version 1.1.1.1, 1999/12/08 05:47:44 |
version 1.2, 1999/12/21 02:47:34 |
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@comment $OpenXM$ |
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\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 |
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\E |
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\BEG |
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@node Univariate polynomials,,, Built-in Function |
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@section Univariate polynomials |
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\E |
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@menu |
@menu |
* umul umul_ff usquare usquare_ff utmul utmul_ff:: |
* umul umul_ff usquare usquare_ff utmul utmul_ff:: |
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* udiv urem urembymul urembymul_precomp ugcd:: |
* udiv urem urembymul urembymul_precomp ugcd:: |
@end menu |
@end menu |
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@node umul umul_ff usquare usquare_ff utmul utmul_ff,,, $B0lJQ?tB?9`<0$N1i;;(B |
\JP @node umul umul_ff usquare usquare_ff utmul utmul_ff,,, $B0lJQ?tB?9`<0$N1i;;(B |
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\EG @node umul umul_ff usquare usquare_ff utmul utmul_ff,,, Univariate polynomials |
@subsection @code{umul}, @code{umul_ff}, @code{usquare}, @code{usquare_ff}, @code{utmul}, @code{utmul_ff} |
@subsection @code{umul}, @code{umul_ff}, @code{usquare}, @code{usquare_ff}, @code{utmul}, @code{utmul_ff} |
@findex umul |
@findex umul |
@findex umul_ff |
@findex umul_ff |
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@table @t |
@table @t |
@item umul(@var{p1},@var{p2}) |
@item umul(@var{p1},@var{p2}) |
@itemx umul_ff(@var{p1},@var{p2}) |
@itemx umul_ff(@var{p1},@var{p2}) |
:: $B0lJQ?tB?9`<0$N9bB.>h;;(B |
\JP :: $B0lJQ?tB?9`<0$N9bB.>h;;(B |
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\EG :: Fast multiplication of univariate polynomials |
@item usquare(@var{p1}) |
@item usquare(@var{p1}) |
@itemx usquare_ff(@var{p1}) |
@itemx usquare_ff(@var{p1}) |
:: $B0lJQ?tB?9`<0$N9bB.(B 2 $B>h;;(B |
\JP :: $B0lJQ?tB?9`<0$N9bB.(B 2 $B>h;;(B |
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\EG :: Fast squaring of a univariate polynomial |
@item utmul(@var{p1},@var{p2},@var{d}) |
@item utmul(@var{p1},@var{p2},@var{d}) |
@itemx utmul_ff(@var{p1},@var{p2},@var{d}) |
@itemx utmul_ff(@var{p1},@var{p2},@var{d}) |
:: $B0lJQ?tB?9`<0$N9bB.>h;;(B ($BBG$A@Z$j<!?t;XDj(B) |
\JP :: $B0lJQ?tB?9`<0$N9bB.>h;;(B ($BBG$A@Z$j<!?t;XDj(B) |
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\EG :: Fast multiplication of univariate polynomials with truncation |
@end table |
@end table |
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@table @var |
@table @var |
@item return |
@item return |
$B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
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\EG univariate polynomial |
@item p1 p2 |
@item p1 p2 |
$B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
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\EG univariate polynomial |
@item d |
@item d |
$BHsIi@0?t(B |
\JP $BHsIi@0?t(B |
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\EG non-negative integer |
@end table |
@end table |
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@itemize @bullet |
@itemize @bullet |
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\BJP |
@item |
@item |
$B0lJQ?tB?9`<0$N>h;;$r(B, $B<!?t$K1~$8$F7h$^$k%"%k%4%j%:%`$rMQ$$$F(B |
$B0lJQ?tB?9`<0$N>h;;$r(B, $B<!?t$K1~$8$F7h$^$k%"%k%4%j%:%`$rMQ$$$F(B |
$B9bB.$K9T$&(B. |
$B9bB.$K9T$&(B. |
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@item |
@item |
@code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()} $B$O(B, |
@code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()} $B$O(B, |
$B78?t$rM-8BBN$N85$H8+$J$7$F(B, $BM-8BBN>e$NB?9`<0$H$7$F(B |
$B78?t$rM-8BBN$N85$H8+$J$7$F(B, $BM-8BBN>e$NB?9`<0$H$7$F(B |
$B@Q$r5a$a$k(B. $B$?$@$7(B, $B0z?t$,A4$/M-8BBN$N85$r4^$^$J$$>l9g$K$O(B, |
$B@Q$r5a$a$k(B. $B$?$@$7(B, $B0z?t$N78?t$,@0?t$N>l9g(B, |
$B@0?t78?t$NB?9`<0$rJV$9>l9g$b$"$k$N$G(B, $B$3$l$i$r8F$S=P$7$?7k2L(B |
$B@0?t78?t$NB?9`<0$rJV$9>l9g$b$"$k$N$G(B, $B$3$l$i$r8F$S=P$7$?7k2L(B |
$B$,M-8BBN78?t$G$"$k$3$H$rJ]>Z$9$k$?$a$K$O(B |
$B$,M-8BBN78?t$G$"$k$3$H$rJ]>Z$9$k$?$a$K$O(B |
$B$"$i$+$8$a(B @code{simp_ff()} $B$G78?t$rM-8BBN$N85$KJQ49$7$F$*$/$H$h$$(B. |
$B$"$i$+$8$a(B @code{simp_ff()} $B$G78?t$rM-8BBN$N85$KJQ49$7$F$*$/$H$h$$(B. |
Line 66 GF(2^n) $B78?t$NB?9`<0$r0z?t$K<h$l$J$$(B. |
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Line 81 GF(2^n) $B78?t$NB?9`<0$r0z?t$K<h$l$J$$(B. |
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@item |
@item |
$B$$$:$l$b(B, @code{set_upkara()} (@code{utmul}, @code{utmul_ff} $B$K$D$$$F$O(B |
$B$$$:$l$b(B, @code{set_upkara()} (@code{utmul}, @code{utmul_ff} $B$K$D$$$F$O(B |
@code{set_uptkara()}) $B$GJV$5$l$kCM0J2<$N<!?t$KBP$7$F$ODL>o$NI.;;(B |
@code{set_uptkara()}) $B$GJV$5$l$kCM0J2<$N<!?t$KBP$7$F$ODL>o$NI.;;(B |
$B7A<0$NJ}K!(B, @code{set_uufft()} $B$GJV$5$l$kCM0J2<$N<!?t$KBP$7$F$O(B Karatsuba |
$B7A<0$NJ}K!(B, @code{set_upfft()} $B$GJV$5$l$kCM0J2<$N<!?t$KBP$7$F$O(B Karatsuba |
$BK!(B, $B$=$l0J>e$G$O(B FFT$B$*$h$SCf9q>jM>DjM}$,MQ$$$i$l$k(B. $B$9$J$o$A(B, |
$BK!(B, $B$=$l0J>e$G$O(B FFT$B$*$h$SCf9q>jM>DjM}$,MQ$$$i$l$k(B. $B$9$J$o$A(B, |
$B@0?t$KBP$9$k(B FFT $B$G$O$J$/(B, $B==J,B?$/$N(B 1 $B%o!<%I0JFb$NK!(B @var{mi} $B$rMQ0U$7(B, |
$B@0?t$KBP$9$k(B FFT $B$G$O$J$/(B, $B==J,B?$/$N(B 1 $B%o!<%I0JFb$NK!(B @var{mi} $B$rMQ0U$7(B, |
@var{p1}, @var{p2} $B$N78?t$r(B @var{mi} $B$G3d$C$?M>$j$H$7$?$b$N$N@Q$r(B, |
@var{p1}, @var{p2} $B$N78?t$r(B @var{mi} $B$G3d$C$?M>$j$H$7$?$b$N$N@Q$r(B, |
FFT $B$G7W;;$7(B, $B:G8e$KCf9q>jM>DjM}$G9g@.$9$k(B. $B$=$N:](B, $BM-8BBNHG$N4X?t$K(B |
FFT $B$G7W;;$7(B, $B:G8e$KCf9q>jM>DjM}$G9g@.$9$k(B. $B$=$N:](B, $BM-8BBNHG$N4X?t$K(B |
$B$*$$$F$O(B, $B:G8e$K4pACBN$rI=$9K!$G3F78?t$N>jM>$r7W;;$9$k$,(B, $B$3$3$G$O(B |
$B$*$$$F$O(B, $B:G8e$K4pACBN$rI=$9K!$G3F78?t$N>jM>$r7W;;$9$k$,(B, $B$3$3$G$O(B |
Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B. |
Shoup $B$K$h$k%H%j%C%/(B @code{[Shoup]} $B$rMQ$$$F9bB.2=$7$F$"$k(B. |
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\E |
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\BEG |
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@item |
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These functions compute products of univariate polynomials |
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by selecting an appropriate algorithm depending on the degrees |
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of inputs. |
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@item |
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@code{umul()}, @code{usquare()}, @code{utmul()} |
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compute products over the integers. |
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Coefficients in GF(p) are regarded as non-negative integers |
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less than p. |
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@item |
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@code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()} |
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compute products over a finite field. However, if some of |
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the coefficients of the inputs are integral, |
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the result may be an integral polynomial. So if one wants |
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to assure that the result is a polynomial over the finite field, |
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apply @code{simp_ff()} to the inputs. |
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@item |
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@code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()} |
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cannot take polynomials over GF(2^n) as their inputs. |
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@item |
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@code{umul()}, @code{umul_ff()} produce @var{p1*p2}. |
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@code{usquare()}, @code{usquare_ff()} produce @var{p1^2}. |
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@code{utmul()}, @code{utmul_ff()} produce @var{p1*p2 mod v^(d+1)}, |
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where @var{v} is the variable of @var{p1}, @var{p2}. |
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@item |
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If the degrees of the inputs are less than or equal to the |
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value returned by @code{set_upkara()} (@code{set_uptkara()} for |
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@code{utmul}, @code{utmul_ff}), usual pencil and paper method is |
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used. If the degrees of the inputs are less than or equall to |
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the value returned by @code{set_upfft()}, Karatsuba algorithm |
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is used. If the degrees of the inputs exceed it, a combination |
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of FFT and Chinese remainder theorem is used. |
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First of all sufficiently many primes @var{mi} |
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within 1 machine word are prepared. |
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Then @var{p1*p2 mod mi} is computed by FFT for each @var{mi}. |
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Finally they are combined by Chinese remainder theorem. |
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The functions over finite fields use an improvement by V. Shoup @code{[Shoup]}. |
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\E |
@end itemize |
@end itemize |
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@example |
@example |
Line 101 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B. |
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Line 156 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B. |
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@end example |
@end example |
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@table @t |
@table @t |
@item $B;2>H(B |
\JP @item $B;2>H(B |
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\EG @item References |
@fref{set_upkara set_uptkara set_upfft}, |
@fref{set_upkara set_uptkara set_upfft}, |
@fref{kmul ksquare ktmul}. |
@fref{kmul ksquare ktmul}. |
@end table |
@end table |
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@node kmul ksquare ktmul,,, $B0lJQ?tB?9`<0$N1i;;(B |
\JP @node kmul ksquare ktmul,,, $B0lJQ?tB?9`<0$N1i;;(B |
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\EG @node kmul ksquare ktmul,,, Univariate polynomials |
@subsection @code{kmul}, @code{ksquare}, @code{ktmul} |
@subsection @code{kmul}, @code{ksquare}, @code{ktmul} |
@findex kmul |
@findex kmul |
@findex ksquare |
@findex ksquare |
Line 114 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B. |
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Line 171 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B. |
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@table @t |
@table @t |
@item kmul(@var{p1},@var{p2}) |
@item kmul(@var{p1},@var{p2}) |
:: $B0lJQ?tB?9`<0$N9bB.>h;;(B |
\JP :: $B0lJQ?tB?9`<0$N9bB.>h;;(B |
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\EG :: Fast multiplication of univariate polynomials |
@item ksquare(@var{p1}) |
@item ksquare(@var{p1}) |
:: $B0lJQ?tB?9`<0$N9bB.(B 2 $B>h;;(B |
\JP :: $B0lJQ?tB?9`<0$N9bB.(B 2 $B>h;;(B |
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\EG :: Fast squaring of a univariate polynomial |
@item ktmul(@var{p1},@var{p2},@var{d}) |
@item ktmul(@var{p1},@var{p2},@var{d}) |
:: $B0lJQ?tB?9`<0$N9bB.>h;;(B ($BBG$A@Z$j<!?t;XDj(B) |
\JP :: $B0lJQ?tB?9`<0$N9bB.>h;;(B ($BBG$A@Z$j<!?t;XDj(B) |
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\EG :: Fast multiplication of univariate polynomials with truncation |
@end table |
@end table |
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@table @var |
@table @var |
@item return |
@item return |
$B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
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\EG univariate polynomial |
@item p1 p2 |
@item p1 p2 |
$B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
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\EG univariate polynomial |
@item d |
@item d |
$BHsIi@0?t(B |
\JP $BHsIi@0?t(B |
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\EG non-negative integer |
@end table |
@end table |
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@itemize @bullet |
@itemize @bullet |
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\BJP |
@item |
@item |
$B0lJQ?tB?9`<0$N>h;;$r(B Karatsuba $BK!$G9T$&(B. |
$B0lJQ?tB?9`<0$N>h;;$r(B Karatsuba $BK!$G9T$&(B. |
@item |
@item |
Line 138 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B. |
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Line 202 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B. |
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FFT $B$rMQ$$$?9bB.2=$O9T$o$J$$(B. |
FFT $B$rMQ$$$?9bB.2=$O9T$o$J$$(B. |
@item |
@item |
GF(2^n) $B78?t$NB?9`<0$K$bMQ$$$k$3$H$,$G$-$k(B. |
GF(2^n) $B78?t$NB?9`<0$K$bMQ$$$k$3$H$,$G$-$k(B. |
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\E |
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\BEG |
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These functions compute products of univariate polynomials by Karatsuba |
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algorithm. |
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@item |
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These functions do not apply FFT for large degree inputs. |
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@item |
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These functions can compute products over GF(2^n). |
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\E |
@end itemize |
@end itemize |
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@example |
@example |
Line 153 return to toplevel |
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Line 226 return to toplevel |
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[37] kmul(A,B)$ |
[37] kmul(A,B)$ |
@end example |
@end example |
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@node set_upkara set_uptkara set_upfft,,, $B0lJQ?tB?9`<0$N1i;;(B |
\JP @node set_upkara set_uptkara set_upfft,,, $B0lJQ?tB?9`<0$N1i;;(B |
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\EG @node set_upkara set_uptkara set_upfft,,, Univariate polynomials |
@subsection @code{set_upkara}, @code{set_uptkara}, @code{set_upfft} |
@subsection @code{set_upkara}, @code{set_uptkara}, @code{set_upfft} |
@findex set_upkara |
@findex set_upkara |
@findex set_uptkara |
@findex set_uptkara |
Line 163 return to toplevel |
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Line 237 return to toplevel |
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@item set_upkara([@var{threshold}]) |
@item set_upkara([@var{threshold}]) |
@itemx set_uptkara([@var{threshold}]) |
@itemx set_uptkara([@var{threshold}]) |
@itemx set_upfft([@var{threshold}]) |
@itemx set_upfft([@var{threshold}]) |
:: 1 $BJQ?tB?9`<0$N@Q1i;;$K$*$1$k(B N^2 , Karatsuba, FFT $B%"%k%4%j%:%`$N@ZBX$($NogCM(B |
\JP :: 1 $BJQ?tB?9`<0$N@Q1i;;$K$*$1$k(B N^2 , Karatsuba, FFT $B%"%k%4%j%:%`$N@ZBX$($NogCM(B |
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\BEG |
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:: Set thresholds in the selection of an algorithm from N^2, Karatsuba, |
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FFT algorithms for univariate polynomial multiplication. |
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\E |
@end table |
@end table |
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@table @var |
@table @var |
@item return |
@item return |
$B@_Dj$5$l$F$$$kCM(B |
\JP $B@_Dj$5$l$F$$$kCM(B |
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\EG value currently set |
@item threshold |
@item threshold |
$BHsIi@0?t(B |
\JP $BHsIi@0?t(B |
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\EG non-negative integer |
@end table |
@end table |
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@itemize @bullet |
@itemize @bullet |
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\BJP |
@item |
@item |
$B$$$:$l$b(B, $B0lJQ?tB?9`<0$N@Q$N7W;;$K$*$1$k(B, $B%"%k%4%j%:%`@ZBX$($NogCM$r(B |
$B$$$:$l$b(B, $B0lJQ?tB?9`<0$N@Q$N7W;;$K$*$1$k(B, $B%"%k%4%j%:%`@ZBX$($NogCM$r(B |
$B@_Dj$9$k(B. |
$B@_Dj$9$k(B. |
Line 183 return to toplevel |
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Line 264 return to toplevel |
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$B$5$l$k(B. $B$3$N@ZBX$($N<!?t$r@_Dj$9$k(B. |
$B$5$l$k(B. $B$3$N@ZBX$($N<!?t$r@_Dj$9$k(B. |
@item |
@item |
$B>\:Y$O(B, $B$=$l$>$l$N@Q4X?t$N9`$r;2>H$N$3$H(B. |
$B>\:Y$O(B, $B$=$l$>$l$N@Q4X?t$N9`$r;2>H$N$3$H(B. |
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\E |
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\BEG |
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@item |
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These functions set thresholds in the selection of an algorithm from N^2, |
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Karatsuba, FFT algorithms for univariate polynomial multiplication. |
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@item |
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Products of univariate polynomials are computed by N^2, Karatsuba, |
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FFT algorithms. The algorithm selection is done according to the degrees of |
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input polynomials and the thresholds. |
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@item |
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See the description of each function for details. |
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\E |
@end itemize |
@end itemize |
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@table @t |
@table @t |
@item $B;2>H(B |
\JP @item $B;2>H(B |
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\EG @item References |
@fref{kmul ksquare ktmul}, |
@fref{kmul ksquare ktmul}, |
@fref{umul umul_ff usquare usquare_ff utmul utmul_ff}. |
@fref{umul umul_ff usquare usquare_ff utmul utmul_ff}. |
@end table |
@end table |
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@node utrunc udecomp ureverse,,, $B0lJQ?tB?9`<0$N1i;;(B |
\JP @node utrunc udecomp ureverse,,, $B0lJQ?tB?9`<0$N1i;;(B |
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\EG @node utrunc udecomp ureverse,,, Univariate polynomials |
@subsection @code{utrunc}, @code{udecomp}, @code{ureverse} |
@subsection @code{utrunc}, @code{udecomp}, @code{ureverse} |
@findex utrunc |
@findex utrunc |
@findex udecomp |
@findex udecomp |
Line 201 return to toplevel |
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Line 296 return to toplevel |
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@item utrunc(@var{p},@var{d}) |
@item utrunc(@var{p},@var{d}) |
@itemx udecomp(@var{p},@var{d}) |
@itemx udecomp(@var{p},@var{d}) |
@itemx ureverse(@var{p}) |
@itemx ureverse(@var{p}) |
:: $BB?9`<0$KBP$9$kA`:n(B |
\JP :: $BB?9`<0$KBP$9$kA`:n(B |
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\EG :: Operations on polynomials |
@end table |
@end table |
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@table @var |
@table @var |
@item return |
@item return |
$B0lJQ?tB?9`<0$"$k$$$O0lJQ?tB?9`<0$N%j%9%H(B |
\JP $B0lJQ?tB?9`<0$"$k$$$O0lJQ?tB?9`<0$N%j%9%H(B |
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\EG univariate polynomial or list of univariate polynomials |
@item p |
@item p |
$B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
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\EG univariate polynomial |
@item d |
@item d |
$BHsIi@0?t(B |
\JP $BHsIi@0?t(B |
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\EG non-negative integer |
@end table |
@end table |
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@itemize @bullet |
@itemize @bullet |
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\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^(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 |
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@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[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[e]}+@var{p[e-1]}x+... $B$rJV$9(B. |
|
\E |
|
\BEG |
|
@item |
|
Let @var{x} be the variable of @var{p}. Then @var{p} can be decomposed |
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as @var{p} = @var{p1}+x^(d+1)@var{p2}, where the degree of @var{p1} |
|
is less than or equal to @var{d}. |
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Under the decomposition, @code{utrunc()} returns |
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@var{p1} and @code{udecomp()} returns [@var{p1},@var{p2}]. |
|
@item |
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Let @var{e} be the degree of @var{p} and @var{p[i]} the coefficient |
|
of @var{p} at degree @var{i}. Then |
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@code{ureverse()} returns @var{p[e]}+@var{p[e-1]}x+.... |
|
\E |
@end itemize |
@end itemize |
|
|
@example |
@example |
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@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
\JP @item $B;2>H(B |
|
\EG @item References |
@fref{udiv urem urembymul urembymul_precomp ugcd}. |
@fref{udiv urem urembymul urembymul_precomp ugcd}. |
@end table |
@end table |
|
|
@node uinv_as_power_series ureverse_inv_as_power_series,,, $B0lJQ?tB?9`<0$N1i;;(B |
\JP @node uinv_as_power_series ureverse_inv_as_power_series,,, $B0lJQ?tB?9`<0$N1i;;(B |
|
\EG @node uinv_as_power_series ureverse_inv_as_power_series,,, Univariate polynomials |
@subsection @code{uinv_as_power_series}, @code{ureverse_inv_as_power_series} |
@subsection @code{uinv_as_power_series}, @code{ureverse_inv_as_power_series} |
@findex uinv_as_power_series |
@findex uinv_as_power_series |
@findex ureverse_inv_as_power_series |
@findex ureverse_inv_as_power_series |
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|
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@table @t |
@table @t |
@item uinv_as_power_series(@var{p},@var{d}) |
@item uinv_as_power_series(@var{p},@var{d}) |
@itemx ureverse_inv_as_power_series(@var{p},@var{d}) |
@itemx ureverse_inv_as_power_series(@var{p},@var{d}) |
:: $BB?9`<0$rQQ5i?t$H$_$F(B, $B5U857W;;(B |
\JP :: $BB?9`<0$rQQ5i?t$H$_$F(B, $B5U857W;;(B |
|
\EG :: Computes the truncated inverse as a power series. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
$B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
|
\EG univariate polynomial |
@item p |
@item p |
$B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
|
\EG univariate polynomial |
@item d |
@item d |
$BHsIi@0?t(B |
\JP $BHsIi@0?t(B |
|
\EG non-negative integer |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
|
\BJP |
@item |
@item |
@code{uinv_as_power_series(@var{p},@var{d})} $B$O(B, $BDj?t9`$,(B 0 $B$G$J$$(B |
@code{uinv_as_power_series(@var{p},@var{d})} $B$O(B, $BDj?t9`$,(B 0 $B$G$J$$(B |
$BB?9`<0(B @var{p} $B$KBP$7(B, @var{p}@var{r}-1 $B$N:GDc<!?t$,(B @var{d}+1 |
$BB?9`<0(B @var{p} $B$KBP$7(B, @var{p}@var{r}-1 $B$N:GDc<!?t$,(B @var{d}+1 |
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$B$KBP$7$F(B @code{uinv_as_power_series(@var{p1},@var{d})} $B$r7W;;$9$k(B. |
$B$KBP$7$F(B @code{uinv_as_power_series(@var{p1},@var{d})} $B$r7W;;$9$k(B. |
@item |
@item |
@code{rembymul_precomp()} $B$N0z?t$H$7$FMQ$$$k>l9g(B, @code{ureverse_inv_as_power_series()} $B$N7k2L$r$=$N$^$^MQ$$$k$3$H$,$G$-$k(B. |
@code{rembymul_precomp()} $B$N0z?t$H$7$FMQ$$$k>l9g(B, @code{ureverse_inv_as_power_series()} $B$N7k2L$r$=$N$^$^MQ$$$k$3$H$,$G$-$k(B. |
|
\E |
|
\BEG |
|
@item |
|
For a polynomial @var{p} with a non zero constant term, |
|
@code{uinv_as_power_series(@var{p},@var{d})} computes |
|
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 |
|
of @var{p}. |
|
@item |
|
Let @var{e} be the degree of @var{p}. |
|
@code{ureverse_inv_as_power_series(@var{p},@var{d})} computes |
|
@code{uinv_as_power_series(@var{p1},@var{d})} for |
|
@var{p1}=@code{ureverse(@var{p},@var{e})}. |
|
@item |
|
The output of @code{ureverse_inv_as_power_series()} can be used |
|
as the input of @code{rembymul_precomp()}. |
|
\E |
@end itemize |
@end itemize |
|
|
@example |
@example |
|
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
\JP @item $B;2>H(B |
|
\EG @item References |
@fref{utrunc udecomp ureverse}, |
@fref{utrunc udecomp ureverse}, |
@fref{udiv urem urembymul urembymul_precomp ugcd}. |
@fref{udiv urem urembymul urembymul_precomp ugcd}. |
@end table |
@end table |
|
|
@node udiv urem urembymul urembymul_precomp ugcd,,, $B0lJQ?tB?9`<0$N1i;;(B |
\JP @node udiv urem urembymul urembymul_precomp ugcd,,, $B0lJQ?tB?9`<0$N1i;;(B |
|
\EG @node udiv urem urembymul urembymul_precomp ugcd,,, Univariate polynomials |
@subsection @code{udiv}, @code{urem}, @code{urembymul}, @code{urembymul_precomp}, @code{ugcd} |
@subsection @code{udiv}, @code{urem}, @code{urembymul}, @code{urembymul_precomp}, @code{ugcd} |
@findex udiv |
@findex udiv |
@findex urem |
@findex urem |
|
|
@item urembymul(@var{p1},@var{p2}) |
@item urembymul(@var{p1},@var{p2}) |
@item urembymul_precomp(@var{p1},@var{p2},@var{inv}) |
@item urembymul_precomp(@var{p1},@var{p2},@var{inv}) |
@item ugcd(@var{p1},@var{p2}) |
@item ugcd(@var{p1},@var{p2}) |
:: $BB?9`<0$KBP$9$kA`:n(B |
\JP :: $B0lJQ?tB?9`<0$N=|;;(B, GCD |
|
\EG :: Division and GCD for univariate polynomials. |
@end table |
@end table |
|
|
@table @var |
@table @var |
@item return |
@item return |
$B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
|
\EG univariate polynomial |
@item p1,p2,inv |
@item p1,p2,inv |
$B0lJQ?tB?9`<0(B |
\JP $B0lJQ?tB?9`<0(B |
|
\EG univariate polynomial |
@end table |
@end table |
|
|
@itemize @bullet |
@itemize @bullet |
|
\BJP |
@item |
@item |
$B0lJQ?tB?9`<0(B @var{p1}, @var{p2} $B$KBP$7(B, |
$B0lJQ?tB?9`<0(B @var{p1}, @var{p2} $B$KBP$7(B, |
@code{udiv} $B$O>&(B, @code{urem}, @code{urembymul} $B$O>jM>(B, |
@code{udiv} $B$O>&(B, @code{urem}, @code{urembymul} $B$O>jM>(B, |
|
|
@code{urembymul} $B$O(B, @var{p2} $B$K$h$k>jM>7W;;$r(B, @var{p2} $B$N(B |
@code{urembymul} $B$O(B, @var{p2} $B$K$h$k>jM>7W;;$r(B, @var{p2} $B$N(B |
$BQQ5i?t$H$7$F$N5U857W;;$*$h$S(B, $B>h;;(B 2 $B2s$KCV$-49$($?$b$N$G(B, |
$BQQ5i?t$H$7$F$N5U857W;;$*$h$S(B, $B>h;;(B 2 $B2s$KCV$-49$($?$b$N$G(B, |
$B<!?t$,Bg$-$$>l9g$KM-8z$G$"$k(B. |
$B<!?t$,Bg$-$$>l9g$KM-8z$G$"$k(B. |
@item @code{urembymul_precomp} $B$O(B, $B8GDj$5$l$?B?9`<0$K$h$k>jM>(B |
@item |
|
@code{urembymul_precomp} $B$O(B, $B8GDj$5$l$?B?9`<0$K$h$k>jM>(B |
$B7W;;$rB??t9T$&>l9g$J$I$K8z2L$rH/4x$9$k(B. |
$B7W;;$rB??t9T$&>l9g$J$I$K8z2L$rH/4x$9$k(B. |
|
$BBh(B 3 $B0z?t$O(B, $B$"$i$+$8$a(B @code{ureverse_inv_as_power_series()} $B$K(B |
|
$B$h$j7W;;$7$F$*$/(B. |
|
\E |
|
\BEG |
|
@item |
|
For univariate polynomials @var{p1} and @var{p2}, |
|
there exist polynomials @var{q} and @var{r} such that |
|
@var{p1=q*p2+r} and the degree of @var{r} is less than that of @var{p2}. |
|
Then @code{udiv} returns @var{q}, @code{urem} and @code{urembymul} return |
|
@var{r}. @code{ugcd} returns the polynomial GCD of @var{p1} and @var{p2}. |
|
These functions are specially tuned up for dense univariate polynomials. |
|
In @code{urembymul} the division by @var{p2} is replaced with |
|
the inverse computation of @var{p2} as a power series and |
|
two polynomial multiplications. It speeds up the computation |
|
when the degrees of inputs are large. |
|
@item |
|
@code{urembymul_precomp} is efficient when one repeats divisions |
|
by a fixed polynomial. |
|
One has to compute the third argument by @code{ureverse_inv_as_power_series()}. |
|
\E |
@end itemize |
@end itemize |
|
|
@example |
@example |
|
|
@end example |
@end example |
|
|
@table @t |
@table @t |
@item $B;2>H(B |
\JP @item $B;2>H(B |
|
\EG @item References |
@fref{uinv_as_power_series ureverse_inv_as_power_series}. |
@fref{uinv_as_power_series ureverse_inv_as_power_series}. |
@end table |
@end table |