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version 1.3, 2001/07/24 09:35:54 version 1.6, 2001/07/28 03:31:10
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 % $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 1.5 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 187  $\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}
Line 183  $\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}
   
 \vskip 1cm  \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  {\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  $BJL$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{\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 211  $\Rightarrow$ {\ec $B:,$rC5$9J}K!$OE,MQ:$Fq(B}
Line 212  $\Rightarrow$ {\ec $B:,$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 $g_1$, $h_1$ $B$,@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 238  $h_1$ $B$r8+$D$1$k(B. 
Line 239  $h_1$ $B$r8+$D$1$k(B. 
 \begin{slide}{}  \begin{slide}{}
 \underline{\uc $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. {\eec $a \equiv b \bmod M$} $B$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,
   
 {\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  \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  {\eec $a \equiv b \bmod M$} $\Leftrightarrow$
 \end{itemize}  {\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}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
Line 278  $p = 3$ $B$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 321  $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$
Line 356  $q$, $r$, $s$, $t$ $B$K(B 0, 1, 2 $B$NCM$rF~$l$F$_$
   
 {\eec $(q,r,s,t) = (0,1,1,2), (1,2,0,1)$}  {\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}  \end{slide}
   
 \begin{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 $b_0 = x^2+1$},
 {\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
   
 \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$b_1$}$))\bmod 3^2$  $(b_0${\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
   
 ${{f-gh}\over 3} \equiv {{a_0-b_0c_0}\over 3}+(a_1-$  ${{f-gh}\over 3} \equiv {{a_0-b_0c_0}\over 3}+(a_1-$
 $(b_0${\ec$c_1$}$+c_0${\ec$b_1$}$))\bmod 3$  $(b_0${\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 372  $2r+t \equiv 0 \bmod 3$}
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 $b_k = qx+r, c_k = sx+t$}  \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 $f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1}) \bmod 3^k$}  \centerline{\eec $f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1}) \bmod 3^k$}
   
Line 396  $2r+t \equiv 0 \bmod 3$}
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 427  $k$ & $g_k$ & $h_k$ \\ \hline
Line 478  $k$ & $g_k$ & $h_k$ \\ \hline
   
 $BI=$G8+$k$H(B, {\eec $k=12 \rightarrow 13$ $B$GJQ2=$,$J$$(B}  $BI=$G8+$k$H(B, {\eec $k=12 \rightarrow 13$ $B$GJQ2=$,$J$$(B}
   
 $\Rightarrow$ {\ec $f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0!}  $\Rightarrow$ {\ec $f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0}
   
 {\eec  {\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...}  \underline{\uc $B<B:]$K$O(B...}
   
Line 444  $(x^2+9235x+342527)$}
Line 495  $(x^2+9235x+342527)$}
 \end{slide}  \end{slide}
   
 \begin{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  $B3F%9%F%C%W$G=P$FMh$k78?t$NJ}Dx<0(B
   
Line 573  $\Rightarrow$ $B7W;;5!$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 586  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
Line 636  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
 \begin{enumerate}  \begin{enumerate}
 \item $0 < s_A < p$ $B$J$k@0?t(B {\eec $s_A$} ($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 $w_A =$ {\eec $g^{s_A} \bmod p$} $B$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 $s =$ {\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}  \end{enumerate}
   
 \item {\eec B $B$5$s$N;E;v(B}  \item {\eec B $B$5$s$N;E;v(B}
Line 594  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
Line 644  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
 \begin{enumerate}  \begin{enumerate}
 \item $0 < s_B < p$ $B$J$k@0?t(B {\eec $s_B$} ($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 $w_B =$ {\eec $g^{s_B} \bmod p$} $B$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 $s =$ {\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}{}
Line 608  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
Line 658  $\Rightarrow$ $B7W;;5!$N%Q%o!<$@$1$G$O%@%a(B.
   
 $B$3$l$G80$,6&M-$G$-$?(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$$(B  $g^{s_A} \bmod p$ $B$+$i(B $s_A$ $B$r5a$a$k$N$OFq$7$$(B
   
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Removed from v.1.3  
changed lines
  Added in v.1.6
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