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version 1.9, 2002/09/03 01:50:58 version 1.11, 2003/04/19 15:44:57
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 @comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.8 2001/03/12 05:01:18 noro Exp $  @comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.10 2003/04/19 10:36:30 noro Exp $
 \BJP  \BJP
 @node $B7?(B,,, Top  @node $B7?(B,,, Top
 @chapter $B7?(B  @chapter $B7?(B
Line 355  This is used for basis conversion in finite fields of 
Line 355  This is used for basis conversion in finite fields of 
 \JP quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(B.  \JP quantifier elimination $B$GMQ$$$i$l$k0l3,=R8lO@M}<0(B.
 \EG This expresses a first order formula used in quantifier elimination.  \EG This expresses a first order formula used in quantifier elimination.
   
 @item 15 @b{matrix over GF(p)}  @item 15 @b{matrix over GF(@var{p})}
 @*  @*
 \JP $B>.I8?tM-8BBN>e$N9TNs(B.  \JP $B>.I8?tM-8BBN>e$N9TNs(B.
 \EG A matrix over a small finite field.  \EG A matrix over a small finite field.
Line 563  g mod f $B$O(B, g, f  $B$r$"$i$o$9(B 2 $B$D$N%S%C
Line 563  g mod f $B$O(B, g, f  $B$r$"$i$o$9(B 2 $B$D$N%S%C
 \E  \E
 \BEG  \BEG
 This type expresses an element of a finite field of characteristic 2.  This type expresses an element of a finite field of characteristic 2.
 Let @var{F} be a finite field of characteristic 2. If @var{[F:GF(2)]}  Let @var{F} be a finite field of characteristic 2. If [F:GF(2)]
 is equal to @var{n}, then @var{F} is expressed as @var{F=GF(2)[t]/(f(t))},  is equal to @var{n}, then @var{F} is expressed as F=GF(2)[t]/(f(t)),
 where @var{f(t)} is an irreducible polynomial over @var{GF(2)}  where f(t) is an irreducible polynomial over GF(2)
 of degree @var{n}.  of degree @var{n}.
 As an element @var{g} of @var{GF(2)[t]} can be expressed by a bit string,  As an element @var{g} of GF(2)[t] can be expressed by a bit string,
 An element @var{g mod f} in @var{F} can be expressed by two bit strings  An element @var{g mod f} in @var{F} can be expressed by two bit strings
 representing @var{g} and @var{f} respectively.  representing @var{g} and @var{f} respectively.
 \E  \E
Line 585  representing @var{g} and @var{f} respectively.
Line 585  representing @var{g} and @var{f} respectively.
 $B$h$C$F(B, @@ $B$NB?9`<0$H$7$F(B F $B$N85$rF~NO$G$-$k(B. (@@^10+@@+1 $B$J$I(B)  $B$h$C$F(B, @@ $B$NB?9`<0$H$7$F(B F $B$N85$rF~NO$G$-$k(B. (@@^10+@@+1 $B$J$I(B)
 \E  \E
 \BEG  \BEG
 @code{@@} represents @var{t mod f} in @var{F=GF(2)[t](f(t))}.  @code{@@} represents @var{t mod f} in F=GF(2)[t](f(t)).
 By using @code{@@} one can input an element of @var{F}. For example  By using @code{@@} one can input an element of @var{F}. For example
 @code{@@^10+@@+1} represents an element of @var{F}.  @code{@@^10+@@+1} represents an element of @var{F}.
 \E  \E
Line 625  coefficients of a polynomial.
Line 625  coefficients of a polynomial.
 \E  \E
   
 @end itemize  @end itemize
   
   
   @item 8
   \JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B}
   \EG @b{element of a finite field of characteristic @var{p^n}}
   
   \BJP
   $B0L?t$,(B @var{p^n} (@var{p} $B$OG$0U$NAG?t(B, @var{n} $B$O@5@0?t(B) $B$O(B,
   $BI8?t(B @var{p} $B$*$h$S(B GF(@var{p}) $B>e4{Ls$J(B @var{n} $B<!B?9`<0(B m(x)
   $B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B.
   $B$3$NBN$N85$O(B m(x) $B$rK!$H$9$k(B GF(@var{p}) $B>e$NB?9`<0$H$7$F(B
   $BI=8=$5$l$k(B.
   \E
   \BEG
   A finite field of order @var{p^n}, where @var{p} is an arbitrary prime
   and @var{n} is a positive integer, is set by @code{setmod_ff}
   by specifying its characteristic @var{p} and an irreducible polynomial
   of degree @var{n} over GF(@var{p}). An element of this field
   is represented by a polynomial over GF(@var{p}) modulo m(x).
   \E
   
   @item 9
   \JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B ($B>.0L?t(B)}
   \EG @b{element of a finite field of characteristic @var{p^n} (small order)}
   
   \BJP
   $B0L?t$,(B @var{p^n} $B$NM-8BBN(B (@var{p^n} $B$,(B @var{2^29} $B0J2<(B, @var{p} $B$,(B @var{2^14} $B0J>e(B
   $B$J$i(B @var{n} $B$O(B 1) $B$O(B,
   $BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n}
   $B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B.
   $B$3$NBN$N(B 0 $B$G$J$$85$O(B, @var{p} $B$,(B @var{2^14} $BL$K~$N>l9g(B,
   GF(@var{p^n}) $B$N>hK!72$N@8@.85$r8GDj$9$k$3$H(B
   $B$K$h$j(B, $B$3$N85$N$Y$-$H$7$FI=$5$l$k(B. $B$3$l$K$h$j(B, $B$3$NBN$N(B 0 $B$G$J$$85(B
   $B$O(B, $B$3$N$Y$-;X?t$H$7$FI=8=$5$l$k(B. @var{p} $B$,(B @var{2^14} $B0J>e(B
   $B$N>l9g$ODL>o$N>jM>$K$h$kI=8=$H$J$k$,(B, $B6&DL$N%W%m%0%i%`$G(B
   $BAPJ}$N>l9g$r07$($k$h$&$K$3$N$h$&$J;EMM$H$J$C$F$$$k(B.
   
   \E
   \BEG
   A finite field of order @var{p^n}, where @var{p^n} must be less than
   @var{2^29} and @var{n} must be equal to 1 if @var{p} is greater or
   equal to @var{2^14}@,
   is set by @code{setmod_ff}
   by specifying its characteristic @var{p} the extension degree
   @var{n}. If @var{p} is less than @var{2^14}, each non-zero element
   of this field
   is a power of a fixed element, which is a generator of the multiplicative
   group of the field, and it is represented by its exponent.
   Otherwise, each element is represented by the redue modulo @var{p}.
   This specification is useful for treating both cases in a single
   program.
   \E
   
 @end table  @end table
   
 \BJP  \BJP
 $BBgI8?tAGBN$NI8?t(B, $BI8?t(B 2 $B$NM-8BBN$NDj5AB?9`<0$O(B, @code{setmod_ff}  $B>.I8?tM-8BAGBN0J30$NM-8BBN$O(B @code{setmod_ff} $B$G@_Dj$9$k(B.
 $B$G@_Dj$9$k(B.  $BM-8BBN$N85$I$&$7$N1i;;$G$O(B,
 $BM-8BBN$N85$I$&$7$N1i;;$G$O(B, @code{setmod_ff} $B$K$h$j@_Dj$5$l$F$$$k(B  
 modulus $B$G(B, $BB0$9$kBN$,J,$+$j(B, $B$=$NCf$G1i;;$,9T$o$l$k(B.  
 $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B  $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B
 $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B.  $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B.
 \E  \E
 \BEG  \BEG
 The characteristic of a large finite prime field and the defining  Finite fields other than small finite prime fields are
 polynomial of a finite field of characteristic 2 are set by @code{setmod_ff}.  set by @code{setmod_ff}.
 Elements of finite fields do not have informations about the modulus.  Elements of finite fields do not have informations about the modulus.
 Upon an arithmetic operation, the modulus set by @code{setmod_ff} is  Upon an arithmetic operation, i
 used. If one of the operands is a rational number, it is automatically  f one of the operands is a rational number, it is automatically
 converted into an element of the finite field currently set and  converted into an element of the finite field currently set and
 the operation is done in the finite field.  the operation is done in the finite field.
 \E  \E
Line 757  sin(x)
Line 808  sin(x)
 \EG @b{functor}  \EG @b{functor}
 @*  @*
 \BJP  \BJP
 $BH!?t8F$S=P$7$O(B, @var{fname(args)} $B$H$$$&7A$G9T$J$o$l$k$,(B, @var{fname} $B$N(B  $BH!?t8F$S=P$7$O(B, @var{fname}(@var{args}) $B$H$$$&7A$G9T$J$o$l$k$,(B, @var{fname} $B$N(B
 $BItJ,$rH!?t;R$H8F$V(B. $BH!?t;R$K$O(B, $BH!?t$N<oN`$K$h$jAH$_9~$_H!?t;R(B,  $BItJ,$rH!?t;R$H8F$V(B. $BH!?t;R$K$O(B, $BH!?t$N<oN`$K$h$jAH$_9~$_H!?t;R(B,
 $B%f!<%6Dj5AH!?t;R(B, $B=iEyH!?t;R$J$I$,$"$k$,(B, $BH!?t;R$OC1FH$GITDj85$H$7$F(B  $B%f!<%6Dj5AH!?t;R(B, $B=iEyH!?t;R$J$I$,$"$k$,(B, $BH!?t;R$OC1FH$GITDj85$H$7$F(B
 $B5!G=$9$k(B.  $B5!G=$9$k(B.
 \E  \E
 \BEG  \BEG
 A function call (or a function form) has a form @var{fname(args)}.  A function call (or a function form) has a form @var{fname}(@var{args}).
 Here, @var{fname} alone is called a @b{functor}.  Here, @var{fname} alone is called a @b{functor}.
 There are several kinds of @b{functor}s: built-in functor, user defined  There are several kinds of @b{functor}s: built-in functor, user defined
 functor and functor for the elementary functions.  A functor alone is  functor and functor for the elementary functions.  A functor alone is
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