bg$-$J@0?t$O07$($J$$$H$$$1$J$$(B}

$\Rightarrow$ $B%W%m%0%i%`$r=q$1$P$h$$(B

$B%a%b%j>e$K@0?t$rJB$Y$F(B, $B%3%s%T%e!<%?$K!VI.;;!W$r$5$;$l$P$h$$(B

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

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

\item $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{\bf $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}

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

$B3F<!?t$N78?t$rJB$l$P$h$$(B

$\Rightarrow$ $B$3$l$G(B, $B@0?t78?t$NB?9`<0$r?t3XE*$K07$($k(B

\end{slide}

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

\begin{enumerate}
\item $B4cNOK!(B ($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 $B0x?tDjM}(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

$x^2+ax+b$ $B$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{\bf $B4cNOK!$OLdBj$rFq$7$/$7$F$$$k(B}

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

$28386587=3581\times 7927$ $B$,$a$N$3$GJ,$+$k?M$O>/$J$$(B($B$H;W$&(B)

\underline{\bf $B2r$N8x<0K!$OM-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
\end{slide}

\begin{slide}{}
\underline{\bf 3 $B<!0J2<$N>l9g(B}

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

$B:,$rC5$9J}K!$,E,MQ$G$-$k(B. 

$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 $BFsJ,K!(B

$B6h4V$rH>J,$:$D69$a$FDI$$9~$`(B

\item Newton $BK!(B

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

\end{slide}

\begin{slide}{}
\underline{\bf 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

$\Rightarrow$ $B:,$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}

$BCf4VCM$NDjM}$O(B, $B<B?t$K$*$1$k(B {\bf $B6a;w(B} $B$NMxMQ(B

$BJL$N6a;w(B $\Rightarrow$ {\bf $B3d$C$?M>$j(B}$B$KCmL\(B
\end{slide}

\begin{slide}{}
\fbox{\bf 4. $p$-$B?J6a;w$K$h$kB?9`<0$N0x?tJ,2r(B}

\underline{\bf $B86M}(B} : {\bf $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}

({\bf $m$ $B$O==J,Bg$-$$@0?t$G3d$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
$B:n$C$F$$$/(B ($k=1,2,\ldots$)

\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. 
\end{enumerate}
\end{slide}

\begin{slide}{}
\underline{\bf $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. 

$f(x) = a_0(x)+p\cdot a_1(x)+p^2\cdot a_2(x)+\cdots$

$B$H!V$Y$-5i?tE83+!W$9$k(B. ( $a_i$ $B$N78?t$O(B $p-1$ $B0J2<(B)

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

$h(x) = c_0(x)+p\cdot c_1(x)+p^2\cdot c_2(x)+\cdots$

$B$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{\bf $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

\begin{itemize}
\item $a,b$ $B$,@0?t$N$H$-(B, 

$a-b$ $B$,(B $M$ $B$G3d$j@Z$l$k$3$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
\end{itemize}

$a$ $B$r(B $M$ $B$G3d$C$?M>$j$b(B $a \bmod M$ $B$H=q$/(B
\end{slide}

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

$f-gh = a_0-b_0c_0$ + ($p$$B$G3d$j@Z$l$kB?9`<0(B)

$B$@$+$i(B, $f=gh$ $B$J$i(B $a_0 \equiv b_0c_0 \bmod p$ $B$N$O$:(B

\underline{$BNc(B}

\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{\bf $B0l<!0x;R$,$"$k$+(B?}

$b_0(x) = x+p$, $h_0(x) = x^3+qx^2+rx+s$ $B$H$*$/(B. 

$a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\

$\left\{
\parbox[c]{6in}{
$p+q \equiv 1 \bmod 3$ \\
$pq+r \equiv 0 \bmod 3$ \\
$pr+s \equiv 1 \bmod 3$ \\
$ps \equiv 2 \bmod 3$}
\right.$\\

$p$, $q$, $r$, $s$ $B$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.

\end{slide}

\begin{slide}{}
\underline{\bf $BFs<!0x;R$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

$a_0 \equiv b_0c_0 \bmod 3$ $B$h$j(B\\

$\left\{
\parbox[c]{6in}{
$p+r \equiv 1 \bmod 3$ \\
$pr+q+s \equiv 0 \bmod 3$ \\
$ps+qr \equiv 1 \bmod 3$ \\
$sq \equiv 2 \bmod 3$}
\right.$\\

$p$, $q$, $r$, $s$ $B$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. 

$B0lJ}$,(B $b_0$, $BB>J}$,(B $c_0$ $B$H$_$J$;$P$3$l$i$OF1$8$b$N(B
\end{slide}

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

$b_0 = x^2+1$, $c_0 = x^2+x+2$ $B$H$9$k$H(B

\centerline{$f-b_0c_0 \equiv 0 \bmod 3$}

$f-gh \equiv a_0-b_0c_0+p(a_1-(b_0c_1+b_1c_0)) \bmod 3^2$

$B$h$j(B, $BN>JU$r(B 3 $B$G3d$C$F(B

${{f-gh} \over 3} \equiv {{a_0-b_0c_0}\over 3} + (a_1-(b_0c_1+b_1c_0)) \bmod 3$

$B:8JU$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 $b_1$, $c_1$ : $x^2$ $B$N78?t$O(B 0 $B$H$7$F$h$$(B

\end{slide}

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

$b_1 = px+q$, $c_1 = rx+s$ $B$H$*$/(B.

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

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

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

$B$3$s$I$OO"N)0l<!J}Dx<0(B($B9gF1<0(B). $B$3$l$r2r$/$H(B

$(p,q,r,s) = (0,1,0,1)$ $B$9$J$o$A(B $b_1 = 1$, $c_1 = 1$

\end{slide}

\begin{slide}{}
\underline{\bf $BFs<!0x;R$D$E$-(B --- $b_k$, $c_k$ $B$bF1MM(B}

$B$3$l$G(B, 

\centerline{$f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$}

$B0J2<F1MM$K(B, 

\centerline{$b_k = px+q, c_k = rx+s$}

$B$H$*$$$F(B, $(p,q,r,s)$ $B$NO"N)0l<!J}Dx<0$r2r$1$P(B

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

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

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

$B$H$J$k(B $g_k, h_k$ $B$,7h$^$k(B. 

\end{slide}

\begin{slide}{}
\underline{\bf $(g_k, h_k)$ $B$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&$x^2+7821x+89347$&$x^2+9235x+342527$\\ \hline
13&$x^2+7821x+89347$&$x^2+9235x+342527$\\ \hline
\end{tabular}}
\end{slide}

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

$BI=$G8+$k$H(B, $k=12$ $B$+$i(B $k=13$ $B$GJQ2=$,$J$$(B

$\Rightarrow$ $f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0 $B$K$J$C$F$$$k(B!

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

$f(x) = $ 

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

\underline{\bf $B<B:]$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{\bf $\bmod p$ $B$G$NJ,2r$,0lHVBg@Z(B}

$B=P$FMh$?(B, $B78?t$NJ}Dx<0(B

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

$BC1$J$kO"N)0l<!J}Dx<0(B

\item $k = 1$

$B0l<!J}Dx<0$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{\bf 5. $BM-8BBN(B $GF(p) = \{0,1,\cdots,p-1\}$ }

$p$ $B$,AG?t$N$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

\begin{enumerate}
\item $B2C8:>h;;$GJD$8$F$$$k(B. 
\item 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
\end{enumerate}

$B$9$J$o$A(B, {\bf $GF(p)$ $B$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. 

\end{slide}

\begin{slide}{}
\underline{\bf $k=1$ $B$G$N7W;;$O(B, $BM-8BBN>e$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

$\Rightarrow$ {\bf $B$h$$%"%k%4%j%:%`$,$?$/$5$s$"$k(B}

\underline{\bf $k > 1$ $B$G$N7W;;$O(B, $BM-8BBN>e$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{\bf $B0x?tJ,2r$N$^$H$a(B (Zassenhaus $B%"%k%4%j%:%`(B)}

\begin{enumerate}
\item $B$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

$GF(p)$ $B$G$N0x;R$,A4$F0[$J$k(B etc.

\item $B<!$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{\bf $BN"$K$O$$$m$$$m?t3X$,1#$l$F$$$k(B}

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

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

\item $BM-8BBN>e$G$N0x?tJ,2r%"%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

Euclid $B$N8_=|K!(B, Hensel $B$NJdBj(B
\end{itemize}

$\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B. $B?t3X$r$&$^$/MxMQ$7$?(B
$B%"%k%4%j%:%`@_7W$,I,MW$H$$$&$3$H(B. 

\end{slide}

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

\underline{\bf $B%M%C%H%o!<%/>e$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

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

\begin{enumerate}
\item $B6&DL$N0E9f2=(B/$BI|9f2=80$r6&M-$9$k(B.

\item $BAw?.B&(B : $B80$G0E9f2=(B $\Rightarrow$ $B<u?.B&(B : $B80$GI|9f2=(B
\end{enumerate}

\underline{\bf $BLdBj(B : $BDL?.O)$,E{H4$1$N$H$-$K(B, $B$I$&$d$C$F80$r6&M-(B?}
\end{slide}

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

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

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

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

\begin{enumerate}
\item $0 < s_A < p$ $B$J$k@0?t(B $s_A$ ($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 $s = w_B^{s_A} \bmod p$ $B$r:n$k(B. 
\end{enumerate}

\item B $B$5$s$N;E;v(B

\begin{enumerate}
\item $0 < s_B < p$ $B$J$k@0?t(B $s_B$ ($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 $s = w_A^{s_B} \bmod p$ $B$r:n$k(B. 
\end{enumerate}

\end{itemize}
\end{slide}

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

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

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

\item $w_A$, $w_B$ $B$O0E9f2=$5$l$J$$(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)

\item $\overline{a^b} = a^b \bmod p$ $B$O(B $p$ $BDxEY$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{\bf $BB>$K$b$$$m$$$m$"$k(B}

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

$BBg$-$J@0?t$NAG0x?tJ,2r$NFq$7$5$rMxMQ(B

\item $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.

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

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

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

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

$B?t3X$,0U30$KLr$KN)$D(B $\cdots$ $BFC$KM-8BBN(B

\item {\bf $B$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. 

\item {\bf $B?t3X$N1|?<$5(B}

$B8e$K$J$C$F$H$s$G$b$J$$$H$3$m$K1~MQ$5$l$k2DG=@-$,$"$k(B
$B$H$$$&3Z$7$5(B, $B1|?<$5(B
\end{enumerate}

\end{slide}

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