| version 1.2, 1999/12/21 02:47:31 |
version 1.6, 2003/04/20 08:01:25 |
|
|
| @comment $OpenXM$ |
@comment $OpenXM: OpenXM/src/asir-doc/parts/ff.texi,v 1.5 2003/04/19 15:44:56 noro Exp $ |
| \BJP |
\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 |
|
|
| \BJP |
\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:: |
| |
* $B>.I8?tM-8BBN>e$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:: |
| \E |
\E |
| \BEG |
\BEG |
| * Representation of finite fields:: |
* Representation of finite fields:: |
| * Univariate polynomials on finite fields:: |
* Univariate polynomials on finite fields:: |
| |
* Polynomials on small finite fields:: |
| * Elliptic curves on finite fields:: |
* Elliptic curves on finite fields:: |
| * Functions for Finite fields:: |
* Functions for Finite fields:: |
| \E |
\E |
|
|
| |
|
| @noindent |
@noindent |
| \BJP |
\BJP |
| @b{Asir} $B$K$*$$$F$O(B, $BM-8BBN$O(B, $B@5I8?tAGBN(B GF(p), $BI8?t(B 2 $B$NM-8BBN(B GF(2^n) |
@b{Asir} $B$K$*$$$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}), |
| |
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. |
| \E |
\E |
| \BEG |
\BEG |
| On @b{Asir} @var{GF(p)} and @var{GF(2^n)} can be defined, where |
On @b{Asir} GF(@var{p}), GF(2^@var{n}), GF(@var{p^n}) can be defined, where |
| @var{GF(p)} is a finite prime field of charateristic @var{p} and |
GF(@var{p}) is a finite prime field of charateristic @var{p}, |
| @var{GF(2^n)} is a finite field of characteristic 2. These are |
GF(2^@var{n}) is a finite field of characteristic 2 and |
| |
GF(@var{p^n}) is a finite extension of GF(@var{p}). These are |
| all defined by @code{setmod_ff()}. |
all defined by @code{setmod_ff()}. |
| \E |
\E |
| |
|
| Line 59 x^50+x^4+x^3+x^2+1 |
|
| Line 63 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 |
| [6] field_type_ff(); |
[6] field_type_ff(); |
| 2 |
2 |
| |
[7] setmod_ff(x^3+x+1,1125899906842679); |
| |
[1*x^3+1*x+1,1125899906842679] |
| |
[8] field_type_ff(); |
| |
3 |
| |
[9] setmod_ff(3,5); |
| |
[3,x^5+2*x+1,x] |
| |
[10] field_type_ff(); |
| |
4 |
| @end example |
@end example |
| \BJP |
\BJP |
| @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 |
@code{setmod_ff()} $B$O(B, $B$5$^$6$^$J%?%$%W$NM-8BBN$r4pACBN$H$7$F%;%C%H$9$k(B. |
| $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 |
$B0z?t$,@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 |
| $B$k(B. @code{setmod_ff()} $B$K$*$$$F$O0z?t$N4{Ls%A%'%C%/$O9T$o$:(B, $B8F$S=P$7B&(B |
$B9g(B, f(x) mod 2 $B$rDj5AB?9`<0$H$9$k(B GF(2^@var{n}) $B$r$=$l$>$l4pACBN$H$7$F%;%C%H$9(B |
| |
$B$k(B. $B$^$?(B, $BM-8BAGBN$NM-8B<!3HBg$bDj5A$G$-$k(B. $B>\$7$/$O(B @xref{$B?t$N7?(B}. |
| |
@code{setmod_ff()} $B$K$*$$$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 73 x^50+x^4+x^3+x^2+1 |
|
| Line 87 x^50+x^4+x^3+x^2+1 |
|
| $B$k(B. |
$B$k(B. |
| |
|
| 0 $B$G$J$$M-8BBN$N85$O(B, $B?t%*%V%8%'%/%H$G$"$j(B, $B<1JL;R$NCM$O(B 1 $B$G$"$k(B. |
0 $B$G$J$$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. |
| |
|
| $BM-8BBN$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. |
| \E |
\E |
| |
|
| \BEG |
\BEG |
| If @var{p} is a positive integer, @code{setmod_ff(@var{p})} sets |
If @var{p} is a positive integer, @code{setmod_ff(@var{p})} sets |
| @var{GF(p)} as the current base field. |
GF(@var{p}) as the current base field. |
| If @var{f} is a univariate polynomial of degree @var{n}, |
If @var{f} is a univariate polynomial of degree @var{n}, |
| @code{setmod_ff(@var{f})} sets @var{GF(2^n)} as the current |
@code{setmod_ff(@var{f})} sets GF(2^@var{n}) as the current |
| base field. @var{GF(2^n)} is represented |
base field. GF(2^@var{n}) is represented |
| as an algebraic extension of @var{GF(2)} with the defining polynomial |
as an algebraic extension of GF(2) with the defining polynomial |
| @var{f mod 2}. In both cases the primality check of the argument is |
@var{f mod 2}. Furthermore, finite extensions of prime finite fields |
| |
can be defined. @xref{Types of numbers}. |
| |
In all cases the primality check of the argument is |
| not done and the caller is responsible for it. |
not done and the caller is responsible for it. |
| |
|
| Correctly speaking there is no actual object corresponding to a 'base field'. |
Correctly speaking there is no actual object corresponding to a 'base field'. |
| Line 99 a finite field, then the other is automatically regard |
|
| Line 115 a finite field, then the other is automatically regard |
|
| same finite field and the operation is done in the finite field. |
same finite field and the operation is done in the finite field. |
| |
|
| A non zero element of a finite field belongs to the number and has object |
A non zero element of a finite field belongs to the number and has object |
| identifier 1. Its number identifier is 6 if the finite field is @var{GF(p)}, |
identifier 1. Its number identifier is 6 if the finite field is GF(@var{p}), |
| 7 if it is @var{GF(2^n)}. |
7 if it is GF(2^@var{n}). |
| |
|
| There are several methods to input an element of a finite field. |
There are several methods to input an element of a finite field. |
| An element of @var{GF(p)} can be input by @code{simp_ff()}. |
An element of GF(@var{p}) can be input by @code{simp_ff()}. |
| \E |
\E |
| |
|
| @example |
@example |
| Line 117 An element of @var{GF(p)} can be input by @code{simp_f |
|
| Line 133 An element of @var{GF(p)} can be input by @code{simp_f |
|
| 6 |
6 |
| @end example |
@end example |
| |
|
| \JP $B$^$?(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. |
| \EG In the case of @var{GF(2^n)} the following methods are available. |
\EG In the case of GF(2^@var{n}) the following methods are available. |
| |
|
| @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); |
| Line 135 x^50+x^4+x^3+x^2+1 |
|
| Line 151 x^50+x^4+x^3+x^2+1 |
|
| |
|
| \BJP |
\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}. |
| \E |
\E |
| \BEG |
\BEG |
| Elements of finite fields are numbers and one can apply field arithmetics |
Elements of finite fields are numbers and one can apply field arithmetics |
| to them. @code{@@} is a generator of @var{GF(2^n)} over @var{GF(2)}. |
to them. @code{@@} is a generator of GF(2^@var{n}) over GF(2). |
| @xref{Types of numbers}. |
@xref{Types of numbers}. |
| \E |
\E |
| |
|
| Line 190 The irreducible factorization proceeds as follows. |
|
| Line 206 The irreducible factorization proceeds as follows. |
|
| @end enumerate |
@end enumerate |
| \E |
\E |
| |
|
| |
@noindent |
| |
|
| \BJP |
\BJP |
| |
@node $B>.I8?tM-8BBN>e$G$NB?9`<0$N1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B |
| |
@section $B>.I8?tM-8BBN>e$G$NB?9`<0$N1i;;(B |
| |
\E |
| |
\BEG |
| |
@node Polynomials on small finite fields,,, Finite fields |
| |
@section Polynomials on small finite fields |
| |
\E |
| |
|
| |
\BJP |
| |
$B>.I8?tM-8BBN78?t$NB?9`<0$K8B$j(B, $BB?JQ?tB?9`<0$N0x?tJ,2r$,(B |
| |
$BAH$_9~$_4X?t$H$7$F<BAu$5$l$F$$$k(B. $B4X?t$O(B @code{sffctr()} |
| |
$B$G$"$k(B. $B$^$?(B, @code{modfctr()} $B$b(B, $BM-8BAGBN>e$GB?JQ?t(B |
| |
$BB?9`<0$N0x?tJ,2r$r9T$&$,(B, $B<B:]$K$O(B, $BFbIt$G==J,Bg$-$J(B |
| |
$B3HBgBN$r@_Dj$7(B, @code{sffctr()} $B$r8F$S=P$7$F(B, |
| |
$B:G=*E*$KAGBN>e$N0x;R$r9=@.$9$k(B, $B$H$$$&J}K!$G7W;;$7$F$$$k(B. |
| |
\E |
| |
|
| |
\BEG |
| |
A multivariate polynomial over small finite field |
| |
set by @code{setmod_ff(p,n)} can be factorized by |
| |
using a builtin function @code{sffctr()}. @code{modfctr()} |
| |
also factorizes a polynomial over a finite prime field. |
| |
Internally, @code{modfctr()} creates a sufficiently large |
| |
field extension of the specified ground field, and |
| |
it calls @code{sffctr()}, then it constructs irreducible |
| |
factors over the ground field from the factors returned by |
| |
@code{sffctr()}. |
| |
\E |
| |
|
| |
\BJP |
| @node $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B |
@node $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B |
| @section $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B |
@section $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B |
| \E |
\E |
| Line 203 The irreducible factorization proceeds as follows. |
|
| Line 251 The irreducible factorization proceeds as follows. |
|
| $BM-8BBN>e$NBJ1_6J@~$K4X$9$k$$$/$D$+$N4pK\E*$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$$$kM-8BBN$,AGBN$N>l9g(B, @var{y^2=x^3+ax+b}, |
| $BI8?t(B 2 $B$NBN$N>l9g(B @var{y^2+xy=x^3+ax^2+b} $B$rI=$9(B. |
$BI8?t(B 2 $B$NBN$N>l9g(B @var{y^2+xy=x^3+ax^2+b} $B$rI=$9(B. |
| Line 215 The irreducible factorization proceeds as follows. |
|
| Line 263 The irreducible factorization proceeds as follows. |
|
| |
|
| @itemize @bullet |
@itemize @bullet |
| @item $BL58B1sE@$O(B 0. |
@item $BL58B1sE@$O(B 0. |
| @item $B$=$l0J30$NE@$O(B, $BD9$5(B 3 $B$N%Y%/%H%k(B @var{[x y z]}. $B$?$@$7(B, @var{z} $B$O(B |
@item $B$=$l0J30$NE@$O(B, $BD9$5(B 3 $B$N%Y%/%H%k(B [@var{x y z}]. $B$?$@$7(B, @var{z} $B$O(B |
| 0 $B$G$J$$(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 |
| Line 232 The irreducible factorization proceeds as follows. |
|
| Line 280 The irreducible factorization proceeds as follows. |
|
| Several fundamental operations on elliptic curves over finite fields |
Several fundamental operations on elliptic curves over finite fields |
| are provided as built-in functions. |
are provided as built-in functions. |
| |
|
| An elliptic curve is specified by a vector @var{[a b]} of length 2, |
An elliptic curve is specified by a vector [@var{a b}] of length 2, |
| where @var{a}, @var{b} are elements of finite fields. |
where @var{a}, @var{b} are elements of finite fields. |
| If the current base field is a prime field, then @var{[a b]} represents |
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 |
@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}. |
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 |
Points on an elliptic curve together with the point at infinity |
| forms an additive group. The addition, the subtraction and the |
forms an additive group. The addition, the subtraction and the |
| Line 246 Here the representation of points are as follows. |
|
| Line 294 Here the representation of points are as follows. |
|
| |
|
| @itemize @bullet |
@itemize @bullet |
| @item 0 denotes the point at infinity. |
@item 0 denotes the point at infinity. |
| @item The other points are represented by vectors @var{[x y z]} of |
@item The other points are represented by vectors [@var{x y z}] of |
| length 3 with non-zero @var{z}. |
length 3 with non-zero @var{z}. |
| @end itemize |
@end itemize |
| |
|
| @var{[x y z]} represents a projective coordinate and |
[@var{x y z}] represents a projective coordinate and |
| it corresponds to @var{[x/z y/z]} in the affine coordinate. |
it corresponds to [@var{x/z y/z}] in the affine coordinate. |
| To apply the above operations to a point @var{[x y]}, |
To apply the above operations to a point [@var{x y}], |
| @var{[x y 1]} should be used instead as an argument. |
[@var{x y 1}] should be used instead as an argument. |
| The result of an operation is also represented by the projective |
The result of an operation is also represented by the projective |
| coordinate. As the third coordinate is not always equal to 1, |
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 has to divide the first and the scond coordinate by the third |
| Line 282 one to obtain the affine coordinate. |
|
| Line 330 one to obtain the affine coordinate. |
|
| * 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 297 one to obtain the affine coordinate. |
|
| Line 347 one to obtain the affine coordinate. |
|
| |
|
| @table @t |
@table @t |
| @item setmod_ff([@var{prime}|@var{poly}]) |
@item setmod_ff([@var{prime}|@var{poly}]) |
| |
@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 |
\JP :: $BM-8BBN$N@_Dj(B, $B@_Dj$5$l$F$$$kM-8BBN$NK!(B, $BDj5AB?9`<0$NI=<((B |
| \EG :: Sets/Gets the current base fields. |
\EG :: Sets/Gets the current base fields. |
| @end table |
@end table |
| Line 311 one to obtain the affine coordinate. |
|
| Line 362 one to obtain the affine coordinate. |
|
| @item poly |
@item poly |
| \JP GF(2) $B>e4{Ls$J(B 1 $BJQ?tB?9`<0(B |
\JP GF(2) $B>e4{Ls$J(B 1 $BJQ?tB?9`<0(B |
| \EG univariate polynomial irreducible over GF(2) |
\EG univariate polynomial irreducible over GF(2) |
| |
@item n |
| |
\JP $B3HBg<!?t(B |
| |
\EG the extension degree |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| Line 322 one to obtain the affine coordinate. |
|
| Line 376 one to obtain the affine coordinate. |
|
| GF(2^deg(@var{poly} mod 2)) = GF(2)[t]/(@var{poly}(t) mod 2) |
GF(2^deg(@var{poly} mod 2)) = GF(2)[t]/(@var{poly}(t) mod 2) |
| $B$r4pACBN$H$7$F@_Dj$9$k(B. |
$B$r4pACBN$H$7$F@_Dj$9$k(B. |
| @item |
@item |
| $BL50z?t$N;~(B, $B@_Dj$5$l$F$$$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 |
\E |
| \BEG |
\BEG |
| Line 337 If the argument is a polynomial @var{poly}, |
|
| Line 401 If the argument is a polynomial @var{poly}, |
|
| 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) mod2) |
| is set as the current base field. |
is set as the current base field. |
| @item |
@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 |
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 |
is returned. If the current base field is GF(@var{prime}), @var{prime} is |
| returned. If it is GF(2^n), the defining polynomial is returned. |
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 |
@item |
| Any irreducible univariate polynomial over GF(2) is available to |
Any irreducible univariate polynomial over GF(2) is available to |
| set GF(2^n). However the use of @code{defpoly_mod2()} is recommended |
set GF(2^@var{n}). However the use of @code{defpoly_mod2()} is recommended |
| for efficiency. |
for efficiency. |
| \E |
\E |
| @end itemize |
@end itemize |
|
|
| 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 |
@item |
| $B@_Dj$5$l$F$$$k4pACBN$N<oN`$rJV$9(B. |
$B@_Dj$5$l$F$$$k4pACBN$N<oN`$rJV$9(B. |
| @item |
@item |
| $B@_Dj$J$7$J$i(B 0, GF(p) $B$J$i(B 1, GF(2^n) $B$J$i(B 2 $B$rJV$9(B. |
$B@_Dj$J$7$J$i(B 0, GF(@var{p}) $B$J$i(B 1, GF(2^@var{n}) $B$J$i(B 2 $B$rJV$9(B. |
| \E |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Returns the type of the current base field. |
Returns the type of the current base field. |
| @item |
@item |
| If no field is set, 0 is returned. If GF(p) is set, 1 is returned. |
If no field is set, 0 is returned. If GF(@var{p}) is set, 1 is returned. |
| If GF(2^n) is set, 2 is returned. |
If GF(2^@var{n}) is set, 2 is returned. |
| \E |
\E |
| @end itemize |
@end itemize |
| |
|
|
|
| @item |
@item |
| $B@_Dj$5$l$F$$$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$$$kBN$,(B GF(@var{q}) $B$J$i$P(B q $B$rJV$9(B. |
| \E |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Returns the order of the current base field. |
Returns the order of the current base field. |
| @item |
@item |
| @var{q} is returned if the current base field is GF(q). |
@var{q} is returned if the current base field is GF(@var{q}). |
| \E |
\E |
| @end itemize |
@end itemize |
| |
|
|
|
| @item |
@item |
| $B@_Dj$5$l$F$$$kBN$NI8?t$rJV$9(B. |
$B@_Dj$5$l$F$$$kBN$NI8?t$rJV$9(B. |
| @item |
@item |
| GF(p) $B$N>l9g(B p, GF(2^n) $B$N>l9g(B 2 $B$rJV$9(B. |
GF(@var{p}) $B$N>l9g(B @var{p}, GF(2^@var{n}) $B$N>l9g(B 2 $B$rJV$9(B. |
| \E |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Returns the characteristic of the current base field. |
Returns the characteristic of the current base field. |
| @item |
@item |
| @var{p} is returned if @var{GF(p)}, where @var{p} is a prime, is set. |
@var{p} is returned if GF(@var{p}), where @var{p} is a prime, is set. |
| @var{2} is returned if @var{GF(2^n)} is set. |
@var{2} is returned if GF(2^@var{n}) is set. |
| \E |
\E |
| @end itemize |
@end itemize |
| |
|
|
|
| @item |
@item |
| $B@_Dj$5$l$F$$$k4pACBN$N(B, $BAGBN$KBP$9$k3HBg<!?t$rJV$9(B. |
$B@_Dj$5$l$F$$$k4pACBN$N(B, $BAGBN$KBP$9$k3HBg<!?t$rJV$9(B. |
| @item |
@item |
| GF(p) $B$N>l9g(B 1, GF(2^n) $B$N>l9g(B n $B$rJV$9(B. |
GF(@var{p}) $B$N>l9g(B 1, GF(2^@var{n}) $B$N>l9g(B @var{n} $B$rJV$9(B. |
| \E |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Returns the extension degree of the current base field over the prime field. |
Returns the extension degree of the current base field over the prime field. |
| @item |
@item |
| GF(p) $B$N>l9g(B 1, GF(2^n) $B$N>l9g(B n $B$rJV$9(B. |
1 is returned if GF(@var{p}), where @var{p} is a prime, is set. |
| 1 is returned if @var{GF(p)}, where @var{p} is a prime, is set. |
@var{n} is returned if GF(2^@var{n}) is set. |
| @var{n} is returned if @var{GF(2^n)} is set. |
|
| \E |
\E |
| @end itemize |
@end itemize |
| |
|
| Line 606 in finite fields. |
|
| Line 681 in finite fields. |
|
| @item |
@item |
| $BM-8BBN$N85$KBP$7(B, $BK!$"$k$$$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 |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Line 616 It is used to convert integers or intrgral polynomials |
|
| Line 695 It is used to convert integers or intrgral polynomials |
|
| elements of finite fields or polynomials over finite fields. |
elements of finite fields or polynomials over finite fields. |
| @item |
@item |
| An element of a finite field may not have the reduced representation. |
An element of a finite field may not have the reduced representation. |
| In such case an application of @code{simp_ff} assures the output has |
In such case an application of @code{simp_ff} ensures that the output has |
| the reduced representation. |
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 |
\E |
| @end itemize |
@end itemize |
| |
|
| Line 630 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 |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item References |
\EG @item References |
| @fref{setmod_ff}, @fref{lmptop}, @fref{gf2nton} |
@fref{setmod_ff}, @fref{lmptop}, @fref{gf2nton}, @fref{ptosfp sfptop} |
| @end table |
@end table |
| |
|
| \JP @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 |
| Line 696 return to toplevel |
|
| Line 787 return to toplevel |
|
| |
|
| @table @t |
@table @t |
| @item lmptop(@var{obj}) |
@item lmptop(@var{obj}) |
| \JP :: 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(p) into integers. |
\EG :: Converts the coefficients of a polynomial over GF(@var{p}) into integers. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| Line 705 return to toplevel |
|
| Line 796 return to toplevel |
|
| \JP $B@0?t78?tB?9`<0(B |
\JP $B@0?t78?tB?9`<0(B |
| \EG integral polynomial |
\EG integral polynomial |
| @item obj |
@item obj |
| \JP GF(p) $B78?tB?9`<0(B |
\JP GF(@var{p}) $B78?tB?9`<0(B |
| \EG polynomial over GF(p) |
\EG polynomial over GF(@var{p}) |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
| \BJP |
\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 |
\E |
| \BEG |
\BEG |
| @item |
@item |
| Converts the coefficients of a polynomial over GF(p) into integers. |
Converts the coefficients of a polynomial over GF(@var{p}) into integers. |
| @item |
@item |
| An element of GF(p) is represented by a non-negative integer @var{r} less than |
An element of GF(@var{p}) is represented by a non-negative integer @var{r} less than |
| @var{p}. |
@var{p}. |
| Each coefficient of a polynomial is converted into an integer object |
Each coefficient of a polynomial is converted into an integer object |
| whose value is @var{r}. |
whose value is @var{r}. |
| Line 738 whose value is @var{r}. |
|
| Line 829 whose value is @var{r}. |
|
| [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 760 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42 |
|
| Line 853 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42 |
|
| |
|
| @table @t |
@table @t |
| @item ntogf2n(@var{m}) |
@item ntogf2n(@var{m}) |
| \JP :: $B<+A3?t$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^n). |
\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 |
| \JP GF(2^n) $B$N85(B |
\JP GF(2^@var{n}) $B$N85(B |
| \EG element of GF(2^n) |
\EG element of GF(2^@var{n}) |
| @item m |
@item m |
| \JP $BHsIi@0?t(B |
\JP $BHsIi@0?t(B |
| \EG non-negative integer |
\EG non-negative integer |
| Line 777 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42 |
|
| Line 870 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42 |
|
| \BJP |
\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 |
| Line 788 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42 |
|
| Line 881 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42 |
|
| Let @var{m} be a non-negative integer. |
Let @var{m} be a non-negative integer. |
| @var{m} has the binary representation |
@var{m} has the binary representation |
| @var{m}=@var{m0}+@var{m1}*2+...+@var{mk}*2^k. |
@var{m}=@var{m0}+@var{m1}*2+...+@var{mk}*2^k. |
| This function returns an element of GF(2^n)=GF(2)[t]/(g(t)), |
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). |
@var{m0}+@var{m1}*t+...+@var{mk}*t^k mod g(t). |
| @item |
@item |
| Apply @code{simp_ff()} to reduce the result. |
Apply @code{simp_ff()} to reduce the result. |
|
|
| |
|
| @table @t |
@table @t |
| @item gf2nton(@var{m}) |
@item gf2nton(@var{m}) |
| \JP :: 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^n) into a non-negative integer. |
\EG :: Converts an element of GF(2^@var{n}) into a non-negative integer. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
|
|
| \JP $BHsIi@0?t(B |
\JP $BHsIi@0?t(B |
| \EG non-negative integer |
\EG non-negative integer |
| @item m |
@item m |
| \JP GF(2^n) $B$N85(B |
\JP GF(2^@var{n}) $B$N85(B |
| \EG element of GF(2^n) |
\EG element of GF(2^@var{n}) |
| @end table |
@end table |
| |
|
| @itemize @bullet |
@itemize @bullet |
|
|
| |
|
| @table @t |
@table @t |
| @item ptogf2n(@var{poly}) |
@item ptogf2n(@var{poly}) |
| \JP :: $B0lJQ?tB?9`<0$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^n). |
\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 |
| \JP GF(2^n) $B$N85(B |
\JP GF(2^@var{n}) $B$N85(B |
| \EG element of GF(2^n) |
\EG element of GF(2^@var{n}) |
| @item poly |
@item poly |
| \JP $B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
| \EG univariate polynomial |
\EG univariate polynomial |
|
|
| @itemize @bullet |
@itemize @bullet |
| @item |
@item |
| \BJP |
\BJP |
| @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 |
@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 |
\E |
| \BEG |
\BEG |
| Generates an element of GF(2^n) represented by @var{poly}. |
Generates an element of GF(2^@var{n}) represented by @var{poly}. |
| The coefficients are reduced modulo 2. |
The coefficients are reduced modulo 2. |
| The output is equal to the result by substituting @code{@@} for |
The output is equal to the result by substituting @code{@@} for |
| the variable of @var{poly}. |
the variable of @var{poly}. |
|
|
| |
|
| @table @t |
@table @t |
| @item gf2ntop(@var{m}[,@var{v}]) |
@item gf2ntop(@var{m}[,@var{v}]) |
| \JP :: 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^n) into a polynomial. |
\EG :: Converts an element of GF(2^@var{n}) into a polynomial. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
|
|
| \JP $B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
| \EG univariate polynomial |
\EG univariate polynomial |
| @item m |
@item m |
| \JP GF(2^n) $B$N85(B |
\JP GF(2^@var{n}) $B$N85(B |
| \EG an element of GF(2^n) |
\EG an element of GF(2^@var{n}) |
| @item v |
@item v |
| \JP $BITDj85(B |
\JP $BITDj85(B |
| \EG indeterminate |
\EG indeterminate |
| Line 963 t^13+t^12+t^11+t^10 |
|
| Line 1056 t^13+t^12+t^11+t^10 |
|
| @fref{ptogf2n} |
@fref{ptogf2n} |
| @end table |
@end table |
| |
|
| |
\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 |
\JP @node defpoly_mod2,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| \EG @node defpoly_mod2,,, Functions for Finite fields |
\EG @node defpoly_mod2,,, Functions for Finite fields |
| @subsection @code{defpoly_mod2} |
@subsection @code{defpoly_mod2} |
| Line 1020 Then fix @var{m3} as small as possible. |
|
| Line 1191 Then fix @var{m3} as small as possible. |
|
| @fref{setmod_ff} |
@fref{setmod_ff} |
| @end table |
@end table |
| |
|
| |
\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 |
| |
[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]] |
| |
@example |
| |
|
| |
@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 |
\JP @node fctr_ff,,, $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B |
| \EG @node fctr_ff,,, Functions for Finite fields |
\EG @node fctr_ff,,, Functions for Finite fields |
| @subsection @code{fctr_ff} |
@subsection @code{fctr_ff} |
| Line 1200 The coefficients are generated by @code{random_ff()}. |
|
| Line 1425 The coefficients are generated by @code{random_ff()}. |
|
| @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}) |
| \JP :: $BBJ1_6J@~>e$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. |
\EG :: Addition, Subtraction and additive inverse for points on an elliptic curve. |
| @end table |
@end table |
| Line 1209 The coefficients are generated by @code{random_ff()}. |
|
| Line 1434 The coefficients are generated by @code{random_ff()}. |
|
| @item return |
@item return |
| \JP $B%Y%/%H%k$^$?$O(B 0 |
\JP $B%Y%/%H%k$^$?$O(B 0 |
| \EG vector or 0 |
\EG vector or 0 |
| @item p1,p2 |
@item p1 p2 |
| \JP $BD9$5(B 3 $B$N%Y%/%H%k$^$?$O(B 0 |
\JP $BD9$5(B 3 $B$N%Y%/%H%k$^$?$O(B 0 |
| \EG vector of length 3 or 0 |
\EG vector of length 3 or 0 |
| @item ec |
@item ec |
| Line 1224 The coefficients are generated by @code{random_ff()}. |
|
| Line 1449 The coefficients are generated by @code{random_ff()}. |
|
| $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 1243 The coefficients are generated by @code{random_ff()}. |
|
| Line 1468 The coefficients are generated by @code{random_ff()}. |
|
| Let @var{p1}, @var{p2} be points on the elliptic curve represented by |
Let @var{p1}, @var{p2} be points on the elliptic curve represented by |
| @var{ec} over the current base field. |
@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}) |
ecm_add_ff(@var{p1},@var{p2},@var{ec}), ecm_sub_ff(@var{p1},@var{p2},@var{ec}) |
| and ecm_chsgn_ff(@var{p1},@var{p2},@var{ec}) returns |
and ecm_chsgn_ff(@var{p1}) returns |
| @var{p1+p2}, @var{p1-p2} and @var{-p1} respectively. |
@var{p1+p2}, @var{p1-p2} and @var{-p1} respectively. |
| @item |
@item |
| If the current base field is a prime field of odd order, then |
If the current base field is a prime field of odd order, then |
| @var{ec} represents @var{y^2=x^3+ec[0]x+ec[1]}. |
@var{ec} represents y^2=x^3+ec[0]x+ec[1]. |
| If the characteristic of the current base field is 2, |
If the characteristic of the current base field is 2, |
| then @var{ec} represents @var{y^2+xy=x^3+ec[0]x^2+ec[1]}. |
then @var{ec} represents y^2+xy=x^3+ec[0]x^2+ec[1]. |
| @item |
@item |
| The point at infinity is represented by 0. |
The point at infinity is represented by 0. |
| @item |
@item |
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