| version 1.10, 2003/04/19 10:36:30 |
version 1.13, 2007/02/15 02:41:38 |
|
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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 1.12 2003/04/20 08:01:27 noro Exp $ |
| \BJP |
\BJP |
| @node $B7?(B,,, Top |
@node $B7?(B,,, Top |
| @chapter $B7?(B |
@chapter $B7?(B |
| Line 207 on the whole value of that vector. |
|
| Line 207 on the whole value of that vector. |
|
| [1] for (I=0;I<3;I++)A3[I] = newvect(3); |
[1] for (I=0;I<3;I++)A3[I] = newvect(3); |
| [2] for (I=0;I<3;I++)for(J=0;J<3;J++)A3[I][J]=newvect(3); |
[2] for (I=0;I<3;I++)for(J=0;J<3;J++)A3[I][J]=newvect(3); |
| [3] A3; |
[3] A3; |
| [ [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] |
[ [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] |
| |
[ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] |
| [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] ] |
[ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] ] |
| [4] A3[0]; |
[4] A3[0]; |
| [ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] |
[ [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] ] |
| Line 355 This is used for basis conversion in finite fields of |
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| Line 356 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. |
| |
|
| @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 476 The default precision is about 9 digits, which can be |
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| Line 477 The default precision is about 9 digits, which can be |
|
| [2] setprec(100); |
[2] setprec(100); |
| 9 |
9 |
| [3] eval(2^(1/2)); |
[3] eval(2^(1/2)); |
| 1.41421356237309504880168872420969807856967187537694807317654396116148 |
1.41421356237309504880168872420969807856967187537694807317... |
| @end example |
@end example |
| |
|
| \BJP |
\BJP |
| 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 564 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 586 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 633 coefficients of a polynomial. |
|
| Line 634 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 |
| |
|
| @item 9 |
@item 9 |
| Line 656 is represented by a polynomial over @var{GF(p)} modulo |
|
| Line 657 is represented by a polynomial over @var{GF(p)} modulo |
|
| $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%`$G(B |
| Line 666 is represented by a polynomial over @var{GF(p)} modulo |
|
| Line 667 is represented by a polynomial over @var{GF(p)} modulo |
|
| \BEG |
\BEG |
| A finite field of order @var{p^n}, where @var{p^n} must be less than |
A finite field of order @var{p^n}, where @var{p^n} must be less than |
| @var{2^29} and @var{n} must be equal to 1 if @var{p} is greater or |
@var{2^29} and @var{n} must be equal to 1 if @var{p} is greater or |
| equal to @var{2^14}@, |
equal to @var{2^14}, |
| is set by @code{setmod_ff} |
is set by @code{setmod_ff} |
| by specifying its characteristic @var{p} the extension degree |
by specifying its characteristic @var{p} the extension degree |
| @var{n}. If @var{p} is less than @var{2^14}, each non-zero element |
@var{n}. If @var{p} is less than @var{2^14}, each non-zero element |
| Line 678 This specification is useful for treating both cases i |
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| Line 679 This specification is useful for treating both cases i |
|
| program. |
program. |
| \E |
\E |
| |
|
| |
@item 10 |
| |
\JP @b{$B0L?t(B @var{p^n} $B$N>.0L?tM-8BBN$NBe?t3HBg$N85(B} |
| |
\EG @b{element of a finite field which is an algebraic extension of a small finite field of characteristic @var{p^n}} |
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|
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\BJP |
| |
$BA09`$N(B, $B0L?t$,(B @var{p^n} $B$N>.0L?tM-8BBN$N(B @var{m} $B<!3HBg$N85$G$"$k(B. |
| |
$BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n}, @var{m} |
| |
$B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B. $B4pACBN>e$N(B @var{m} |
| |
$B<!4{LsB?9`<0$,<+F0@8@.$5$l(B, $B$=$NBe?t3HBg$N@8@.85$NDj5AB?9`<0$H$7$FMQ$$$i$l$k(B. |
| |
$B@8@.85$O(B @code{@@s} $B$G$"$k(B. |
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|
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\E |
| |
\BEG |
| |
An extension field @var{K} of the small finite field @var{F} of order @var{p^n} |
| |
is set by @code{setmod_ff} |
| |
by specifying its characteristic @var{p} the extension degree |
| |
@var{n} and @var{m}=[@var{K}:@var{F}]. An irreducible polynomial of degree @var{m} |
| |
over @var{K} is automatically generated and used as the defining polynomial of |
| |
the generator of the extension @var{K/F}. The generator is denoted by @code{@@s}. |
| |
\E |
| |
|
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@item 11 |
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\JP @b{$BJ,;6I=8=$NBe?tE*?t(B} |
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\EG @b{algebraic number represented by a distributed polynomial} |
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@* |
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\JP @xref{$BBe?tE*?t$K4X$9$k1i;;(B}. |
| |
\EG @xref{Algebraic numbers}. |
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|
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\BJP |
| |
|
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\E |
| |
\BEG |
| |
\E |
| @end table |
@end table |
| |
|
| \BJP |
\BJP |
|
|
| \EG @b{functor} |
\EG @b{functor} |
| @* |
@* |
| \BJP |
\BJP |
| $BH!?t8F$S=P$7$O(B, @var{fname(args)} $B$H$$$&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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