version 1.1, 2001/11/16 10:33:25 |
version 1.3, 2001/11/26 08:41:14 |
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% $OpenXM$ |
% $OpenXM: OpenXM/doc/Papers/rims2001-noro.tex,v 1.2 2001/11/19 00:53:58 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 |
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$B2<;3(B-$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 |
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$\sqrt{I}$ $B$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$,I,MW(B |
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$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 |
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\item Reed-Solomon $BId9f$N(B list decoding $B$X$N1~MQ$"$j(B |
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\item $B$=$l<+BN$*$b$7$m$$(B |
\item $B$=$l<+BN$*$b$7$m$$(B |
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$BI8?t$,>.$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. |
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$BL5J?J}J,2r$G$N:$Fq(B |
$BL5J?J}J,2r$G$N:$Fq(B |
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$BFC$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 |
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\end{itemize} |
\end{itemize} |
\end{slide} |
\end{slide} |
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Line 180 EEZ $B%"%k%4%j%:%`$r=q$1$F$$$J$$$?$a(B |
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Line 180 EEZ $B%"%k%4%j%:%`$r=q$1$F$$$J$$$?$a(B |
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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$,I,MW(B |
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$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 $F_m = F[x]/(h(x))$ |
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$B$GI=8=(B |
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$\Rightarrow$ $B$3$l$G$O7W;;$,BgJQ(B |
$\Rightarrow$ $B7W;;$,BgJQ(B |
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$q$ $B$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 |
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Line 193 $\Rightarrow$ $F_m$ $B$r86;O:,I=8=$9$l$P$h$$(B |
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Line 194 $\Rightarrow$ $F_m$ $B$r86;O:,I=8=$9$l$P$h$$(B |
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\begin{slide}{} |
\begin{slide}{} |
\begin{center} |
\begin{center} |
\fbox{\fbc \large $B86;O:,I=8=(B $F_m^{\times} = \{\alpha^i | ( 0 \le i \le q-2) \}$} |
\fbox{\fbc \large $B86;O:,I=8=(B $F_m^{\times} = \{\alpha^i | ( 0 \le i \le q^m-2) \}$} |
\end{center} |
\end{center} |
\begin{itemize} |
\begin{itemize} |
\item $B$+$1;;(B, $B3d;;(B, $B$Y$->h$OMF0W(B |
\item $B$+$1;;(B, $B3d;;(B, $B$Y$->h$OMF0W(B |
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$\alpha^i \cdot \alpha^j = \alpha^{i+j \bmod q-1}$ |
$\alpha^i \cdot \alpha^j = \alpha^{i+j \bmod q^m-1}$ |
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\item $BB-$7;;(B, $B0z$-;;$O%F!<%V%k;2>H(B (Faug\`ere GB $B$GMQ$$$i$l$?(B) |
\item $BB-$7;;(B, $B0z$-;;$O%F!<%V%k;2>H(B (Faug\`ere GB $B$GMQ$$$i$l$?(B) |
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Line 207 $(i,a_i)$ $B$r%F!<%V%k$GJ];}(B |
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Line 208 $(i,a_i)$ $B$r%F!<%V%k$GJ];}(B |
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$\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 |
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\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$i<BMQE*(B |
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$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. |
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Line 311 $k > tdeg(f)^2/\deg_x(g_k)$ $B$J$i(B deterministic |
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Line 312 $k > tdeg(f)^2/\deg_x(g_k)$ $B$J$i(B deterministic |
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$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 |
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$f \in F[x_1,\ldots,x_n]$, $f$ : $F$ $B>e4{Ls$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. |
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$F_m/F$ $B$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. |
$\sigma : \beta \mapsto \beta^q$ |
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$F_m/F$ $B$O(B Galois $B3HBg$G(B, $G=Gal(F_m/F) = \langle \sigma \rangle$ |
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$B$?$@$7(B $\sigma : \beta \mapsto \beta^q$ |
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$S$ $B$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 |
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$f = \prod_{s\in S}s$. |
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$B$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$$(B. |
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$f$ $B$O(B $F$ $B>e4{Ls$@$+$i(B $f = \prod_{s\in S}s$. |
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$\Rightarrow$ $F$ $B>e$N4{Ls0x;R(B = $F_m$ $B>e$N4{Ls0x;R$N(B $G$-orbit |
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$\sigma(h)$ $B$O78?t$r(B $q$ $B>h$9$l$P$h$$$+$iMF0W(B. |
$\sigma(h)$ $B$O78?t$r(B $q$ $B>h$9$l$P$h$$$+$iMF0W(B. |
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\end{slide} |
\end{slide} |
Line 409 $f_{17,y\rightarrow y^2}$ & 0.57 & 0.78 & 0.55 & 2.3 & |
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Line 413 $f_{17,y\rightarrow y^2}$ & 0.57 & 0.78 & 0.55 & 2.3 & |
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\fbox{\fbc \large $BAH$_9g$o$;GzH/$r5/$3$9>l9g(B} |
\fbox{\fbc \large $BAH$_9g$o$;GzH/$r5/$3$9>l9g(B} |
\end{center} |
\end{center} |
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$f(x,y) = f_{17,y\rightarrow y^2}(x,y)f_{17,y\rightarrow y^2}(x+1,y^2)$ |
$f(x,y) = f_{17}(x,y^2)f_{17}(x+1,y^2)$ |
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$B??$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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