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version 1.1, 1999/12/08 05:47:44

version 1.4, 2003/04/19 10:36:30

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/ff.texi,v 1.3 2000/01/13 08:29:56 noro Exp $

\BJP

@node $BM-8BBN$K4X$9$k1i;;(B,,, Top

@chapter $BM-8BBN$K4X$9$k1i;;(B

\BEG

@node Finite fields,,, Top

@chapter Finite fields

@menu

\BJP

* $BM-8BBN$NI=8=$*$h$S1i;;(B::

* $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B::

* $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B::

* $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B::

\BEG

* Representation of finite fields::

* Univariate polynomials on finite fields::

* Elliptic curves on finite fields::

* Functions for Finite fields::

@end menu

\BJP

@node $BM-8BBN$NI=8=$*$h$S1i;;(B,,, $BM-8BBN$K4X$9$k1i;;(B

@section $BM-8BBN$NI=8=$*$h$S1i;;(B

\BEG

@node Representation of finite fields,,, Finite fields

@section Representation of finite fields

@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(p), $BI8?t(B 2 $B$NM-8BBN(B GF(2^n),

GF(p) $B$N(B n $B<!3HBg(B GF(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.

\BEG

On @b{Asir} @var{GF(p)}, @var{GF(2^n)}, @var{GF(p^n} can be defined, where

@var{GF(p)} is a finite prime field of charateristic @var{p},

@var{GF(2^n)} is a finite field of characteristic 2 and

@var{GF(p^n} is a finite extension of @var{GF(p)}. These are

all defined by @code{setmod_ff()}.

@example

[0] P=pari(nextprime,2^50);

Line 30 x^50+x^4+x^3+x^2+1

Line 61 x^50+x^4+x^3+x^2+1

x^50+x^4+x^3+x^2+1

[6] field_type_ff();

[7] setmod_ff(x^3+x+1,1125899906842679);

[1*x^3+1*x+1,1125899906842679]

[8] field_type_ff();

[9] setmod_ff(3,5);

[3,x^5+2*x+1,x]

[10] field_type_ff();

@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

@code{setmod_ff()} $B$O(B, $B$5$^$6$^$J%?%$%W$NM-8BBN$r4pACBN$H$7$F%;%C%H$9$k(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

$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

$B$k(B. @code{setmod_ff()} $B$K$*$$$F$O0z?t$N4{Ls%A%'%C%/$O9T$o$:(B, $B8F$S=P$7B&(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.

$B4pACBN$H$O(B, $B$"$/$^$GM-8BBN$N85$H$7$F@k8@$"$k$$$ODj5A$5$l$?%*%V%8%'%/%H$,(B,

Line 48 x^50+x^4+x^3+x^2+1

Line 90 x^50+x^4+x^3+x^2+1

$BM-8BBN$N85$NF~NOJ}K!$O(B, $BM-8BBN$N<oN`$K$h$jMM!9$G$"$k(B. GF(p) $B$N>l9g(B,

@code{simp_ff()} $B$K$h$k(B.

\BEG

If @var{p} is a positive integer, @code{setmod_ff(@var{p})} sets

@var{GF(p)} as the current base field.

If @var{f} is a univariate polynomial of degree @var{n},

@code{setmod_ff(@var{f})} sets @var{GF(2^n)} as the current

base field. @var{GF(2^n)} is represented

as an algebraic extension of @var{GF(2)} with the defining polynomial

@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.

Correctly speaking there is no actual object corresponding to a 'base field'.

Setting a base field means that operations on elements of finite fields

are done according to the arithmetics of the base field. Thus, if

operands of an arithmetic operation are both rational numbers, then the result

is also a rational number. However, if one of the operands is in

a finite field, then the other is automatically regarded as in the

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

identifier 1. Its number identifier is 6 if the finite field is @var{GF(p)},

7 if it is @var{GF(2^n)}.

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()}.

@example

[0] P=pari(nextprime,2^50);

1125899906842679

Line 60 x^50+x^4+x^3+x^2+1

Line 131 x^50+x^4+x^3+x^2+1

@end example

$B$^$?(B, GF(2^n) $B$N>l9g$$$/$D$+$NJ}K!$,$"$k(B.

\JP $B$^$?(B, GF(2^n) $B$N>l9g$$$/$D$+$NJ}K!$,$"$k(B.

\EG In the case of @var{GF(2^n)} the following methods are available.

@example

[0] setmod_ff(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 147 x^50+x^4+x^3+x^2+1

(@@^9+@@^8+@@^7+@@^6+@@^5+@@^4+@@^3+@@^2+@@+1)

@end example

\BJP

$BM-8BBN$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}.

\BEG

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)}.

@xref{Types of numbers}.

@noindent

\BJP

@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

\BEG

@node Univariate polynomials on finite fields,,, Finite fields

@section Univariate polynomials on finite fields

@noindent

\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,

$BB?9`<0$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

$B$H$J$j(B, $BF~NOB?9`<0$N<g78?t$O<N$F$i$l$k(B.

@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

$B%"%k%4%j%:%`$r:NMQ$7$F$$$k(B.

@example

@end example

@noindent

$BM-8BBN>e$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

$B$r(B Cantor-Zassenhaus $B%"%k%4%j%:%`$K$h$j5a$a$k(B, $B$H$$$&J}K!$r<BAu$7$F$$$k(B.

\BEG

In @samp{fff} square-free factorization, DDF (distinct degree factorization),

irreducible factorization and primality check are implemented for

univariate polynomials over finite fields.

@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

input polynomial is abandoned.

The algorithm used in square-free factorization is the most primitive one.

The irreducible factorization proceeds as follows.

@enumerate

@item DDF

@item Nullspace computation by Berlekamp algorithm

@item Root finding of minimal polynomials of bases of the nullspace

@item Separation of irreducible factors by the roots

@end enumerate

\BJP

@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

\BEG

@node Elliptic curves on finite fields,,, Finite fields

@section Elliptic curves on finite fields

\BJP

$BM-8BBN>e$NBJ1_6J@~$K4X$9$k$$$/$D$+$N4pK\E*$J1i;;$,(B, $BAH$_9~$_4X?t$H$7$F(B

$BDs6!$5$l$F$$$k(B.

Line 133 GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7

Line 240 GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7

$B1i;;7k2L$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

$B3d$kI,MW$,$"$k(B.

\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.

\BJP

@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

\BEG

@node Functions for Finite fields,,, Finite fields

@section Functions for Finite fields

@menu

* setmod_ff::

Line 158 GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7

Line 304 GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7

* extdeg_ff::

@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}

@findex setmod_ff

@table @t

@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

\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

@table @var

@item return

$B?t$^$?$OB?9`<0(B

\JP $B?t$^$?$OB?9`<0(B

\EG number or polynomial

@item prime

$BAG?t(B

\JP $BAG?t(B

\EG prime

@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)

@end table

@itemize @bullet

\BJP

@item

$B0z?t$,@5@0?t(B @var{prime} $B$N;~(B, GF(@var{prime}) $B$r4pACBN$H$7$F@_Dj$9$k(B.

@item

$B0z?t$,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.

@item

$BL50z?t$N;~(B, $B@_Dj$5$l$F$$$k4pACBN$,(B GF(@var{prime}) $B$N>l9g(B @var{prime},

Line 189 GF(2^n) $B$N>l9gDj5AB?9`<0$rJV$9(B.

Line 341 GF(2^n) $B$N>l9gDj5AB?9`<0$rJV$9(B.

@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

$B1F6A$9$k$?$a(B, @code{defpoly_mod2()} $B$G@8@.$9$k$N$,$h$$(B.

\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 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^n), the defining polynomial is returned.

@item

Any irreducible univariate polynomial over GF(2) is available to

set GF(2^n). However the use of @code{defpoly_mod2()} is recommended

for efficiency.

@end itemize

@example

Line 201 x^100+x^15+1

Line 371 x^100+x^15+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{defpoly_mod2}

@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}

@findex field_type_ff

@table @t

@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

@table @var

@item return

$B?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

$B@_Dj$5$l$F$$$k4pACBN$N<oN`$rJV$9(B.

@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.

\BEG

@item

Returns the type of the current base field.

@item

If no field is set, 0 is returned. If GF(p) is set, 1 is returned.

If GF(2^n) is set, 2 is returned.

@end itemize

@example

Line 240 x^2+x+1

Line 423 x^2+x+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}

@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}

@findex field_order_ff

@table @t

@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

@table @var

@item return

$B?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

$B@_Dj$5$l$F$$$k4pACBN$N0L?t(B ($B85$N8D?t(B) $B$rJV$9(B.

@item

$B@_Dj$5$l$F$$$kBN$,(B GF(q) $B$J$i$P(B q $B$rJV$9(B.

\BEG

@item

Returns the order of the current base field.

@item

@var{q} is returned if the current base field is GF(q).

@end itemize

@example

Line 280 x^2+x+1

Line 475 x^2+x+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}

@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}

@findex characteristic_ff

@table @t

@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

@table @var

@item return

$B?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

$B@_Dj$5$l$F$$$kBN$NI8?t$rJV$9(B.

@item

GF(p) $B$N>l9g(B p, GF(2^n) $B$N>l9g(B 2 $B$rJV$9(B.

\BEG

@item

Returns the characteristic of the current base field.

@item

@var{p} is returned if @var{GF(p)}, where @var{p} is a prime, is set.

@var{2} is returned if @var{GF(2^n)} is set.

@end itemize

@example

Line 320 x^2+x+1

Line 528 x^2+x+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}

@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}

@findex extdeg_ff

@table @t

@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

@table @var

@item return

$B?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

\BJP

@item

$B@_Dj$5$l$F$$$k4pACBN$N(B, $BAGBN$KBP$9$k3HBg<!?t$rJV$9(B.

@item

GF(p) $B$N>l9g(B 1, GF(2^n) $B$N>l9g(B n $B$rJV$9(B.

\BEG

@item

Returns the extension degree of the current base field over the prime field.

@item

GF(p) $B$N>l9g(B 1, GF(2^n) $B$N>l9g(B n $B$rJV$9(B.

1 is returned if @var{GF(p)}, where @var{p} is a prime, is set.

@var{n} is returned if @var{GF(2^n)} is set.

@end itemize

@example

Line 360 x^2+x+1

Line 582 x^2+x+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}

@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}

@findex simp_ff

@table @t

@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.

@end table

@table @var

@item return

$B?t$^$?$OB?9`<0(B

\JP $B?t$^$?$OB?9`<0(B

\EG number or polynomial

@item obj

$B?t$^$?$OB?9`<0(B

\JP $B?t$^$?$OB?9`<0(B

\EG number or polynomial

@end table

@itemize @bullet

\BJP

@item

$B?t(B, $B$"$k$$$OB?9`<0$N78?t$rM-8BBN$N85$KJQ49$9$k(B.

@item

Line 389 x^2+x+1

Line 620 x^2+x+1

@item

$BM-8BBN$N85$KBP$7(B, $BK!$"$k$$$ODj5AB?9`<0$K$h$k(B reduction $B$r9T$&>l9g$K$b(B

$BMQ$$$k(B.

\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} assures the output has

the reduced representation.

@end itemize

@example

Line 403 x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120

Line 647 x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}, @fref{lmptop}, @fref{gf2nton}

@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}

@findex random_ff

@table @t

@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

@table @var

@item return

$BM-8BBN$N85(B

\JP $BM-8BBN$N85(B

\EG element of a finite field

@end table

@itemize @bullet

\BJP

@item

$BM-8BBN$N85$rMp?t@8@.$9$k(B.

@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.

\BEG

@item

Generates an element of the current base field randomly.

@item

The same random generator as in @code{random()}, @code{lrandom()}

is used.

@end itemize

@example

Line 445 return to toplevel

Line 698 return to toplevel

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}, @fref{random}, @fref{lrandom}

@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}

@findex lmptop

@table @t

@item lmptop(@var{obj})

:: GF(p) $B78?tB?9`<0$N78?t$r@0?t$KJQ49(B

\JP :: GF(p) $B78?tB?9`<0$N78?t$r@0?t$KJQ49(B

\EG :: Converts the coefficients of a polynomial over GF(p) into integers.

@end table

@table @var

@item return

$B@0?t78?tB?9`<0(B

\JP $B@0?t78?tB?9`<0(B

\EG integral polynomial

@item obj

GF(p)$B78?tB?9`<0(B

\JP GF(p) $B78?tB?9`<0(B

\EG polynomial over GF(p)

@end table

@itemize @bullet

\BJP

@item

GF(p) $B78?tB?9`<0$N78?t$r@0?t$KJQ49$9$k(B.

@item

GF(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

$BJQ49$5$l$k(B.

\BEG

@item

GF(p) $B$N85$O(B, $B@0?t$KJQ49$5$l$k(B.

Converts the coefficients of a polynomial over GF(p) into integers.

@item

An element of GF(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}.

@end itemize

@example

Line 495 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42

Line 762 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{simp_ff}

@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}

@findex ntogf2n

@table @t

@item ntogf2n(@var{m})

:: $B<+A3?t$r(B GF(2^n) $B$N85$KJQ49(B

\JP :: $B<+A3?t$r(B GF(2^n) $B$N85$KJQ49(B

\EG :: Converts a non-negative integer into an element of GF(2^n).

@end table

@table @var

@item return

GF(2^n) $B$N85(B

\JP GF(2^n) $B$N85(B

\EG element of GF(2^n)

@item m

$BHsIi@0?t(B

\JP $BHsIi@0?t(B

\EG non-negative integer

@end table

@itemize @bullet

\BJP

@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

$B$KBP$7(B, GF(2^n)=GF(2)[t]/(g(t)) $B$N85(B

Line 523 GF(2^n) $B$N85(B

Line 796 GF(2^n) $B$N85(B

@item

$BDj5AB?9`<0$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.

\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^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.

@end itemize

@example

Line 535 x^30+x+1

Line 819 x^30+x+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{gf2nton}

@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}

@findex gf2nton

@table @t

@item gf2nton(@var{m})

:: GF(2^n) $B$N85$r<+A3?t$KJQ49(B

\JP :: GF(2^n) $B$N85$r<+A3?t$KJQ49(B

\EG :: Converts an element of GF(2^n) into a non-negative integer.

@end table

@table @var

@item return

$BHsIi@0?t(B

\JP $BHsIi@0?t(B

\EG non-negative integer

@item m

GF(2^n) $B$N85(B

\JP GF(2^n) $B$N85(B

\EG element of GF(2^n)

@end table

@itemize @bullet

@item

@code{gf2nton} $B$N5UJQ49$G$"$k(B.

\JP @code{gf2nton} $B$N5UJQ49$G$"$k(B.

\EG The inverse of @code{gf2nton}.

@end itemize

@example

Line 574 x^30+x+1

Line 864 x^30+x+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{gf2nton}

@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}

@findex ptogf2n

@table @t

@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^n) $B$N85$KJQ49(B

\EG :: Converts a univariate polynomial into an element of GF(2^n).

@end table

@table @var

@item return

GF(2^n) $B$N85(B

\JP GF(2^n) $B$N85(B

\EG element of GF(2^n)

@item poly

$B0lJQ?tB?9`<0(B

\JP $B0lJQ?tB?9`<0(B

\EG univariate polynomial

@end table

@itemize @bullet

@item

\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

$BJQ49$5$l$k(B.

@var{poly} $B$NJQ?t$K(B @code{@@} $B$rBeF~$7$?7k2L$HEy$7$$(B.

\BEG

Generates an element of GF(2^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}.

@end itemize

@example

Line 609 x^30+x+1

Line 912 x^30+x+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{gf2ntop}

@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}

@findex gf2ntop

@table @t

@item gf2ntop(@var{m}[,@var{v}])

:: GF(2^n) $B$N85$rB?9`<0$KJQ49(B

\JP :: GF(2^n) $B$N85$rB?9`<0$KJQ49(B

\EG :: Converts an element of GF(2^n) into a polynomial.

@end table

@table @var

@item return

$B0lJQ?tB?9`<0(B

\JP $B0lJQ?tB?9`<0(B

\EG univariate polynomial

@item m

GF(2^n) $B$N85(B

\JP GF(2^n) $B$N85(B

\EG an element of GF(2^n)

@item v

$BITDj85(B

\JP $BITDj85(B

\EG indeterminate

@end table

@itemize @bullet

\BJP

@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.

@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;XDj$5$l$?ITDj85$rJQ?t$H$9$kB?9`<0$rJV$9(B.

\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}.

@end itemize

@example

Line 652 t^13+t^12+t^11+t^10

Line 972 t^13+t^12+t^11+t^10

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{ptogf2n}

@end table

@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

@subsection @code{defpoly_mod2}

@findex defpoly_mod2

@table @t

@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

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG univariate polynomial

@item d

$B@5@0?t(B

\JP $B@5@0?t(B

\EG positive integer

@end table

@itemize @bullet

\BJP

@item

@samp{fff} $B$GDj5A$5$l$F$$$k(B.

@item

Line 682 t^13+t^12+t^11+t^10

Line 1008 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,

$B$=$NCf$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.

\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.

@end itemize

@example

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}

@end table

@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

@subsection @code{fctr_ff}

@findex fctr_ff

@table @t

@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

@table @var

@item return

$B%j%9%H(B

\JP $B%j%9%H(B

\EG list

@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

@itemize @bullet

\BJP

@item

@samp{fff} $B$GDj5A$5$l$F$$$k(B.

@item

Line 719 t^13+t^12+t^11+t^10

Line 1066 t^13+t^12+t^11+t^10

$B=EJ#EY$G$"$k(B.

@item

@var{poly} $B$N<g78?t$O<N$F$i$l$k(B.

\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.

@end itemize

@example

Line 730 t^13+t^12+t^11+t^10

Line 1090 t^13+t^12+t^11+t^10

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}

@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}

@findex irredcheck_ff

@table @t

@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

@table @var

@item return

0|1

@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

@itemize @bullet

\BJP

@item

@samp{fff} $B$GDj5A$5$l$F$$$k(B.

@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.

\BEG

@item

Defined in @samp{fff}.

@item

Returns 1 if @var{poly} is irreducible over the current base field.

Returns 0 otherwise.

@end itemize

@example

Line 767 x^10+14687973587364016969

Line 1140 x^10+14687973587364016969

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}

@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}

@findex randpoly_ff

@table @t

@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

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item d

$B@5@0?t(B

\JP $B@5@0?t(B

\EG positive integer

@item v

$BITDj85(B

\JP $BITDj85(B

\EG indeterminate

@end table

@itemize @bullet

\BJP

@item

@samp{fff} $B$GDj5A$5$l$F$$$k(B.

@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

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.

\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()}.

@end itemize

@example

Line 810 x^10+14687973587364016969

Line 1199 x^10+14687973587364016969

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}, @fref{random_ff}

@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}

@findex ecm_add_ff

@findex ecm_sub_ff

Line 823 x^10+14687973587364016969

Line 1214 x^10+14687973587364016969

@table @t

@item ecm_add_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

@table @var

@item return

$B%Y%/%H%k$^$?$O(B 0

\JP $B%Y%/%H%k$^$?$O(B 0

\EG vector or 0

@item p1,p2

$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

@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

@itemize @bullet

\BJP

@item

$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.

Line 855 x^10+14687973587364016969

Line 1251 x^10+14687973587364016969

$B$G3d$kI,MW$,$"$k(B.

@item

@var{p1}, @var{p2} $B$,BJ1_6J@~>e$NE@$+$I$&$+$N%A%'%C%/$O$7$J$$(B.

\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 @var{y^2=x^3+ec[0]x+ec[1]}.

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]}.

@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.

@end itemize

@example

Line 878 x^10+14687973587364016969

Line 1301 x^10+14687973587364016969

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{setmod_ff}

@end table

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