| version 1.6, 2000/09/23 07:53:25 |
version 1.11, 2003/04/19 15:44:57 |
|
|
| @comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.5 2000/01/26 01:37:33 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 83 x afo (2.3*x+y)^10 |
|
| Line 83 x afo (2.3*x+y)^10 |
|
| |
|
| \BJP |
\BJP |
| $BB?9`<0$O(B, $BA4$FE83+$5$l(B, $B$=$N;~E@$K$*$1$kJQ?t=g=x$K=>$C$F(B, $B:F5"E*$K(B |
$BB?9`<0$O(B, $BA4$FE83+$5$l(B, $B$=$N;~E@$K$*$1$kJQ?t=g=x$K=>$C$F(B, $B:F5"E*$K(B |
| 1 $BJQ?tB?9`<0$H$7$F9_QQ$N=g$K@0M}$5$l$k(B (@xref{$BJ,;6I=8=B?9`<0(B}). |
1 $BJQ?tB?9`<0$H$7$F9_QQ$N=g$K@0M}$5$l$k(B. (@xref{$BJ,;6I=8=B?9`<0(B}.) |
| $B$3$N;~(B, $B$=$NB?9`<0$K8=$l$k=g=x:GBg$NJQ?t$r(B @b{$B<gJQ?t(B} $B$H8F$V(B. |
$B$3$N;~(B, $B$=$NB?9`<0$K8=$l$k=g=x:GBg$NJQ?t$r(B @b{$B<gJQ?t(B} $B$H8F$V(B. |
| \E |
\E |
| \BEG |
\BEG |
| Every polynomial is maintained internally in its full expanded form, |
Every polynomial is maintained internally in its full expanded form, |
| represented as a nested univariate polynomial, according to the current |
represented as a nested univariate polynomial, according to the current |
| variable ordering, arranged by the descending order of exponents. |
variable ordering, arranged by the descending order of exponents. |
| (@xref{Distributed polynomial}). |
(@xref{Distributed polynomial}.) |
| In the representation, the indeterminate (or variable), appearing in |
In the representation, the indeterminate (or variable), appearing in |
| the polynomial, with maximum ordering is called the @b{main variable}. |
the polynomial, with maximum ordering is called the @b{main variable}. |
| Moreover, we call the coefficient of the maximum degree term of |
Moreover, we call the coefficient of the maximum degree term of |
| 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(@var{p})} |
| |
@* |
| |
\JP $B>.I8?tM-8BBN>e$N9TNs(B. |
| |
\EG A matrix over a small finite field. |
| |
|
| |
@item 16 @b{byte array} |
| |
@* |
| |
\JP $BId9f$J$7(B byte $B$NG[Ns(B |
| |
\EG An array of unsigned bytes. |
| |
|
| \JP @item -1 @b{VOID $B%*%V%8%'%/%H(B} |
\JP @item -1 @b{VOID $B%*%V%8%'%/%H(B} |
| \EG @item -1 @b{VOID object} |
\EG @item -1 @b{VOID object} |
| @* |
@* |
| Line 482 not guarantee the accuracy of the result, |
|
| Line 492 not guarantee the accuracy of the result, |
|
| but it indicates the representation size of numbers with which internal |
but it indicates the representation size of numbers with which internal |
| operations of @b{PARI} are performed. |
operations of @b{PARI} are performed. |
| \E |
\E |
| (@ref{eval}, @xref{pari}) |
(@xref{eval deval}, @ref{pari}.) |
| |
|
| @item 4 |
@item 4 |
| \JP @b{$BJ#AG?t(B} |
\JP @b{$BJ#AG?t(B} |
| Line 553 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 575 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$I(B) |
| \E |
\E |
| \BEG |
\BEG |
| @code{@@} represents @var{t mod f} in @var{F=GF(2)[t](f(t))}. |
@code{@@} represents @var{t mod f} in F=GF(2)[t](f(t)). |
| By using @code{@@} one can input an element of @var{F}. For example |
By using @code{@@} one can input an element of @var{F}. For example |
| @code{@@^10+@@+1} represents an element of @var{F}. |
@code{@@^10+@@+1} represents an element of @var{F}. |
| \E |
\E |
| Line 615 coefficients of a polynomial. |
|
| Line 625 coefficients of a polynomial. |
|
| \E |
\E |
| |
|
| @end itemize |
@end itemize |
| |
|
| |
|
| |
@item 8 |
| |
\JP @b{$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 |
|
|
| \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 |
![[BACK]](/icons/cvsweb/back.gif){kind=icon}
Return to type.texi CVS log ![[TXT]](/icons/cvsweb/text.gif){kind=icon}
![[DIR]](/icons/cvsweb/dir.gif){kind=icon}
Up to [local] / OpenXM / src / asir-doc / parts