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Diff for /OpenXM/src/asir-doc/parts/type.texi between version 1.9 and 1.10

version 1.9, 2002/09/03 01:50:58 version 1.10, 2003/04/19 10:36:30
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 @comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.8 2001/03/12 05:01:18 noro Exp $  @comment $OpenXM: OpenXM/src/asir-doc/parts/type.texi,v 1.9 2002/09/03 01:50:58 noro Exp $
 \BJP  \BJP
 @node $B7?(B,,, Top  @node $B7?(B,,, Top
 @chapter $B7?(B  @chapter $B7?(B
Line 625  coefficients of a polynomial.
Line 625  coefficients of a polynomial.
 \E  \E
   
 @end itemize  @end itemize
   
   
   @item 8
   \JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B}
   \EG @b{element of a finite field of characteristic @var{p^n}}
   
   \BJP
   $B0L?t$,(B @var{p^n} (@var{p} $B$OG$0U$NAG?t(B, @var{n} $B$O@5@0?t(B) $B$O(B,
   $BI8?t(B @var{p} $B$*$h$S(B @var{GF(p)} $B>e4{Ls$J(B @var{n} $B<!B?9`<0(B @var{m(x)}
   $B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B.
   $B$3$NBN$N85$O(B @var{m(x)} $B$rK!$H$9$k(B @var{GF(p)} $B>e$NB?9`<0$H$7$F(B
   $BI=8=$5$l$k(B.
   \E
   \BEG
   A finite field of order @var{p^n}, where @var{p} is an arbitrary prime
   and @var{n} is a positive integer, is set by @code{setmod_ff}
   by specifying its characteristic @var{p} and an irreducible polynomial
   of degree @var{n} over @var{GF(p)}. An element of this field
   is represented by a polynomial over @var{GF(p)} modulo @var{m(x)}.
   \E
   
   @item 9
   \JP @b{$B0L?t(B @var{p^n} $B$NM-8BBN$N85(B ($B>.0L?t(B)}
   \EG @b{element of a finite field of characteristic @var{p^n} (small order)}
   
   \BJP
   $B0L?t$,(B @var{p^n} $B$NM-8BBN(B (@var{p^n} $B$,(B @var{2^29} $B0J2<(B, @var{p} $B$,(B @var{2^14} $B0J>e(B
   $B$J$i(B @var{n} $B$O(B 1) $B$O(B,
   $BI8?t(B @var{p} $B$*$h$S3HBg<!?t(B @var{n}
   $B$r(B @code{setmod_ff} $B$K$h$j;XDj$9$k$3$H$K$h$j@_Dj$9$k(B.
   $B$3$NBN$N(B 0 $B$G$J$$85$O(B, @var{p} $B$,(B @var{2^14} $BL$K~$N>l9g(B,
   @var{GF(p^n)} $B$N>hK!72$N@8@.85$r8GDj$9$k$3$H(B
   $B$K$h$j(B, $B$3$N85$N$Y$-$H$7$FI=$5$l$k(B. $B$3$l$K$h$j(B, $B$3$NBN$N(B 0 $B$G$J$$85(B
   $B$O(B, $B$3$N$Y$-;X?t$H$7$FI=8=$5$l$k(B. @var{p} $B$,(B @var{2^14} $B0J>e(B
   $B$N>l9g$ODL>o$N>jM>$K$h$kI=8=$H$J$k$,(B, $B6&DL$N%W%m%0%i%`$G(B
   $BAPJ}$N>l9g$r07$($k$h$&$K$3$N$h$&$J;EMM$H$J$C$F$$$k(B.
   
   \E
   \BEG
   A finite field of order @var{p^n}, where @var{p^n} must be less than
   @var{2^29} and @var{n} must be equal to 1 if @var{p} is greater or
   equal to @var{2^14}@,
   is set by @code{setmod_ff}
   by specifying its characteristic @var{p} the extension degree
   @var{n}. If @var{p} is less than @var{2^14}, each non-zero element
   of this field
   is a power of a fixed element, which is a generator of the multiplicative
   group of the field, and it is represented by its exponent.
   Otherwise, each element is represented by the redue modulo @var{p}.
   This specification is useful for treating both cases in a single
   program.
   \E
   
 @end table  @end table
   
 \BJP  \BJP
 $BBgI8?tAGBN$NI8?t(B, $BI8?t(B 2 $B$NM-8BBN$NDj5AB?9`<0$O(B, @code{setmod_ff}  $B>.I8?tM-8BAGBN0J30$NM-8BBN$O(B @code{setmod_ff} $B$G@_Dj$9$k(B.
 $B$G@_Dj$9$k(B.  $BM-8BBN$N85$I$&$7$N1i;;$G$O(B,
 $BM-8BBN$N85$I$&$7$N1i;;$G$O(B, @code{setmod_ff} $B$K$h$j@_Dj$5$l$F$$$k(B  
 modulus $B$G(B, $BB0$9$kBN$,J,$+$j(B, $B$=$NCf$G1i;;$,9T$o$l$k(B.  
 $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B  $B0lJ}$,M-M}?t$N>l9g$K$O(B, $B$=$NM-M}?t$O<+F0E*$K8=:_@_Dj$5$l$F$$$k(B
 $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B.  $BM-8BBN$N85$KJQ49$5$l(B, $B1i;;$,9T$o$l$k(B.
 \E  \E
 \BEG  \BEG
 The characteristic of a large finite prime field and the defining  Finite fields other than small finite prime fields are
 polynomial of a finite field of characteristic 2 are set by @code{setmod_ff}.  set by @code{setmod_ff}.
 Elements of finite fields do not have informations about the modulus.  Elements of finite fields do not have informations about the modulus.
 Upon an arithmetic operation, the modulus set by @code{setmod_ff} is  Upon an arithmetic operation, i
 used. If one of the operands is a rational number, it is automatically  f one of the operands is a rational number, it is automatically
 converted into an element of the finite field currently set and  converted into an element of the finite field currently set and
 the operation is done in the finite field.  the operation is done in the finite field.
 \E  \E
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