| version 1.6, 2000/09/23 07:53:25 |
version 1.15, 2019/09/13 09:31:00 |
|
|
| @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 1.14 2016/03/22 07:25:14 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 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 |
|
| Line 356 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. |
| |
|
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@item 15 @b{matrix over GF(@var{p})} |
| |
@* |
| |
\JP $B>.I8?tM-8BBN>e$N9TNs(B. |
| |
\EG A matrix over a small finite field. |
| |
|
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@item 16 @b{byte array} |
| |
@* |
| |
\JP $BId9f$J$7(B byte $B$NG[Ns(B |
| |
\EG An array of unsigned bytes. |
| |
|
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\JP @item 26 @b{$BJ,;6I=8=2C72B?9`<0(B} |
| |
\EG @item 26 @b{distributed module polynomial} |
| |
|
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@example |
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2*<<0,1,2,3:1>>-3*<<1,2,3,4:2>> |
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@end example |
| |
|
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\BJP |
| |
$B$3$l$O(B, $BB?9`<04D>e$N<+M32C72$N85$r(B, $B2C72C19`<0$NOB$H$7$FFbItI=8=$7$?$b$N$G$"$k(B. |
| |
$B$3$3$G(B, $B2C72C19`<0$H$OC19`<0$H2C72$NI8=`4pDl$N@Q$G$"$k(B. |
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$B$3$l$K$D$$$F$O(B @xref{$B%0%l%V%J4pDl$N7W;;(B}. |
| |
\E |
| |
\BEG |
| |
This represents an element in a free module over a polynomial ring |
| |
as a linear sum of module monomials, where a module monomial is |
| |
the product of a monomial in the polynomial ring and a standard base of the free module. |
| |
For details @xref{Groebner basis computation}. |
| |
\E |
| \JP @item -1 @b{VOID $B%*%V%8%'%/%H(B} |
\JP @item -1 @b{VOID $B%*%V%8%'%/%H(B} |
| \EG @item -1 @b{VOID object} |
\EG @item -1 @b{VOID object} |
| @* |
@* |
| Line 440 result shall be computed as a double float number. |
|
| Line 469 result shall be computed as a double float number. |
|
| @b{bigfloat} |
@b{bigfloat} |
| @* |
@* |
| \BJP |
\BJP |
| @b{bigfloat} $B$O(B, @b{Asir} $B$G$O(B @b{PARI} $B%i%$%V%i%j$K$h$j(B |
@b{bigfloat} $B$O(B, @b{Asir} $B$G$O(B @b{MPFR} $B%i%$%V%i%j$K$h$j(B |
| $B<B8=$5$l$F$$$k(B. @b{PARI} $B$K$*$$$F$O(B, @b{bigfloat} $B$O(B, $B2>?tIt(B |
$B<B8=$5$l$F$$$k(B. @b{MPFR} $B$K$*$$$F$O(B, @b{bigfloat} $B$O(B, $B2>?tIt(B |
| $B$N$_G$0UB?G\D9$G(B, $B;X?tIt$O(B 1 $B%o!<%I0JFb$N@0?t$K8B$i$l$F$$$k(B. |
$B$N$_G$0UB?G\D9$G(B, $B;X?tIt$O(B 64bit $B@0?t$G$"$k(B. |
| @code{ctrl()} $B$G(B @b{bigfloat} $B$rA*Br$9$k$3$H$K$h$j(B, $B0J8e$NIbF0>.?t(B |
@code{ctrl()} $B$G(B @b{bigfloat} $B$rA*Br$9$k$3$H$K$h$j(B, $B0J8e$NIbF0>.?t(B |
| $B$NF~NO$O(B @b{bigfloat} $B$H$7$F07$o$l$k(B. $B@:EY$O%G%U%)%k%H$G$O(B |
$B$NF~NO$O(B @b{bigfloat} $B$H$7$F07$o$l$k(B. $B@:EY$O%G%U%)%k%H$G$O(B |
| 10 $B?J(B 9 $B7eDxEY$G$"$k$,(B, @code{setprec()} $B$K$h$j;XDj2DG=$G$"$k(B. |
10 $B?J(B 9 $B7eDxEY$G$"$k$,(B, @code{setprec()}, @code{setbprec()} $B$K$h$j;XDj2DG=$G$"$k(B. |
| \E |
\E |
| \BEG |
\BEG |
| The @b{bigfloat} numbers of @b{Asir} is realized by @b{PARI} library. |
The @b{bigfloat} numbers of @b{Asir} is realized by @b{MPFR} library. |
| A @b{bigfloat} number of @b{PARI} has an arbitrary precision mantissa |
A @b{bigfloat} number of @b{MPFR} has an arbitrary precision mantissa |
| part. However, its exponent part admits only an integer with a single |
part. However, its exponent part admits only a 64bit integer. |
| word precision. |
|
| Floating point operations will be performed all in @b{bigfloat} after |
Floating point operations will be performed all in @b{bigfloat} after |
| activating the @b{bigfloat} switch by @code{ctrl()} command. |
activating the @b{bigfloat} switch by @code{ctrl()} command. |
| The default precision is about 9 digits, which can be specified by |
The default precision is 53 bits (about 15 digits), which can be specified by |
| @code{setprec()} command. |
@code{setbprec()} and @code{setprec()} command. |
| \E |
\E |
| |
|
| @example |
@example |
| [0] ctrl("bigfloat",1); |
[0] ctrl("bigfloat",1); |
| 1 |
1 |
| [1] eval(2^(1/2)); |
[1] eval(2^(1/2)); |
| 1.414213562373095048763788073031 |
1.4142135623731 |
| [2] setprec(100); |
[2] setprec(100); |
| 9 |
15 |
| [3] eval(2^(1/2)); |
[3] eval(2^(1/2)); |
| 1.41421356237309504880168872420969807856967187537694807317654396116148 |
1.41421356237309504880168872420969807856967187537694...764157 |
| |
[4] setbprec(100); |
| |
332 |
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[5] 1.41421356237309504880168872421 |
| @end example |
@end example |
| |
|
| \BJP |
\BJP |
| @code{eval()} $B$O(B, $B0z?t$K4^$^$l$kH!?tCM$r2DG=$J8B$j?tCM2=$9$kH!?t$G$"$k(B. |
@code{eval()} $B$O(B, $B0z?t$K4^$^$l$kH!?tCM$r2DG=$J8B$j?tCM2=$9$kH!?t$G$"$k(B. |
| @code{setprec()} $B$G;XDj$5$l$?7e?t$O(B, $B7k2L$N@:EY$rJ]>Z$9$k$b$N$G$O$J$/(B, |
@code{setbprec()} $B$G;XDj$5$l$?(B2 $B?J7e?t$O(B, $B4]$a%b!<%I$K1~$8$?7k2L$N@:EY$rJ]>Z$9$k(B. @code{setprec()} $B$G;XDj$5$l$k(B10$B?J7e?t$O(B 2 $B?J7e?t$KJQ49$5$l$F@_Dj$5$l$k(B. |
| @b{PARI} $BFbIt$GMQ$$$i$l$kI=8=$N%5%$%:$r<($9$3$H$KCm0U$9$Y$-$G$"$k(B. |
|
| \E |
\E |
| \BEG |
\BEG |
| Function @code{eval()} evaluates numerically its argument as far as |
Function @code{eval()} evaluates numerically its argument as far as |
| possible. |
possible. |
| Notice that the integer given for the argument of @code{setprec()} does |
Notice that the integer given for the argument of @code{setbprec()} |
| not guarantee the accuracy of the result, |
guarantees the accuracy of the result according to the current rounding mode. |
| but it indicates the representation size of numbers with which internal |
The argument of @code{setbprec()} is converted to the corresonding bit length |
| operations of @b{PARI} are performed. |
and set. |
| \E |
\E |
| (@ref{eval}, @xref{pari}) |
(@xref{eval deval}.) |
| |
|
| @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 584 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 606 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 646 coefficients of a polynomial. |
|
| \E |
\E |
| |
|
| @end itemize |
@end itemize |
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|
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|
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@item 8 |
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\JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B} |
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\EG @b{element of a finite field of characteristic @var{p^n}} |
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|
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\BJP |
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$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, |
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$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) |
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$B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B. |
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$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 |
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$BI=8=$5$l$k(B. |
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\E |
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\BEG |
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A finite field of order @var{p^n}, where @var{p} is an arbitrary prime |
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and @var{n} is a positive integer, is set by @code{setmod_ff} |
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by specifying its characteristic @var{p} and an irreducible polynomial |
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of degree @var{n} over GF(@var{p}). An element of this field |
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is represented by a polynomial over GF(@var{p}) modulo m(x). |
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\E |
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|
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@item 9 |
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\JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B ($B>.0L?t(B)} |
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\EG @b{element of a finite field of characteristic @var{p^n} (small order)} |
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|
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\BJP |
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$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 |
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$B$J$i(B @var{n} $B$O(B 1) $B$O(B, |
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$BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n} |
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$B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B. |
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$B$3$NBN$N(B 0 $B$G$J$$85$O(B, @var{p} $B$,(B @var{2^14} $BL$K~$N>l9g(B, |
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GF(@var{p^n}) $B$N>hK!72$N@8@.85$r8GDj$9$k$3$H(B |
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$B$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$$85(B |
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$B$O(B, $B$3$N$Y$-;X?t$H$7$FI=8=$5$l$k(B. @var{p} $B$,(B @var{2^14} $B0J>e(B |
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$B$N>l9g$ODL>o$N>jM>$K$h$kI=8=$H$J$k$,(B, $B6&DL$N%W%m%0%i%`$G(B |
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$BAPJ}$N>l9g$r07$($k$h$&$K$3$N$h$&$J;EMM$H$J$C$F$$$k(B. |
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|
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\E |
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\BEG |
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A finite field of order @var{p^n}, where @var{p^n} must be less than |
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@var{2^29} and @var{n} must be equal to 1 if @var{p} is greater or |
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equal to @var{2^14}, |
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is set by @code{setmod_ff} |
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by specifying its characteristic @var{p} the extension degree |
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@var{n}. If @var{p} is less than @var{2^14}, each non-zero element |
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of this field |
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is a power of a fixed element, which is a generator of the multiplicative |
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group of the field, and it is represented by its exponent. |
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Otherwise, each element is represented by the redue modulo @var{p}. |
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This specification is useful for treating both cases in a single |
| |
program. |
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\E |
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|
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@item 10 |
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\JP @b{$B0L?t(B @var{p^n} $B$N>.0L?tM-8BBN$NBe?t3HBg$N85(B} |
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\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 |
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$BA09`$N(B, $B0L?t$,(B @var{p^n} $B$N>.0L?tM-8BBN$N(B @var{m} $B<!3HBg$N85$G$"$k(B. |
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$BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n}, @var{m} |
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$B$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} |
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$B<!4{LsB?9`<0$,<+F0@8@.$5$l(B, $B$=$NBe?t3HBg$N@8@.85$NDj5AB?9`<0$H$7$FMQ$$$i$l$k(B. |
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$B@8@.85$O(B @code{@@s} $B$G$"$k(B. |
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|
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\E |
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\BEG |
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An extension field @var{K} of the small finite field @var{F} of order @var{p^n} |
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is set by @code{setmod_ff} |
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by specifying its characteristic @var{p} the extension degree |
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@var{n} and @var{m}=[@var{K}:@var{F}]. An irreducible polynomial of degree @var{m} |
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over @var{K} is automatically generated and used as the defining polynomial of |
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the generator of the extension @var{K/F}. The generator is denoted by @code{@@s}. |
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\E |
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|
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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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|
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\E |
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\BEG |
| |
\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 |
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