% $OpenXM: OpenXM/doc/Papers/rims2002-noro.tex,v 1.2 2002/12/09 04:23:05 noro Exp $
\documentclass{slides}
\usepackage{color}
\usepackage{rgb}
\usepackage{graphicx}
\usepackage{epsfig}
\newcommand{\qed}{$\Box$}
\newcommand{\mred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}}}
\newcommand{\tmred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}\limits^{\scriptstyle *}}}
\def\gr{Gr\"obner basis }
\def\st{\, s.t. \,}
\def\ni{\noindent} 
\def\ve{\vfill\eject} 
\textwidth 9.2in
\textheight 7.2in
\columnsep 0.33in
\topmargin -1in
\def\tc{\color{red}}
\def\fbc{\bf\color{MediumBlue}}
\def\itc{\color{brown}}
\def\urlc{\bf\color{DarkGreen}}
\def\bc{\color{LightGoldenrod1}}

\def\HT{{\rm HT}}
\def\HC{{\rm HC}}
\def\GCD{{\rm GCD}}
\def\tdeg{{\rm tdeg}}
\def\pp{{\rm pp}}
\def\lc{{\rm lc}}
\def\Z{{\bf Z}}

\title{\tc $B>.I8?tM-8BBN>e$NB?JQ?tB?9`<0$N0x?tJ,2r$K$D$$$F(B ($B$=$N(B 2)}

\author{$BLnO$(B $B@59T(B ($B?@8MBg!&M}(B)}
\begin{document}
\large
\setlength{\parskip}{0pt}
\maketitle

\begin{slide}{}
\begin{center}
\fbox{\fbc \large $BL5J?J}J,2r(B ($BI|=,(B)}
\end{center}

modification of Bernardin's algorithm [Ber97]

$f \in F[x_1,\ldots,x_n]$, $F$ : $BM-8BBN(B $Char(F) = p$

$f = FGH$, where $F=\prod f_i^{a_i}$, 
$G=\prod g_j^{b_j}$, 
$H=\prod h_k^{c_k}$ 

$f_i, g_j, h_k$ : $BL5J?J}(B, $B8_$$$KAG(B.

$'$ $B$r(B $d/dx_1$ $B$H$7$F(B
$f_i' \neq 0$, $p \not{|}a_j$, $p | b_j$, $h_k' = 0$
$B$H=q$/$H(B

$f' = F'GH$ $B$9$k$H(B $GCD(f,f') = GCD(F,F')GH$

$GCD(F,F') = \prod f_i^{a_i-1}$ $B$@$+$i(B $f/GCD(f,f')=\prod f_i$

$\prod f_i$ $B$G(B $f$ $B$r7+$jJV$73d$k$3$H$G(B, $f_1$ ($B=EJ#EY:G>.(B)
$B$,5a$^$k(B

$\Rightarrow$ $F$ $B$,A4$FJ,2r$G$-$k(B

\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large $BL5J?J}J,2r(B ($B$D$E$-(B) }
\end{center}

$B;D$j(B $f = GH$ $B$G(B,  $f' = 0$

$B$3$l$r(B $x_i$ $B$K$D$$$F7+$jJV$7$F;D$C$?(B $f$

$\Rightarrow$ $df/dx_1 = \ldots = df/dx_n = 0$ 

$\Rightarrow$ $B$3$l$O(B, $BA4$F$N;X?t$,(B $p$ $B$G3d$j@Z$l$k$3$H$r0UL#$9$k(B

$\Rightarrow$ $F$ $B$OM-8BBN$@$+$i(B $f = g^p$ $B$H=q$1$k(B

$\Rightarrow$ $g$ $B$KBP$7$F%"%k%4%j%:%`$rE,MQ(B
\end{slide}


\begin{slide}{}
\begin{center}
\fbox{\fbc \large $B<BAu>e$N:$Fq(B : $\GCD(f,f')$ $B$N7W;;(B}
\end{center}

$BI8?t$,(B 0 $B$N>l9g(B $\GCD(g,f'/g)=1$ $\Rightarrow$ $BB?JQ?t$N(B Hensel $B9=@.$,(B
$B;H$($k(B

$B@5I8?t$N>l9g(B $$\GCD(g,f'/g) = \GCD(\GCD(F,F')GH,F'/\GCD(F,F'))$$
$\Rightarrow$ $GH$ $B$NB8:_$N$?$a(B GCD $B$,(B 1 $B$H$O8B$i$J$$(B.

$\Rightarrow$ $B$d$`$J$/(B Brown $B$N%"%k%4%j%:%`(B ($BCf9q>jM>DjM}$K$h$k(B GCD $B$N(B
$B7W;;(B) $B$rMQ$$$F$$$k(B.
\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large GCD $B$N7W;;(B}
\end{center}

\begin{tabbing}
$BF~NO(B : $f_1,\ldots,f_m \in K[X]$ ($K$ $B$OBN(B, $X$ $B$OJQ?t$N=89g(B)\\
$B=PNO(B : $\GCD(f_1,\ldots,f_m)$\\
$y \leftarrow$ $BE,Ev$JJQ?t(B; $Z \leftarrow X\setminus \{y\}$\\
$< \leftarrow K[Z]$ $B$NE,Ev$J9`=g=x(B; $B0J2<(B $f_i \in K[y][Z]$ $B$H$_$J$9(B\\
$h_i(y) \leftarrow \HT_<(f_i)$; $h_g(y) \leftarrow \GCD(h_1,\ldots,h_m)$\\
$g \leftarrow 0$; $M \leftarrow 1$\\
do \= \\
   \> $a \leftarrow $ $BL$;HMQ$N(B $K$ $B$N85(B\\
   \> $g_a \leftarrow \GCD(f_1|_{y=a},\ldots,f_m|_{y=a})$\\
   \> if \= $g \neq 0$ $B$+$D(B $\HT_<(g) = \HT_<(g_a)$ then \\
   \>    \> $adj \leftarrow h_g(a)/\HC_<(g_a)\cdot g_a - g(a)$
\end{tabbing}
\end{slide}

\begin{slide}{}
\begin{tabbing}
do \= if \= \kill
   \>    \> if \= $adj = 0$ $B$+$D(B, $B$9$Y$F$N(B $f_i$ $B$KBP$7(B $g | h_g\cdot f_i$  then \\
   \>    \>    \> return $\pp(g)$\\
   \>    \> endif\\
   \>    \> $g \leftarrow g+adj \cdot M(a)^{-1} \cdot M$; $M \leftarrow M\cdot (y-a)$\\
   \> else if $\tdeg(\HT_<(g)) > \tdeg(\HT_<(g_a))$ then \\
   \>    \> $g \leftarrow g_a$; $M \leftarrow  y-a$\\
   \> else if $\tdeg(\HT_<(g)) = \tdeg(\HT_<(g_a))$ then \\
   \>    \> $g \leftarrow 0$; $M \leftarrow 1$\\
   \> endif\\
end do
\end{tabbing}
\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large $BFsJQ?t$X$N5"Ce(B}
\end{center}

\begin{itemize}
\item $B<gJQ?t(B $x$ $B$NA*Br(B

$x$ $B$K4X$9$kHyJ,$,>C$($J$$$h$&$KA*$V(B. 

\item $B=>JQ?t(B $y$ $B$NA*Br(B

$K[x,y]$ $B$G0x?tJ,2r$7$F(B, $Z=X\setminus \{x,y\}$ $B$K4X$7$F(B Hensel $B9=@.(B

($B0lJQ?t$^$GMn$9$H%K%;0x;R$,BgNLH/@8(B)

\item $B$=$l0J30$NJQ?t(B $Z$ $B$X$NBeF~CM$NA*Br(B

$f_a(x,y) = f(x,y,a)$ $B$,L5J?J}$K$J$k$h$&$KA*$V(B. 

$B$5$i$K(B, $f_a|_{y=0}$ $B$bL5J?J}$K$J$k$h$&$K(B, $y\leftarrow y+c$ $B$H(B
$BJ?9T0\F0(B.
\end{itemize}

\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large $K[y]$ $B>e$G$N(B Hensel $B9=@.(B ($BA0=hM}(B)}
\end{center}

$f_a(x,y)$ $B$N0x;R$r(B 2 $BAH$K$o$1(B $f_a(x,y) = g_0(x,y)h_0(x,y)$
$B$+$i(B $K[y]$ $B>e$G(B Hensel $B9=@.(B

$g_0$, $h_0$ $B$N(B $x$ $B$K4X$9$k<g78?t$N7h$aJ}(B

$B<g78?tLdBj$N2sHr(B : $B??$N0x;R$N<g78?t$H$J$k$Y$/6a$$$b$N$r$"$i$+$8$a8GDj(B

$B??$N0x;R$N<g78?t$O(B $\lc_x(f)$ $B$N0x;R$G$"$k$3$H$r;H$C$?8+@Q$j(B
\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large $B<g78?t$N8+@Q$j(B}
\end{center}
\begin{enumerate}
\item $\lc_x(f) = \prod u_i^{n_i}$ $B$H0x?tJ,2r$9$k(B ($u_i \in K[y,Z]$ : $B4{Ls(B).

\item $B3F(B $i$ $B$KBP$7(B, $u_i(a) \in K[y]$ $B$,(B $\lc_x(g_0)$ $B$r3d$j@Z$k2s?t$r(B
$B?t$($k(B. $B$=$l$r(B $m_i$ $B$H$7$?$H$-(B, $\lc_g = \prod u_i^{m_i}$ $B$H$9$k(B. 
$BF1MM$K(B $h_0$ $B$KBP$7$F$b(B $\lc_h$ $B$r5a$a$k(B. 

$B$b$7(B, 
$\lc_x(g_0) \not{|}\, \lc_g(a)$ $B$^$?$O(B $\lc_x(h_0) \not{|}\, \lc_h(a)$
$B$^$?$O(B, 
$\lc_x(f) \not{|}\, \lc_g \cdot \lc_h$
$B$J$i$P(B, $B$=$l$O(B, $f_a$ $B$N0x;R$NAH9g$;$,@5$7$/$J$$$3$H$r0UL#$9$k$N$G(B, 
$g_0$, $h_0$ $B$r$H$jD>$9(B. 

\item

$g_0 \leftarrow \lc_g(a)/\lc_x(g_0)\cdot g_0$ $B$N<g78?t$r(B $\lc_g$ $B$GCV$-49$($?$b$N(B

$h_0 \leftarrow \lc_h(a)/\lc_x(h_0)\cdot h_0$ $B$N<g78?t$r(B $\lc_h$ $B$GCV$-49$($?$b$N(B

$f \leftarrow \lc_g\cdot \lc_h/\lc_x(f) \cdot f$

$B$H$9$k(B. $B$3$N;~(B, $f = g_0h_0$ $B$H$J$C$F$$$k(B. 
\end{enumerate}
\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large $K[y]$ $B>e$G$N(B Hensel $B9=@.(B}
\end{center}

$f=g_0h_0$ $B$G(B, $g_0$ $B$,@5$7$$0x;R$N<M1F(B, $\lc_x(g_0)$ $B$H2>Dj(B

$g_0$, $h_0$ $B$+$i(B $K[y]$ $B>e$N(BHensel $B9=@.$K$h$j(B, 
$$f=g_kh_k \bmod I^{k+1}$$
$B$H;}$A>e$2$k(B ($I = \langle z_1-a_1,\ldots,z_{n-2}-a_{n-2} \rangle$).

$z_i \rightarrow z_i+a_i$ $B$J$kJ?9T0\F0$K$h$j(B, 
$I=\langle z_1,\ldots,z_{n-2} \rangle$

$BDL>o$N(B EZ $BK!(B : $\Z/(p^l)$ $B$G7W;;(B ($B78?t$KJ,?t$,8=$l$k$N$rHr$1$k(B)

$B:#2s$N<BAu(B : $\deg_y(f) > d$ $B$J$k(B $d$ $B$KBP$7(B, $K[y]/(y^d)$ 
$B>e$G7W;;$9$k(B. ($K[y]$ $B$G$N>&BN$G$N1i;;$rHr$1$k(B)

$B$3$l$O(B, $u g_0(a)+v h_0(a)=1 \bmod y^d$ $B$H$J$k(B $u, v \in K[y]$ 
$B$K$h$j2DG=(B

Hensel $B9=@.$O(B $\bmod\,  y^d$ $B$G9T$&(B. 
\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large Hensel $B9=@.(B}
\end{center}
$f = g_kh_k \bmod (I^{k+1},y^d)$ $B$@$,(B, 
$B==J,Bg$-$$(B $k$ $B$KBP$7(B $f = g_kh_k$

$u$, $v$ $B$N7W;;(B : Hensel $B9=@.(B
($g_0(a)|_{y=0}$, $h_0(a)_{y=0}$ $B$,8_$$$KAG(B)

$K[y]$ $B>e$N(B Hensel $B9=@.$O<!$N$h$&$K9T$&(B. 

\begin{enumerate}
\item $f-g_{k-1}h_{k-1} = \sum_t F_t t \bmod (I^k,y^d)$ $B$H=q$/(B. 
$B$3$3$G(B $t \in I^k$ $B$OC19`<0(B, $F_t \in K[y][x]$.
\item $G_th_0+H_tg_0 = F_t \bmod y^d$ $B$H$J$k(B $G_t, H_t \in K[y][x]$ $B$r7W;;$9$k(B. 
$B$3$l$O(B $u$, $v$ $B$r;H$C$F:n$l$k(B. 
\item $g_{k+1} \leftarrow g_k + \sum_t G_t t$,
$h_{k+1} \leftarrow h_k + \sum_t H_t t$ $B$H$9$l$P(B $f = g_{k+1}h_{k+1} \bmod (I^{k+1},y^d)$.
\end{enumerate}

$\sum_t G_t t = 0$ $B$^$?$O(B $\sum_t H_t t = 0$ $\Rightarrow$ $B;n$73d$j(B

$g_k$ $B$^$?$O(B $h_k$ $B$G(B $f$ $B$r3d$C$F$_$k$3$H$G(B, $B<!?t$N>e8B$^$G(B Hensel $B9=@.(B
$B$;$:$K(B, $B??$N0x;R$r8!=P$G$-$k(B. 
\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large $BM-8BBN$NI=8=$K$D$$$F(B}
\end{center}

$B3F%"%k%4%j%:%`$K$*$$$F(B, $B78?tBN$N0L?t$,==J,Bg$-$$I,MW$"$j(B. 

$BBeF~$9$kE@$N?t$,ITB-$9$k(B $\Rightarrow$ $BBe?t3HBg(B

$B7W;;8zN($,Mn$A$J$$$h$&(B, $B86;O:,$rMQ$$$?I=8=$r<BAu(B

$B7gE@(B : $B<BMQE*$J0L?t$,(B $2^{16}$ $BDxEY$K8B$i$l$k(B

$B2~NI(B : $BI8?t$,(B $2^{14}$ $B0J2<$N>l9g$K$O(B, $B86;O:,I=8=(B

$B$=$l0J>e$N>l9g$K$O(B, $BDL>o$NI=8=(B
($B<BMQ>e==J,$KBeF~$9$kE@$,F@$i$l$k$+$i(B)

$\Rightarrow$ $B0L?t$,(B $2^{29}$ $BDxEY$^$G$NAGBN>e$G(B, $BB?JQ?tB?9`<0(B
$B$N0x?tJ,2r$,2DG=(B
\end{slide}

\begin{slide}
\begin{center}
\fbox{\fbc \large $B78?t4D$H$7$F$N(B $K[y]/(y^d)$ $B$K$D$$$F(B}
\end{center}

Hensel $B9=@.$K$*$$$F(B $K[y]/(y^d)$ $B$r78?t4D$H$7$F07$&(B

Asir : $B>.0L?tM-8BBN(B $K$ $B$NBe?t3HBg$rI=8=$9$k7?(B (GFSN) $B$,$"$k(B

$K[y]/(m(y))$ $B$H$7$FI=8=(B $\Rightarrow$ $m(y)=y^d$ $B$H$7$FN.MQ(B

$B5U857W;;$K$D$$$F$O(B, 0 $B$G$J$$Dj?t9`$r;}$DB?9`<0$O2D5U(B ($B8_=|K!(B)

$\lc_x \neq 0$ $B$h$j(B $K[y]/(y^d)$ $B$,$3$NJ}K!$G$G$-$k(B. 

$BB?JQ?tB?9`<0(B : Hensel $B9=@.$N:G=i$G(B, $B$3$N7?$N78?t$r;}$DB?9`<0$KJQ49(B

$d$ $B$r%;%C%H$7$F$*$/(B $\Rightarrow$ $B7k2L$O<+F0E*$K(B $\bmod \, y^d$ $B$5$l$k(B

$\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d)$ $B78?t$N(B
$BB?9`<01i;;$,<B9T$G$-$k(B. 
\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large Timing data (Wang $B$NNc(B)}
\end{center}

{\tt OpenXM\_contrib2/asir2000/lib/fctrdata} $B$N(B

 {\tt Wang[1],\ldots,Wang[15]} $B$G%F%9%H(B. 

$B%^%7%s(B : Athlon 1900+

Maple7 $B$HHf3S(B --- Maple7 $B$b(B Kernel $B$G=t1i;;$r%5%]!<%H$7$F$$$k$N$G(B, $B%"%s%U%'%"$G$O$J$$$@$m$&(B

$B7k2L(B : $B0lIt(B ({\tt Wang[8]}) $B$r=|$$$FNI9%(B
\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large Timing data (Maple7)}
\end{center}

\begin{center}
% &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \hline
{\small
\begin{tabular}{c|ccccccccc} \hline
$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
2 & N & F & F & F & N & N & 0.01 & 1 & 0.01 \\ \hline
3 & 0.07 & 0.1 & 0.07 & N & 0.4 & N & 0.01 & 0.02 & 0.06 \\ \hline
5 & N & 0.05 & 0.08 & 3.5 & 0.2 & 0.4 & 0.01 & 0.6 & 0.1 \\ \hline
7 & 0.08  & 0.1  & 0.1  & 0.25 & 0.6  & 0.5  & 0.02  & 1  & F \\ \hline
\end{tabular}

\begin{tabular}{c|cccccc} \hline
$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
2 & F & N & 0.005 & 0.006 & 0.008 & F \\ \hline
3 & 4 & N & 0.004 & 0.007 & 0.14  & 0.02 \\ \hline
5 & 0.2  & F  & 0.005 & 0.006 & 0.03  &  0.4 \\ \hline
7 & 0.6  & 14  & 0.005  & 0.16  & 0.04 & 0.6 \\ \hline
\end{tabular}

\begin{tabular}{c|ccccccccc} \hline
$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
547 & 0.2& 0.2& 0.1& 0.3& 1& 1.2& 0.02& 6& F\\ \hline
32003&  0.2& 0.2&  0.2& 0.4 & 1 & 1 & 0.02 & 4.2 & F  \\ \hline
99981793 & 0.5 & 0.6 & 0.5 & 3 & 3 & 4.5 & 0.02 & N &  F\\ \hline
\end{tabular}

\begin{tabular}{c|cccccc} \hline
$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
547 & 0.9 &3.3  & 0.005 & 0.2 & 0.1 & 0.4  \\ \hline
32003 & 1.8 & 4.9 &0.006  & 0.3 & 0.1  & 0.4 \\ \hline
99981793 & 2.6  & 11  & 0.006 & 0.9 & 0.5  & 1.4  \\ \hline
\end{tabular}
}
\end{center}
\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large Timing data (Asir)}
\end{center}

\begin{center}
% &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \hline
{\small
\begin{tabular}{c|ccccccccc} \hline
$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
2(5) & 0.003 & 0.003 & 0.004 & 0.01 & 0.02 & 0.05 & 0.001 & 0.01 & 0.0003 \\ \hline
3(5) & 0.003 & 0.002 & 0.005 & 0.003 & 0.003 & 0.1 & 0.002 & 0.001 & 0.003  \\ \hline
5(2) & 0.004 & 0.003 & 0.004 & 0.02 & 0.06 & 0.4 & 0.002 & 0.4 & 0.005 \\ \hline
7(2) & 0.004  & 0.004  & 0.005 & 0.03 & 0.1  & 0.1  & 0.004  & 1.8  & 0.2 \\ \hline
\end{tabular}

\begin{tabular}{c|cccccc} \hline
$p$ & 10 & 11 & 12 & 13 & 14 &  15 \\ \hline
2(5) & 0.03 & 0.07 & 0.0006 & 0.001 & 0.002 & 0.001  \\ \hline
3(5) & 0.04 & 0.2 & 0.0001  & 0.0005 & 0.02  & 0.001 \\ \hline
5(2) & 0.01  & 0.2  & 0.001 & 0.001 & 0.004  & 0.01  \\ \hline
7(2) & 0.02  & 0.6  & 0.001  & 0.007  & 0.005 & 0.01 \\ \hline
\end{tabular}

\begin{tabular}{c|ccccccccc} \hline
$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
547 & 0.004 & 0.004 & 0.005 & 0.03 & 0.05 & 0.2 & 0.02& 2& 0.2\\ \hline
32003 & 0.004  &  0.004 & 0.005  &0.04  &0.07  & 0.2 & 0.004 & 2 & 0.2 \\ \hline
99981793& 0.004 & 0.004& 0.005 & 0.03 & 0.03 & 0.2 & 0.004 & 4 & 0.2  \\ \hline
\end{tabular}

\begin{tabular}{c|cccccc} \hline
$p$ & 10 & 11 & 12 & 13 & 14 &  15 \\ \hline
547 & 0.04 & 0.3 & 0.001 &0.006  & 0.006 & 0.01  \\ \hline
32003 & 0.04 &0.2  &0.001  &0.007  & 0.006  & 0.03 \\ \hline
99981793 & 0.04  & 0.3  &0.001  & 0.008 & 0.008  & 0.01  \\ \hline
\end{tabular}
}
\end{center}
\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\fbc \large $B:#8e$NM=Dj(B}
\end{center}

\begin{itemize}
\item $B@5I8?t$N=`AGJ,2r$N<BAu(B. 

\item $BBN$N0L?t$,B-$j$J$$>l9g$K(B, $B<+F0E*$K4pACBN$r3HBg$9$k(B. 

\item $BBN$NI8?t$,==J,Bg$-$$>l9g$K(B, $BL5J?J}J,2r$rI8?t(B 0 $B$H(B
$BF1MM$N(B Hensel $B9=@.$G9T$&$h$&$K$9$k(B. 

\item 2 $BJQ?t$N0x?tJ,2r$K$*$$$F(B, \cite{funny01} $B$G=R$Y$?(B, $BB?9`<0(B
$B;~4V%"%k%4%j%:%`$r<+F0E*$KA*Br$7$F<B9T$9$k(B. 
\end{itemize}
\end{slide}

\end{document}

\begin{thebibliography}{99}
\bibitem{B97-2}
Bernardin, L. (1997).
On square-free factorization of multivariate polynomials over a finite
field.
{\em Theoret.\ Comput.\ Sci.\/} {\bf 187}, 105--116. 

\bibitem{funny01}
M. Noro and K. Yokoyama (2002).
Yet Another Practical Implementation of Polynomial Factorization
 over Finite Fields.
Proceedings of ISSAC2002, ACM Press, 200--206.
\end{thebibliography}

