Annotation of OpenXM/doc/Papers/rims2001-noro.tex, Revision 1.1
1.1 ! noro 1: % $OpenXM$
! 2: \documentclass{slides}
! 3: \usepackage{color}
! 4: \usepackage{rgb}
! 5: \usepackage{graphicx}
! 6: \usepackage{epsfig}
! 7: \newcommand{\qed}{$\Box$}
! 8: \newcommand{\mred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}}}
! 9: \newcommand{\tmred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}\limits^{\scriptstyle *}}}
! 10: \def\gr{Gr\"obner basis }
! 11: \def\st{\, s.t. \,}
! 12: \def\ni{\noindent}
! 13: \def\ve{\vfill\eject}
! 14: \textwidth 9.2in
! 15: \textheight 7.2in
! 16: \columnsep 0.33in
! 17: \topmargin -1in
! 18: \def\tc{\color{red}}
! 19: \def\fbc{\bf\color{MediumBlue}}
! 20: \def\itc{\color{brown}}
! 21: \def\urlc{\bf\color{DarkGreen}}
! 22: \def\bc{\color{LightGoldenrod1}}
! 23:
! 24: \title{\tc $B>.I8?tM-8BBN>e$NB?JQ?tB?9`<0$N0x?tJ,2r$K$D$$$F(B}
! 25:
! 26: \author{$BLnO$(B $B@59T(B ($B?@8MBg!&M}(B)\\ $B2#;3(B $BOB90(B ($B6e=#Bg!&?tM}(B)}
! 27: \begin{document}
! 28: \large
! 29: \setlength{\parskip}{0pt}
! 30: \maketitle
! 31:
! 32: \begin{slide}{}
! 33: \begin{center}
! 34: \fbox{\fbc \large $B$J$<(B ($B$$$^$5$i(B)$BM-8BBN>e$NB?JQ?tB?9`<0$N0x?tJ,2r(B?}
! 35: \end{center}
! 36:
! 37: \begin{itemize}
! 38: \item Risa/Asir $B$K<BAu$5$l$F$$$J$$(B
! 39:
! 40: $B$J$<$J$$$N$+(B, $B$H<+J,$G;W$&$3$H$O$7$P$7$P$"$C$?(B
! 41:
! 42: \item $B@5I8?t=`AGJ,2r$KI,MW(B
! 43:
! 44: $B2<;3(B-$B2#;3%"%k%4%j%:%`$G$O(B, $\sqrt{I}$ $B$NAG%$%G%"%kJ,2r(B
! 45: $B$+$i(B $I$ $B$N=`AGJ,2r$rF3$/(B
! 46:
! 47: $\sqrt{I}$ $B$NJ,2r$K$O(B, $BB?JQ?t$N0x?tJ,2r$,I,MW(B
! 48:
! 49: $B$R$g$C$H$7$?$iBe?t4v2?Id9f$X$N1~MQ$,$"$k$+$b$7$l$J$$(B
! 50:
! 51: \item $B$=$l<+BN$*$b$7$m$$(B
! 52:
! 53: $BI8?t$,>.$5$$>l9g(B (2,3,5,7 $B$J$I(B) $BFCM-$N:$Fq$,$"$k(B.
! 54:
! 55: $BL5J?J}J,2r$G$N:$Fq(B
! 56:
! 57: $BFC$K(B, $BJQ?t$N8D?t$r8:$i$9>l9g$N(B evaluation point
! 58: $B$,B-$j$J$$>l9g(B
! 59: \end{itemize}
! 60: \end{slide}
! 61:
! 62: \begin{slide}{}
! 63: \begin{center}
! 64: \fbox{\fbc \large $B%"%k%4%j%:%`$N35MW(B I : $BL5J?J}J,2r(B}
! 65: \end{center}
! 66:
! 67: modification of Bernardin's algorithm [Ber97]
! 68:
! 69: $f \in F[x_1,\ldots,x_n]$, $F$ : $BM-8BBN(B $Char(F) = p$
! 70:
! 71: $f = FGH$, where $F=\prod f_i^{a_i}$,
! 72: $G=\prod g_j^{b_j}$,
! 73: $H=\prod h_k^{c_k}$
! 74:
! 75: $f_i, g_j, h_k$ : $BL5J?J}(B, $B8_$$$KAG(B.
! 76:
! 77: $'$ $B$r(B $d/dx_1$ $B$H$7$F(B
! 78: $f_i' \neq 0$, $p \not{|}a_j$, $p | b_j$, $h_k' = 0$
! 79: $B$H=q$/$H(B
! 80:
! 81: $f' = F'GH$ $B$9$k$H(B $GCD(f,f') = GCD(F,F')GH$
! 82:
! 83: $GCD(F,F') = \prod f_i^{a_i-1}$ $B$@$+$i(B $f/GCD(f,f')=\prod f_i$
! 84:
! 85: $\prod f_i$ $B$G(B $f$ $B$r7+$jJV$73d$k$3$H$G(B, $f_1$ ($B=EJ#EY:G>.(B)
! 86: $B$,5a$^$k(B
! 87:
! 88: $\Rightarrow$ $F$ $B$,A4$FJ,2r$G$-$k(B
! 89:
! 90: \end{slide}
! 91:
! 92: \begin{slide}{}
! 93: \begin{center}
! 94: \fbox{\fbc \large $BL5J?J}J,2r(B $B$D$E$-(B}
! 95: \end{center}
! 96:
! 97: $B;D$j(B $f = GH$ $B$G(B, $f' = 0$
! 98:
! 99: $B$3$l$r(B $x_i$ $B$K$D$$$F7+$jJV$7$F;D$C$?(B $f$
! 100:
! 101: $\Rightarrow$ $df/dx_1 = \ldots = df/dx_n = 0$
! 102:
! 103: $\Rightarrow$ $B$3$l$O(B, $BA4$F$N;X?t$,(B $p$ $B$G3d$j@Z$l$k$3$H$r0UL#$9$k(B
! 104:
! 105: $\Rightarrow$ $F$ $B$OM-8BBN$@$+$i(B $f = g^p$ $B$H=q$1$k(B
! 106:
! 107: $\Rightarrow$ $g$ $B$KBP$7$F%"%k%4%j%:%`$rE,MQ(B
! 108: \end{slide}
! 109:
! 110:
! 111: \begin{slide}{}
! 112: \begin{center}
! 113: \fbox{\fbc \large Bernardin $B$NJ}K!$H$NHf3S(B}
! 114: \end{center}
! 115: $BI8?t(B 0 $B$K$*$1$k(B Yun $B$NJ}K!(B : $B=EJ#EY$N8zN(E*7W;;(B
! 116:
! 117: $B=EJ#EY$,9b$$>l9g$KM-Mx(B
! 118:
! 119: Bernardin : $B@5I8?t$N>l9g$K$b(B Yun $B$NJ}K!$r=$@5$7$FE,MQ(B
! 120:
! 121: $\Rightarrow$ $B=EJ#EY$,(B $p$ $B$h$jBg$-$$>l9g$KJ#;(2=(B
! 122:
! 123: $\Rightarrow$ $p$ $B$,>.$5$$>l9g$K$O(B, $BC1=c$K=|;;$N7+$jJV$7$G=EJ#EY$r5a(B
! 124: $B$a$k(B
! 125:
! 126: ($B>\:Y$JHf3S$O$^$@9T$C$F$$$J$$(B)
! 127: \end{slide}
! 128:
! 129: \begin{slide}{}
! 130: \begin{center}
! 131: \fbox{\fbc \large $B%"%k%4%j%:%`$N35MW(B II : $BFsJQ?t$X$N5"Ce(B}
! 132: \end{center}
! 133:
! 134: \begin{itemize}
! 135: \item $B@0?t78?tB?JQ?tB?9`<0$N>l9g(B
! 136:
! 137: $n$ $B$NFb(B $n-1$ $B8D$NJQ?t$r(B fix $\Rightarrow$ 1 $BJQ?t$G0x;R$N%?%M$r:n$k(B
! 138:
! 139: $B%K%;0x;R$O$[$H$s$I=P$J$$(B
! 140:
! 141: \item $BM-8BBN78?t$N>l9g(B
! 142:
! 143: $B0lJQ?t$KMn$9$H%K%;0x;R$@$i$1(B
! 144:
! 145: $\Rightarrow$ $Z[x]$ $B$KAjEv$9$k$N$O(B $F[y][x]$
! 146:
! 147: \end{itemize}
! 148:
! 149: \end{slide}
! 150:
! 151: \begin{slide}{}
! 152: \begin{center}
! 153: \fbox{\fbc \large $B8=:_$N<BAu(B}
! 154:
! 155: \begin{itemize}
! 156: \item $BL5J?J}J,2r$N%\%H%k%M%C%/$O(B $GCD(f,f')$
! 157:
! 158: Brown's algorithm $B$G7W;;(B $\Rightarrow$ evaluation point $B$NITB-(B
! 159: $B$,LdBj(B $\Rightarrow$ $F$ $B$N3HBgBN$r;H$&(B ($B8e=R(B)
! 160:
! 161: \item 2 $BJQ?t0J30$OA4$F(B Asir $B$G5-=R(B
! 162:
! 163: \item 2 $BJQ?t$N$_(B builtin
! 164:
! 165: $B8zN(>e(B critical $B$N$?$a(B $\Rightarrow$ $B$d$O$j(B $F$ $B$N3HBgBN$,I,MW(B
! 166:
! 167: $B@0?t>e0lJQ?tB?9`<0$N0x?tJ,2r$H6K$a$FN`;w(B
! 168:
! 169: \item 2 $BJQ?t$G%?%M$r:n$C$F(B, $B0lJQ?t$+$i2~$a$F(B Hensel $B9=@.(B
! 170:
! 171: EEZ $B%"%k%4%j%:%`$r=q$1$F$$$J$$$?$a(B
! 172: \end{itemize}
! 173: \end{center}
! 174: \end{slide}
! 175:
! 176: \begin{slide}{}
! 177: \begin{center}
! 178: \fbox{\fbc \large $BM-8BBN$NBe?t3HBg$N8zN(E*I=8=(B}
! 179: \end{center}
! 180:
! 181: evaluation point $B$N3NJ]$N$?$a(B, $BM-8BBN$NBe?t3HBg$,I,MW(B
! 182:
! 183: $F=GF(q)$ $B$N(B $m$ $B<!3HBg(B $F_m$ $\cdots$ $h(x) \in F[x]$ : $m$ $B<!4{Ls(B
! 184: $B$K$h$j(B $F_m = F[x]/(h(x))$
! 185:
! 186: $\Rightarrow$ $B$3$l$G$O7W;;$,BgJQ(B
! 187:
! 188: $q$ $B$O>.$G(B, $\#(F_m)$ $B$,$=$l$J$j$KBg$-$1$l$P$h$$(B
! 189:
! 190: $\Rightarrow$ $F_m$ $B$r86;O:,I=8=$9$l$P$h$$(B
! 191:
! 192: \end{slide}
! 193:
! 194: \begin{slide}{}
! 195: \begin{center}
! 196: \fbox{\fbc \large $B86;O:,I=8=(B $F_m^{\times} = \{\alpha^i | ( 0 \le i \le q-2) \}$}
! 197: \end{center}
! 198: \begin{itemize}
! 199: \item $B$+$1;;(B, $B3d;;(B, $B$Y$->h$OMF0W(B
! 200:
! 201: $\alpha^i \cdot \alpha^j = \alpha^{i+j \bmod q-1}$
! 202:
! 203: \item $BB-$7;;(B, $B0z$-;;$O%F!<%V%k;2>H(B (Faug\`ere GB $B$GMQ$$$i$l$?(B)
! 204:
! 205: $\alpha^i + 1 = \alpha^{a_i}$ $B$J$k(B $a_i$ $B$r7W;;$7(B,
! 206: $(i,a_i)$ $B$r%F!<%V%k$GJ];}(B
! 207:
! 208: $\alpha^i+\alpha^j = \alpha^j(\alpha^{i-j}+1)$ $B$H$7$F7W;;(B
! 209:
! 210: \item $F_m$ $B$N%5%$%:$,(B $2^16$ $BDxEY$^$G$J$i<BMQE*(B
! 211:
! 212: $BBN$r3HBg$7$F$b(B, $B7W;;B.EY$O$[$H$s$IJQ$o$i$J$$(B.
! 213:
! 214: \end{itemize}
! 215: \end{slide}
! 216:
! 217: \begin{slide}{}
! 218: \begin{center}
! 219: \fbox{\fbc \large 2 $BJQ?t$N(B Hensel $B9=@.(B }
! 220: \end{center}
! 221:
! 222: $f(x,y) = \prod_{i=1}^l f_i(x) \bmod y$
! 223:
! 224: $\Rightarrow$ $f(x,y) = \prod_{i=1}^l f_{k,i}(x) \bmod y^k$ $B$X$N(B Hensel $B9=@.$O(B
! 225:
! 226: $f(x,y) = f_{k,1} \cdot F_1 \bmod y^k$
! 227:
! 228: $\Rightarrow$ $F_1(x,y) = f_{k,2} \cdot F_2 \bmod y^k$
! 229:
! 230: $\Rightarrow$ $F_2(x,y) = f_{k,3} \cdot F_3 \bmod y^k$
! 231:
! 232: $\cdots$
! 233:
! 234: $B$H7W;;$7$F$$$/(B $\cdots$ $BC1=c$+$D9bB.(B
! 235:
! 236:
! 237: \end{slide}
! 238:
! 239: \begin{slide}{}
! 240: \begin{center}
! 241: \fbox{\fbc \large Finding true factors}
! 242: \end{center}
! 243:
! 244: \begin{itemize}
! 245: \item combination $\Rightarrow$ trial division
! 246:
! 247: bound $B$h$j>/$7B?$a$K(B Hensel $B9=@.(B
! 248:
! 249: $\Rightarrow$ $d-1$ test ($B:4!9LZ(B, Abbott et al.) $B$,$h$/8z$/(B
! 250:
! 251: $B$7$+$7(B, $BAH$_9g$o$;GzH/$O9nI~$G$-$J$$(B
! 252:
! 253: \item ``funny factorization''
! 254:
! 255: change of ordering $B$K$h$j(B true factor $B$r8+$D$1$k(B
! 256:
! 257: $B%K%;0x;R$,B?$$>l9g$K8z2LE*(B
! 258:
! 259: \end{itemize}
! 260: \end{slide}
! 261:
! 262: \begin{slide}{}
! 263: \begin{center}
! 264: \fbox{\fbc \large Funny factorization}
! 265: \end{center}
! 266:
! 267: $f(x,y)$ : $x$ $B$K$D$$$F(B monic $B$H$9$k(B.
! 268:
! 269: $f(x,y) \simeq g_k(x,y) h_k(x,y) \bmod y^k$
! 270:
! 271: $B$KBP$7(B, $I=Id(g_k,y^k)$ $B$r9M$($k$H(B $\{g_k,y^k\}$ $B$O(B
! 272: $x<_l y$ $B$J$k(B lex order $B$G$N%0%l%V%J4pDl(B
! 273:
! 274: $g_k| g \bmod y^k$, $g|f$ $B$J$k4{Ls0x;R(B $g$ $B$O(B $g\in I$.
! 275:
! 276: \underline{$BDjM}(B}
! 277:
! 278: $k$ $B$,==J,Bg$-$$$H$-(B, $g$ $B$O(B $I$ $B$N(B degree compatible order $<$ $B$G$N%0(B
! 279: $B%l%V%J4pDl$N=g=x:G>.$N85(B
! 280:
! 281: [$B>ZL@(B] $B$b$7(B $g' < g$, $g'\in I$ $B$,$"$l$P(B $Id(g',g)$ $B$O(B 0 $B<!85$G(B,
! 282:
! 283: $\#V(Id(g',g)) \le tdeg(g')\cdot tdeg(g)$ (B\'ezout) $B$@$,(B,
! 284:
! 285: $\#V(I) = k\deg_x(g_k) \le \#V(Id(g',g))$
! 286:
! 287: $B$h$j(B $k$ $B$rBg$-$/$H$l$PL7=b(B.
! 288: \end{slide}
! 289:
! 290: \begin{slide}{}
! 291: \begin{center}
! 292: \fbox{\fbc \large Funny factorization $B$D$E$-(B}
! 293: \end{center}
! 294:
! 295: $I$ $B$N(B $<_l$ $B$G$N%0%l%V%J4pDl$,J,$+$C$F$$$k(B
! 296:
! 297: $\Rightarrow$ change of ordering $B$G(B $g$ $B$,7W;;$G$-$k(B
! 298:
! 299: $k > tdeg(f)^2/\deg_x(g_k)$ $B$J$i(B deterministic
! 300:
! 301: $B>.$5$$<!?t$N0x;R$r4|BT$7$F(B $k$ $B$r>.$5$/$H$k(B
! 302:
! 303: $BFC$K(B, $\bmod y$ $B$G$N0x;R$,B?$$>l9g$KM-8z(B
! 304:
! 305: \end{slide}
! 306:
! 307: \begin{slide}{}
! 308: \begin{center}
! 309: \fbox{\fbc \large $B85$NBN$G$N0x;R$N7W;;(B}
! 310: \end{center}
! 311:
! 312: $F = GF(q)$ $B>e$N4{Ls0x;R$,(B $F_m$ $B>e$GJ,2r$9$k2DG=@-$"$j(B
! 313:
! 314: $f \in F[x_1,\ldots,x_n]$, $f$ : $F$ $B>e4{Ls$G(B $f = \prod f_i$,
! 315: $f_i$ : $F_m$ $B>e4{Ls$H$9$k(B.
! 316:
! 317: $F_m/F$ $B$O(B Galois $B3HBg$G(B, $G=Gal(F_m/F) = \langle \sigma \rangle$ $B$?$@$7(B
! 318: $\sigma : \beta \mapsto \beta^q$
! 319:
! 320: $S$ $B$r(B $f_1$ $B$N(B $G$-orbit $B$H$9$k$H(B $\prod_{s\in S}s$ $B$O(B $G$-$BITJQ$@$+$i(B
! 321: $\prod_{s\in S}s \in F[x_1,\ldots,x_n]$. $f$ $B$O(B $F$ $B>e4{Ls$@$+$i(B
! 322: $f = \prod_{s\in S}s$.
! 323:
! 324: $B$h$C$F(B, $F$ $B>e$N4{Ls0x;R$O(B, $F_m$ $B>e$N4{Ls0x;R$N(B $G$-orbit $B$r5a$a$l$P(B
! 325: $B$h$$(B.
! 326:
! 327: $\sigma(h)$ $B$O78?t$r(B $q$ $B>h$9$l$P$h$$$+$iMF0W(B.
! 328:
! 329: \end{slide}
! 330:
! 331: \begin{slide}{}
! 332: \begin{center}
! 333: \fbox{\fbc \large $B%?%$%_%s%0%G!<%?(B --- [BM97] $B$+$i$NNc(B}
! 334: \end{center}
! 335:
! 336: \underline{ 2 $BJQ?tB?9`<0(B}
! 337: \begin{center}
! 338: {\normalsize
! 339: \begin{tabular}{|c|c|c|c|c|c|c|} \hline
! 340: & $f_2$ & $f_7$ & $f_{11}$ & $f_{13}$ & $f_{17}$ & $f_{17,y\rightarrow y^2}$ \\ \hline
! 341: $\deg_x,\deg_y$ & 7,5 & 100,100 & 300,300 & $10^3,10^3$ & 32,16 & 32,32 \\ \hline
! 342: \#factors & 2 & 2 & 1 & 2 & 1 & 2 \\ \hline
! 343: Maple7 & 4.2 & 30 & 68 & $>$1day & 36 & 32 \\ \hline
! 344: Asir & 0 ($2^3$) & 2.6($7^2$) & 34 & 5040 & 0.24 & 0.57 \\ \hline
! 345: %Hensel & 0 & & & 2965 & & \\ \hline
! 346: \end{tabular}
! 347: }
! 348: \end{center}
! 349: $f_p$ $B$r(B $GF(p)$ $B>e$G0x?tJ,2r(B
! 350:
! 351: Asir $B$N(B $(p^m)$ : $GF(p^m)$ $B$^$G3HBg$9$kI,MW$,$"$C$?(B
! 352: \end{slide}
! 353:
! 354: \begin{slide}{}
! 355: \fbox{\fbc \large Maple $B$H$NHf3S(B}
! 356:
! 357: \begin{itemize}
! 358: \item Hensel $B9=@.$N@-G=Hf3S(B
! 359:
! 360: $f_{11}$ $B$O(B Hensel $B9=@.$,$[$H$s$I(B $\Rightarrow$ Hensel $B9=@.(B
! 361: $B<+BN$N@-G=$OBg:9$J$$(B
! 362:
! 363: [BM97] $B$K$h$l$P(B, $BAGBN>e$NB?9`<0$O(B kernel $B$GFCJL$K<BAu$5$l$F$$$k(B
! 364: (Asir $B$HF1$8(B)
! 365:
! 366: \item $B2rC5:wIt$N@-G=Hf3S(B
! 367:
! 368: $f_{17}$ $B$N>l9g(B, Maple $B$N7W;;;~4V$N$[$H$s$I$O(B bad combination $B$N(B
! 369: $BGS=|(B $\Rightarrow$ $B2?$+FCJL$J$3$H$r$7$F$$$FCY$$(B?
! 370:
! 371: \item $B3HBg$,I,MW$J>l9g$NHf3S(B
! 372:
! 373: Maple $B$G$O(B, $BBN$N3HBg$,I,MW$J>l9g$K$O(B, Domains package ($B0lHL$NM-8BBN$r(B
! 374: $B07$&(B generic $B$J<BAu(B) $B$r;H$&$?$a(B, $f_2$, $f_7$ $B$G:9$,=P$?(B
! 375: \end{itemize}
! 376: \end{slide}
! 377:
! 378:
! 379: \begin{slide}{}
! 380: \begin{center}
! 381: \fbox{\fbc \large $B%?%$%_%s%0%G!<%?(B --- $BBN$N3HBg$H8zN((B}
! 382: \end{center}
! 383:
! 384: \begin{center}
! 385: {\normalsize
! 386: \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
! 387: $B3HBg<!?t(B & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline
! 388: $f_7$ & --- & 2.6 & 2.9 & 2.7 & 2.6 & 5.8 & 9.1 \\ \hline
! 389: $f_{17,y\rightarrow y^2}$ & 0.57 & 0.78 & 0.55 & 2.3 & --- & --- & --- \\ \hline
! 390: \end{tabular}
! 391: }
! 392: \end{center}
! 393:
! 394: \underline{$B9M;!(B}
! 395:
! 396: \begin{itemize}
! 397: \item $BBN$,>.$5$$$&$A$O(B, $B3HBg$7$F$b8zN($OMn$A$J$$(B
! 398:
! 399: $B$b$A$m$s(B, $B3HBg$K$h$j0x;R$,A}$($k>l9g$O=|$/(B
! 400:
! 401: \item $BBN$N%5%$%:$,Bg$-$/$J$k$H(B, $B5^7c$K8zN($,Mn$A$k(B
! 402:
! 403: $B;2>H$9$k%F!<%V%k$,Bg$-$/$J$k(B $\Rightarrow$ $B%-%c%C%7%e%5%$%:$K4X78(B?
! 404: \end{itemize}
! 405: \end{slide}
! 406:
! 407: \begin{slide}{}
! 408: \begin{center}
! 409: \fbox{\fbc \large $BAH$_9g$o$;GzH/$r5/$3$9>l9g(B}
! 410: \end{center}
! 411:
! 412: $f(x,y) = f_{17,y\rightarrow y^2}(x,y)f_{17,y\rightarrow y^2}(x+1,y^2)$
! 413:
! 414: $B??$N0x;R$O(B 4 $B8D(B, $\bmod y$ $B$G$N0x;R(B 32 $B8D(B
! 415:
! 416: \underline{$BDL>o$N(B Belrekamp-Zassenhaus $B7?$G7W;;$7$?>l9g(B}
! 417:
! 418: \begin{itemize}
! 419: \item $B=hM}$7$?AH$_9g$o$;(B : 3852356
! 420: \item $B<!?t%A%'%C%/$GGS=|(B : 3734707
! 421: \item $B7W;;;~4V(B : 162sec
! 422: \end{itemize}
! 423:
! 424: \underline{Funny factorization $B$G7W;;$7$?>l9g(B}
! 425: \begin{itemize}
! 426: \item bound = 16 ($B$3$l$O4E$9$.(B) 2.6sec
! 427: \item bound = 32 ($B>/$J$/$H$b0x;R$O=P$k(B) 17sec
! 428: \end{itemize}
! 429: \end{slide}
! 430:
! 431: \begin{slide}{}
! 432: \begin{center}
! 433: \fbox{\fbc \large $B:#8e$NM=Dj(B}
! 434: \end{center}
! 435:
! 436: \begin{itemize}
! 437: \item $BL5J?J}J,2rIt$N2~NI(B, engine $BAH$_9~$_(B
! 438:
! 439: \item 3 $BJQ?t0J>e$X$N(B Hensel $B9=@.$r(B EEZ $B2=(B
! 440:
! 441: \item 2 $BJQ?tB?9`<0$N@Q$N(B, Karatsuba $B$K$h$k9bB.2=(B
! 442:
! 443: \item Zassenhaus, Funny $B$N(B OpenXM $B$K$h$kJBNs2=(B
! 444:
! 445: \item knapsack $B7?$N%"%k%4%j%:%`$NE,MQ2DG=@-$N8!F$(B
! 446: \end{itemize}
! 447: \end{slide}
! 448:
! 449: \end{document}
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