| version 1.1, 1999/12/08 05:47:44 |
version 1.7, 2003/04/21 03:07:32 |
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@comment $OpenXM: OpenXM/src/asir-doc/parts/ff.texi,v 1.6 2003/04/20 08:01:25 noro Exp $ |
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\BJP |
| @node $BM-8BBN$K4X$9$k1i;;(B,,, Top |
@node $BM-8BBN$K4X$9$k1i;;(B,,, Top |
| @chapter $BM-8BBN$K4X$9$k1i;;(B |
@chapter $BM-8BBN$K4X$9$k1i;;(B |
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\E |
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\BEG |
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@node Finite fields,,, Top |
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@chapter Finite fields |
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\E |
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|
| @menu |
@menu |
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\BJP |
| * $BM-8BBN$NI=8=$*$h$S1i;;(B:: |
* $BM-8BBN$NI=8=$*$h$S1i;;(B:: |
| * $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B:: |
* $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B:: |
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* $B>.I8?tM-8BBN>e$G$NB?9`<0$N1i;;(B:: |
| * $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B:: |
* $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B:: |
| * $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B:: |
* $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B:: |
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\E |
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\BEG |
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* Representation of finite fields:: |
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* Univariate polynomials on finite fields:: |
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* Polynomials on small finite fields:: |
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* Elliptic curves on finite fields:: |
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* Functions for Finite fields:: |
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\E |
| @end menu |
@end menu |
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|
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\BJP |
| @node $BM-8BBN$NI=8=$*$h$S1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B |
@node $BM-8BBN$NI=8=$*$h$S1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B |
| @section $BM-8BBN$NI=8=$*$h$S1i;;(B |
@section $BM-8BBN$NI=8=$*$h$S1i;;(B |
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\E |
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\BEG |
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@node Representation of finite fields,,, Finite fields |
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@section Representation of finite fields |
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\E |
| |
|
| @noindent |
@noindent |
| @b{Asir} $B$K$*$$$F$O(B, $BM-8BBN$O(B, $B@5I8?tAGBN(B GF(p), $BI8?t(B 2 $B$NM-8BBN(B GF(2^n) |
\BJP |
| |
@b{Asir} $B$K$*$$$F$O(B, $BM-8BBN$O(B, $B@5I8?tAGBN(B GF(@var{p}), $BI8?t(B 2 $B$NM-8BBN(B GF(2^@var{n}), |
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GF(@var{p}) $B$N(B @var{n} $B<!3HBg(B GF(@var{p^n}) |
| $B$,Dj5A$G$-$k(B. $B$3$l$i$OA4$F(B, @code{setmod_ff()} $B$K$h$jDj5A$5$l$k(B. |
$B$,Dj5A$G$-$k(B. $B$3$l$i$OA4$F(B, @code{setmod_ff()} $B$K$h$jDj5A$5$l$k(B. |
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\E |
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\BEG |
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On @b{Asir} GF(@var{p}), GF(2^@var{n}), GF(@var{p^n}) can be defined, where |
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GF(@var{p}) is a finite prime field of charateristic @var{p}, |
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GF(2^@var{n}) is a finite field of characteristic 2 and |
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GF(@var{p^n}) is a finite extension of GF(@var{p}). These are |
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all defined by @code{setmod_ff()}. |
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\E |
| |
|
| @example |
@example |
| [0] P=pari(nextprime,2^50); |
[0] P=pari(nextprime,2^50); |
| Line 30 x^50+x^4+x^3+x^2+1 |
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| Line 63 x^50+x^4+x^3+x^2+1 |
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| x^50+x^4+x^3+x^2+1 |
x^50+x^4+x^3+x^2+1 |
| [6] field_type_ff(); |
[6] field_type_ff(); |
| 2 |
2 |
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[7] setmod_ff(x^3+x+1,1125899906842679); |
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[1*x^3+1*x+1,1125899906842679] |
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[8] field_type_ff(); |
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3 |
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[9] setmod_ff(3,5); |
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[3,x^5+2*x+1,x] |
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[10] field_type_ff(); |
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4 |
| @end example |
@end example |
| @code{setmod_ff()} $B$O(B, $B0z?t$,@5@0?t(B p $B$N>l9g(B GF(p), n $B<!B?9`<0(B f(x) $B$N>l(B |
\BJP |
| $B9g(B, f(x) mod 2 $B$rDj5AB?9`<0$H$9$k(B GF(2^n) $B$r$=$l$>$l4pACBN$H$7$F%;%C%H$9(B |
@code{setmod_ff()} $B$O(B, $B$5$^$6$^$J%?%$%W$NM-8BBN$r4pACBN$H$7$F%;%C%H$9$k(B. |
| $B$k(B. @code{setmod_ff()} $B$K$*$$$F$O0z?t$N4{Ls%A%'%C%/$O9T$o$:(B, $B8F$S=P$7B&(B |
$B0z?t$,@5@0?t(B @var{p} $B$N>l9g(B GF(@var{p}), @var{n} $B<!B?9`<0(B f(x) $B$N>l(B |
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$B9g(B, f(x) mod 2 $B$rDj5AB?9`<0$H$9$k(B GF(2^@var{n}) $B$r$=$l$>$l4pACBN$H$7$F%;%C%H$9(B |
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$B$k(B. $B$^$?(B, $BM-8BAGBN$NM-8B<!3HBg$bDj5A$G$-$k(B. $B>\$7$/$O(B @xref{$B?t$N7?(B}. |
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@code{setmod_ff()} $B$K$*$$$F$O0z?t$N4{Ls%A%'%C%/$O9T$o$:(B, $B8F$S=P$7B&(B |
| $B$,@UG$$r;}$D(B. |
$B$,@UG$$r;}$D(B. |
| |
|
| $B4pACBN$H$O(B, $B$"$/$^$GM-8BBN$N85$H$7$F@k8@$"$k$$$ODj5A$5$l$?%*%V%8%'%/%H$,(B, |
$B4pACBN$H$O(B, $B$"$/$^$GM-8BBN$N85$H$7$F@k8@$"$k$$$ODj5A$5$l$?%*%V%8%'%/%H$,(B, |
| Line 43 x^50+x^4+x^3+x^2+1 |
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| Line 87 x^50+x^4+x^3+x^2+1 |
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| $B$k(B. |
$B$k(B. |
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|
| 0 $B$G$J$$M-8BBN$N85$O(B, $B?t%*%V%8%'%/%H$G$"$j(B, $B<1JL;R$NCM$O(B 1 $B$G$"$k(B. |
0 $B$G$J$$M-8BBN$N85$O(B, $B?t%*%V%8%'%/%H$G$"$j(B, $B<1JL;R$NCM$O(B 1 $B$G$"$k(B. |
| $B$5$i$K(B, 0 $B$G$J$$M-8BBN$N85$N?t<1JL;R$O(B, GF(p) $B$N>l9g(B 6, GF(2^n) $B$N>l9g(B 7 |
$B$5$i$K(B, 0 $B$G$J$$M-8BBN$N85$N?t<1JL;R$O(B, GF(@var{p}) $B$N>l9g(B 6, GF(2^@var{n}) $B$N>l9g(B 7 |
| $B$H$J$k(B. |
$B$H$J$k(B. |
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|
| $BM-8BBN$N85$NF~NOJ}K!$O(B, $BM-8BBN$N<oN`$K$h$jMM!9$G$"$k(B. GF(p) $B$N>l9g(B, |
$BM-8BBN$N85$NF~NOJ}K!$O(B, $BM-8BBN$N<oN`$K$h$jMM!9$G$"$k(B. GF(@var{p}) $B$N>l9g(B, |
| @code{simp_ff()} $B$K$h$k(B. |
@code{simp_ff()} $B$K$h$k(B. |
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\E |
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|
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\BEG |
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If @var{p} is a positive integer, @code{setmod_ff(@var{p})} sets |
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GF(@var{p}) as the current base field. |
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If @var{f} is a univariate polynomial of degree @var{n}, |
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@code{setmod_ff(@var{f})} sets GF(2^@var{n}) as the current |
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base field. GF(2^@var{n}) is represented |
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as an algebraic extension of GF(2) with the defining polynomial |
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@var{f mod 2}. Furthermore, finite extensions of prime finite fields |
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can be defined. @xref{Types of numbers}. |
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In all cases the primality check of the argument is |
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not done and the caller is responsible for it. |
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|
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Correctly speaking there is no actual object corresponding to a 'base field'. |
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Setting a base field means that operations on elements of finite fields |
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are done according to the arithmetics of the base field. Thus, if |
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operands of an arithmetic operation are both rational numbers, then the result |
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is also a rational number. However, if one of the operands is in |
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a finite field, then the other is automatically regarded as in the |
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same finite field and the operation is done in the finite field. |
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|
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A non zero element of a finite field belongs to the number and has object |
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identifier 1. Its number identifier is 6 if the finite field is GF(@var{p}), |
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7 if it is GF(2^@var{n}). |
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|
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There are several methods to input an element of a finite field. |
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An element of GF(@var{p}) can be input by @code{simp_ff()}. |
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\E |
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|
| @example |
@example |
| [0] P=pari(nextprime,2^50); |
[0] P=pari(nextprime,2^50); |
| 1125899906842679 |
1125899906842679 |
| Line 60 x^50+x^4+x^3+x^2+1 |
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| Line 133 x^50+x^4+x^3+x^2+1 |
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| 6 |
6 |
| @end example |
@end example |
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|
| $B$^$?(B, GF(2^n) $B$N>l9g$$$/$D$+$NJ}K!$,$"$k(B. |
\JP $B$^$?(B, GF(2^@var{n}) $B$N>l9g$$$/$D$+$NJ}K!$,$"$k(B. |
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\EG In the case of GF(2^@var{n}) the following methods are available. |
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|
| @example |
@example |
| [0] setmod_ff(x^50+x^4+x^3+x^2+1); |
[0] setmod_ff(x^50+x^4+x^3+x^2+1); |
| x^50+x^4+x^3+x^2+1 |
x^50+x^4+x^3+x^2+1 |
| Line 74 x^50+x^4+x^3+x^2+1 |
|
| Line 149 x^50+x^4+x^3+x^2+1 |
|
| (@@^9+@@^8+@@^7+@@^6+@@^5+@@^4+@@^3+@@^2+@@+1) |
(@@^9+@@^8+@@^7+@@^6+@@^5+@@^4+@@^3+@@^2+@@+1) |
| @end example |
@end example |
| |
|
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\BJP |
| $BM-8BBN$N85$O?t$G$"$j(B, $BBN1i;;$,2DG=$G$"$k(B. @code{@@} $B$O(B |
$BM-8BBN$N85$O?t$G$"$j(B, $BBN1i;;$,2DG=$G$"$k(B. @code{@@} $B$O(B |
| GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7$/$O(B @xref{$B?t$N7?(B}. |
GF(2^@var{n}) $B$N(B, GF(2) $B>e$N@8@.85$G$"$k(B. $B>\$7$/$O(B @xref{$B?t$N7?(B}. |
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\E |
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\BEG |
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Elements of finite fields are numbers and one can apply field arithmetics |
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to them. @code{@@} is a generator of GF(2^@var{n}) over GF(2). |
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@xref{Types of numbers}. |
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\E |
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|
| @noindent |
@noindent |
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|
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\BJP |
| @node $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B |
@node $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B |
| @section $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B |
@section $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B |
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\E |
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\BEG |
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@node Univariate polynomials on finite fields,,, Finite fields |
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@section Univariate polynomials on finite fields |
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\E |
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|
| @noindent |
@noindent |
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\BJP |
| @samp{fff} $B$G$O(B, $BM-8BBN>e$N(B 1 $BJQ?tB?9`<0$KBP$7(B, $BL5J?J}J,2r(B, DDF, $B0x?tJ,2r(B, |
@samp{fff} $B$G$O(B, $BM-8BBN>e$N(B 1 $BJQ?tB?9`<0$KBP$7(B, $BL5J?J}J,2r(B, DDF, $B0x?tJ,2r(B, |
| $BB?9`<0$N4{LsH=Dj$J$I$N4X?t$,Dj5A$5$l$F$$$k(B. |
$BB?9`<0$N4{LsH=Dj$J$I$N4X?t$,Dj5A$5$l$F$$$k(B. |
| |
|
| $B$$$:$l$b(B, $B7k2L$O(B [@b{$B0x;R(B}, @b{$B=EJ#EY(B}] $B$N%j%9%H$H$J$k$,(B, $B0x;R$O(B monic |
$B$$$:$l$b(B, $B7k2L$O(B [@b{$B0x;R(B}, @b{$B=EJ#EY(B}] $B$N%j%9%H$H$J$k$,(B, $B0x;R$O(B monic |
| $B$H$J$j(B, $BF~NOB?9`<0$N<g78?t$O<N$F$i$l$k(B. |
$B$H$J$j(B, $BF~NOB?9`<0$N<g78?t$O<N$F$i$l$k(B. |
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| @noindent |
|
| $BL5J?J}J,2r$O(B, $BB?9`<0$H$=$NHyJ,$H$N(B GCD $B$N7W;;$+$i;O$^$k$b$C$H$b0lHLE*$J(B |
$BL5J?J}J,2r$O(B, $BB?9`<0$H$=$NHyJ,$H$N(B GCD $B$N7W;;$+$i;O$^$k$b$C$H$b0lHLE*$J(B |
| $B%"%k%4%j%:%`$r:NMQ$7$F$$$k(B. |
$B%"%k%4%j%:%`$r:NMQ$7$F$$$k(B. |
| |
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| @example |
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| @end example |
|
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| @noindent |
|
| $BM-8BBN>e$G$N0x?tJ,2r$O(B, DDF $B$N8e(B, $B<!?tJL0x;R$NJ,2r$N:]$K(B, Berlekamp |
$BM-8BBN>e$G$N0x?tJ,2r$O(B, DDF $B$N8e(B, $B<!?tJL0x;R$NJ,2r$N:]$K(B, Berlekamp |
| $B%"%k%4%j%:%`$GNm6u4V$r5a$a(B, $B4pDl%Y%/%H%k$N:G>.B?9`<0$r5a$a(B, $B$=$N:,(B |
$B%"%k%4%j%:%`$GNm6u4V$r5a$a(B, $B4pDl%Y%/%H%k$N:G>.B?9`<0$r5a$a(B, $B$=$N:,(B |
| $B$r(B Cantor-Zassenhaus $B%"%k%4%j%:%`$K$h$j5a$a$k(B, $B$H$$$&J}K!$r<BAu$7$F$$$k(B. |
$B$r(B Cantor-Zassenhaus $B%"%k%4%j%:%`$K$h$j5a$a$k(B, $B$H$$$&J}K!$r<BAu$7$F$$$k(B. |
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\E |
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\BEG |
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In @samp{fff} square-free factorization, DDF (distinct degree factorization), |
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irreducible factorization and primality check are implemented for |
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univariate polynomials over finite fields. |
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|
| @example |
Factorizers return lists of [@b{factor},@b{multiplicity}]. The factor |
| @end example |
part is monic and the information on the leading coefficient of the |
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input polynomial is abandoned. |
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|
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The algorithm used in square-free factorization is the most primitive one. |
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|
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The irreducible factorization proceeds as follows. |
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|
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@enumerate |
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@item DDF |
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@item Nullspace computation by Berlekamp algorithm |
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@item Root finding of minimal polynomials of bases of the nullspace |
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@item Separation of irreducible factors by the roots |
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@end enumerate |
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\E |
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|
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@noindent |
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|
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\BJP |
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@node $B>.I8?tM-8BBN>e$G$NB?9`<0$N1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B |
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@section $B>.I8?tM-8BBN>e$G$NB?9`<0$N1i;;(B |
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\E |
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\BEG |
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@node Polynomials on small finite fields,,, Finite fields |
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@section Polynomials on small finite fields |
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\E |
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|
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\BJP |
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$B>.I8?tM-8BBN78?t$NB?9`<0$K8B$j(B, $BB?JQ?tB?9`<0$N0x?tJ,2r$,(B |
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$BAH$_9~$_4X?t$H$7$F<BAu$5$l$F$$$k(B. $B4X?t$O(B @code{sffctr()} |
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$B$G$"$k(B. $B$^$?(B, @code{modfctr()} $B$b(B, $BM-8BAGBN>e$GB?JQ?t(B |
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$BB?9`<0$N0x?tJ,2r$r9T$&$,(B, $B<B:]$K$O(B, $BFbIt$G==J,Bg$-$J(B |
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$B3HBgBN$r@_Dj$7(B, @code{sffctr()} $B$r8F$S=P$7$F(B, |
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$B:G=*E*$KAGBN>e$N0x;R$r9=@.$9$k(B, $B$H$$$&J}K!$G7W;;$7$F$$$k(B. |
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\E |
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|
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\BEG |
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A multivariate polynomial over small finite field |
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set by @code{setmod_ff(p,n)} can be factorized by |
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using a builtin function @code{sffctr()}. @code{modfctr()} |
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also factorizes a polynomial over a finite prime field. |
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Internally, @code{modfctr()} creates a sufficiently large |
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field extension of the specified ground field, and |
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it calls @code{sffctr()}, then it constructs irreducible |
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factors over the ground field from the factors returned by |
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@code{sffctr()}. |
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\E |
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|
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\BJP |
| @node $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B |
@node $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B |
| @section $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B |
@section $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B |
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\E |
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\BEG |
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@node Elliptic curves on finite fields,,, Finite fields |
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@section Elliptic curves on finite fields |
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\E |
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|
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\BJP |
| $BM-8BBN>e$NBJ1_6J@~$K4X$9$k$$$/$D$+$N4pK\E*$J1i;;$,(B, $BAH$_9~$_4X?t$H$7$F(B |
$BM-8BBN>e$NBJ1_6J@~$K4X$9$k$$$/$D$+$N4pK\E*$J1i;;$,(B, $BAH$_9~$_4X?t$H$7$F(B |
| $BDs6!$5$l$F$$$k(B. |
$BDs6!$5$l$F$$$k(B. |
| |
|
| $BBJ1_6J@~$N;XDj$O(B, $BD9$5(B 2 $B$N%Y%/%H%k(B @var{[a b]} $B$G9T$&(B. @var{a}, @var{b} |
$BBJ1_6J@~$N;XDj$O(B, $BD9$5(B 2 $B$N%Y%/%H%k(B [@var{a b}] $B$G9T$&(B. @var{a}, @var{b} |
| $B$OM-8BBN$N85$G(B, |
$B$OM-8BBN$N85$G(B, |
| @code{setmod_ff} $B$GDj5A$5$l$F$$$kM-8BBN$,AGBN$N>l9g(B, @var{y^2=x^3+ax+b}, |
@code{setmod_ff} $B$GDj5A$5$l$F$$$kM-8BBN$,AGBN$N>l9g(B, @var{y^2=x^3+ax+b}, |
| $BI8?t(B 2 $B$NBN$N>l9g(B @var{y^2+xy=x^3+ax^2+b} $B$rI=$9(B. |
$BI8?t(B 2 $B$NBN$N>l9g(B @var{y^2+xy=x^3+ax^2+b} $B$rI=$9(B. |
| Line 122 GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7 |
|
| Line 263 GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7 |
|
| |
|
| @itemize @bullet |
@itemize @bullet |
| @item $BL58B1sE@$O(B 0. |
@item $BL58B1sE@$O(B 0. |
| @item $B$=$l0J30$NE@$O(B, $BD9$5(B 3 $B$N%Y%/%H%k(B @var{[x y z]}. $B$?$@$7(B, @var{z} $B$O(B |
@item $B$=$l0J30$NE@$O(B, $BD9$5(B 3 $B$N%Y%/%H%k(B [@var{x y z}]. $B$?$@$7(B, @var{z} $B$O(B |
| 0 $B$G$J$$(B. |
0 $B$G$J$$(B. |
| @end itemize |
@end itemize |
| |
|
| $B$H$$$&E@$G$"$k(B. @var{[x y z]} $B$O@F<!:BI8$K$h$kI=8=$G$"$j(B, $B%"%U%#%s:BI8(B |
$B$H$$$&E@$G$"$k(B. [@var{x y z}] $B$O@F<!:BI8$K$h$kI=8=$G$"$j(B, $B%"%U%#%s:BI8(B |
| $B$G$O(B @var{[x/z y/z]} $B$J$kE@$rI=$9(B. $B$h$C$F(B, $B%"%U%#%s:BI8(B @var{[x y]} $B$G(B |
$B$G$O(B [@var{x/z y/z}] $B$J$kE@$rI=$9(B. $B$h$C$F(B, $B%"%U%#%s:BI8(B [@var{x y}] $B$G(B |
| $BI=8=$5$l$?E@$r1i;;BP>]$H$9$k$K$O(B, @var{[x y 1]} $B$J$k%Y%/%H%k$r(B |
$BI=8=$5$l$?E@$r1i;;BP>]$H$9$k$K$O(B, [@var{x y 1}] $B$J$k%Y%/%H%k$r(B |
| $B@8@.$9$kI,MW$,$"$k(B. |
$B@8@.$9$kI,MW$,$"$k(B. |
| $B1i;;7k2L$b@F<!:BI8$GF@$i$l$k$,(B, @var{z} $B:BI8$,(B 1 $B$H$O8B$i$J$$$?$a(B, |
$B1i;;7k2L$b@F<!:BI8$GF@$i$l$k$,(B, @var{z} $B:BI8$,(B 1 $B$H$O8B$i$J$$$?$a(B, |
| $B%"%U%#%s:BI8$r5a$a$k$?$a$K$O(B @var{x}, @var{y} $B:BI8$r(B @var{z} $B:BI8$G(B |
$B%"%U%#%s:BI8$r5a$a$k$?$a$K$O(B @var{x}, @var{y} $B:BI8$r(B @var{z} $B:BI8$G(B |
| $B3d$kI,MW$,$"$k(B. |
$B3d$kI,MW$,$"$k(B. |
| |
\E |
| |
|
| |
\BEG |
| |
Several fundamental operations on elliptic curves over finite fields |
| |
are provided as built-in functions. |
| |
|
| |
An elliptic curve is specified by a vector [@var{a b}] of length 2, |
| |
where @var{a}, @var{b} are elements of finite fields. |
| |
If the current base field is a prime field, then [@var{a b}] represents |
| |
@var{y^2=x^3+ax+b}. If the current base field is a finite field of |
| |
characteristic 2, then [@var{a b}] represents @var{y^2+xy=x^3+ax^2+b}. |
| |
|
| |
Points on an elliptic curve together with the point at infinity |
| |
forms an additive group. The addition, the subtraction and the |
| |
additive inverse operation are provided as @code{ecm_add_ff()}, |
| |
@code{ecm_sub_ff()} and @code{ecm_chsgn_ff()} respectively. |
| |
Here the representation of points are as follows. |
| |
|
| |
@itemize @bullet |
| |
@item 0 denotes the point at infinity. |
| |
@item The other points are represented by vectors [@var{x y z}] of |
| |
length 3 with non-zero @var{z}. |
| |
@end itemize |
| |
|
| |
[@var{x y z}] represents a projective coordinate and |
| |
it corresponds to [@var{x/z y/z}] in the affine coordinate. |
| |
To apply the above operations to a point [@var{x y}], |
| |
[@var{x y 1}] should be used instead as an argument. |
| |
The result of an operation is also represented by the projective |
| |
coordinate. As the third coordinate is not always equal to 1, |
| |
one has to divide the first and the scond coordinate by the third |
| |
one to obtain the affine coordinate. |
| |
\E |
| |
|
| |
\BJP |
| @node $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B,,, $BM-8BBN$K4X$9$k1i;;(B |
@node $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B,,, $BM-8BBN$K4X$9$k1i;;(B |
| @section $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
@section $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\E |
| |
\BEG |
| |
@node Functions for Finite fields,,, Finite fields |
| |
@section Functions for Finite fields |
| |
\E |
| |
|
| @menu |
@menu |
| * setmod_ff:: |
* setmod_ff:: |
| Line 150 GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7 |
|
| Line 330 GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7 |
|
| * gf2nton:: |
* gf2nton:: |
| * ptogf2n:: |
* ptogf2n:: |
| * gf2ntop:: |
* gf2ntop:: |
| |
* ptosfp sfptop:: |
| * defpoly_mod2:: |
* defpoly_mod2:: |
| |
* sffctr:: |
| * fctr_ff:: |
* fctr_ff:: |
| * irredcheck_ff:: |
* irredcheck_ff:: |
| * randpoly_ff:: |
* randpoly_ff:: |
| Line 158 GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7 |
|
| Line 340 GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7 |
|
| * extdeg_ff:: |
* extdeg_ff:: |
| @end menu |
@end menu |
| |
|
| @node setmod_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node setmod_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node setmod_ff,,, Functions for Finite fields |
| @subsection @code{setmod_ff} |
@subsection @code{setmod_ff} |
| @findex setmod_ff |
@findex setmod_ff |
| |
|
| @table @t |
@table @t |
| @item setmod_ff([@var{prime}|@var{poly}]) |
@item setmod_ff([@var{prime}|@var{poly}]) |
| :: $BM-8BBN$N@_Dj(B, $B@_Dj$5$l$F$$$kM-8BBN$NK!(B, $BDj5AB?9`<0$NI=<((B |
@itemx setmod_ff(@var{prime},@var{n}]) |
| |
\JP :: $BM-8BBN$N@_Dj(B, $B@_Dj$5$l$F$$$kM-8BBN$NK!(B, $BDj5AB?9`<0$NI=<((B |
| |
\EG :: Sets/Gets the current base fields. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B?t$^$?$OB?9`<0(B |
\JP $B?t$^$?$OB?9`<0(B |
| |
\EG number or polynomial |
| @item prime |
@item prime |
| $BAG?t(B |
\JP $BAG?t(B |
| |
\EG prime |
| @item poly |
@item poly |
| GF(2) $B>e4{Ls$J(B 1 $BJQ?tB?9`<0(B |
\JP GF(2) $B>e4{Ls$J(B 1 $BJQ?tB?9`<0(B |
| |
\EG univariate polynomial irreducible over GF(2) |
| |
@item n |
| |
\JP $B3HBg<!?t(B |
| |
\EG the extension degree |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B0z?t$,@5@0?t(B @var{prime} $B$N;~(B, GF(@var{prime}) $B$r4pACBN$H$7$F@_Dj$9$k(B. |
$B0z?t$,@5@0?t(B @var{prime} $B$N;~(B, GF(@var{prime}) $B$r4pACBN$H$7$F@_Dj$9$k(B. |
| @item |
@item |
| $B0z?t$,B?9`<0(B @var{poly} $B$N;~(B, |
$B0z?t$,B?9`<0(B @var{poly} $B$N;~(B, |
| GF(2^deg(@var{poly} mod 2)) = GF(2)[t]/(@var{poly}(t) mod2) |
GF(2^deg(@var{poly} mod 2)) = GF(2)[t]/(@var{poly}(t) mod 2) |
| $B$r4pACBN$H$7$F@_Dj$9$k(B. |
$B$r4pACBN$H$7$F@_Dj$9$k(B. |
| @item |
@item |
| $BL50z?t$N;~(B, $B@_Dj$5$l$F$$$k4pACBN$,(B GF(@var{prime}) $B$N>l9g(B @var{prime}, |
$B0z?t$,(B @var{p} $B$H(B @var{n} $B$N;~(B, |
| GF(2^n) $B$N>l9gDj5AB?9`<0$rJV$9(B. |
GF(@var{p^n}) $B$r4pACBN$H$7$F@_Dj$9$k(B. @var{p^n} $B$O(B @var{2^29} $BL$K~$G(B |
| |
$B$J$1$l$P$J$i$J$$(B. $B$^$?(B, @var{p} $B$,(B @var{2^14} $B0J>e$N$H$-(B, |
| |
@var{n} $B$O(B 1 $B$G$J$1$l$P$J$i$J$$(B. |
| @item |
@item |
| GF(2^n) $B$NDj5AB?9`<0$O(B, GF(2) $B>e(B n $B<!4{Ls$J$i$J$s$G$bNI$$$,(B, $B8zN($K(B |
$BL50z?t$N;~(B, $B@_Dj$5$l$F$$$k4pACBN$,(B GF(@var{prime})$B$N>l9g(B @var{prime}, |
| |
GF(2^@var{n}) $B$N>l9gDj5AB?9`<0$rJV$9(B. |
| |
$B4pACBN$,(B GF(p^@var{n}) |
| |
(@var{p^n} $B$,(B @var{2^14} $BL$K~(B) $B$N>l9g(B, |
| |
[@var{p},@var{defpoly},@var{prim_elem}] $B$rJV$9(B. $B$3$3$G(B, @var{defpoly} |
| |
$B$O(B, @var{n} $B<!3HBg$NDj5AB?9`<0(B, @var{prim_elem} $B$O(B, GF(@var{p^n}) |
| |
$B>hK!72$N@8@.85$r0UL#$9$k(B. |
| |
@item |
| |
GF(2^@var{n}) $B$NDj5AB?9`<0$O(B, GF(2) $B>e(B n $B<!4{Ls$J$i$J$s$G$bNI$$$,(B, $B8zN($K(B |
| $B1F6A$9$k$?$a(B, @code{defpoly_mod2()} $B$G@8@.$9$k$N$,$h$$(B. |
$B1F6A$9$k$?$a(B, @code{defpoly_mod2()} $B$G@8@.$9$k$N$,$h$$(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
If the argument is a non-negative integer @var{prime}, GF(@var{prime}) |
| |
is set as the current base field. |
| |
@item |
| |
If the argument is a polynomial @var{poly}, |
| |
GF(2^deg(@var{poly} mod 2)) = GF(2)[t]/(@var{poly}(t) mod2) |
| |
is set as the current base field. |
| |
@item |
| |
If the arguments are a prime @var{p} and an extension degree @var{n}, |
| |
GF(@var{p^n}) is set as the current base field. @var{p^n} must be |
| |
less than @var{2^29} and if @var{p} is greater than or equal to @var{2^14}, |
| |
then @var{n} must be equal to 1. |
| |
@item |
| |
If no argument is specified, the modulus indicating the current base field |
| |
is returned. If the current base field is GF(@var{prime}), @var{prime} is |
| |
returned. If it is GF(2^@var{n}), the defining polynomial is returned. |
| |
If it is GF(@var{p^n}), where @var{p^n} is less than @var{2^14}, |
| |
[@var{p},@var{defpoly},@var{prim_elem}] is returned. Here, @var{defpoly} |
| |
is the defining polynomial of the @var{n}-th extension, |
| |
and @var{prim_elem} is the generator of the multiplicative group |
| |
of GF(@var{p^n}). |
| |
@item |
| |
Any irreducible univariate polynomial over GF(2) is available to |
| |
set GF(2^@var{n}). However the use of @code{defpoly_mod2()} is recommended |
| |
for efficiency. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| x^100+x^15+1 |
x^100+x^15+1 |
| [176] setmod_ff(); |
[176] setmod_ff(); |
| x^100+x^15+1 |
x^100+x^15+1 |
| |
[177] setmod_ff(2,5); |
| |
[2,x^5+x^2+1,x] |
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{defpoly_mod2} |
@fref{defpoly_mod2} |
| @end table |
@end table |
| |
|
| @node field_type_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node field_type_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node field_type_ff,,, Functions for Finite fields |
| @subsection @code{field_type_ff} |
@subsection @code{field_type_ff} |
| @findex field_type_ff |
@findex field_type_ff |
| |
|
| @table @t |
@table @t |
| @item field_type_ff() |
@item field_type_ff() |
| :: $B@_Dj$5$l$F$$$k4pACBN$N<oN`(B |
\JP :: $B@_Dj$5$l$F$$$k4pACBN$N<oN`(B |
| |
\EG :: Type of the current base field. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B?t(B |
\JP $B@0?t(B |
| |
\EG integer |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B@_Dj$5$l$F$$$k4pACBN$N<oN`$rJV$9(B. |
$B@_Dj$5$l$F$$$k4pACBN$N<oN`$rJV$9(B. |
| @item |
@item |
| $B@_Dj$J$7$J$i(B 0, GF(p) $B$J$i(B 1, GF(2^n) $B$J$i(B 2 $B$rJV$9(B. |
$B@_Dj$J$7$J$i(B 0, GF(@var{p}) $B$J$i(B 1, GF(2^@var{n}) $B$J$i(B 2 $B$rJV$9(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Returns the type of the current base field. |
| |
@item |
| |
If no field is set, 0 is returned. If GF(@var{p}) is set, 1 is returned. |
| |
If GF(2^@var{n}) is set, 2 is returned. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{setmod_ff} |
@fref{setmod_ff} |
| @end table |
@end table |
| |
|
| @node field_order_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node field_order_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node field_order_ff,,, Functions for Finite fields |
| @subsection @code{field_order_ff} |
@subsection @code{field_order_ff} |
| @findex field_order_ff |
@findex field_order_ff |
| |
|
| @table @t |
@table @t |
| @item field_order_ff() |
@item field_order_ff() |
| :: $B@_Dj$5$l$F$$$k4pACBN$N0L?t(B |
\JP :: $B@_Dj$5$l$F$$$k4pACBN$N0L?t(B |
| |
\EG :: Order of the current base field. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B?t(B |
\JP $B@0?t(B |
| |
\EG integer |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B@_Dj$5$l$F$$$k4pACBN$N0L?t(B ($B85$N8D?t(B) $B$rJV$9(B. |
$B@_Dj$5$l$F$$$k4pACBN$N0L?t(B ($B85$N8D?t(B) $B$rJV$9(B. |
| @item |
@item |
| $B@_Dj$5$l$F$$$kBN$,(B GF(q) $B$J$i$P(B q $B$rJV$9(B. |
$B@_Dj$5$l$F$$$kBN$,(B GF(@var{q}) $B$J$i$P(B q $B$rJV$9(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Returns the order of the current base field. |
| |
@item |
| |
@var{q} is returned if the current base field is GF(@var{q}). |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{setmod_ff} |
@fref{setmod_ff} |
| @end table |
@end table |
| |
|
| @node characteristic_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node characteristic_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node characteristic_ff,,, Functions for Finite fields |
| @subsection @code{characteristic_ff} |
@subsection @code{characteristic_ff} |
| @findex characteristic_ff |
@findex characteristic_ff |
| |
|
| @table @t |
@table @t |
| @item characteristic_ff() |
@item characteristic_ff() |
| :: $B@_Dj$5$l$F$$$kBN$NI8?t(B |
\JP :: $B@_Dj$5$l$F$$$kBN$NI8?t(B |
| |
\EG :: Characteristic of the current base field. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B?t(B |
\JP $B@0?t(B |
| |
\EG integer |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B@_Dj$5$l$F$$$kBN$NI8?t$rJV$9(B. |
$B@_Dj$5$l$F$$$kBN$NI8?t$rJV$9(B. |
| @item |
@item |
| GF(p) $B$N>l9g(B p, GF(2^n) $B$N>l9g(B 2 $B$rJV$9(B. |
GF(@var{p}) $B$N>l9g(B @var{p}, GF(2^@var{n}) $B$N>l9g(B 2 $B$rJV$9(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Returns the characteristic of the current base field. |
| |
@item |
| |
@var{p} is returned if GF(@var{p}), where @var{p} is a prime, is set. |
| |
@var{2} is returned if GF(2^@var{n}) is set. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{setmod_ff} |
@fref{setmod_ff} |
| @end table |
@end table |
| |
|
| @node extdeg_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node extdeg_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node extdeg_ff,,, Functions for Finite fields |
| @subsection @code{extdeg_ff} |
@subsection @code{extdeg_ff} |
| @findex extdeg_ff |
@findex extdeg_ff |
| |
|
| @table @t |
@table @t |
| @item extdeg_ff() |
@item extdeg_ff() |
| :: $B@_Dj$5$l$F$$$k4pACBN$N(B, $BAGBN$KBP$9$k3HBg<!?t(B |
\JP :: $B@_Dj$5$l$F$$$k4pACBN$N(B, $BAGBN$KBP$9$k3HBg<!?t(B |
| |
\EG :: Extension degree of the current base field over the prime field. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B?t(B |
\JP $B@0?t(B |
| |
\EG integer |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B@_Dj$5$l$F$$$k4pACBN$N(B, $BAGBN$KBP$9$k3HBg<!?t$rJV$9(B. |
$B@_Dj$5$l$F$$$k4pACBN$N(B, $BAGBN$KBP$9$k3HBg<!?t$rJV$9(B. |
| @item |
@item |
| GF(p) $B$N>l9g(B 1, GF(2^n) $B$N>l9g(B n $B$rJV$9(B. |
GF(@var{p}) $B$N>l9g(B 1, GF(2^@var{n}) $B$N>l9g(B @var{n} $B$rJV$9(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Returns the extension degree of the current base field over the prime field. |
| |
@item |
| |
1 is returned if GF(@var{p}), where @var{p} is a prime, is set. |
| |
@var{n} is returned if GF(2^@var{n}) is set. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{setmod_ff} |
@fref{setmod_ff} |
| @end table |
@end table |
| |
|
| @node simp_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node simp_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node simp_ff,,, Functions for Finite fields |
| @subsection @code{simp_ff} |
@subsection @code{simp_ff} |
| @findex simp_ff |
@findex simp_ff |
| |
|
| @table @t |
@table @t |
| @item simp_ff(@var{obj}) |
@item simp_ff(@var{obj}) |
| :: $B?t(B, $B$"$k$$$OB?9`<0$N78?t$rM-8BBN$N85$KJQ49(B |
\JP :: $B?t(B, $B$"$k$$$OB?9`<0$N78?t$rM-8BBN$N85$KJQ49(B |
| |
\BEG |
| |
:: Converts numbers or coefficients of polynomials into elements |
| |
in finite fields. |
| |
\E |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B?t$^$?$OB?9`<0(B |
\JP $B?t$^$?$OB?9`<0(B |
| |
\EG number or polynomial |
| @item obj |
@item obj |
| $B?t$^$?$OB?9`<0(B |
\JP $B?t$^$?$OB?9`<0(B |
| |
\EG number or polynomial |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B?t(B, $B$"$k$$$OB?9`<0$N78?t$rM-8BBN$N85$KJQ49$9$k(B. |
$B?t(B, $B$"$k$$$OB?9`<0$N78?t$rM-8BBN$N85$KJQ49$9$k(B. |
| @item |
@item |
|
|
| @item |
@item |
| $BM-8BBN$N85$KBP$7(B, $BK!$"$k$$$ODj5AB?9`<0$K$h$k(B reduction $B$r9T$&>l9g$K$b(B |
$BM-8BBN$N85$KBP$7(B, $BK!$"$k$$$ODj5AB?9`<0$K$h$k(B reduction $B$r9T$&>l9g$K$b(B |
| $BMQ$$$k(B. |
$BMQ$$$k(B. |
| |
@item |
| |
$B>.I8?tM-8BBN$N85$KJQ49$9$k>l9g(B, $B0lC6AGBN>e$K<M1F$7$F$+$i(B, $B3HBgBN$N(B |
| |
$B85$KJQ49$5$l$k(B. $B3HBgBN$N85$KD>@\JQ49$9$k$K$O(B @code{ptosfp()} $B$r(B |
| |
$BMQ$$$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Converts numbers or coefficients of polynomials into elements in finite |
| |
fields. |
| |
@item |
| |
It is used to convert integers or intrgral polynomials int |
| |
elements of finite fields or polynomials over finite fields. |
| |
@item |
| |
An element of a finite field may not have the reduced representation. |
| |
In such case an application of @code{simp_ff} ensures that the output has |
| |
the reduced representation. |
| |
If a small finite field is set as a ground field, |
| |
an integer is projected the finite prime field, then |
| |
it is embedded into the ground field. @code{ptosfp()} |
| |
can be used for direct projection to the ground field. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 400 x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120 |
|
| Line 713 x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120 |
|
| 1*x^10+1*x^9+1*x+1 |
1*x^10+1*x^9+1*x+1 |
| [3] ntype(coef(@@@@,10)); |
[3] ntype(coef(@@@@,10)); |
| 6 |
6 |
| |
[4] setmod_ff(2,3); |
| |
[2,x^3+x+1,x] |
| |
[5] simp_ff(1); |
| |
@@_0 |
| |
[6] simp_ff(2); |
| |
0 |
| |
[7] ptosfp(2); |
| |
@@_1 |
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| @fref{setmod_ff}, @fref{lmptop}, @fref{gf2nton} |
\EG @item References |
| |
@fref{setmod_ff}, @fref{lmptop}, @fref{gf2nton}, @fref{ptosfp sfptop} |
| @end table |
@end table |
| |
|
| @node random_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node random_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node random_ff,,, Functions for Finite fields |
| @subsection @code{random_ff} |
@subsection @code{random_ff} |
| @findex random_ff |
@findex random_ff |
| |
|
| @table @t |
@table @t |
| @item random_ff() |
@item random_ff() |
| :: $BM-8BBN$N85$NMp?t@8@.(B |
\JP :: $BM-8BBN$N85$NMp?t@8@.(B |
| |
\EG :: Random generation of an element of a finite field. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BM-8BBN$N85(B |
\JP $BM-8BBN$N85(B |
| |
\EG element of a finite field |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $BM-8BBN$N85$rMp?t@8@.$9$k(B. |
$BM-8BBN$N85$rMp?t@8@.$9$k(B. |
| @item |
@item |
| GF(p) $B$N>l9g(B, 0 $B0J>e(B p $BL$K~$N@0?t$G$"$i$o$5$l$k(B GF(p) $B$N85(B, |
|
| GF(2^n) $B$N>l9g(B, n $B<!L$K~$N(B GF(2) $B>e$NB?9`<0$GI=$5$l$k(B GF(2^n) $B$r(B |
|
| $BJV$9(B. |
|
| @item |
|
| @code{random()}, @code{lrandom()} $B$HF1$8(B 32bit $BMp?tH/@84o$r;HMQ$7$F$$$k(B. |
@code{random()}, @code{lrandom()} $B$HF1$8(B 32bit $BMp?tH/@84o$r;HMQ$7$F$$$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Generates an element of the current base field randomly. |
| |
@item |
| |
The same random generator as in @code{random()}, @code{lrandom()} |
| |
is used. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 445 return to toplevel |
|
| Line 775 return to toplevel |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{setmod_ff}, @fref{random}, @fref{lrandom} |
@fref{setmod_ff}, @fref{random}, @fref{lrandom} |
| @end table |
@end table |
| |
|
| @node lmptop,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node lmptop,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node lmptop,,, Functions for Finite fields |
| @subsection @code{lmptop} |
@subsection @code{lmptop} |
| @findex lmptop |
@findex lmptop |
| |
|
| @table @t |
@table @t |
| @item lmptop(@var{obj}) |
@item lmptop(@var{obj}) |
| :: GF(p) $B78?tB?9`<0$N78?t$r@0?t$KJQ49(B |
\JP :: GF(@var{p}) $B78?tB?9`<0$N78?t$r@0?t$KJQ49(B |
| |
\EG :: Converts the coefficients of a polynomial over GF(@var{p}) into integers. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B@0?t78?tB?9`<0(B |
\JP $B@0?t78?tB?9`<0(B |
| |
\EG integral polynomial |
| @item obj |
@item obj |
| GF(p)$B78?tB?9`<0(B |
\JP GF(@var{p}) $B78?tB?9`<0(B |
| |
\EG polynomial over GF(@var{p}) |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| GF(p) $B78?tB?9`<0$N78?t$r@0?t$KJQ49$9$k(B. |
GF(@var{p}) $B78?tB?9`<0$N78?t$r@0?t$KJQ49$9$k(B. |
| @item |
@item |
| GF(p) $B$N85$O(B, 0 $B0J>e(B p $BL$K~$N@0?t$GI=8=$5$l$F$$$k(B. |
GF(@var{p}) $B$N85$O(B, 0 $B0J>e(B p $BL$K~$N@0?t$GI=8=$5$l$F$$$k(B. |
| $BB?9`<0$N3F78?t$O(B, $B$=$NCM$r@0?t%*%V%8%'%/%H(B($B?t<1JL;R(B 0)$B$H$7$?$b$N$K(B |
$BB?9`<0$N3F78?t$O(B, $B$=$NCM$r@0?t%*%V%8%'%/%H(B($B?t<1JL;R(B 0)$B$H$7$?$b$N$K(B |
| $BJQ49$5$l$k(B. |
$BJQ49$5$l$k(B. |
| |
\E |
| |
\BEG |
| @item |
@item |
| GF(p) $B$N85$O(B, $B@0?t$KJQ49$5$l$k(B. |
Converts the coefficients of a polynomial over GF(@var{p}) into integers. |
| |
@item |
| |
An element of GF(@var{p}) is represented by a non-negative integer @var{r} less than |
| |
@var{p}. |
| |
Each coefficient of a polynomial is converted into an integer object |
| |
whose value is @var{r}. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 485 GF(p) $B$N85$O(B, $B@0?t$KJQ49$5$l$k(B. |
|
| Line 829 GF(p) $B$N85$O(B, $B@0?t$KJQ49$5$l$k(B. |
|
| [2] setmod_ff(547); |
[2] setmod_ff(547); |
| 547 |
547 |
| [3] F=simp_ff((x-1)^10); |
[3] F=simp_ff((x-1)^10); |
| 1*x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+427*x^3+45*x^2+537*x+1 |
1*x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+427*x^3 |
| |
+45*x^2+537*x+1 |
| [4] lmptop(F); |
[4] lmptop(F); |
| x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+427*x^3+45*x^2+537*x+1 |
x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+427*x^3 |
| |
+45*x^2+537*x+1 |
| [5] lmptop(coef(F,1)); |
[5] lmptop(coef(F,1)); |
| 537 |
537 |
| [6] ntype(@@@@); |
[6] ntype(@@@@); |
| Line 495 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42 |
|
| Line 841 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{simp_ff} |
@fref{simp_ff} |
| @end table |
@end table |
| |
|
| @node ntogf2n,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node ntogf2n,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node ntogf2n,,, Functions for Finite fields |
| @subsection @code{ntogf2n} |
@subsection @code{ntogf2n} |
| @findex ntogf2n |
@findex ntogf2n |
| |
|
| @table @t |
@table @t |
| @item ntogf2n(@var{m}) |
@item ntogf2n(@var{m}) |
| :: $B<+A3?t$r(B GF(2^n) $B$N85$KJQ49(B |
\JP :: $B<+A3?t$r(B GF(2^@var{n}) $B$N85$KJQ49(B |
| |
\EG :: Converts a non-negative integer into an element of GF(2^@var{n}). |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| GF(2^n) $B$N85(B |
\JP GF(2^@var{n}) $B$N85(B |
| |
\EG element of GF(2^@var{n}) |
| @item m |
@item m |
| $BHsIi@0?t(B |
\JP $BHsIi@0?t(B |
| |
\EG non-negative integer |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B<+A3?t(B @var{m} $B$N(B 2 $B?JI=8=(B @var{m}=@var{m0}+@var{m1}*2+...+@var{mk}*2^k |
$B<+A3?t(B @var{m} $B$N(B 2 $B?JI=8=(B @var{m}=@var{m0}+@var{m1}*2+...+@var{mk}*2^k |
| $B$KBP$7(B, GF(2^n)=GF(2)[t]/(g(t)) $B$N85(B |
$B$KBP$7(B, GF(2^@var{n})=GF(2)[t]/(g(t)) $B$N85(B |
| @var{m0}+@var{m1}*t+...+@var{mk}*t^k mod g(t) $B$rJV$9(B. |
@var{m0}+@var{m1}*t+...+@var{mk}*t^k mod g(t) $B$rJV$9(B. |
| @item |
@item |
| $BDj5AB?9`<0$K$h$k>jM>$O<+F0E*$K$O7W;;$5$l$J$$$?$a(B, @code{simp_ff()} $B$r(B |
$BDj5AB?9`<0$K$h$k>jM>$O<+F0E*$K$O7W;;$5$l$J$$$?$a(B, @code{simp_ff()} $B$r(B |
| $BE,MQ$9$kI,MW$,$"$k(B. |
$BE,MQ$9$kI,MW$,$"$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Let @var{m} be a non-negative integer. |
| |
@var{m} has the binary representation |
| |
@var{m}=@var{m0}+@var{m1}*2+...+@var{mk}*2^k. |
| |
This function returns an element of GF(2^@var{n}) = GF(2)[t]/(g(t)), |
| |
@var{m0}+@var{m1}*t+...+@var{mk}*t^k mod g(t). |
| |
@item |
| |
Apply @code{simp_ff()} to reduce the result. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{gf2nton} |
@fref{gf2nton} |
| @end table |
@end table |
| |
|
| @node gf2nton,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node gf2nton,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node gf2nton,,, Functions for Finite fields |
| @subsection @code{gf2nton} |
@subsection @code{gf2nton} |
| @findex gf2nton |
@findex gf2nton |
| |
|
| @table @t |
@table @t |
| @item gf2nton(@var{m}) |
@item gf2nton(@var{m}) |
| :: GF(2^n) $B$N85$r<+A3?t$KJQ49(B |
\JP :: GF(2^@var{n}) $B$N85$r<+A3?t$KJQ49(B |
| |
\EG :: Converts an element of GF(2^@var{n}) into a non-negative integer. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BHsIi@0?t(B |
\JP $BHsIi@0?t(B |
| |
\EG non-negative integer |
| @item m |
@item m |
| GF(2^n) $B$N85(B |
\JP GF(2^@var{n}) $B$N85(B |
| |
\EG element of GF(2^@var{n}) |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| @item |
@item |
| @code{gf2nton} $B$N5UJQ49$G$"$k(B. |
\JP @code{gf2nton} $B$N5UJQ49$G$"$k(B. |
| |
\EG The inverse of @code{gf2nton}. |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{gf2nton} |
@fref{gf2nton} |
| @end table |
@end table |
| |
|
| @node ptogf2n,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node ptogf2n,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node ptogf2n,,, Functions for Finite fields |
| @subsection @code{ptogf2n} |
@subsection @code{ptogf2n} |
| @findex ptogf2n |
@findex ptogf2n |
| |
|
| @table @t |
@table @t |
| @item ptogf2n(@var{poly}) |
@item ptogf2n(@var{poly}) |
| :: $B0lJQ?tB?9`<0$r(B GF(2^n) $B$N85$KJQ49(B |
\JP :: $B0lJQ?tB?9`<0$r(B GF(2^@var{n}) $B$N85$KJQ49(B |
| |
\EG :: Converts a univariate polynomial into an element of GF(2^@var{n}). |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| GF(2^n) $B$N85(B |
\JP GF(2^@var{n}) $B$N85(B |
| |
\EG element of GF(2^@var{n}) |
| @item poly |
@item poly |
| $B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
| |
\EG univariate polynomial |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| @item |
@item |
| @var{poly} $B$NI=$9(B GF(2^n) $B$N85$r@8@.$9$k(B. $B78?t$O(B, 2 $B$G3d$C$?M>$j$K(B |
\BJP |
| |
@var{poly} $B$NI=$9(B GF(2^@var{n}) $B$N85$r@8@.$9$k(B. $B78?t$O(B, 2 $B$G3d$C$?M>$j$K(B |
| $BJQ49$5$l$k(B. |
$BJQ49$5$l$k(B. |
| @var{poly} $B$NJQ?t$K(B @code{@@} $B$rBeF~$7$?7k2L$HEy$7$$(B. |
@var{poly} $B$NJQ?t$K(B @code{@@} $B$rBeF~$7$?7k2L$HEy$7$$(B. |
| |
\E |
| |
\BEG |
| |
Generates an element of GF(2^@var{n}) represented by @var{poly}. |
| |
The coefficients are reduced modulo 2. |
| |
The output is equal to the result by substituting @code{@@} for |
| |
the variable of @var{poly}. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
|
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{gf2ntop} |
@fref{gf2ntop} |
| @end table |
@end table |
| |
|
| @node gf2ntop,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node gf2ntop,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node gf2ntop,,, Functions for Finite fields |
| @subsection @code{gf2ntop} |
@subsection @code{gf2ntop} |
| @findex gf2ntop |
@findex gf2ntop |
| |
|
| @table @t |
@table @t |
| @item gf2ntop(@var{m}[,@var{v}]) |
@item gf2ntop(@var{m}[,@var{v}]) |
| :: GF(2^n) $B$N85$rB?9`<0$KJQ49(B |
\JP :: GF(2^@var{n}) $B$N85$rB?9`<0$KJQ49(B |
| |
\EG :: Converts an element of GF(2^@var{n}) into a polynomial. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
| |
\EG univariate polynomial |
| @item m |
@item m |
| GF(2^n) $B$N85(B |
\JP GF(2^@var{n}) $B$N85(B |
| |
\EG an element of GF(2^@var{n}) |
| @item v |
@item v |
| $BITDj85(B |
\JP $BITDj85(B |
| |
\EG indeterminate |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| @var{m} $B$rI=$9B?9`<0$r(B, $B@0?t78?t$NB?9`<0%*%V%8%'%/%H$H$7$FJV$9(B. |
@var{m} $B$rI=$9B?9`<0$r(B, $B@0?t78?t$NB?9`<0%*%V%8%'%/%H$H$7$FJV$9(B. |
| @item @var{v} $B$N;XDj$,$J$$>l9g(B, $BD>A0$N(B @code{ptogf2n()} $B8F$S=P$7(B |
@item |
| |
@var{v} $B$N;XDj$,$J$$>l9g(B, $BD>A0$N(B @code{ptogf2n()} $B8F$S=P$7(B |
| $B$K$*$1$k0z?t$NJQ?t(B ($B%G%U%)%k%H$O(B @code{x}), $B;XDj$,$"$k>l9g$K$O(B |
$B$K$*$1$k0z?t$NJQ?t(B ($B%G%U%)%k%H$O(B @code{x}), $B;XDj$,$"$k>l9g$K$O(B |
| $B;XDj$5$l$?ITDj85$rJQ?t$H$9$kB?9`<0$rJV$9(B. |
$B;XDj$5$l$?ITDj85$rJQ?t$H$9$kB?9`<0$rJV$9(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Returns a polynomial representing @var{m}. |
| |
@item |
| |
If @var{v} is used as the variable of the output. |
| |
If @var{v} is not specified, the variable of the argument |
| |
of the latest @code{ptogf2n()} call. The default variable is @code{x}. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 652 t^13+t^12+t^11+t^10 |
|
| Line 1051 t^13+t^12+t^11+t^10 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{ptogf2n} |
@fref{ptogf2n} |
| @end table |
@end table |
| |
|
| @node defpoly_mod2,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node ptosfp sfptop,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node ptosfp sfptop,,, Functions for Finite fields |
| |
@subsection @code{ptosfp}, @code{sfptop} |
| |
@findex ptosfp |
| |
@findex sfptop |
| |
|
| |
@table @t |
| |
@item ptosfp(@var{p}) |
| |
@itemx sfptop(@var{p}) |
| |
\JP :: $B>.I8?tM-8BBN$X$NJQ49(B, $B5UJQ49(B |
| |
\EG :: Transformation to/from a small finite field |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| |
@item p |
| |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
@code{ptosfp()} $B$O(B, $BB?9`<0$N78?t$r(B, $B8=:_@_Dj$5$l$F$$$k>.I8?tM-8BBN(B |
| |
GF(p^@var{n}) $B$N85$KD>@\JQ49$9$k(B. $B78?t$,4{$KM-8BBN$N85$N>l9g$OJQ2=$7$J$$(B. |
| |
$B@5@0?t$N>l9g(B, $B$^$:0L?t$G>jM>$r7W;;$7$?$"$H(B, $BI8?t(B @var{p} $B$K$h$j(B @var{p} |
| |
$B?JE83+$7(B, @var{p} $B$r(B x $B$KCV$-49$($?B?9`<0$r(B, $B86;O85I=8=$KJQ49$9$k(B. |
| |
$BNc$($P(B, GF(3^5) $B$O(B GF(3)[x]/(x^5+2*x+1) $B$H$7$FI=8=$5$l(B, $B$=$N3F(B |
| |
$B85$O86;O85(B x $B$K4X$9$k$Y$-;X?t(B @var{k} $B$K$h$j(B @var{@@_k} $B$H$7$F(B |
| |
$BI=<($5$l$k(B. $B$3$N$H$-(B, $BNc$($P(B @var{23 = 2*3^2+3+2} $B$O(B, 2*x^2+x+2 |
| |
$B$HI=8=$5$l(B, $B$3$l$O7k6I(B x^17 $B$HK!(B x^5+2*x+1 $B$GEy$7$$$N$G(B, |
| |
@var{@@_17} $B$HJQ49$5$l$k(B. |
| |
@item |
| |
@code{sfptop()} $B$O(B @code{ptosfp()} $B$N5UJQ49$G$"$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
@code{ptosfp()} converts coefficients of a polynomial to |
| |
elements in a small finite field GF(@var{p^n}) set as a ground field. |
| |
If a coefficient is already an element of the field, |
| |
no conversion is done. If a coefficient is a positive integer, |
| |
then its residue modulo @var{p^n} is expanded as @var{p}-adic integer, |
| |
then @var{p} is substituted by @var{x}, finally the polynomial |
| |
is converted to its correspoding logarithmic representation |
| |
with respect to the primitive element. |
| |
For example, GF(3^5) is represented as F(3)[@var{x}]/(@var{x^5+2*x+1}), |
| |
and each element of the field is represented as @var{@@_k} |
| |
by its exponent @var{k} with respect to the primitive element @var{x}. |
| |
@var{23 = 2*3^2+3+2} is represented as @var{2*x^2+x+2} and |
| |
it is equivalent to @var{x^17} modulo @var{x^5+2*x+1}. |
| |
Therefore an integer @var{23} is conterted to @var{@@_17}. |
| |
@item |
| |
@code{sfptop()} is the inverse of @code{ptosfp()}. |
| |
\E |
| |
@end itemize |
| |
|
| |
@example |
| |
[196] setmod_ff(3,5); |
| |
[3,x^5+2*x+1,x] |
| |
[197] A = ptosfp(23); |
| |
@@_17 |
| |
[198] 9*2+3+2; |
| |
23 |
| |
[199] x^17-(2*x^2+x+2); |
| |
x^17-2*x^2-x-2 |
| |
[200] sremm(@@,x^5+2*x+1,3); |
| |
0 |
| |
[201] sfptop(A); |
| |
23 |
| |
@end example |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
@fref{setmod_ff}, @fref{simp_ff} |
| |
@end table |
| |
\JP @node defpoly_mod2,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node defpoly_mod2,,, Functions for Finite fields |
| @subsection @code{defpoly_mod2} |
@subsection @code{defpoly_mod2} |
| @findex defpoly_mod2 |
@findex defpoly_mod2 |
| |
|
| @table @t |
@table @t |
| @item defpoly_mod2(@var{d}) |
@item defpoly_mod2(@var{d}) |
| :: GF(2) $B>e4{Ls$J0lJQ?tB?9`<0$N@8@.(B |
\JP :: GF(2) $B>e4{Ls$J0lJQ?tB?9`<0$N@8@.(B |
| |
\EG :: Generates an irreducible univariate polynomial over GF(2). |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
| |
\EG univariate polynomial |
| @item d |
@item d |
| $B@5@0?t(B |
\JP $B@5@0?t(B |
| |
\EG positive integer |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| @samp{fff} $B$GDj5A$5$l$F$$$k(B. |
@samp{fff} $B$GDj5A$5$l$F$$$k(B. |
| @item |
@item |
| Line 682 t^13+t^12+t^11+t^10 |
|
| Line 1165 t^13+t^12+t^11+t^10 |
|
| 3 $B9`<0$,B8:_$7$J$1$l$P(B, $B4{Ls(B 5 $B9`<0$NCf$G(B, $BBh(B 2 $B9`$N<!?t$,$b$C$H$b>.$5$/(B, |
3 $B9`<0$,B8:_$7$J$1$l$P(B, $B4{Ls(B 5 $B9`<0$NCf$G(B, $BBh(B 2 $B9`$N<!?t$,$b$C$H$b>.$5$/(B, |
| $B$=$NCf$GBh(B 3 $B9`$N<!?t$,$b$C$H$b>.$5$/(B, $B$=$NCf$GBh(B 4 $B9`$N<!?t$,$b$C$H$b(B |
$B$=$NCf$GBh(B 3 $B9`$N<!?t$,$b$C$H$b>.$5$/(B, $B$=$NCf$GBh(B 4 $B9`$N<!?t$,$b$C$H$b(B |
| $B>.$5$$$b$N$rJV$9(B. |
$B>.$5$$$b$N$rJV$9(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Defined in @samp{fff}. |
| |
@item |
| |
An irreducible univariate polynomial of degree @var{d} is returned. |
| |
@item |
| |
If an irreducible trinomial @var{x^d+x^m+1} exists, then the one |
| |
with the smallest @var{m} is returned. |
| |
Otherwise, an irreducible pentanomial @var{x^d+x^m1+x^m2+x^m3+1} |
| |
(@var{m1>m2>m3} is returned. |
| |
@var{m1}, @var{m2} and @var{m3} are determined as follows: |
| |
Fix @var{m1} as small as possible. Then fix @var{m2} as small as possible. |
| |
Then fix @var{m3} as small as possible. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{setmod_ff} |
@fref{setmod_ff} |
| @end table |
@end table |
| |
|
| @node fctr_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node sffctr,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node sffctr,,, Functions for Finite fields |
| |
@subsection @code{sffctr} |
| |
@findex sffctr |
| |
|
| |
@table @t |
| |
@item sffctr(@var{poly}) |
| |
\JP :: $BB?9`<0$N>.I8?tM-8BBN>e$G$N4{LsJ,2r(B |
| |
\EG :: Irreducible factorization over a small finite field. |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $B%j%9%H(B |
| |
\EG list |
| |
@item poly |
| |
\JP $BM-8BBN>e$N(B $BB?9`<0(B |
| |
\EG polynomial over a finite field |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
$BB?9`<0$r(B, $B8=:_@_Dj$5$l$F$$$k>.I8?tM-8BBN>e$G4{LsJ,2r$9$k(B. |
| |
@item |
| |
$B7k2L$O(B, [[@var{f1},@var{m1}],[@var{f2},@var{m2}],...] $B$J$k(B |
| |
$B%j%9%H$G$"$k(B. $B$3$3$G(B, @var{fi} $B$O(B monic $B$J4{Ls0x;R(B, @var{mi} $B$O$=$N(B |
| |
$B=EJ#EY$G$"$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Factorize @var{poly} into irreducible factors over |
| |
a small finite field currently set. |
| |
@item |
| |
The result is a list [[@var{f1},@var{m1}],[@var{f2},@var{m2}],...], |
| |
where @var{fi} is a monic irreducible factor and @var{mi} is its |
| |
multiplicity. |
| |
\E |
| |
@end itemize |
| |
|
| |
@example |
| |
[0] setmod_ff(2,10); |
| |
[2,x^10+x^3+1,x] |
| |
[1] sffctr((z*y^3+z*y)*x^3+(y^5+y^3+z*y^2+z)*x^2+z^11*y*x+z^10*y^3+z^11); |
| |
[[@@_0,1],[@@_0*z*y*x+@@_0*y^3+@@_0*z,1],[(@@_0*y+@@_0)*x+@@_0*z^5,2]] |
| |
@end example |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
\EG @item References |
| |
@fref{setmod_ff}, |
| |
@fref{modfctr} |
| |
@end table |
| |
|
| |
\JP @node fctr_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node fctr_ff,,, Functions for Finite fields |
| @subsection @code{fctr_ff} |
@subsection @code{fctr_ff} |
| @findex fctr_ff |
@findex fctr_ff |
| |
|
| @table @t |
@table @t |
| @item fctr_ff(@var{poly}) |
@item fctr_ff(@var{poly}) |
| :: 1 $BJQ?tB?9`<0$NM-8BBN>e$G$N4{LsJ,2r(B |
\JP :: 1 $BJQ?tB?9`<0$NM-8BBN>e$G$N4{LsJ,2r(B |
| |
\EG :: Irreducible univariate factorization over a finite field. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B%j%9%H(B |
\JP $B%j%9%H(B |
| |
\EG list |
| @item poly |
@item poly |
| $BM-8BBN>e$N(B 1 $BJQ?tB?9`<0(B |
\JP $BM-8BBN>e$N(B 1 $BJQ?tB?9`<0(B |
| |
\EG univariate polynomial over a finite field |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| @samp{fff} $B$GDj5A$5$l$F$$$k(B. |
@samp{fff} $B$GDj5A$5$l$F$$$k(B. |
| @item |
@item |
| Line 719 t^13+t^12+t^11+t^10 |
|
| Line 1277 t^13+t^12+t^11+t^10 |
|
| $B=EJ#EY$G$"$k(B. |
$B=EJ#EY$G$"$k(B. |
| @item |
@item |
| @var{poly} $B$N<g78?t$O<N$F$i$l$k(B. |
@var{poly} $B$N<g78?t$O<N$F$i$l$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Defined in @samp{fff}. |
| |
@item |
| |
Factorize @var{poly} into irreducible factors over the current base field. |
| |
@item |
| |
The result is a list [[@var{f1},@var{m1}],[@var{f2},@var{m2}],...], |
| |
where @var{fi} is a monic irreducible factor and @var{mi} is its |
| |
multiplicity. |
| |
@item |
| |
The leading coefficient of @var{poly} is abandoned. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 730 t^13+t^12+t^11+t^10 |
|
| Line 1301 t^13+t^12+t^11+t^10 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{setmod_ff} |
@fref{setmod_ff} |
| @end table |
@end table |
| |
|
| @node irredcheck_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node irredcheck_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node irredcheck_ff,,, Functions for Finite fields |
| @subsection @code{irredcheck_ff} |
@subsection @code{irredcheck_ff} |
| @findex irredcheck_ff |
@findex irredcheck_ff |
| |
|
| @table @t |
@table @t |
| @item irredcheck_ff(@var{poly}) |
@item irredcheck_ff(@var{poly}) |
| :: 1 $BJQ?tB?9`<0$NM-8BBN>e$G$N4{LsH=Dj(B |
\JP :: 1 $BJQ?tB?9`<0$NM-8BBN>e$G$N4{LsH=Dj(B |
| |
\EG :: Primality check of a univariate polynomial over a finite field. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| 0|1 |
0|1 |
| @item poly |
@item poly |
| $BM-8BBN>e$N(B 1 $BJQ?tB?9`<0(B |
\JP $BM-8BBN>e$N(B 1 $BJQ?tB?9`<0(B |
| |
\EG univariate polynomial over a finite field |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| @samp{fff} $B$GDj5A$5$l$F$$$k(B. |
@samp{fff} $B$GDj5A$5$l$F$$$k(B. |
| @item |
@item |
| $BM-8BBN>e$N(B 1 $BJQ?tB?9`<0$N4{LsH=Dj$r9T$$(B, $B4{Ls$N>l9g(B 1, $B$=$l0J30$O(B 0 $B$rJV$9(B. |
$BM-8BBN>e$N(B 1 $BJQ?tB?9`<0$N4{LsH=Dj$r9T$$(B, $B4{Ls$N>l9g(B 1, $B$=$l0J30$O(B 0 $B$rJV$9(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Defined in @samp{fff}. |
| |
@item |
| |
Returns 1 if @var{poly} is irreducible over the current base field. |
| |
Returns 0 otherwise. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 767 x^10+14687973587364016969 |
|
| Line 1351 x^10+14687973587364016969 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{setmod_ff} |
@fref{setmod_ff} |
| @end table |
@end table |
| |
|
| @node randpoly_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node randpoly_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node randpoly_ff,,, Functions for Finite fields |
| @subsection @code{randpoly_ff} |
@subsection @code{randpoly_ff} |
| @findex randpoly_ff |
@findex randpoly_ff |
| |
|
| @table @t |
@table @t |
| @item randpoly_ff(@var{d},@var{v}) |
@item randpoly_ff(@var{d},@var{v}) |
| :: $BM-8BBN>e$N(B $BMp?t78?t(B 1 $BJQ?tB?9`<0$N@8@.(B |
\JP :: $BM-8BBN>e$N(B $BMp?t78?t(B 1 $BJQ?tB?9`<0$N@8@.(B |
| |
\EG :: Generation of a random univariate polynomial over a finite field. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $BB?9`<0(B |
\JP $BB?9`<0(B |
| |
\EG polynomial |
| @item d |
@item d |
| $B@5@0?t(B |
\JP $B@5@0?t(B |
| |
\EG positive integer |
| @item v |
@item v |
| $BITDj85(B |
\JP $BITDj85(B |
| |
\EG indeterminate |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| @samp{fff} $B$GDj5A$5$l$F$$$k(B. |
@samp{fff} $B$GDj5A$5$l$F$$$k(B. |
| @item |
@item |
| @var{d} $B<!L$K~(B, $BJQ?t$,(B @var{v}, $B78?t$,8=:_@_Dj$5$l$F$$$kM-8BBN$KB0$9$k(B |
@var{d} $B<!L$K~(B, $BJQ?t$,(B @var{v}, $B78?t$,8=:_@_Dj$5$l$F$$$kM-8BBN$KB0$9$k(B |
| 1 $BJQ?tB?9`<0$r@8@.$9$k(B. $B78?t$O(B @code{random_ff()} $B$K$h$j@8@.$5$l$k(B. |
1 $BJQ?tB?9`<0$r@8@.$9$k(B. $B78?t$O(B @code{random_ff()} $B$K$h$j@8@.$5$l$k(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Defined in @samp{fff}. |
| |
@item |
| |
Generates a polynomial of @var{v} such that the degree is less than @var{d} |
| |
and the coefficients are in the current base field. |
| |
The coefficients are generated by @code{random_ff()}. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 810 x^10+14687973587364016969 |
|
| Line 1410 x^10+14687973587364016969 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{setmod_ff}, @fref{random_ff} |
@fref{setmod_ff}, @fref{random_ff} |
| @end table |
@end table |
| |
|
| @node ecm_add_ff ecm_sub_ff ecm_chsgn_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
\JP @node ecm_add_ff ecm_sub_ff ecm_chsgn_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| |
\EG @node ecm_add_ff ecm_sub_ff ecm_chsgn_ff,,, Functions for Finite fields |
| @subsection @code{ecm_add_ff}, @code{ecm_sub_ff}, @code{ecm_chsgn_ff} |
@subsection @code{ecm_add_ff}, @code{ecm_sub_ff}, @code{ecm_chsgn_ff} |
| @findex ecm_add_ff |
@findex ecm_add_ff |
| @findex ecm_sub_ff |
@findex ecm_sub_ff |
| Line 823 x^10+14687973587364016969 |
|
| Line 1425 x^10+14687973587364016969 |
|
| @table @t |
@table @t |
| @item ecm_add_ff(@var{p1},@var{p2},@var{ec}) |
@item ecm_add_ff(@var{p1},@var{p2},@var{ec}) |
| @itemx ecm_sub_ff(@var{p1},@var{p2},@var{ec}) |
@itemx ecm_sub_ff(@var{p1},@var{p2},@var{ec}) |
| @itemx ecm_chsgn_ff(@var{p1},@var{p2},@var{ec}) |
@itemx ecm_chsgn_ff(@var{p1}) |
| :: $BBJ1_6J@~>e$NE@$N2C;;(B, $B8:;;(B, $B5U85(B |
\JP :: $BBJ1_6J@~>e$NE@$N2C;;(B, $B8:;;(B, $B5U85(B |
| |
\EG :: Addition, Subtraction and additive inverse for points on an elliptic curve. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| @item return |
@item return |
| $B%Y%/%H%k$^$?$O(B 0 |
\JP $B%Y%/%H%k$^$?$O(B 0 |
| @item p1,p2 |
\EG vector or 0 |
| $BD9$5(B 3 $B$N%Y%/%H%k$^$?$O(B 0 |
@item p1 p2 |
| |
\JP $BD9$5(B 3 $B$N%Y%/%H%k$^$?$O(B 0 |
| |
\EG vector of length 3 or 0 |
| @item ec |
@item ec |
| $BD9$5(B 2 $B$N%Y%/%H%k(B |
\JP $BD9$5(B 2 $B$N%Y%/%H%k(B |
| |
\EG vector of length 2 |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| |
\BJP |
| @item |
@item |
| $B8=:_@_Dj$5$l$F$$$kM-8BBN>e$G(B, @var{ec} $B$GDj5A$5$l$kBJ1_6J@~>e$N(B |
$B8=:_@_Dj$5$l$F$$$kM-8BBN>e$G(B, @var{ec} $B$GDj5A$5$l$kBJ1_6J@~>e$N(B |
| $BE@(B @var{p1}, @var{p2} $B$NOB(B @var{p1+p2}, $B:9(B @var{p1-p2}, $B5U85(B @var{-p1} $B$rJV$9(B. |
$BE@(B @var{p1}, @var{p2} $B$NOB(B @var{p1+p2}, $B:9(B @var{p1-p2}, $B5U85(B @var{-p1} $B$rJV$9(B. |
| @item |
@item |
| @var{ec} $B$O(B, $B@_Dj$5$l$F$$$kM-8BBN$,4qI8?tAGBN$N>l9g(B, |
@var{ec} $B$O(B, $B@_Dj$5$l$F$$$kM-8BBN$,4qI8?tAGBN$N>l9g(B, |
| @var{y^2=x^3+ec[0]x+ec[1]}, $BI8?t(B 2 $B$N>l9g(B @var{y^2+xy=x^3+ec[0]x^2+ec[1]} |
y^2=x^3+ec[0]x+ec[1], $BI8?t(B 2 $B$N>l9g(B y^2+xy=x^3+ec[0]x^2+ec[1] |
| $B$rI=$9(B. |
$B$rI=$9(B. |
| @item |
@item |
| $B0z?t(B, $B7k2L$H$b$K(B, $BL58B1sE@$O(B 0 $B$GI=$5$l$k(B. |
$B0z?t(B, $B7k2L$H$b$K(B, $BL58B1sE@$O(B 0 $B$GI=$5$l$k(B. |
| Line 855 x^10+14687973587364016969 |
|
| Line 1462 x^10+14687973587364016969 |
|
| $B$G3d$kI,MW$,$"$k(B. |
$B$G3d$kI,MW$,$"$k(B. |
| @item |
@item |
| @var{p1}, @var{p2} $B$,BJ1_6J@~>e$NE@$+$I$&$+$N%A%'%C%/$O$7$J$$(B. |
@var{p1}, @var{p2} $B$,BJ1_6J@~>e$NE@$+$I$&$+$N%A%'%C%/$O$7$J$$(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
Let @var{p1}, @var{p2} be points on the elliptic curve represented by |
| |
@var{ec} over the current base field. |
| |
ecm_add_ff(@var{p1},@var{p2},@var{ec}), ecm_sub_ff(@var{p1},@var{p2},@var{ec}) |
| |
and ecm_chsgn_ff(@var{p1}) returns |
| |
@var{p1+p2}, @var{p1-p2} and @var{-p1} respectively. |
| |
@item |
| |
If the current base field is a prime field of odd order, then |
| |
@var{ec} represents y^2=x^3+ec[0]x+ec[1]. |
| |
If the characteristic of the current base field is 2, |
| |
then @var{ec} represents y^2+xy=x^3+ec[0]x^2+ec[1]. |
| |
@item |
| |
The point at infinity is represented by 0. |
| |
@item |
| |
If an argument denoting a point is a vector of length 3, |
| |
then it is the projective coordinate. In such a case |
| |
the third coordinate must not be 0. |
| |
@item |
| |
If the result is a vector of length 3, then the third coordinate |
| |
is not equal to 0 but not necessarily 1. To get the result by |
| |
the affine coordinate, the first and the second coordinates should |
| |
be divided by the third coordinate. |
| |
@item |
| |
The check whether the arguments are on the curve is omitted. |
| |
\E |
| @end itemize |
@end itemize |
| |
|
| @example |
@example |
| Line 878 x^10+14687973587364016969 |
|
| Line 1512 x^10+14687973587364016969 |
|
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| @item $B;2>H(B |
\JP @item $B;2>H(B |
| |
\EG @item References |
| @fref{setmod_ff} |
@fref{setmod_ff} |
| @end table |
@end table |
| |
|
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