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Diff for /OpenXM/src/asir-doc/parts/ff.texi between version 1.5 and 1.8

version 1.5, 2003/04/19 15:44:56

version 1.8, 2005/09/08 07:40:49

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/ff.texi,v 1.4 2003/04/19 10:36:30 noro Exp $

@comment $OpenXM: OpenXM/src/asir-doc/parts/ff.texi,v 1.7 2003/04/21 03:07:32 noro Exp $

\BJP

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

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

Line 332 one to obtain the affine coordinate.

* gf2ntop::

* ptosfp sfptop::

* defpoly_mod2::

* sffctr::

* fctr_ff::

* irredcheck_ff::

* randpoly_ff::

Line 345 one to obtain the affine coordinate.

Line 346 one to obtain the affine coordinate.

@findex setmod_ff

@table @t

@item setmod_ff([@var{prime}|@var{poly}])

@item setmod_ff([@var{p}|@var{defpoly2}])

@itemx setmod_ff(@var{prime},@var{n}])

@itemx setmod_ff([@var{defpolyp},@var{p}])

@itemx setmod_ff([@var{p},@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

Line 355 one to obtain the affine coordinate.

Line 357 one to obtain the affine coordinate.

@item return

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

\EG number or polynomial

@item prime

@item p

\JP $BAG?t(B

\EG prime

@item poly

@item defpoly2

\JP GF(2) $B>e4{Ls$J(B 1 $BJQ?tB?9`<0(B

\EG univariate polynomial irreducible over GF(2)

@item defpolyp

\JP GF(@var{p}) $B>e4{Ls$J(B 1 $BJQ?tB?9`<0(B

\EG univariate polynomial irreducible over GF(@var{p})

@item n

\JP $B3HBg<!?t(B

\EG the extension degree

Line 369 one to obtain the affine coordinate.

Line 374 one to obtain the affine coordinate.

@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.

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

@item

$B0z?t$,B?9`<0(B @var{poly} $B$N;~(B,

$B0z?t$,B?9`<0(B @var{defpoly2} $B$N;~(B,

GF(2^deg(@var{poly} mod 2)) = GF(2)[t]/(@var{poly}(t) mod 2)

GF(2^deg(@var{defpoly2} mod 2)) = GF(2)[t]/(@var{defpoly2}(t) mod 2)

$B$r4pACBN$H$7$F@_Dj$9$k(B.

@item

$B0z?t$,(B @var{defpolyp} $B$H(B @var{p} $B$N;~(B,

GF(@var{p^deg(defpolyp)}) $B$r4pACBN$H$7$F@_Dj$9$k(B.

@item

$B0z?t$,(B @var{p} $B$H(B @var{n} $B$N;~(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

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

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

GF(2^@var{n}) $B$N>l9gDj5AB?9`<0$rJV$9(B.

$B4pACBN$,(B GF(p^@var{n})

$B4pACBN$,(B @code{setmod_ff(@var{defpoly},@var{p})} $B$GDj5A$5$l$?(B

(@var{p^n} $B$,(B @var{2^14} $BL$K~(B) $B$N>l9g(B,

GF(@var{p}^@var{n}) $B$N>l9g(B, [@var{defpoly},@var{p}] $B$rJV$9(B.

$B4pACBN$,(B @code{setmod_ff(@var{p},@var{n})} $B$GDj5A$5$l$?(B

GF(p^@var{n}) $B$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})

$B$O(B, @var{n} $B<!3HBg$NDj5AB?9`<0(B, @var{prim_elem} $B$O(B, GF(@var{p^n})$B$N(B

$B>hK!72$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

Line 393 GF(2^@var{n}) $B$NDj5AB?9`<0$O(B, GF(2) $B>e(B n

Line 403 GF(2^@var{n}) $B$NDj5AB?9`<0$O(B, GF(2) $B>e(B n

\BEG

@item

If the argument is a non-negative integer @var{prime}, GF(@var{prime})

If the argument is a non-negative integer @var{p}, GF(@var{p})

is set as the current base field.

@item

If the argument is a polynomial @var{poly},

If the argument is a polynomial @var{defpoly2},

GF(2^deg(@var{poly} mod 2)) = GF(2)[t]/(@var{poly}(t) mod2)

GF(2^deg(@var{defpoly2} mod 2)) = GF(2)[t]/(@var{defpoly2}(t) mod2)

is set as the current base field.

@item

If the arguments are a polynomial @var{defpolyp} and a prime @var{p},

GF(@var{p}^deg(@var{defpolyp})) = GF(@var{p})[t]/(@var{defpolyp}(t))

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

is returned. If the current base field is GF(@var{p}), @var{p} 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},

If it is GF(@var{p^n}) defined by @code{setmod_ff(@var{defpoly},@var{p})},

[@var{defpolyp},@var{p}] is returned.

If it is GF(@var{p^n}) defined by @code{setmod_ff(@var{p},@var{n})},

[@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

Line 427 x^100+x^15+1

Line 443 x^100+x^15+1

x^100+x^15+1

[176] setmod_ff();

x^100+x^15+1

[177] setmod_ff(2,5);

[177] setmod_ff(x^4+x+1,547);

[1*x^4+1*x+1,547]

[178] setmod_ff(2,5);

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

@end example

Line 828 whose value is @var{r}.

Line 846 whose value is @var{r}.

[2] setmod_ff(547);

547

[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);

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));

537

[6] ntype(@@@@);

Line 878 x^10+537*x^9+45*x^8+427*x^7+210*x^6+295*x^5+210*x^4+42

Line 898 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.

@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)),

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.

Line 1186 Then fix @var{m3} as small as possible.

Line 1206 Then fix @var{m3} as small as possible.

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

\EG @item References

@fref{setmod_ff}

@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.

\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.

@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

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