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Diff for /OpenXM/src/asir-doc/parts/builtin/upoly.texi between version 1.2 and 1.3

version 1.2, 1999/12/21 02:47:34 version 1.3, 2003/04/19 15:44:59
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 @comment $OpenXM$  @comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/upoly.texi,v 1.2 1999/12/21 02:47:34 noro Exp $
 \BJP  \BJP
 @node $B0lJQ?tB?9`<0$N1i;;(B,,, $BAH$_9~$_H!?t(B  @node $B0lJQ?tB?9`<0$N1i;;(B,,, $BAH$_9~$_H!?t(B
 @section $B0lJQ?tB?9`<0$N1i;;(B  @section $B0lJQ?tB?9`<0$N1i;;(B
Line 112  cannot take polynomials over GF(2^n) as their inputs.
Line 112  cannot take polynomials over GF(2^n) as their inputs.
 @item  @item
 @code{umul()}, @code{umul_ff()} produce @var{p1*p2}.  @code{umul()}, @code{umul_ff()} produce @var{p1*p2}.
 @code{usquare()}, @code{usquare_ff()} produce @var{p1^2}.  @code{usquare()}, @code{usquare_ff()} produce @var{p1^2}.
 @code{utmul()}, @code{utmul_ff()} produce @var{p1*p2 mod v^(d+1)},  @code{utmul()}, @code{utmul_ff()} produce @var{p1*p2 mod} @var{v}^(@var{d}+1),
 where @var{v} is the variable of @var{p1}, @var{p2}.  where @var{v} is the variable of @var{p1}, @var{p2}.
 @item  @item
 If the degrees of the inputs are less than or equal to the  If the degrees of the inputs are less than or equal to the
Line 315  See the description of each function for details.
Line 315  See the description of each function for details.
 @itemize @bullet  @itemize @bullet
 \BJP  \BJP
 @item  @item
 @var{p} $B$NJQ?t$r(B x $B$H$9$k(B. $B$3$N$H$-(B @var{p} = @var{p1}+x^(d+1)@var{p2}  @var{p} $B$NJQ?t$r(B x $B$H$9$k(B. $B$3$N$H$-(B @var{p} = @var{p1}+x^(@var{d}+1)@var{p2}
 (@var{p1} $B$N<!?t$O(B @var{d} $B0J2<(B) $B$HJ,2r$G$-$k(B. @code{utrunc()} $B$O(B  (@var{p1} $B$N<!?t$O(B @var{d} $B0J2<(B) $B$HJ,2r$G$-$k(B. @code{utrunc()} $B$O(B
 @var{p1} $B$rJV$7(B, @code{udecomp()} $B$O(B [@var{p1},@var{p2}] $B$rJV$9(B.  @var{p1} $B$rJV$7(B, @code{udecomp()} $B$O(B [@var{p1},@var{p2}] $B$rJV$9(B.
 @item  @item
 @var{p} $B$N<!?t$r(B @var{e} $B$H$7(B, @var{i} $B<!$N78?t$r(B @var{p[i]} $B$H$9$l$P(B,  @var{p} $B$N<!?t$r(B @var{e} $B$H$7(B, @var{i} $B<!$N78?t$r(B @var{p}[@var{i}] $B$H$9$l$P(B,
 @code{ureverse()} $B$O(B @var{p[e]}+@var{p[e-1]}x+... $B$rJV$9(B.  @code{ureverse()} $B$O(B @var{p}[@var{e}]+@var{p}[@var{e}-1]x+... $B$rJV$9(B.
 \E  \E
 \BEG  \BEG
 @item  @item
 Let @var{x} be the variable of @var{p}. Then @var{p} can be decomposed  Let @var{x} be the variable of @var{p}. Then @var{p} can be decomposed
 as @var{p} = @var{p1}+x^(d+1)@var{p2}, where the degree of @var{p1}  as @var{p} = @var{p1}+x^(@var{d}+1)@var{p2}, where the degree of @var{p1}
 is less than or equal to @var{d}.  is less than or equal to @var{d}.
 Under the decomposition, @code{utrunc()} returns  Under the decomposition, @code{utrunc()} returns
 @var{p1} and  @code{udecomp()} returns [@var{p1},@var{p2}].  @var{p1} and  @code{udecomp()} returns [@var{p1},@var{p2}].
 @item  @item
 Let @var{e} be the degree of @var{p} and @var{p[i]} the coefficient  Let @var{e} be the degree of @var{p} and @var{p}[@var{i}] the coefficient
 of @var{p} at degree @var{i}. Then  of @var{p} at degree @var{i}. Then
 @code{ureverse()} returns @var{p[e]}+@var{p[e-1]}x+....  @code{ureverse()} returns @var{p}[@var{e}]+@var{p}[@var{e}-1]x+....
 \E  \E
 @end itemize  @end itemize
   
Line 394  of @var{p} at degree @var{i}. Then 
Line 394  of @var{p} at degree @var{i}. Then 
 For a polynomial @var{p} with a non zero constant term,  For a polynomial @var{p} with a non zero constant term,
 @code{uinv_as_power_series(@var{p},@var{d})} computes  @code{uinv_as_power_series(@var{p},@var{d})} computes
 a polynomial @var{r} whose degree is at most @var{d}  a polynomial @var{r} whose degree is at most @var{d}
 such that @var{p*r = 1 mod x^(d+1)}, where @var{x} is the variable  such that @var{p*r = 1 mod} x^(@var{d}+1), where @var{x} is the variable
 of @var{p}.  of @var{p}.
 @item  @item
 Let @var{e} be the degree of @var{p}.  Let @var{e} be the degree of @var{p}.
Line 452  x^10+x^9
Line 452  x^10+x^9
 @item return  @item return
 \JP $B0lJQ?tB?9`<0(B  \JP $B0lJQ?tB?9`<0(B
 \EG univariate polynomial  \EG univariate polynomial
 @item p1,p2,inv  @item p1 p2 inv
 \JP $B0lJQ?tB?9`<0(B  \JP $B0lJQ?tB?9`<0(B
 \EG univariate polynomial  \EG univariate polynomial
 @end table  @end table
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