===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/ff.texi,v
retrieving revision 1.5
retrieving revision 1.8
diff -u -p -r1.5 -r1.8
--- OpenXM/src/asir-doc/parts/ff.texi	2003/04/19 15:44:56	1.5
+++ OpenXM/src/asir-doc/parts/ff.texi	2005/09/08 07:40:49	1.8
@@ -1,4 +1,4 @@
-@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 有限体に関する演算,,, Top
 @chapter 有限体に関する演算
@@ -332,6 +332,7 @@ one to obtain the affine coordinate.
 * gf2ntop::
 * ptosfp sfptop::
 * defpoly_mod2::
+* sffctr::
 * fctr_ff::
 * irredcheck_ff::
 * randpoly_ff::
@@ -345,8 +346,9 @@ one to obtain the affine coordinate.
 @findex setmod_ff
 
 @table @t
-@item setmod_ff([@var{prime}|@var{poly}])
-@itemx setmod_ff(@var{prime},@var{n}])
+@item setmod_ff([@var{p}|@var{defpoly2}])
+@itemx setmod_ff([@var{defpolyp},@var{p}])
+@itemx setmod_ff([@var{p},@var{n}])
 \JP :: 有限体の設定, 設定されている有限体の法, 定義多項式の表示
 \EG :: Sets/Gets the current base fields.
 @end table
@@ -355,12 +357,15 @@ one to obtain the affine coordinate.
 @item return
 \JP 数または多項式
 \EG number or polynomial
-@item prime
+@item p
 \JP 素数
 \EG prime
-@item poly
+@item defpoly2
 \JP GF(2) 上既約な 1 変数多項式
 \EG univariate polynomial irreducible over GF(2)
+@item defpolyp
+\JP GF(@var{p}) 上既約な 1 変数多項式
+\EG univariate polynomial irreducible over GF(@var{p})
 @item n
 \JP 拡大次数
 \EG the extension degree
@@ -369,23 +374,28 @@ one to obtain the affine coordinate.
 @itemize @bullet
 \BJP
 @item
-引数が正整数 @var{prime} の時, GF(@var{prime}) を基礎体として設定する. 
+引数が正整数 @var{p} の時, GF(@var{p}) を基礎体として設定する. 
 @item
-引数が多項式 @var{poly} の時, 
-GF(2^deg(@var{poly} mod 2)) = GF(2)[t]/(@var{poly}(t) mod 2)
+引数が多項式 @var{defpoly2} の時, 
+GF(2^deg(@var{defpoly2} mod 2)) = GF(2)[t]/(@var{defpoly2}(t) mod 2)
 を基礎体として設定する.
 @item
+引数が @var{defpolyp} と @var{p} の時, 
+GF(@var{p^deg(defpolyp)}) を基礎体として設定する.
+@item
 引数が @var{p} と @var{n} の時, 
 GF(@var{p^n}) を基礎体として設定する. @var{p^n} は @var{2^29} 未満で
 なければならない. また, @var{p} が @var{2^14} 以上のとき, 
 @var{n} は 1 でなければならない. 
 @item
-無引数の時, 設定されている基礎体が GF(@var{prime})の場合 @var{prime},
+無引数の時, 設定されている基礎体が GF(@var{p})の場合 @var{p},
 GF(2^@var{n}) の場合定義多項式を返す. 
-基礎体が GF(p^@var{n}) 
-(@var{p^n} が @var{2^14} 未満) の場合,
+基礎体が @code{setmod_ff(@var{defpoly},@var{p})} で定義された
+GF(@var{p}^@var{n}) の場合, [@var{defpoly},@var{p}] を返す.
+基礎体が @code{setmod_ff(@var{p},@var{n})} で定義された
+GF(p^@var{n}) の場合,
 [@var{p},@var{defpoly},@var{prim_elem}] を返す. ここで, @var{defpoly}
-は, @var{n} 次拡大の定義多項式, @var{prim_elem} は, GF(@var{p^n})
+は, @var{n} 次拡大の定義多項式, @var{prim_elem} は, GF(@var{p^n})の
 乗法群の生成元を意味する.
 @item
 GF(2^@var{n}) の定義多項式は, GF(2) 上 n 次既約ならなんでも良いが, 効率に
@@ -393,22 +403,28 @@ GF(2^@var{n}) の定義多項式は, GF(2) 上 n 
 \E
 \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},
-GF(2^deg(@var{poly} mod 2)) = GF(2)[t]/(@var{poly}(t) mod2)
+If the argument is a polynomial @var{defpoly2},
+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
@@ -427,7 +443,9 @@ 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
 
@@ -828,9 +846,11 @@ 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(@@@@);
@@ -878,7 +898,7 @@ 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.
@@ -1186,6 +1206,60 @@ Then fix @var{m3} as small as possible.
 \JP @item 参照
 \EG @item References
 @fref{setmod_ff}
+@end table
+
+\JP @node sffctr,,, 有限体に関する函数のまとめ
+\EG @node sffctr,,, Functions for Finite fields
+@subsection @code{sffctr}
+@findex sffctr
+
+@table @t
+@item sffctr(@var{poly})
+\JP :: 多項式の小標数有限体上での既約分解
+\EG :: Irreducible factorization over a small finite field.
+@end table
+
+@table @var
+@item return
+\JP リスト
+\EG list
+@item poly
+\JP 有限体上の 多項式
+\EG polynomial over a finite field
+@end table
+
+@itemize @bullet
+\BJP
+@item
+多項式を, 現在設定されている小標数有限体上で既約分解する. 
+@item
+結果は, [[@var{f1},@var{m1}],[@var{f2},@var{m2}],...] なる
+リストである. ここで, @var{fi} は monic な既約因子, @var{mi} はその
+重複度である. 
+\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
+
+@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 参照
+\EG @item References
+@fref{setmod_ff},
+@fref{modfctr}
 @end table
 
 \JP @node fctr_ff,,, 有限体に関する函数のまとめ