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version 1.1, 2001/11/16 10:33:25 version 1.2, 2001/11/19 00:53:58
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 % $OpenXM$  % $OpenXM: OpenXM/doc/Papers/rims2001-noro.tex,v 1.1 2001/11/16 10:33:25 noro Exp $
 \documentclass{slides}  \documentclass{slides}
 \usepackage{color}  \usepackage{color}
 \usepackage{rgb}  \usepackage{rgb}
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 \item $B@5I8?t=`AGJ,2r$KI,MW(B  \item $B@5I8?t=`AGJ,2r$KI,MW(B
   
 $B2<;3(B-$B2#;3%"%k%4%j%:%`$G$O(B, $\sqrt{I}$ $B$NAG%$%G%"%kJ,2r(B  $B2<;3(B-$B2#;3;;K!(B : $\sqrt{I}$ $B$NAG%$%G%"%kJ,2r(B $\Rightarrow$ $I$ $B$N=`AGJ,2r(B
 $B$+$i(B $I$ $B$N=`AGJ,2r$rF3$/(B  
   
 $\sqrt{I}$ $B$NJ,2r$K$O(B, $BB?JQ?t$N0x?tJ,2r$,I,MW(B  $\sqrt{I}$ $B$NJ,2r$K$O(B, $BB?JQ?t$N0x?tJ,2r$,I,MW(B
   
 $B$R$g$C$H$7$?$iBe?t4v2?Id9f$X$N1~MQ$,$"$k$+$b$7$l$J$$(B  $B$R$g$C$H$7$?$iBe?t4v2?Id9f$X$N1~MQ$,$"$k$+$b$7$l$J$$(B
   
   \item Reed-Solomon $BId9f$N(B list decoding $B$X$N1~MQ$"$j(B
   
 \item $B$=$l<+BN$*$b$7$m$$(B  \item $B$=$l<+BN$*$b$7$m$$(B
   
 $BI8?t$,>.$5$$>l9g(B (2,3,5,7 $B$J$I(B) $BFCM-$N:$Fq$,$"$k(B.  $BI8?t$,>.$5$$>l9g(B (2,3,5,7 $B$J$I(B) $BFCM-$N:$Fq$,$"$k(B.
   
 $BL5J?J}J,2r$G$N:$Fq(B  $BL5J?J}J,2r$G$N:$Fq(B
   
 $BFC$K(B, $BJQ?t$N8D?t$r8:$i$9>l9g$N(B evaluation point  evaluation point $B$,B-$j$J$$>l9g(B
 $B$,B-$j$J$$>l9g(B  
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
Line 180  EEZ $B%"%k%4%j%:%`$r=q$1$F$$$J$$$?$a(B
Line 180  EEZ $B%"%k%4%j%:%`$r=q$1$F$$$J$$$?$a(B
   
 evaluation point $B$N3NJ]$N$?$a(B, $BM-8BBN$NBe?t3HBg$,I,MW(B  evaluation point $B$N3NJ]$N$?$a(B, $BM-8BBN$NBe?t3HBg$,I,MW(B
   
 $F=GF(q)$ $B$N(B $m$ $B<!3HBg(B $F_m$ $\cdots$ $h(x) \in F[x]$ : $m$ $B<!4{Ls(B  $F=GF(q)$ $B$N(B $m$ $B<!3HBg(B $F_m$ $B$r(B
 $B$K$h$j(B $F_m = F[x]/(h(x))$  $h(x) \in F[x]$ : $m$ $B<!4{Ls(B $B$K$h$j(B $F_m = F[x]/(h(x))$
   $B$GI=8=(B
   
 $\Rightarrow$ $B$3$l$G$O7W;;$,BgJQ(B  $\Rightarrow$ $B7W;;$,BgJQ(B
   
 $q$ $B$O>.$G(B, $\#(F_m)$ $B$,$=$l$J$j$KBg$-$1$l$P$h$$(B  $q$ $B$O>.$G(B, $\#(F_m)$ $B$,$=$l$J$j$KBg$-$1$l$P$h$$(B
   
Line 207  $(i,a_i)$ $B$r%F!<%V%k$GJ];}(B
Line 208  $(i,a_i)$ $B$r%F!<%V%k$GJ];}(B
   
 $\alpha^i+\alpha^j = \alpha^j(\alpha^{i-j}+1)$ $B$H$7$F7W;;(B  $\alpha^i+\alpha^j = \alpha^j(\alpha^{i-j}+1)$ $B$H$7$F7W;;(B
   
 \item $F_m$ $B$N%5%$%:$,(B $2^16$ $BDxEY$^$G$J$i<BMQE*(B  \item $F_m$ $B$N%5%$%:$,(B $2^{16}$ $BDxEY$^$G$J$i<BMQE*(B
   
 $BBN$r3HBg$7$F$b(B, $B7W;;B.EY$O$[$H$s$IJQ$o$i$J$$(B.  $BBN$r3HBg$7$F$b(B, $B7W;;B.EY$O$[$H$s$IJQ$o$i$J$$(B.
   
Line 311  $k > tdeg(f)^2/\deg_x(g_k)$ $B$J$i(B deterministic
Line 312  $k > tdeg(f)^2/\deg_x(g_k)$ $B$J$i(B deterministic
   
 $F = GF(q)$ $B>e$N4{Ls0x;R$,(B $F_m$ $B>e$GJ,2r$9$k2DG=@-$"$j(B  $F = GF(q)$ $B>e$N4{Ls0x;R$,(B $F_m$ $B>e$GJ,2r$9$k2DG=@-$"$j(B
   
 $f \in F[x_1,\ldots,x_n]$, $f$ : $F$ $B>e4{Ls$G(B $f = \prod f_i$,  $f \in F[x_1,\ldots,x_n]$, $f$ : $F$ $B>e4{Ls$G(B
 $f_i$ : $F_m$ $B>e4{Ls$H$9$k(B.  
   
 $F_m/F$ $B$O(B Galois $B3HBg$G(B, $G=Gal(F_m/F) = \langle \sigma \rangle$ $B$?$@$7(B  $f = \prod f_i$, $f_i$ : $F_m$ $B>e4{Ls$H$9$k(B.
 $\sigma : \beta \mapsto \beta^q$  
   
   $F_m/F$ $B$O(B Galois $B3HBg$G(B, $G=Gal(F_m/F) = \langle \sigma \rangle$
   
   $B$?$@$7(B $\sigma : \beta \mapsto \beta^q$
   
 $S$ $B$r(B $f_1$ $B$N(B $G$-orbit $B$H$9$k$H(B $\prod_{s\in S}s$ $B$O(B $G$-$BITJQ$@$+$i(B  $S$ $B$r(B $f_1$ $B$N(B $G$-orbit $B$H$9$k$H(B $\prod_{s\in S}s$ $B$O(B $G$-$BITJQ$@$+$i(B
 $\prod_{s\in S}s \in F[x_1,\ldots,x_n]$. $f$ $B$O(B $F$ $B>e4{Ls$@$+$i(B  
 $f = \prod_{s\in S}s$.  
   
 $B$h$C$F(B, $F$ $B>e$N4{Ls0x;R$O(B, $F_m$ $B>e$N4{Ls0x;R$N(B $G$-orbit $B$r5a$a$l$P(B  $\prod_{s\in S}s \in F[x_1,\ldots,x_n]$.
 $B$h$$(B.  
   
   $f$ $B$O(B $F$ $B>e4{Ls$@$+$i(B $f = \prod_{s\in S}s$.
   
   $\Rightarrow$ $F$ $B>e$N4{Ls0x;R(B = $F_m$ $B>e$N4{Ls0x;R$N(B $G$-orbit
   
 $\sigma(h)$ $B$O78?t$r(B $q$ $B>h$9$l$P$h$$$+$iMF0W(B.  $\sigma(h)$ $B$O78?t$r(B $q$ $B>h$9$l$P$h$$$+$iMF0W(B.
   
 \end{slide}  \end{slide}
Line 409  $f_{17,y\rightarrow y^2}$ & 0.57 & 0.78 & 0.55 & 2.3 &
Line 413  $f_{17,y\rightarrow y^2}$ & 0.57 & 0.78 & 0.55 & 2.3 &
 \fbox{\fbc \large $BAH$_9g$o$;GzH/$r5/$3$9>l9g(B}  \fbox{\fbc \large $BAH$_9g$o$;GzH/$r5/$3$9>l9g(B}
 \end{center}  \end{center}
   
 $f(x,y) = f_{17,y\rightarrow y^2}(x,y)f_{17,y\rightarrow y^2}(x+1,y^2)$  $f(x,y) = f_{17}(x,y^2)f_{17}(x+1,y^2)$
   
 $B??$N0x;R$O(B 4 $B8D(B, $\bmod y$ $B$G$N0x;R(B 32 $B8D(B  $B??$N0x;R$O(B 4 $B8D(B, $\bmod y$ $B$G$N0x;R(B 32 $B8D(B
   
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