OpenXM/src/asir-doc/parts/ff.texi - diff

Return to ff.texi CVS log

Up to [local] / OpenXM / src / asir-doc / parts

Diff for /OpenXM/src/asir-doc/parts/ff.texi between version 1.2 and 1.4

version 1.2, 1999/12/21 02:47:31

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

Line 1

@comment $OpenXM$

@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

Line 34

@noindent

\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(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)} and @var{GF(2^n)} can be defined, where

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} and

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

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

@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()}.

Line 59 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

\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.

$B0z?t$,@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 87 If @var{f} is a univariate polynomial of degree @var{n

Line 99 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}. 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.

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

Line 1200 The coefficients are generated by @code{random_ff()}.

Line 1214 The coefficients are generated by @code{random_ff()}.

@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})

\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

Line 1243 The coefficients are generated by @code{random_ff()}.

Line 1257 The coefficients are generated by @code{random_ff()}.

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},@var{p2},@var{ec}) returns

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

FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>