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version 1.3, 2001/10/12 02:22:17 version 1.5, 2001/10/12 05:11:36
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 % $OpenXM: OpenXM/doc/Papers/dag-noro.tex,v 1.2 2001/10/10 06:32:10 noro Exp $  % $OpenXM: OpenXM/doc/Papers/dag-noro.tex,v 1.4 2001/10/12 02:58:35 noro Exp $
 \documentclass{slides}  \documentclass{slides}
 \usepackage{color}  \usepackage{color}
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   \def\fbc{\bf\color{MediumBlue}}
   \def\itc{\color{brown}}
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   \def\bc{\color{LightGoldenrod1}}
   
 \title{A computer algebra system Risa/Asir and OpenXM}  \title{\tc A computer algebra system Risa/Asir and OpenXM}
   
 \author{Masayuki Noro\\ Kobe University, Japan}  \author{Masayuki Noro\\ Kobe University, Japan}
 \begin{document}  \begin{document}
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 %\begin{slide}{}  %\begin{slide}{}
 %\begin{center}  %\begin{center}
 %\fbox{\large Part I : OpenXM and Risa/Asir --- overview and history}  %\fbox{\fbc \large Part I : OpenXM and Risa/Asir --- overview and history}
 %\end{center}  %\end{center}
 %\end{slide}  %\end{slide}
   
 %\begin{slide}{}  %\begin{slide}{}
 %\fbox{Integration of mathematical software systems}  %\fbox{\fbc Integration of mathematical software systems}
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item Data integration  %\item Data integration
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item OpenMath ({\tt http://www.openmath.org}) , MP [GRAY98]  %\item OpenMath ({\urlc \tt http://www.openmath.org}) , MP [GRAY98]
 %\end{itemize}  %\end{itemize}
 %  %
 %Standards for representing mathematical objects  %Standards for representing mathematical objects
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 %\begin{itemize}  %\begin{itemize}
 %\item MathLink, OpenMath+MCP, MP+MCP  %\item MathLink, OpenMath+MCP, MP+MCP
 %  %
 %and OpenXM ({\tt http://www.openxm.org})  %and OpenXM ({\urlc \tt http://www.openxm.org})
 %\end{itemize}  %\end{itemize}
 %  %
 %Both are necessary for practical implementation  %Both are necessary for practical implementation
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 %\end{itemize}  %\end{itemize}
 %\end{slide}  %\end{slide}
 \begin{slide}{}  \begin{slide}{}
 \fbox{A computer algebra system Risa/Asir}  \fbox{\fbc A computer algebra system Risa/Asir}
   
 ({\tt http://www.math.kobe-u.ac.jp/Asir/asir.html})  
   
 \begin{itemize}  \begin{itemize}
 \item Software mainly for polynomial computation  \item {\itc Software mainly for polynomial computation}
   
 \item User language with C-like syntax  Polynomial factorization, Groebner basis computation
   \item {\itc User language with C-like syntax}
   
 C language without type declaration, with list processing  C language without type declaration, with list processing
   
 \item Builtin {\tt gdb}-like debugger for user programs  \item {\itc Builtin {\tt gdb}-like debugger for user programs}
   
 \item Open source  \item {\itc Open source} ({\urlc \tt http://www.math.kobe-u.ac.jp/Asir/asir.html})
   
 Whole source tree is available via CVS  The source and binaries are available via ftp or CVS
   
 The latest version : see {\tt http://www.openxm.org}  See {\urlc \tt http://www.openxm.org} to get the latest version
   
 \item OpenXM interface  \item {\itc OpenXM interface}
   
 \begin{itemize}  \begin{itemize}
 \item OpenXM  \item OpenXM ({\urlc \tt http://www.openxm.org})
   
 An infrastructure for exchanging mathematical data  An infrastructure for exchanging mathematical data
 \item Risa/Asir is a main client in OpenXM package.  \item Risa/Asir is a main client in OpenXM package
 \item An OpenXM server {\tt ox\_asir}  \item {\tt ox\_asir} is an OpenXM server
 \item A library with OpenXM library interface {\tt libasir.a}  \item {\tt libasir.a} provides OpenXM interface via function call
 \end{itemize}  \end{itemize}
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Goal of developing Risa/Asir}  \fbox{\fbc Goal of developing Risa/Asir}
   
 \begin{itemize}  \begin{itemize}
 \item Testing new algorithms  \item {\itc Testing new algorithms}
   
 \begin{itemize}  \begin{itemize}
 \item Development started in Fujitsu labs  \item Development started in Fujitsu labs
Line 107  Polynomial factorization, Groebner basis related compu
Line 111  Polynomial factorization, Groebner basis related compu
 cryptosystems , quantifier elimination , $\ldots$  cryptosystems , quantifier elimination , $\ldots$
 \end{itemize}  \end{itemize}
   
 \item To be a general purpose, open system  \item {\itc To be a general purpose, open system}
   
 Since 1997, we have been developing OpenXM package  Since 1997, we have been developing OpenXM package
 containing various servers and clients  containing various servers and clients
   
 Risa/Asir is a component of OpenXM  Risa/Asir is a component of OpenXM
   
 \item Environment for parallel and distributed computation  \item {\itc Environment for parallel and distributed computation}
   
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 %\begin{slide}{}  %\begin{slide}{}
 %\fbox{Capability for polynomial computation}  %\fbox{\fbc Capability for polynomial computation}
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item Fundamental polynomial arithmetics  %\item Fundamental polynomial arithmetics
Line 150  Risa/Asir is a component of OpenXM
Line 154  Risa/Asir is a component of OpenXM
 %\end{slide}  %\end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{History of development : Polynomial factorization}  \fbox{\fbc History of development : Polynomial factorization}
   
 \begin{itemize}  \begin{itemize}
 \item 1989  \item {\itc 1989}
   
 Start of Risa/Asir with Boehm's conservative GC  Start of Risa/Asir with Boehm's conservative GC
   
 ({\tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc})  ({\urlc \tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc})
   
 \item 1989-1992  \item {\itc 1989-1992}
   
 Univariate and multivariate factorizers over {\bf Q}  Univariate and multivariate factorizers over {\bf Q}
   
 \item 1992-1994  \item {\itc 1992-1994}
   
 Univariate factorization over algebraic number fields  Univariate factorization over algebraic number fields
   
 Intensive use of successive extension, non-squarefree norms  Intensive use of successive extension, non-squarefree norms
   
 \item 1996-1998  Application to splitting field and Galois group computation
   
   \item {\itc 1996-1998}
   
 Univariate factorization over large finite fields  Univariate factorization over large finite fields
   
 Motivated by a reseach project in Fujitsu on cryptography  Motivated by a reseach project in Fujitsu on cryptography
   
 \item 2000-current  \item {\itc 2000-current}
   
 Multivariate factorization over small finite fields (in progress)  Multivariate factorization over small finite fields (in progress)
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{History of development : Groebner basis}  \fbox{\fbc History of development : Groebner basis}
   
 \begin{itemize}  \begin{itemize}
 \item 1992-1994  \item {\itc 1992-1994}
   
 User language $\Rightarrow$ C version; trace lifting [TRAV88]  User language $\Rightarrow$ C version; trace lifting [TRAV88]
   
 \item 1994-1996  \item {\itc 1994-1996}
   
 Trace lifting with homogenization  Trace lifting with homogenization
   
Line 199  Modular change of ordering/RUR[ROUI96] [NOYO99]
Line 205  Modular change of ordering/RUR[ROUI96] [NOYO99]
   
 Primary ideal decomposition [SHYO96]  Primary ideal decomposition [SHYO96]
   
 \item 1996-1998  \item {\itc 1996-1998}
   
 Efficient content reduction during NF computation [NORO97]  Efficient content reduction during NF computation [NORO97]
 Solved {\it McKay} system for the first time  Solved {\it McKay} system for the first time
   
 \item 1998-2000  \item {\itc 1998-2000}
   
 Test implementation of $F_4$ [FAUG99]  Test implementation of $F_4$ [FAUG99]
   
 \item 2000-current  \item {\itc 2000-current}
   
 Buchberger algorithm in Weyl algebra  Buchberger algorithm in Weyl algebra
   
Line 217  Efficient $b$-function computation[OAKU97] by a modula
Line 223  Efficient $b$-function computation[OAKU97] by a modula
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Timing data --- Factorization}  \fbox{\fbc Timing data --- Factorization}
   
 \underline{Univariate; over {\bf Q}}  \underline{\itc Univariate; over {\bf Q}}
   
 $N_i$ : a norm of a polynomial, $\deg(N_i) = i$  $N_{i,j}$ : a norm of a polynomial, $\deg(N_i) = i$ with $j$ modular factor
 \begin{center}  \begin{center}
 \begin{tabular}{|c||c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|} \hline
                 & $N_{105}$ & $N_{120}$ & $N_{168}$ & $N_{210}$ \\ \hline                  & $N_{105,23}$ & $N_{120,20}$ & $N_{168,24}$ & $N_{210,54}$ \\ \hline
 Asir    & 0.86  & 59 & 840 & hard \\ \hline  {\tc Asir}      & {\tc 0.86}    & {\tc 59} & {\tc 840} & {\tc hard} \\ \hline
 Asir NormFactor & 1.6   & 2.2& 6.1& hard \\ \hline  Asir NormFactor & 1.6   & 2.2& 6.1& hard \\ \hline
 %Singular& hard?        & hard?& hard? & hard? \\ \hline  %Singular& hard?        & hard?& hard? & hard? \\ \hline
 CoCoA 4 & 0.2   & 7.1   & 16 & 0.5 \\ \hline\hline  CoCoA 4 & 0.2   & 7.1   & 16 & 0.5 \\ \hline\hline
Line 233  NTL-5.2 & 0.16  & 0.9  & 1.4 & 0.4 \\ \hline
Line 239  NTL-5.2 & 0.16  & 0.9  & 1.4 & 0.4 \\ \hline
 \end{tabular}  \end{tabular}
 \end{center}  \end{center}
   
 \underline{Multivariate; over {\bf Q}}  \underline{\itc Multivariate; over {\bf Q}}
   
 $W_{i,j,k} = Wang[i]\cdot Wang[j]\cdot Wang[k]$ in {\tt asir2000/lib/fctrdata}  $W_{i,j,k} = Wang[i]\cdot Wang[j]\cdot Wang[k]$ in {\tt asir2000/lib/fctrdata}
 \begin{center}  \begin{center}
 \begin{tabular}{|c||c|c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|c|} \hline
         & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline          & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline
 Asir    & 0.2 & 4.7 & 14 & 17 & 0.4 \\ \hline  variables & 3 & 5 & 5 & 5 & 4 \\ \hline
   monomials & 905 & 41369 & 51940 & 30988 & 3344 \\ \hline\hline
   {\tc Asir}      & {\tc 0.2} & {\tc 4.7} & {\tc 14} & {\tc 17} & {\tc 0.4} \\ \hline
 %Singular& $>$15min     & ---   & ---& ---& ---\\ \hline  %Singular& $>$15min     & ---   & ---& ---& ---\\ \hline
 CoCoA 4 & 5.2 & $>$15min        & $>$15min & $>$15min & 117 \\ \hline\hline  CoCoA 4 & 5.2 & $>$15min        & $>$15min & $>$15min & 117 \\ \hline\hline
 Mathematica 4& 0.2      & 16    & 23 & 36 & 1.1 \\ \hline  Mathematica 4& 0.2      & 16    & 23 & 36 & 1.1 \\ \hline
Line 251  Maple 7& 0.5  & 18  & 967  & 48 & 1.3 \\ \hline
Line 259  Maple 7& 0.5  & 18  & 967  & 48 & 1.3 \\ \hline
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Timing data --- DRL Groebner basis computation}  \fbox{\fbc Timing data --- DRL Groebner basis computation}
   
 \underline{Over $GF(32003)$}  \underline{\itc Over $GF(32003)$}
 \begin{center}  \begin{center}
 \begin{tabular}{|c||c|c|c|c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
                 & $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline                  & $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline
 Asir $Buchberger$       & 31 & 1687  & 2.6  & 27 & 294  & 4309 & --- \\ \hline  {\tc Asir $Buchberger$} & {\tc 31} & {\tc 1687}  & {\tc 2.6}  & {\tc 27} & {\tc 294}  & {\tc 4309} & --- \\ \hline
 Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline  Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline
 CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021  & --- \\ \hline\hline  CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021  & --- \\ \hline\hline
 Asir $F_4$      & 5.3 & 129 & 0.5  & 4.5 & 31  & 273 & 2641 \\ \hline  {\tc Asir $F_4$}        & {\tc 5.3} & {\tc 129} & {\tc 0.5}  & {\tc 4.5} & {\tc 31}  & {\tc 273} & {\tc 2641} \\ \hline
 FGb(estimated)  & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline  FGb(estimated)  & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline
 \end{tabular}  \end{tabular}
 \end{center}  \end{center}
   
 \underline{Over {\bf Q}}  \underline{\itc Over {\bf Q}}
   
 \begin{center}  \begin{center}
 \begin{tabular}{|c||c|c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|c|} \hline
                 & $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline                  & $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline
 Asir $Buchberger$       & 389 & 594 & 29 & 299 & 34950 \\ \hline  {\tc Asir $Buchberger$}         & {\tc 389} & {\tc 594} & {\tc 29} & {\tc 299} & {\tc 34950} \\ \hline
 Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline  Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline
 CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline  CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline
 Asir $F_4$      &  989 & 456 & 90 & 991 & 4939 \\ \hline  {\tc Asir $F_4$}        &  {\tc 989} & {\tc 456} & {\tc 90} & {\tc 991} & {\tc 4939} \\ \hline
 FGb(estimated)  & 8 &11 & 0.6 & 5 & 10 \\ \hline  FGb(estimated)  & 8 &11 & 0.6 & 5 & 10 \\ \hline
 \end{tabular}  \end{tabular}
 \end{center}  \end{center}
Line 281  FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
Line 289  FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Summary of performance}  \fbox{\fbc Summary of performance}
   
 \begin{itemize}  \begin{itemize}
 \item Factorizer  \item {\itc Factorizer}
   
 \begin{itemize}  \begin{itemize}
 \item Multivariate : reasonable performance  \item Multivariate : reasonable performance
Line 292  FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
Line 300  FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
 \item Univariate : obsoleted by M. van Hoeij's new algorithm [HOEI00]  \item Univariate : obsoleted by M. van Hoeij's new algorithm [HOEI00]
 \end{itemize}  \end{itemize}
   
 \item Groebner basis computation  \item {\itc Groebner basis computation}
   
 \begin{itemize}  \begin{itemize}
 \item Buchberger  \item Buchberger
Line 312  But we observe that {\it McKay} is computed efficientl
Line 320  But we observe that {\it McKay} is computed efficientl
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{What is the merit to use Risa/Asir?}  \fbox{\fbc What is the merit to use Risa/Asir?}
   
 \begin{itemize}  \begin{itemize}
 \item Total performance is not excellent, but not bad  \item {\itc Total performance is not excellent, but not bad}
   
 \item A completely open system  \item {\itc A completely open system}
   
 The whole source is available  The whole source is available
   
 \item Interface compliant to OpenXM RFC-100  \item {\itc It serves as a test bench to try new ideas}
   
 The interface is fully documented  Interactive debugger is very useful
   
 \item It serves as a test bench to try new ideas  \item {\itc Interface compliant to OpenXM RFC-100}
   
 Interactive debugger is very useful  The interface is fully documented
   
 \end{itemize}  \end{itemize}
   
 \end{slide}  \end{slide}
   
   
 %\begin{slide}{}  %\begin{slide}{}
 %\fbox{CMO = Serialized representation of mathematical object}  %\fbox{\fbc CMO = Serialized representation of mathematical object}
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item primitive data  %\item primitive data
Line 361  Interactive debugger is very useful
Line 370  Interactive debugger is very useful
 %\end{slide}  %\end{slide}
 %  %
 %\begin{slide}{}  %\begin{slide}{}
 %\fbox{Stack based communication}  %\fbox{\fbc Stack based communication}
 %  %
 %\begin{itemize}  %\begin{itemize}
 %\item Data arrived a client  %\item Data arrived a client
Line 387  Interactive debugger is very useful
Line 396  Interactive debugger is very useful
 %\end{slide}  %\end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{OpenXM (Open message eXchange protocol for Mathematics) }  \fbox{\fbc OpenXM (Open message eXchange protocol for Mathematics) }
   
 \begin{itemize}  \begin{itemize}
 \item An environment for parallel distributed computation  \item {\itc An environment for parallel distributed computation}
   
 Both for interactive, non-interactive environment  Both for interactive, non-interactive environment
   
 \item OpenXM RFC-100 = Client-server architecture  \item {\itc OpenXM RFC-100 = Client-server architecture}
   
 Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server  Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server
   
 OX (OpenXM) message : command and data  OX (OpenXM) message : command and data
   
 \item Data  \item {\itc Data}
   
 Encoding : CMO (Common Mathematical Object format)  Encoding : CMO (Common Mathematical Object format)
   
Line 408  Serialized representation of mathematical object
Line 417  Serialized representation of mathematical object
   
 --- Main idea was borrowed from OpenMath  --- Main idea was borrowed from OpenMath
   
 ({\tt http://www.openmath.org})  ({\urlc \tt http://www.openmath.org})
   
 \item Command  \item {\itc Command}
   
 stack machine command --- server is a stackmachine  stack machine command --- server is a stackmachine
   
Line 419  stack machine command --- server is a stackmachine
Line 428  stack machine command --- server is a stackmachine
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Example of distributed computation --- $F_4$ vs. $Buchberger$ }  \fbox{\fbc Example of distributed computation --- $F_4$ vs. $Buchberger$ }
   
 \begin{verbatim}  \begin{verbatim}
 /* competitive Gbase computation over GF(M) */  /* competitive Gbase computation over GF(M) */
 /* Cf. A.28 in SINGULAR Manual */  /* Cf. A.28 in SINGULAR Manual */
 /* Process list is specified as an option : grvsf4(...|proc=P) */  /* Process list is specified as buch_vs_f4_mod(...|proc=P) */
 def grvsf4(G,V,M,O)  def buch_vs_f4_mod(G,V,M,O)
 {  {
   P = getopt(proc);    P = getopt(proc);
   if ( type(P) == -1 ) return dp_f4_mod_main(G,V,M,O);    if ( type(P) == -1 ) return dp_f4_mod_main(G,V,M,O);
   P0 = P[0]; P1 = P[1]; P = [P0,P1];    P0 = P[0]; P1 = P[1]; P = [P0,P1];
   map(ox_reset,P);    map(ox_reset,P); /* resets the both servers */
   ox_cmo_rpc(P0,"dp_f4_mod_main",G,V,M,O);    ox_cmo_rpc(P0,"dp_f4_mod_main",G,V,M,O);  /* for F4 */
   ox_cmo_rpc(P1,"dp_gr_mod_main",G,V,0,M,O);    ox_cmo_rpc(P1,"dp_gr_mod_main",G,V,0,M,O); /* for Buchberger */
   map(ox_push_cmd,P,262); /* 262 = OX_popCMO */    map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
   F = ox_select(P); R = ox_get(F[0]);    F = ox_select(P); /* waits a server to return the result */
     R = ox_get(F[0]); /* gets the result from the winner */
   if ( F[0] == P0 ) { Win = "F4"; Lose = P1;}    if ( F[0] == P0 ) { Win = "F4"; Lose = P1;}
   else { Win = "Buchberger"; Lose = P0; }    else { Win = "Buchberger"; Lose = P0; }
   ox_reset(Lose); /* simply resets the loser */    ox_reset(Lose); /* simply resets the loser */
Line 444  def grvsf4(G,V,M,O)
Line 454  def grvsf4(G,V,M,O)
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{References}  \fbox{\fbc Real speedup by parallelism --- polynomial multiplication}
   
   {\itc Product of dense univariate polynomials with 3000bit coefficients}
   
   {\itc Algorithm : FFT+Chinese remainder (by Shoup)}
   
   \epsfxsize=20cm
   \epsffile{3k.ps}
   
   {\itc Communication cost}
   
   \begin{itemize}
   \item $O(n{\color{red}\log L})$ with server-server communication (OX-RFC102)
   \item $O(n{\color{red}L})$ without server-server communication (OX-RFC100)
   \end{itemize}
   ($L$: number of processes, $n$: degree)
   
   \end{slide}
   
   \begin{slide}{}
   \fbox{\fbc References}
   
 [BERN97] L. Bernardin, On square-free factorization of  [BERN97] L. Bernardin, On square-free factorization of
 multivariate polynomials over a finite field, Theoretical  multivariate polynomials over a finite field, Theoretical
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