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version 1.1, 1999/12/08 05:47:44

version 1.3, 2003/04/19 15:44:59

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/upoly.texi,v 1.2 1999/12/21 02:47:34 noro Exp $

\BJP

@node $B0lJQ?tB?9`<0$N1i;;(B,,, $BAH$_9~$_H!?t(B

@section $B0lJQ?tB?9`<0$N1i;;(B

\BEG

@node Univariate polynomials,,, Built-in Function

@section Univariate polynomials

@menu

* umul umul_ff usquare usquare_ff utmul utmul_ff::

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* udiv urem urembymul urembymul_precomp ugcd::

@end menu

@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

\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}

@findex umul

@findex umul_ff

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@table @t

@item umul(@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

\EG :: Fast multiplication of univariate polynomials

@item usquare(@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

\EG :: Fast squaring of a univariate polynomial

@item utmul(@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)

\EG :: Fast multiplication of univariate polynomials with truncation

@end table

@table @var

@item return

$B0lJQ?tB?9`<0(B

\JP $B0lJQ?tB?9`<0(B

\EG univariate polynomial

@item p1 p2

$B0lJQ?tB?9`<0(B

\JP $B0lJQ?tB?9`<0(B

\EG univariate polynomial

@item d

$BHsIi@0?t(B

\JP $BHsIi@0?t(B

\EG non-negative integer

@end table

@itemize @bullet

\BJP

@item

$B0lJQ?tB?9`<0$N>h;;$r(B, $B<!?t$K1~$8$F7h$^$k%"%k%4%j%:%`$rMQ$$$F(B

$B9bB.$K9T$&(B.

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@item

@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

$B@Q$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

$B$,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.

Line 66 GF(2^n) $B78?t$NB?9`<0$r0z?t$K<h$l$J$$(B.

Line 81 GF(2^n) $B78?t$NB?9`<0$r0z?t$K<h$l$J$$(B.

@item

$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

$B7A<0$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,

$B@0?t$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,

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

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.

\BEG

@item

These functions compute products of univariate polynomials

by selecting an appropriate algorithm depending on the degrees

of inputs.

@item

@code{umul()}, @code{usquare()}, @code{utmul()}

compute products over the integers.

Coefficients in GF(p) are regarded as non-negative integers

less than p.

@item

@code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()}

compute products over a finite field. However, if some of

the coefficients of the inputs are integral,

the result may be an integral polynomial. So if one wants

to assure that the result is a polynomial over the finite field,

apply @code{simp_ff()} to the inputs.

@item

@code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()}

cannot take polynomials over GF(2^n) as their inputs.

@item

@code{umul()}, @code{umul_ff()} produce @var{p1*p2}.

@code{usquare()}, @code{usquare_ff()} produce @var{p1^2}.

@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}.

@item

If the degrees of the inputs are less than or equal to the

value returned by @code{set_upkara()} (@code{set_uptkara()} for

@code{utmul}, @code{utmul_ff}), usual pencil and paper method is

used. If the degrees of the inputs are less than or equall to

the value returned by @code{set_upfft()}, Karatsuba algorithm

is used. If the degrees of the inputs exceed it, a combination

of FFT and Chinese remainder theorem is used.

First of all sufficiently many primes @var{mi}

within 1 machine word are prepared.

Then @var{p1*p2 mod mi} is computed by FFT for each @var{mi}.

Finally they are combined by Chinese remainder theorem.

The functions over finite fields use an improvement by V. Shoup @code{[Shoup]}.

@end itemize

@example

Line 101 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B.

Line 156 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B.

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{set_upkara set_uptkara set_upfft},

@fref{kmul ksquare ktmul}.

@end table

@node kmul ksquare ktmul,,, $B0lJQ?tB?9`<0$N1i;;(B

\JP @node kmul ksquare ktmul,,, $B0lJQ?tB?9`<0$N1i;;(B

\EG @node kmul ksquare ktmul,,, Univariate polynomials

@subsection @code{kmul}, @code{ksquare}, @code{ktmul}

@findex kmul

@findex ksquare

Line 114 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B.

Line 171 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B.

@table @t

@item kmul(@var{p1},@var{p2})

:: $B0lJQ?tB?9`<0$N9bB.>h;;(B

\JP :: $B0lJQ?tB?9`<0$N9bB.>h;;(B

\EG :: Fast multiplication of univariate polynomials

@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

\EG :: Fast squaring of a univariate polynomial

@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)

\EG :: Fast multiplication of univariate polynomials with truncation

@end table

@table @var

@item return

$B0lJQ?tB?9`<0(B

\JP $B0lJQ?tB?9`<0(B

\EG univariate polynomial

@item p1 p2

$B0lJQ?tB?9`<0(B

\JP $B0lJQ?tB?9`<0(B

\EG univariate polynomial

@item d

$BHsIi@0?t(B

\JP $BHsIi@0?t(B

\EG non-negative integer

@end table

@itemize @bullet

\BJP

@item

$B0lJQ?tB?9`<0$N>h;;$r(B Karatsuba $BK!$G9T$&(B.

@item

Line 138 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B.

Line 202 Shoup $B$K$h$k%H%j%C%/$rMQ$$$F9bB.2=$7$F$"$k(B.

FFT $B$rMQ$$$?9bB.2=$O9T$o$J$$(B.

@item

GF(2^n) $B78?t$NB?9`<0$K$bMQ$$$k$3$H$,$G$-$k(B.

\BEG

These functions compute products of univariate polynomials by Karatsuba

algorithm.

@item

These functions do not apply FFT for large degree inputs.

@item

These functions can compute products over GF(2^n).

@end itemize

@example

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[37] kmul(A,B)$

@end example

@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

\EG @node set_upkara set_uptkara set_upfft,,, Univariate polynomials

@subsection @code{set_upkara}, @code{set_uptkara}, @code{set_upfft}

@findex set_upkara

@findex set_uptkara

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@item set_upkara([@var{threshold}])

@itemx set_uptkara([@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

\BEG

:: Set thresholds in the selection of an algorithm from N^2, Karatsuba,

FFT algorithms for univariate polynomial multiplication.

@end table

@table @var

@item return

$B@_Dj$5$l$F$$$kCM(B

\JP $B@_Dj$5$l$F$$$kCM(B

\EG value currently set

@item threshold

$BHsIi@0?t(B

\JP $BHsIi@0?t(B

\EG non-negative integer

@end table

@itemize @bullet

\BJP

@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@_Dj$9$k(B.

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$B$5$l$k(B. $B$3$N@ZBX$($N<!?t$r@_Dj$9$k(B.

@item

$B>\:Y$O(B, $B$=$l$>$l$N@Q4X?t$N9`$r;2>H$N$3$H(B.

\BEG

@item

These functions set thresholds in the selection of an algorithm from N^2,

Karatsuba, FFT algorithms for univariate polynomial multiplication.

@item

Products of univariate polynomials are computed by N^2, Karatsuba,

FFT algorithms. The algorithm selection is done according to the degrees of

input polynomials and the thresholds.

@item

See the description of each function for details.

@end itemize

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{kmul ksquare ktmul},

@fref{umul umul_ff usquare usquare_ff utmul utmul_ff}.

@end table

@node utrunc udecomp ureverse,,, $B0lJQ?tB?9`<0$N1i;;(B

\JP @node utrunc udecomp ureverse,,, $B0lJQ?tB?9`<0$N1i;;(B

\EG @node utrunc udecomp ureverse,,, Univariate polynomials

@subsection @code{utrunc}, @code{udecomp}, @code{ureverse}

@findex utrunc

@findex udecomp

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@item utrunc(@var{p},@var{d})

@itemx udecomp(@var{p},@var{d})

@itemx ureverse(@var{p})

:: $BB?9`<0$KBP$9$kA`:n(B

\JP :: $BB?9`<0$KBP$9$kA`:n(B

\EG :: Operations on polynomials

@end table

@table @var

@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

\EG univariate polynomial or list of univariate polynomials

@item p

$B0lJQ?tB?9`<0(B

\JP $B0lJQ?tB?9`<0(B

\EG univariate polynomial

@item d

$BHsIi@0?t(B

\JP $BHsIi@0?t(B

\EG non-negative integer

@end table

@itemize @bullet

\BJP

@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} $B$rJV$7(B, @code{udecomp()} $B$O(B [@var{p1},@var{p2}] $B$rJV$9(B.

@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.

\BEG

@item

Let @var{x} be the variable of @var{p}. Then @var{p} can be decomposed

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}.

Under the decomposition, @code{utrunc()} returns

@var{p1} and @code{udecomp()} returns [@var{p1},@var{p2}].

@item

Let @var{e} be the degree of @var{p} and @var{p}[@var{i}] the coefficient

of @var{p} at degree @var{i}. Then

@code{ureverse()} returns @var{p}[@var{e}]+@var{p}[@var{e}-1]x+....

@end itemize

@example

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@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{udiv urem urembymul urembymul_precomp ugcd}.

@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}

@findex uinv_as_power_series

@findex ureverse_inv_as_power_series

Line 245 return to toplevel

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@table @t

@item uinv_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

@table @var

@item return

$B0lJQ?tB?9`<0(B

\JP $B0lJQ?tB?9`<0(B

\EG univariate polynomial

@item p

$B0lJQ?tB?9`<0(B

\JP $B0lJQ?tB?9`<0(B

\EG univariate polynomial

@item d

$BHsIi@0?t(B

\JP $BHsIi@0?t(B

\EG non-negative integer

@end table

@itemize @bullet

\BJP

@item

@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

Line 268 return to toplevel

Line 388 return to toplevel

$B$KBP$7$F(B @code{uinv_as_power_series(@var{p1},@var{d})} $B$r7W;;$9$k(B.

@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.

\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^(@var{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()}.

@end itemize

@example

Line 286 x^10+x^9

Line 423 x^10+x^9

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{utrunc udecomp ureverse},

@fref{udiv urem urembymul urembymul_precomp ugcd}.

@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}

@findex udiv

@findex urem

Line 305 x^10+x^9

Line 444 x^10+x^9

@item urembymul(@var{p1},@var{p2})

@item urembymul_precomp(@var{p1},@var{p2},@var{inv})

@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

@table @var

@item return

$B0lJQ?tB?9`<0(B

\JP $B0lJQ?tB?9`<0(B

@item p1,p2,inv

\EG univariate polynomial

$B0lJQ?tB?9`<0(B

@item p1 p2 inv

\JP $B0lJQ?tB?9`<0(B

\EG univariate polynomial

@end table

@itemize @bullet

\BJP

@item

$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,

Line 324 x^10+x^9

Line 467 x^10+x^9

@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,

$B<!?t$,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.

$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.

\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()}.

@end itemize

@example

Line 348 x^10+x^9

Line 512 x^10+x^9

@end example

@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}.

@end table

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