version 1.10, 2003/04/19 10:36:30 |
version 1.11, 2003/04/19 15:44:57 |
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@comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.9 2002/09/03 01:50:58 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 |
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Line 355 This is used for basis conversion in finite fields of |
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\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. |
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@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 |
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Line 563 g mod f $B$O(B, g, f $B$r$"$i$o$9(B 2 $B$D$N%S%C |
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\E |
\E |
\BEG |
\BEG |
This type expresses an element of a finite field of characteristic 2. |
This type expresses an element of a finite field of characteristic 2. |
Let @var{F} be a finite field of characteristic 2. If @var{[F:GF(2)]} |
Let @var{F} be a finite field of characteristic 2. If [F:GF(2)] |
is equal to @var{n}, then @var{F} is expressed as @var{F=GF(2)[t]/(f(t))}, |
is equal to @var{n}, then @var{F} is expressed as F=GF(2)[t]/(f(t)), |
where @var{f(t)} is an irreducible polynomial over @var{GF(2)} |
where f(t) is an irreducible polynomial over GF(2) |
of degree @var{n}. |
of degree @var{n}. |
As an element @var{g} of @var{GF(2)[t]} can be expressed by a bit string, |
As an element @var{g} of GF(2)[t] can be expressed by a bit string, |
An element @var{g mod f} in @var{F} can be expressed by two bit strings |
An element @var{g mod f} in @var{F} can be expressed by two bit strings |
representing @var{g} and @var{f} respectively. |
representing @var{g} and @var{f} respectively. |
\E |
\E |
Line 585 representing @var{g} and @var{f} respectively. |
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Line 585 representing @var{g} and @var{f} respectively. |
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$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 633 coefficients of a polynomial. |
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Line 633 coefficients of a polynomial. |
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\BJP |
\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, |
$B0L?t$,(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 @var{GF(p)} $B>e4{Ls$J(B @var{n} $B<!B?9`<0(B @var{m(x)} |
$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. |
$B$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 @var{m(x)} $B$rK!$H$9$k(B @var{GF(p)} $B>e$NB?9`<0$H$7$F(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. |
$BI=8=$5$l$k(B. |
\E |
\E |
\BEG |
\BEG |
A finite field of order @var{p^n}, where @var{p} is an arbitrary prime |
A finite field of order @var{p^n}, where @var{p} is an arbitrary prime |
and @var{n} is a positive integer, is set by @code{setmod_ff} |
and @var{n} is a positive integer, is set by @code{setmod_ff} |
by specifying its characteristic @var{p} and an irreducible polynomial |
by specifying its characteristic @var{p} and an irreducible polynomial |
of degree @var{n} over @var{GF(p)}. An element of this field |
of degree @var{n} over GF(@var{p}). An element of this field |
is represented by a polynomial over @var{GF(p)} modulo @var{m(x)}. |
is represented by a polynomial over GF(@var{p}) modulo m(x). |
\E |
\E |
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@item 9 |
@item 9 |
Line 656 is represented by a polynomial over @var{GF(p)} modulo |
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Line 656 is represented by a polynomial over @var{GF(p)} modulo |
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$BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n} |
$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. |
$B$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$$85$O(B, @var{p} $B$,(B @var{2^14} $BL$K~$N>l9g(B, |
$B$3$NBN$N(B 0 $B$G$J$$85$O(B, @var{p} $B$,(B @var{2^14} $BL$K~$N>l9g(B, |
@var{GF(p^n)} $B$N>hK!72$N@8@.85$r8GDj$9$k$3$H(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$$85(B |
$B$K$h$j(B, $B$3$N85$N$Y$-$H$7$FI=$5$l$k(B. $B$3$l$K$h$j(B, $B$3$NBN$N(B 0 $B$G$J$$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 |
$B$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 |
$B$N>l9g$ODL>o$N>jM>$K$h$kI=8=$H$J$k$,(B, $B6&DL$N%W%m%0%i%`$G(B |
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\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 |