===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/ff.texi,v
retrieving revision 1.1.1.1
retrieving revision 1.5
diff -u -p -r1.1.1.1 -r1.5
--- OpenXM/src/asir-doc/parts/ff.texi	1999/12/08 05:47:44	1.1.1.1
+++ OpenXM/src/asir-doc/parts/ff.texi	2003/04/19 15:44:56	1.5
@@ -1,19 +1,52 @@
+@comment $OpenXM: OpenXM/src/asir-doc/parts/ff.texi,v 1.4 2003/04/19 10:36:30 noro Exp $
+\BJP
 @node $BM-8BBN$K4X$9$k1i;;(B,,, Top
 @chapter $BM-8BBN$K4X$9$k1i;;(B
+\E
+\BEG
+@node Finite fields,,, Top
+@chapter Finite fields
+\E
 
 @menu
+\BJP
 * $BM-8BBN$NI=8=$*$h$S1i;;(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::
 * $BM-8BBN$K4X$9$kH!?t$N$^$H$a(B::
+\E
+\BEG
+* Representation of finite fields::
+* Univariate polynomials on finite fields::
+* Polynomials on small finite fields::
+* Elliptic curves on finite fields::
+* Functions for Finite fields::
+\E
 @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
+\E
+\BEG
+@node Representation of finite fields,,, Finite fields
+@section Representation of finite fields
+\E
 
 @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}),
+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. 
+\E
+\BEG
+On @b{Asir} GF(@var{p}), GF(2^@var{n}), GF(@var{p^n}) can be defined, where
+GF(@var{p}) is a finite prime field of charateristic @var{p},
+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()}.
+\E
 
 @example
 [0] P=pari(nextprime,2^50);
@@ -30,10 +63,21 @@ x^50+x^4+x^3+x^2+1
 x^50+x^4+x^3+x^2+1
 [6] field_type_ff();
 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
-@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
-$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$*$$$F$O0z?t$N4{Ls%A%'%C%/$O9T$o$:(B, $B8F$S=P$7B&(B
+\BJP
+@code{setmod_ff()} $B$O(B, $B$5$^$6$^$J%?%$%W$NM-8BBN$r4pACBN$H$7$F%;%C%H$9$k(B. 
+$B0z?t$,@5@0?t(B @var{p} $B$N>l9g(B GF(@var{p}), @var{n} $B<!B?9`<0(B f(x) $B$N>l(B
+$B9g(B, f(x) mod 2 $B$rDj5AB?9`<0$H$9$k(B GF(2^@var{n}) $B$r$=$l$>$l4pACBN$H$7$F%;%C%H$9(B
+$B$k(B. $B$^$?(B, $BM-8BAGBN$NM-8B<!3HBg$bDj5A$G$-$k(B. $B>\$7$/$O(B @xref{$B?t$N7?(B}.
+@code{setmod_ff()} $B$K$*$$$F$O0z?t$N4{Ls%A%'%C%/$O9T$o$:(B, $B8F$S=P$7B&(B
 $B$,@UG$$r;}$D(B.
 
 $B4pACBN$H$O(B, $B$"$/$^$GM-8BBN$N85$H$7$F@k8@$"$k$$$ODj5A$5$l$?%*%V%8%'%/%H$,(B, 
@@ -43,12 +87,41 @@ x^50+x^4+x^3+x^2+1
 $B$k(B.
 
 0 $B$G$J$$M-8BBN$N85$O(B, $B?t%*%V%8%'%/%H$G$"$j(B, $B<1JL;R$NCM$O(B 1 $B$G$"$k(B. 
-$B$5$i$K(B, 0 $B$G$J$$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. 
 
-$BM-8BBN$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. 
+\E
 
+\BEG
+If @var{p} is a positive integer, @code{setmod_ff(@var{p})} sets
+GF(@var{p}) as the current base field. 
+If @var{f} is a univariate polynomial of degree @var{n},
+@code{setmod_ff(@var{f})} sets GF(2^@var{n}) as the current
+base field.  GF(2^@var{n}) is represented
+as an algebraic extension of 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 GF(@var{p}),
+7 if it is GF(2^@var{n}).
+
+There are several methods to input an element of a finite field.
+An element of GF(@var{p}) can be input by @code{simp_ff()}.
+\E
+
 @example
 [0] P=pari(nextprime,2^50);
 1125899906842679
@@ -60,7 +133,9 @@ x^50+x^4+x^3+x^2+1
 6
 @end example
 
-$B$^$?(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 GF(2^@var{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
@@ -74,43 +149,109 @@ 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}.
+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
+\BEG
+Elements of finite fields are numbers and one can apply field arithmetics
+to them. @code{@@} is a generator of GF(2^@var{n}) over GF(2).
+@xref{Types of numbers}.
+\E
 
 @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
+\E
+\BEG
+@node Univariate polynomials on finite fields,,, Finite fields
+@section Univariate polynomials on finite fields
+\E
 
 @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. 
+\E
+\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
-@end example
+Factorizers return lists of [@b{factor},@b{multiplicity}]. The factor
+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
+\E
+
+@noindent
+
+\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
 @section $BM-8BBN>e$NBJ1_6J@~$K4X$9$k1i;;(B
+\E
+\BEG
+@node Elliptic curves on finite fields,,, Finite fields
+@section Elliptic curves on finite fields
+\E
 
+\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. 
 
-$BBJ1_6J@~$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, 
 @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. 
@@ -122,20 +263,59 @@ GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7
 
 @itemize @bullet
 @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. 
 @end itemize
 
-$B$H$$$&E@$G$"$k(B. @var{[x y z]} $B$O@F<!:BI8$K$h$kI=8=$G$"$j(B, $B%"%U%#%s:BI8(B
-$B$G$O(B @var{[x/z y/z]} $B$J$kE@$rI=$9(B. $B$h$C$F(B, $B%"%U%#%s:BI8(B @var{[x y]} $B$G(B
-$BI=8=$5$l$?E@$r1i;;BP>]$H$9$k$K$O(B, @var{[x y 1]} $B$J$k%Y%/%H%k$r(B
+$B$H$$$&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
+$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. 
 $B1i;;7k2L$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. 
+\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
 @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
 * setmod_ff::
@@ -150,6 +330,7 @@ GF(2^n) $B$N(B, GF(2)$B>e$N@8@.85$G$"$k(B. $B>\$7
 * gf2nton::
 * ptogf2n::
 * gf2ntop::
+* ptosfp sfptop::
 * defpoly_mod2::
 * fctr_ff::
 * irredcheck_ff::
@@ -158,37 +339,85 @@ 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$$$kM-8BBN$NK!(B, $BDj5AB?9`<0$NI=<((B
+@itemx setmod_ff(@var{prime},@var{n}])
+\JP :: $BM-8BBN$N@_Dj(B, $B@_Dj$5$l$F$$$kM-8BBN$NK!(B, $BDj5AB?9`<0$NI=<((B
+\EG :: Sets/Gets the current base fields.
 @end table
 
 @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)
+@item n
+\JP $B3HBg<!?t(B
+\EG the extension degree
 @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$$$k4pACBN$,(B GF(@var{prime}) $B$N>l9g(B @var{prime},
-GF(2^n) $B$N>l9gDj5AB?9`<0$rJV$9(B. 
+$B0z?t$,(B @var{p} $B$H(B @var{n} $B$N;~(B, 
+GF(@var{p^n}) $B$r4pACBN$H$7$F@_Dj$9$k(B. @var{p^n} $B$O(B @var{2^29} $BL$K~$G(B
+$B$J$1$l$P$J$i$J$$(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$$(B. 
 @item
-GF(2^n) $B$NDj5AB?9`<0$O(B, GF(2) $B>e(B n $B<!4{Ls$J$i$J$s$G$bNI$$$,(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. 
+\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
 
 @example
@@ -198,32 +427,47 @@ x^100+x^15+1
 x^100+x^15+1
 [176] setmod_ff();
 x^100+x^15+1
+[177] setmod_ff(2,5);
+[2,x^5+x^2+1,x]
 @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$9(B. 
+$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
 
 @example
@@ -240,29 +484,41 @@ 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$9(B. 
+$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
 
 @example
@@ -280,29 +536,42 @@ 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. 
+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
 
 @example
@@ -320,29 +589,42 @@ 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. 
+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
 
 @example
@@ -360,27 +642,36 @@ 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.
+\E
 @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$$$OB?9`<0$N78?t$rM-8BBN$N85$KJQ49$9$k(B. 
 @item
@@ -389,6 +680,27 @@ x^2+x+1
 @item
 $BM-8BBN$N85$KBP$7(B, $BK!$"$k$$$ODj5AB?9`<0$K$h$k(B reduction $B$r9T$&>l9g$K$b(B
 $BMQ$$$k(B. 
+@item
+$B>.I8?tM-8BBN$N85$KJQ49$9$k>l9g(B, $B0lC6AGBN>e$K<M1F$7$F$+$i(B, $B3HBgBN$N(B
+$B85$KJQ49$5$l$k(B. $B3HBgBN$N85$KD>@\JQ49$9$k$K$O(B @code{ptosfp()} $B$r(B
+$BMQ$$$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
 
 @example
@@ -400,36 +712,53 @@ 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
 [3] ntype(coef(@@@@,10));
 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
 
 @table @t
-@item $B;2>H(B
-@fref{setmod_ff}, @fref{lmptop}, @fref{gf2nton}
+\JP @item $B;2>H(B
+\EG @item References
+@fref{setmod_ff}, @fref{lmptop}, @fref{gf2nton}, @fref{ptosfp sfptop}
 @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. 
+\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
 
 @example
@@ -445,35 +774,49 @@ 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(@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
 
 @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(@var{p}) $B78?tB?9`<0(B
+\EG polynomial over GF(@var{p})
 @end table
 
 @itemize @bullet
+\BJP
 @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
-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
 $BJQ49$5$l$k(B. 
+\E
+\BEG
 @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
 
 @example
@@ -495,34 +838,51 @@ 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^@var{n}) $B$N85$KJQ49(B
+\EG :: Converts a non-negative integer into an element of GF(2^@var{n}).
 @end table
 
 @table @var
 @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
-$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 
+$B$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. 
 @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. 
+\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
 
 @example
@@ -535,29 +895,35 @@ 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^@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
 
 @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^@var{n}) $B$N85(B
+\EG element of GF(2^@var{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
@@ -574,31 +940,44 @@ 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^@var{n}) $B$N85$KJQ49(B
+\EG :: Converts a univariate polynomial into an element of GF(2^@var{n}).
 @end table
 
 @table @var
 @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
-$B0lJQ?tB?9`<0(B
+\JP $B0lJQ?tB?9`<0(B
+\EG univariate polynomial
 @end table
 
 @itemize @bullet
 @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. 
 @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
 
 @example
@@ -609,34 +988,51 @@ 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^@var{n}) $B$N85$rB?9`<0$KJQ49(B
+\EG :: Converts an element of GF(2^@var{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^@var{n}) $B$N85(B
+\EG an element of GF(2^@var{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. 
+\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
 
 @example
@@ -652,27 +1048,111 @@ 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 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}
 @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$$$k(B. 
 @item
@@ -682,33 +1162,54 @@ t^13+t^12+t^11+t^10
 3 $B9`<0$,B8:_$7$J$1$l$P(B, $B4{Ls(B 5 $B9`<0$NCf$G(B, $BBh(B 2 $B9`$N<!?t$,$b$C$H$b>.$5$/(B, 
 $B$=$NCf$GBh(B 3 $B9`$N<!?t$,$b$C$H$b>.$5$/(B, $B$=$NCf$GBh(B 4 $B9`$N<!?t$,$b$C$H$b(B
 $B>.$5$$$b$N$rJV$9(B. 
+\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
 
 @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
@@ -719,6 +1220,19 @@ 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. 
+\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
 
 @example
@@ -730,31 +1244,44 @@ 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. 
+\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
 
 @example
@@ -767,34 +1294,50 @@ 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. 
+\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
 
 @example
@@ -810,11 +1353,13 @@ 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
@@ -823,26 +1368,31 @@ 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})
-:: $BBJ1_6J@~>e$NE@$N2C;;(B, $B8:;;(B, $B5U85(B
+@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
 
 @table @var
 @item return
-$B%Y%/%H%k$^$?$O(B 0
-@item p1,p2
-$BD9$5(B 3 $B$N%Y%/%H%k$^$?$O(B 0
+\JP $B%Y%/%H%k$^$?$O(B 0
+\EG vector or 0
+@item p1 p2
+\JP $BD9$5(B 3 $B$N%Y%/%H%k$^$?$O(B 0
+\EG vector of length 3 or 0
 @item ec
-$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$$$kM-8BBN>e$G(B,  @var{ec} $B$GDj5A$5$l$kBJ1_6J@~>e$N(B
 $BE@(B @var{p1}, @var{p2} $B$NOB(B @var{p1+p2}, $B:9(B @var{p1-p2}, $B5U85(B @var{-p1} $B$rJV$9(B. 
 @item
 @var{ec} $B$O(B, $B@_Dj$5$l$F$$$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. 
 @item
 $B0z?t(B, $B7k2L$H$b$K(B, $BL58B1sE@$O(B 0 $B$GI=$5$l$k(B. 
@@ -855,6 +1405,33 @@ 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. 
+\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
 
 @example
@@ -878,7 +1455,8 @@ 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