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Diff for /OpenXM/doc/sci-semi2001/factorb.tex between version 1.4 and 1.6

version 1.4, 2001/07/25 05:44:01 version 1.6, 2001/07/28 03:31:10
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 % $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.3 2001/07/24 09:35:54 noro Exp $  % $OpenXM: OpenXM/doc/sci-semi2001/factorb.tex,v 1.5 2001/07/26 07:55:05 noro Exp $
   
 \Large  \Large
 \parskip 0pt  \parskip 0pt
Line 115  $\Rightarrow$ $B$3$l$G(B{\ec $B@0?t78?t$NB?9`<0$r?t
Line 115  $\Rightarrow$ $B$3$l$G(B{\ec $B@0?t78?t$NB?9`<0$r?t
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{\sc 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 {\eec $B4cNOK!(B} ($B2r$H78?t$N4X78(B)  \item {\eec $B4cNOK!(B} ($B2r$H78?t$N4X78(B)
   
Line 135  $x^2+ax+b=0$ $B$N:,(B ${-b \pm \sqrt{a^2-4b}} \over 
Line 132  $x^2+ax+b=0$ $B$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  $\Rightarrow$ $a^2-4b = t^2$ ($t$ : $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}{}
Line 161  $\Rightarrow$ {\bf \ec $x^2-t=0$ $B$N@0?t:,$rC5$9J}K!
Line 157  $\Rightarrow$ {\bf \ec $x^2-t=0$ $B$N@0?t:,$rC5$9J}K!
   
 $\Rightarrow$ {\ec $B:,$rC5$9J}K!$,E,MQ$G$-$k(B}  $\Rightarrow$ {\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}  \begin{itemize}
 \item {\eec $BFsJ,K!(B}  \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$$9~$`(B  $B6h4V$rH>J,$:$D69$a$FDI$$9~$`(B
   
 \item {\eec Newton $BK!(B}  \item {\eec Newton $BK!(B}
Line 204  $\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}
Line 200  $\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{\sc 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{\uc $B86M}(B} : {\eec $B@0?t(B $m$ $B$,(B 0} $\Leftrightarrow$  \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$I$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,  $B$?$H$($P(B,
   
 \begin{enumerate}  \begin{enumerate}
Line 222  $h_1$ $B$r8+$D$1$k(B. 
Line 215  $h_1$ $B$r8+$D$1$k(B. 
 $B=g<!:n$C$F$$$/(B ($k=2,3,\ldots$)  $B=g<!:n$C$F$$$/(B ($k=2,3,\ldots$)
   
 \item $g_1$, $h_1$ $B$,@52r$KBP1~$7$F$$$l$P(B, $BE,Ev$J(B $k$ $B$N$H$3$m$G$[$s$H$K3d$j@Z$l$k$@$m$&(B.  \item $g_1$, $h_1$ $B$,@52r$KBP1~$7$F$$$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}{}
Line 289  $p = 3$ $B$H$9$k$H(B $a_0(x)=x^4+x^3+x+2$
Line 282  $p = 3$ $B$H$9$k$H(B $a_0(x)=x^4+x^3+x+2$
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
   \underline{\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?}  \underline{\uc $B0l<!0x;R$,$"$k$+(B?}
   
 {\ec $b_0(x) = x+q$},  {\ec $b_0(x) = x+q$},
Line 338  $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$
Line 362  $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$
 \begin{slide}{}  \begin{slide}{}
 \underline{\uc $BFs<!0x;R$D$E$-(B --- $b_1$, $c_1$ $B$,K~$?$9>r7o(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 $b_0 = x^2+1$},
 {\eec $c_0 = x^2+x+2$} $B$H$9$k$H(B  {\eec $c_0 = x^2+x+2$} $B$H$9$k$H(B
   
Line 356  $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3$
Line 379  $(b_0${\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 {\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}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \underline{\uc $BFs<!0x;R$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}
   
 {\Large\parskip 0pt  
 {\ec $b_1 = qx+r$},  {\ec $b_1 = qx+r$},
 {\ec $c_1 = sx+t$} $B$H$*$/(B.  {\ec $c_1 = sx+t$} $B$H$*$/(B.
   
Line 385  $2r+t \equiv 0 \bmod 3$}
Line 407  $2r+t \equiv 0 \bmod 3$}
   
 $B$3$s$I$OO"N)0l<!J}Dx<0(B($B9gF1<0(B). $B$3$l$r2r$/$H(B  $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}  \end{slide}
   
 \begin{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, {\eec $f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$}
 $B$3$l$G(B,  
   
 \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 $c_2=sx+t$}
   
   $\Rightarrow$ $k=1$ $B$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,  $B0J2<F1MM$K(B,
   
 \centerline{\ec $b_i = qx+r, c_i = sx+t$}  \centerline{\ec $b_i = qx+r, c_i = sx+t$}
Line 410  $2r+t \equiv 0 \bmod 3$}
Line 448  $2r+t \equiv 0 \bmod 3$}
   
 \centerline{\eec $f \equiv g_kh_k \bmod 3^k$}  \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}{}
Line 587  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
Line 624  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 {\Large\parskip 0pt  
 \underline{\uc A $B$5$s$H(B B $B$5$s$,80$r6&M-(B --- Diffie-Hellman}  \underline{\uc A $B$5$s$H(B B $B$5$s$,80$r6&M-(B --- Diffie-Hellman}
   
 \begin{itemize}  \begin{itemize}
Line 611  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
Line 647  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(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.  \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}{}
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