A Binary Recursive Gcd Algorithm

(1)

HAL Id: inria-00071533

https://hal.inria.fr/inria-00071533

Submitted on 23 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents

A Binary Recursive Gcd Algorithm

Damien Stehlé, Paul Zimmermann

To cite this version:

Damien Stehlé, Paul Zimmermann. A Binary Recursive Gcd Algorithm. [Research Report] RR-5050, INRIA. 2002. �inria-00071533�

(2)

ISRN INRIA/RR--5050--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 2

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A Binary Recursive Gcd Algorithm

Damien Stehlé and Paul Zimmermann

N° 5050

Décembre 2003

(3)

(4)

Unité de recherche INRIA Lorraine

')(+*-,/.1032546.17"8:9;(+0"<>=?(A@B8?CD,:*-*E.GFH*I(A0B0

JLKNMHOQPSRUT VXWHYNZ[PX\^]`_aZ^bHZ[PH\

Pcbedf\^bHgh\ikj1Ol+]`\^Z[m1ghP

npok]rqkPctsGuvdfbHPHi

w?dauhux]`oycLzNP o{Pb|KNPHokb|KNP}YA~pafT tWbHPOlNo{PURa`a!T RaQuNda_`PHi

BerNa JLKhP!lhZ^Yvdaoyj)da\[_`]`okZ[cKhOZ[idadfo{ZdfYc]a"c{KNP pgNbH\[Z^zNPHdaY&da\^_a]`o{ZcKNOcKNdfc

uxPok/]`okOitDPH\[\Z^YuNodabcZ[bHPaxPuNokPHikPHYcSdm1gNdai{Z\^Z[YNPedfo?cZ[OPSo{PbHgNoki{Z[aPQda\^_`]ao{Z[c{KNO

cKvdfcbH]ÒQuNghc{PHicKNP)_ao{PedrcPHiyc bH]ÒQOQ]`YEzNZ[1Z[i{]òS]a?cD]¡Z^YcP_`PHokiljEi{Z[OgN\drcZ^Yh_;d

i{\^Z[_`Kc\j¢OQ]GzNZ£vPHz `Po{ikZ^]`Y]fhcKhP¤lNZ[Yvdaokj!df\^_`]`okZ[c{KNO¥JLKNP¤iycokgNbc{gNo{PD]aNcKhPDokPHbHgho{i{Z`P

da\^_`]ao{Z[c{KNO¦Z^i`PokjQbH\[]`i{PLc{] c{KNP§]`YNP]a¨c{KNPL¤P\^\©YN]6Y«ªXYghc{KGysGb|Kh¬`YNKvda_aP?:daiycD_abHz

da\^_`]ao{Z[c{KNO&lhghccKNP)zhPHi{bo{Z[uhcZ[]`Y®daYNzEcKNP)uho{]1]a§]atb]`o{okPHbcYNPi{idao{P)ikZ^_`YNZ£vbedfYc{\[j

i{Z^OQuN\[PHo§Z^Y¯]`gNo?bedaikPaJLKNZ[i¢\^PHdazNitc{])di{Z[Ouh\^Z[£Nbedfc{Z^]`Y&]apcKhPZ^OQuN\^POPYcdfcZ[]`Y¥daYNz

c] lxPc{c{PHoo{gNYhYNZ^YN_c{Z^OQPHiH

°¥±²¨³k´µ r¶"a VXbHzQbH]`OQuNghcdfcZ[]`Y5`pgNb\^Z[zNPedaYda\[_`]`o{ZcKNO&fª Yghc{KGys1b|KN¬`YNKvdf_`Ptdf\^_`]f

o{Z[c{KNO&hmgvdaikZ\[Z^YNPHdaoDc{Z^OQP¢bH]aOuN\[P·GZ[cja

(5)

da\[_`]ò{ZcKNOQP&lNZ^YvdfZô{PPHikc&gNYhP;adfo{ZdfYc{P¯zNP¡\ da\^_a]ò{ZcKNOQPz pgNb\^Z[zNP

uvdaokc{Z^bgN\^Z[MHo{POPYcPQbHdabHP§PHY uNo{dfcZ[m1ghPa§]`gNi zNWHbo{Z`]`YNi gNYda\[_`]`o{ZcKNOQP o{WHbgNo{ikZ[xzNP

bH]ÒQuN\^P·GZ[cWmgvdfi{Z \^Z[YNWedaZ[o{PmgNZ bHda\^bgN\^P \^P uN\[gNi _òdfYNzb]ÒOUgNY¡zNZ[1ZîkPHgNo!zNPQzNPHg1·

PHYcZ[PHo{iXPY¯i{Z[OgN\dfYctgNYNP\[WH_`Mo{POQ]GzhZ[£vbHdfcZ[]`Y¥zhP\ da\^_`]ao{Z[c{KNOQP!lhZ^YvdaZ[o{Pf dikc{o{gNb

cgNokPXzNP bHPc?da\[_`]`okZ[cKhOP§PHiyc§c{o{Mi§uho{]1b|KNP zNPXbHP\^\^PXzhg&bWH\[MHlNokPSdf\^_`]`okZ[c{KNOP§zNP bedf\^bHgh\

odauNZ[zNPLzNPLuN_abHzQzNP§ª Yghc{KPcDsGb|KN¬`YhKvda_`PfO«daZ[iid zNPi{bHokZ^uhc{Z^]`YQdaZ[YNi{ZGmgNP?i{dXuNo{Pgh`P

i{]`Yc YNPc{cPOPYc uN\^gNiikZ^OQuN\^PiH¤PH\^d d;u+]`gNoUbH]`YNikWHmgNPYNbHP¥gNYNP)i{Z[Ouh\^Z[£Nbedfc{Z^]`Y zNP

\ Z^OQuN\daYcdfcZ[]`YPc§zNP OPZ^\^\[PHgNokiDc{PHOQuNizNP beda\[bHgN\

µ ³ 3 da\[bHgN\?zNP¯uN_`bz5Lda\[_`]`o{ZcKNOQP&z pgNb\^Z[zNPa¤da\^_`]ao{Z[c{KNOQP¥zNP¡ªXYghc{KG

sGb|KN¬`YNKNda_`Pavb]`OQuN\^P·hZcWXmgvdaikZ\^Z^YhWedaZ[o{Pa

(6)

( )*,+#-/.1023+45.6*

V bzIb]`Ouhghc|dfc{Z^]`Y-Z[id¡bHPYcodf\cdai{©EZ[YIb]`OQuNghcPoQda\^_`PlNodGpZ^Y-uvdaoycZ^bgN\dfoKNPHY

bH]`OQuNghc{Z^YN_;]6aPHo odfc{Z^]`YNda\DYgNOUlxPo{i]`o]6`Po O]1zNgN\^daoSZ[YcPH_aPHo{iJLKNP)DPH\^\©YN]6Y

ghbH\^Z[zNPedfYdf\^_`]òkZ[c{KNO i{]`\`PHipc{KNZîpuNok]`lN\[PHO Z[YcZÔQPmgvdazhodfc{Z^bLZ^Yc{KNPi{Z87PL]a+cKNPZ^Y1

uNghci`s1P`Poda\1fdao{Z^daYci ]aNc{KNZ^i"da\^_a]`o{ZcKNO Kvde`Pl+PHPYuNo{Pi{PYc{PHzZ[YScKhP¤\dfikczNPHbHdazNPiH

9A]aoSP·NdfOuN\[Pa lj cda©Z^YN_&bHPYcPHokPHz-mgN]ac{Z^PHYc{iUZ^YhikcPHdaz ]aux]`ikZ[c{Z[`PQmgN]ac{Z^PHYc{iHB]aYNP

da\ôkPedazhj;_`Pc{iUdi{Z[_`YNZ[£NbedaYc!ZÔQuNo{]6aPHOQPHYc;::i{PP=<R$> /]ò!dzNPcdaZ^\[PHz daYvda\j1i{Zî ]apgG

bH\^Z[zNPedfY da\^_`]ao{Z[c{KNOQixZ[c{K zNZ[aPHo{ikPDzhZ[1Z^ikZ^]`YNi?A@YBDCFEah6ªXYghcK<$>uNo{Pi{PYc{PHz!cKNPp£vokikc

m1gNdai{Z\^Z[YNPedfo"cZ[OP¤_`bHzQda\[_`]`o{ZcKNO& lvdaikPHzQ]`Y:daikc Ogh\[cZ[uN\^Z[bedfc{Z^]`Y<GE$> aJLKNPbH]`OQuN\^P·

Z[cj ]aLcKhZîdf\^_`]òkZ[c{KNO¤daiZÔQuNok] aPHz lj sGb|KN¬`YhKvda_`PH<!IJ>Dc]LK;MONQPSRFTFUVNQP8RFTWPSRFT6NYX

Z

bH]ÒQuNokPHKNPYNi{Z`P zNPHikbHokZûhc{Z^]`Y)]a c{KNP ªXY1gGcKG sGb|KN¬`YNKNda_`P da\[_`]ò{ZcKNObHdaYlxP¢/]agNYNz

Z^YH<[BFB\> hJLKNPXb]ò{okPHbcYNPi{i]acKhZîLda\[_`]òkZ[cKhOZî¤mgNZcP¢KNdao{zc{]PHikcdalN\[Zî{K¨GPi{i{PYcZdf\^\[j

lxPbedagNikP ]a cKNPXcPb|KNYNZ[beda\5zhPc|dfZ^\^idao{]`ghYNzcKNP ik]fbHda\^\[PHz^] £h·gNuuNok]GbPHzNgNokP_a`NJLKNZ[i

]£h·ghu®uho{]1bHPHzhgNo{P_)Z^iUodfc{KNPHozNZQbHgN\cc]¡Z^OQuN\[PHOQPHYcdfYNz-O«da©fPHicKhPda\^_`]ao{Z[c{KNO

gNYNZ^Yc{PHo{Pikc{Z^YN_!gNYN\[PHi{ip/]`oaPHoyj\^dao{_aPYgNOl+PHo{i::]f¨\[PHYN_ac{KikZ^_`YhZ[£vbHdaYc\[jSKNZ[_`KNPHopc{KvdaY

bDcd lNZ[c{i?

@YcKNZ[iDuvdfuxPoH¤P?uNo{Pi{PHYcd fdao{Z^daYc¤]a¨cKNP?ªXY1gGcKG sGb|KN¬`YNKNda_`PXda\[_`]`okZ[cKhO cKNdfc

zN]GPiSYh]ac!Kvde`Pc{KNP]£h·ghu uNok]GbPHzNgho{P_Qzhodelvdab|©AJ"]dab|KNZ^P`PQcKNZ[iH5DPzNPi{bo{Z^l+P

dYhPzNZ1Z^i{Z[]`Ye: Vgf>/]`o!V PYNPHo{da\^ZS7HPzhfDZ^Yvdaoyji?¨KNZ[b|KbHdaY l+P i{PPHY dfiSdYvdfc{gNodf\

_`PHYNPoda\[Z87HdfcZ[]`Y¥]fpcKNPSlNZ^Yvdfokj&zNZ[1Z[i{Z^]aY5V@ cXzN]1PHitYh]acXi{PPHO ux]ai{i{Z[lN\^P!c]ghi{PUc{KNPlNZ

YvdaokjzNZ[1Z^ikZ^]`YZcikPH\[+Z[Yd!ªXYghcKG sGb|KN¬`YhKvda_`P\^Z[©aPda\[_`]`okZ[cKhO¥rlxPbedaghi{P§Z[cipzhP£vYNZcZ[]`Y

ZîQdaikj1OQOPcokZ^bjBZ[cQPH\[ZÔQZ^Yvdfc{PHi\[PedaiycQi{Z^_aYNZ[£vbHdaYcQlNZ[c{iQlhghc«Zcda\îk] b]`YNikZ^zNPo{iQO]ìyc

i{Z^_aYNZ[£vbHdaYclNZciUc{]¯uxPok/]`okO b]`OQuvdao{Z[i{]`YhiHBJLKhPHo{P)Z[iYN]¯ikgNb|K dfikj1OOQPc{okj Z^Yc{KNP

VkfzNZGZ[i{Z[]`Y5BJLKhP¥okPHbHgho{i{Z`PVkf}pgNb\^Z^zhPedaY®da\[_`]`o{ZcKNO Z[iOgNb|K ikZ^OQuN\^PoUc]¡zNP

i{bHokZ^l+PtdfYNz«c]SuNo{]6aP?cKNdaY«cKhP¢ªXYghcKG sGb|KN¬`YhKvda_`Ptda\^_`]ao{Z[c{KNO&`daYNz«Z[c Kvdai cKNP?idaOQP

daikj1OQuhc]ac{Z^b¢b]`OQuN\^P·hZcj`

JLKhZîi{Z[OuN\[Z[£vbHdfcZ[]`YQ]a"c{KNP zNPHikbHokZûhc{Z^]`Yc{gNo{YNi]`gGc§c{]lxP!daYZ[Ou+]òkcdaYcLdazGadfYG

c|da_`PUZ^Y;uNo{dabc{Z^bPjxDPZ^OQuN\[PHOQPHYcPzcKNP df\^_`]`okZ[c{KNO Z^Y¡V lmn"dfYNz¯Z[cto{PaPeda\[PHz

lxPc¤PPHY¡cKNokPHPQdaYNz;/]`ghotcZ[OPit:daikc{PHo¢c{KvdaY;cKhPZ^OQuN\^POPYcdfcZ[]`YNi§]a cKNP ªXYghc{KG

sGb|KN¬`YNKNda_`P§da\^_a]`o{ZcKNO Z^Y^m¥da_`OdXdaYNzmdfc{KNPHOdfcZ[bedhJ@YV lnm¥n"`cKhPda\^_`]ao{Z[c{KNO

gNi{Pz/]òD_abHzb]Òuhghc|dfc{Z^]`YhiD]a \^dao{_aP§Z[YcPH_aPHo{i¤Zî cKNP¢]aYNPt]a sG]ò{PYNi{]aYo<GpJ> KNZ^b|KZ[i

YN]ac mgvdai{Z\[Z^YNPHdaoHfJLKNPtbghc]q)ZcK«sG]`okPHYNik]`Y

ida\[_`]`o{ZcKNO¦Z^i daok]`gNYNzRa``XOdab|KNZ[YNP

rstVuwvyx{zAu}|W~\xtxx{Ou}J|WtOu}xtu}vyDxYv\}!{x,vAx~A}w/[auwtVuwvyDx\txxu5V\x

x\~\uwxY~$w

Y

t

v ¡

xA¡¢}tO£u}¡\8 \ uw~¢}DA5x{Ou}Ou

|W

(7)

¤]`okzNiH¨Z PaI ``«lhZ[ci Z i¢:dfo¢dai¢DP©YN]6 ]`gNo¢b]GzNPZ[itcKhP:daikc{PHikc _abHz¯o{]aghcZ[YNP

/]`oL\daok_`P¢Z[Yc{PH_`Po{i

JLKhP¤uvdau+PHoZ^i ]`ok_daYNZS7HPzUdai /]`\[\^]6iHD@YsGPHbcZ^]aYR¤PDZ^Ycok]GzNghbHP cKNP¤Vkf zNZ[1Z[i{Z^]aY

daYNz_aZ[`PQi{]aOPlvdai{Z[bokPHikgN\[c{iUdal+]`ghcSZ[cH @YEsGPbcZ[]`Y GDP«zNPi{bo{Z^l+P«uho{PHbZ^ikPH\[j¯c{KNP

YNP o{PbHgNoki{Z[aP da\^_a]`o{ZcKNO&`PXuho{]6`PXZcib]`o{okPHbc{YNPHikiZ^YsGPbc{Z^]`Y daYhzdaYvda\j7P¢Z[c{i

bH]ÒQuN\^P·GZ[cjXZ[YUs1PHbc{Z^]`YGrsG]ÒQP ZÔQuN\^POPYc|dfc{Z^]`Y¢Z[i{ikgNPHiBdao{P zNZîkbHgNiki{PzZ^YUsGPHbcZ[]`Y IG

¥µ 6h µ f JLKNP§YN]acdfcZ[]`YNi6K MXiMXpdaYhz;MX dfo{PikcdaYNzvdaokz5JLKNP§bH]`OQuN\^P·GZ[cjZ[i

OPHdai{gNokPHz;Z[YP\^PHOQPHYcdaokj]`u+PHo{dfcZ[]`YNi§]`YlhZ[cil§YN\^Pi{i?i{u+PHbHZ£vPHzP·GuN\^Z[bHZc\[jaAda\^\5c{KNP

\^]`_dfo{Z[c{KNOQidao{Ppcda©aPYUZ^Y!lvdaikPR1$@ ¢Z^idYN]`YG7HPHok]tZ[Yc{PH_`PoHFM#XzNPHYh]acPic{KNPD\[PHYN_fcK

]aBc{KNPUlNZ[Yvdaoyjo{PHuho{PHikPHYc|drcZ^]aY]avAZ PaFM#XP8RFT! " #%$ b '&M#X§zNPYN]ac{PHi?c{KNPR6

dazNZ^b fda\^gvdrcZ^]aYU]a(vaZ PffcKhPYgNOl+PHop]aAb]`YNi{PbHghc{Z[aP 7PHok]GPipZ^YUc{KNPL\^PHdaikcpikZ^_`YhZ[£vbHdaYc

lNZ[c{i§]fBcKhPSlNZ[Yvdaoyjo{PuNo{Pi{PYcdfcZ[]`Y;]a)+*AljzhP£vYNZcZ[]`Y5&M c X-, +.0/12«bHOQ]1z43

zNPHYN]fcPHic{KNPXbHPYcPHokPHz&o{PO«daZ[YNzNPo]a5OQ]1zNgN\^]636NZ Pa7.UidrcZ^iy£vPHi89:.OQ]Gz43?daYNz

;=<

U?>

.@

<U

` P§okPHbHda\^\hc{KvdfcBAMNYXC MNQP8RFT NQP8RFT6P8RTWNYXZ^ic{KNP§daikj1OQuhc]ac{Z^bc{Z^OQP

o{PHmgNZ[o{Pzc]XOgN\cZ[uN\[j¢cD]gN+ lNZ[cBZ^YcP_`PHokiBZ[c{KQsGb|KN¬`YNKNda_`P s1co{dai{ikPHYQOgN\cZ^uh\^Z^bHdfcZ[]`Y

:/DP!okP/Poc{] <B6$>/]òdbH]ÒQuN\^PcPXzNPi{bo{ZûGcZ^]aY&]fBlvdfi{Z^b dfo{Z[c{KNOQPcZ[bt]ù+PHo{dfcZ[]`YNi?

D EGFIHKJLHY*IH-NMO}4P(HY0Qh45*RM6-7SUTn4Vy4W#45.6*

@Y-c{KNZ^iikPHbcZ^]aY-DP£vo{iyc o{PHbHda\^\DcKNPlNZ[Yvdaokj da\^_a]`o{ZcKNO& JLKNPHY-¤PzNP£vYNPcKNP&Vgf

zNZ[1ZîkZ^]`Y&daYhz¥_`Z`P!i{]ÒQP!lvdai{Z[b uNo{]ù+PHoycZ[PHi?dalx]aghc§Z[cevsGgNlNikPHbc{Z^]`Y¯RG P·GuN\^daZ^Yhi§Kh]

c]tb]ÒQuNghcPDO]1zNgN\^daoZ^YaPHo{ikPHiB/ok]Ò>cKNPD]`ghc{uNghc"]avcKNP¤pgNbH\[Z^zNPHdaYda\[_`]òkZ[cKhO lvdaikPHz

]`YcKNP!Vgf zNZGZ[i{Z[]`Y51JLKNZîikgNlNikPHbc{Z^]`Y&Zî¤Z^YNzNPuxPYNzNPYcL/ok]Ò}cKNPXokPHOdaZ^YNzhPHo¤]fc{KNP

uvdau+PHolNghcZ[ivqkghikcZ£vPHzSlj!c{KNPD:dfbccKvdfc"bH]aOuNgGcZ^Yh_?OQ]GzhgN\daoZ[Y`PHoki{PipZ^id?iyc|daYNzNdao{z

dauNuN\[Z^bedrcZ^]aY]acKNP!pgNbH\[Z^zNPHdaY&da\^_a]`o{ZcKNO&

XZY\[ ]6^

± ) N ²`_ "\ ¶ ± \a µ b/

^

JLKNP¤lNZ[Yvdaoyj zNZ[1Z^ikZ^]`Y!o{P\^Z[PHi ]`YUc{KNPD/]`\[\^]6Z^Yh_LuNo{]`u+PHoycZ[PHi\jr_`bz Mdce+fgch3AXiCc_abHz M+fj3X

_`bHz Mcek$

b

fgce3X _`bHzYMdcek$

b

fj3XvdaYNz)_`bz MdchR$

b

fgce35$

b

Xi _`bHz Mce35$

b

fj

;

3AX

@ cXbH]`Yhi{Z^iyci¢]f PH\^Z[OZ[Yvdfc{Z^YN_c{KNP\^PHdaikc¢i{Z^_aYNZ[£vbHdaYc¢lNZc¢drcXPedab|K¯\[]G]auZ[c{PHo{dfcZ[]`Y5l?Po{P

ZîdzNPHikbHokZûhc{Z^]`Y]acKNP lNZ[Yvdaoyj)da\[_`]ò{ZcKNO j

JLKhPUl+PHKvde1Z^]ò§]a"cKNZ[itda\[_`]òkZ[cKhO ZîLaPHokj&DPH\^\ gNYhzNPHokikc]1]1z :/DPUokP/Po?c] <R$> /]ò

daYEde`Poda_`P)b]ÒQuN\^P·hZcj daYvda\jGikZî? Z \[c{KN]`gN_aKZ[cUZîSikcZ[\^\ mgvdazNo{dfcZ[bZ^Y cKNP«i{Z87P]a

(8)

v¢u5

"!$#&%('

) *

+

"!,#.-

m

0/

#

/213/

/x{¡Av

v¢u5 4+

#!$5-

6 m

#87:9

Ax¡Av

; m u}v¢

#

u}xYDw,xxv \xv x{¡Av=<2>

v¢u5 2,+

5?

<

!,#.?

< -

@ m

tVxxvuwv\

#

tV\6\xv x{¡Av

v¢u} 24+

?

<

!A#B-

§ m

tV\,uwv¢

#

tVxxv \xv x{¡Av

v¢u} 24+

C!,#.?

< -

D

FE

\x{

txx{¡v

v\u} 2 +$+

HGI#B-$?

<

!A#.-

J KMLK4N

\xY\v\u} PO#¡A!Axu}v1uw}\|

cKNPZ[YNuNghc{iHAcKNPo{PZ[i¢d«ikZ^_`YNZ£vbedfYct_daZ[Y¥c]6¤dao{zhi¢cKNPghi{gvda\BpgNbH\[Z^zNPHdaY¯da\[_`]`o{ZcKNO&

Z^YuvdaoycZ^bgN\dfo¤l+PHbedfgNi{P cKNPo{P Z^iLYh]YNPPHzc]QbH]`OQuNghc{P daYj)mgN]ac{Z^PHYcH

XZYX ]6^

± a ±7 ± AQ ± ¶I N ²`_ "\:¶ ± \a µ b/

^

@Y cKNPbHdai{P)]f§c{KNPiyc|daYNzNdao{z-pgNb\^Z^zhPedaYEzNZ1Z^i{Z[]`Y]fI¡lj 3 Z[c{KC " SR 3 B]aYNP

bH]`OQuNghc{PHidmgN]ac{Z^PHYcUTSi{gNb|KcKNdfcKNPYVTh3?Z[ii{ghlNikc{odabcPHzc]0v1cKNPXOQ]`ikci{Z^_aYNZ[£h

bedaYc?lNZ[c{i§]fi«fdaYNZ[i{K5vJLKNPUVgf zNZGZ[i{Z[]`Y)Z^iLcKhPSokP`Po{ikPj¨Z[Y¥]ao{zNPo§c{]Vgf zhZ[1Z^zNP

lj3§Z[cK &M\X%@:&M\3AXA]`YNP bH]`OQuNghc{PHi?dQmgN]fcZ^PYcWTikgNb|K¥c{KvdfcKNPYXT

<

UY[Z]\_^a`CY[Zb^

Z[i

dazNzNPz&c{] vNc{KNP \^Pedfikci{Z[_`YNZ£vbedaYclNZciL]a)fdaYNZ[i{K5

c ±

[ed2fhg i

$j : µ(lk

Ynm

% +fj3po "$8qsr$ " %{ # 3tW%& &M\X @ &M3Xu

v

&L%& wF8% ;#sx\V=yi "zQ%{ #

MsT f .X

& %&i%B{

| M b X-. $}T

<

U YZ]\^~`CY[Zb.^

|

McFX T @ c 5

<A$

|

Ms XK&M\.X=RL&M\3AX5u

v &^%5{ # T . 8= .y%O£ %&=,x\V"$%O{%g=%&L, q

'^# "z

M+fj3Xu

#A6,

M+fj3X

$k%&Uyi

MAT7f .Xu

"D"z uTXZ^ipP·hdabc\j

;

UY[Zb^

8M

<

UY[Z]\_^

X

8

bO]1z c 5

<A$

_&

1sGZ[YNbHPTXZ^ip]1zNz5`c{KNP§i{PHb]`YNz

bH]`YNzhZ[cZ[]`Y®Z^i/gh\[£v\[\^PHzIdaYNz _`Z[aPHi«c{KNP;gNYNZ[mgNPHYNPi{i)]anTG

Z

i)d bH]`YNikPHmgNPYNbHPf%.

$T

<

UY[Z]\_^a`"Y[Zb^

Z^iLghYNZ^mgNPf

(9)

I

PYN]6_`Z`P«cD]¡da\^_a]ò{ZcKNOQiXc];b]ÒuhghcP T¥daYNz .1JLKNP«£vokikc]`YhPZîSZîSc{KNP

PHmgNZ[fda\[PHYc ]a cKhPQYNdaZ[aPzNZ[1Z^ikZ^]`Y¡df\^_`]`okZ[c{KNO¥AdfYNz Z[iXdaiyjGOQuhc{]acZ[beda\[\[j&i{\[] DPHoXc{KvdaY

cKNPLikPHbH]aYNz«]`YhPaaKNZ[b|KZ[icKNPLPmgNZ[fda\[PHYc ]a+cKNP:daiyc zNZ[1ZîkZ^]`Yda\^_`]ao{Z[c{KNO&J9v]òBO]ìyc

]a5cKhPtuvdaZ[o{iM+fj3XcKhP¢pgNbH\[Z^zNPHdaYdf\^_`]`okZ[c{KNOlvdaikPHz)]`YQcKNP¢Vkf®zNZ[1Z[i{Z^]aYu+PHoy/]`o{OQi

da\ÔQ]ìkcda\^\`ZcizNZ1Zî{Z[]`YNi]`Y!uvdaZ[o{i M

f#Xx/]`o5KhZ^b|K&M#X

;

&M

XZ^i i{Oda\[\ JLKNZ^iPHYN\[Z^_`Kc{i

cKNPX:dabcLcKvdfcLc{KNPX£vo{iyc?da\^_`]ao{Z[c{KNOi{g QbHPi?Z[YuNodabcZ[bHPa

O#x|Wx{vOu5 2,

N

vAv5x vx{x{t

"!$#

tu5!tG !vA +

5- +

#B-

) *

+

! - 7

+

C!,#.-

7:9

7

6

\x +

*- +

#B-

A

;

7 G <

Y[Z ^~`CY[Zb.^

@

7 G <

Y[Z ^~`CY[Z]\_^

#

§

7 |W(<

Y[Z\^a`"Y[Zb^

7 \

!#"%$'&)(*!+"-,&/.

D ¤

x¡Av +

!*-

J

K104K

s!ww\|32

4554

#687:9<;>=

@ c§Z[iLbH\^PHdaocKNdfcLcKNPX/]`\[\^]6Z^Yh_o{Pi{gN\cKN]`\^zhi\j

c ± X Y v

& ! #"$ %&#'

_ ± ± 6N ²¨³ f g

"z@?6 # BA 8"$ %,z %&

y%

M+fj3X

\%O8Ay

FM#Xgf FM3X

> N

%&L% 68&QH%';

K;MON Md&M\3AX

;

&M#XXwXu

JLKhPi{PbH]`YNz df\^_`]òkZ[c{KNO b]ÒuhghcPiDCFE M+fj3XZ[YUmgvdaikZ \^Z^YhPedao5c{ZÔQPa Z^Y!cKNPDbedaikP¤]a

sGb|KN¬`YNKNda_`P s1co{dai{ikPHY;OUgN\[c{Z^uN\[Z^bedrcZ^]aY5ljgNi{Z[YN_ §PLc{]`Y i?ZcPHo{dfcZ[]`Y5/@ c§Z[i?dfzvdauhc{PHz

/o{]`O < >daYNz¤P!o{P/PHoLc] <J>5/]`oduNok]G]a"]acKNPX/]`\[\^]6Z^Yh_\^POOdh

c ± HG

Y v & ! #"$ %&#'JI N ³ f g

"zK?6 # ML 8"$ %;H%&$D"z

&/N$i&iD #q V% w$D '£%Oy£{%O{"$

z

3

\%8Azg%&"$%O{"$i

FM#Xgf FM3X

>

cJN &M3X

;

&M#X

> N

%&L%&68& H%';O

U A MONYX $ iMAMONXXu

Z Vgf pgNb\^Z[zNPedaY!da\[_`]`okZ[cKhO bHdaYUl+P zNPHokZ[`PzSaPHokj YNdfcgNo{da\^\jt/ok]`O c{KNPDzhP£vYNZcZ[]`Y

]a"c{KNPSVkf zNZ1Z^i{Z[]`Y5Gi{PHP 9BZ[_`gNo{P

(10)

uwt$

N

vAv5x vx{x{t

"!$#

tu5!tG !vA +

5- +

#B-

)

* +

! - 7

+

C!,#.-

73G

b

!+"-,&

7 \

!#"%$'&

7 4+

#.-G 4+

5-

.

6

7

;

w w|

G

@

7 . 5+

G@

-

|W(<

§

7 |W(<

D 7 . 5+

G@

-

|W=<

7 .

(

#

¤

x¡Av +

! -

J K!4K

s¥}\|

J 6"B#9;>=

N

v\wv5x{ !vxwx{t

"!$#

tu5!tG !vA +

5- +

#.-

) *$#

!+"%&

\x{x'&6t\xwxu5xt{|W|Wv,Atww

u}v¢

#

m # 7 9

Ax{¡v

b

!#"%,<&

6 +

!*-

7

+

"!$#.-

; ¤ x{¡v=

Sw +

#!*-

J

K)( KN

Ax

O#¡A!Axu}v1uw}\|

(11)

p

c ±

Y v & ,,V£O#L! #"$ %&#' "z ?6 # =8"$ %

^zntQD

%&1! #"$ %&#'

_ ± ± 6N ²¨³ f g

"z ?6 # A

%& z¢"$ yy#%

M+fj3X

\%O8AzJ#

M\#X f FM3X

> N

%068&QH%O'; K;MON

U

Xu

"D"z u Pck. :+f .

C3 f . U f LlxPXc{KNPXi{Pm1ghPHYNbPU]a"o{PO«daZ[YNzNPo{icKNdfc§dauNu+PedaoLZ[Y

cKNPSP·GPHbghcZ[]`Y¯]ac{KNPda\[_`]ò{ZcKNO&h PS£vokikcti{KN]6¦cKvdfc§cKhZîtikPHmgNPYNbHP Zî£vYNZcPUdaYNz

cKgNiLcKvdrc§c{KNP!da\^_a]`o{ZcKNOcPo{OQZ^YvdrcPHi

P!Kvde`PS/]ao?dfYj

c

.

U >

.

&

$K . vi{]c{Kvdfc0 .7

>

c

&

d

U

m¥]`o{P]6`PHo c

zNZ1Z^zNPik . 6KNZ^b|KS_`Z[aPHi"c{KNP Z^YNPm1gNda\^ZcZ^Pi5c >

.

> c

&

d

U

daYNz M b ; P8RFT

d

U

X

>

N"aJLKNPo{P/]`o{Pc{KNPHokPdao{P1K;MONXo{PO«dfZ^YNzNPo{i"Z^YUc{KNPLo{PO«daZ[YNzNPo

i{PHmgNPYNbHPf

Pc K;MONXQl+P¥cKhP;\^PYN_acK ]aXcKNP¥o{PO«daZ[YNzNPo«i{PmgNPHYNbPa§sGghuNux]ai{P

cKvdfc .tZ^ic{KNP¥\^daikc Yh]`YG7PHok]o{PO«dfZ^YNzNPoHW9"o{]`O POOd¡RGDP©YN]6cKNdfcQPedab|K

]a cKNP;bHda\^\[i«c]-d-Vgf zNZ[1Z^ikZ^]`Y®Z^Ya]`\[aPHidEYgNOl+PHo]a!lNZc)]`u+PHo{dfcZ[]`YNi«l+]`gNYhzNPHz

lj K MOPSRFT . £Md&M .

&

X ;

&M\.¢XXwX K MNWM&M\.

&

X ;

&M .DXwXXi{]¡cKvdrccKNP]6`Poda\[\

bH]`OQuN\^P·GZ[cjZ[iLlx]agNYNzNPz¥lj K;MONZ&M\.XwX

JLKhP¯bH]`oko{PHbcYNPi{ibH]aOPi«/o{]`O cKhP:dabc«cKvdfcZ[ _`bHz M\+fj3X«daYNz MAT7f .X

CME M\+fj3XNcKhPHY&_`bHz M\3 f .FXZ^i§dux]6DPHo]a cG

Z

lxPc{cPo lx]`ghYNz-]`YK . daYNzEcKgNiQ]`YUZ[iuNo{]6aPHz Z[YIsGPHbcZ^]aY 1SB` ?]`YNPcKNP\^Pi{i

cKNPuNokPHikPHYc!lx]`ghYNz¡Z^iXi{g bZ^PYc c{])_`gvdao{daYcPHPc{KNP mgvdai{Z\[Z^YNPHdao§c{Z^OQPdaiyjGOQuhc{]acZ[b

bH]`OQuN\^P·GZ[cj]fxc{KNP?okPHbgNo{ikZ[`P?da\[_`]`o{ZcKNO&rJLKNP?Z[Ouho{]6`Pzl+]`gNYNzQ]`YN\jzNPbHokPedaikPHiDc{KNP

OgN\[c{Z^uN\j1Z^YN_SbH]`YNiyc|daYc§]acKNP!daiyj1Ouhc{]acZ[bXbH]`OQuN\[P·GZ[cja

9v]`oN

b

daYhzXT! #"

; c $ b fgc ;

b%$

¤PSzNP£vYhP!c{KNPSOdfc{o{Z·&"T

$ ('

c c

c}T)

Pc.6 .

l+PcD]!YN]`YG7HPHok]SZ[YcPH_aPHo{ipZcK

c

2&M\.XB@ &M\.

XdaYhz .f . f . U f l+P

cKNPZ^oSVkf o{PO«dfZ^YNzNPoXi{Pm1ghPHYNbPadfYNz T f T U

f lxP cKhPb]`o{okPHi{u+]`YNzhZ^YN_&mgN]acZ[PHYciAj

.*

&

:.*

8

$ T%* +-,

U

YZ-

,

^a`CYZ- ,` ^

/]`oLdaYj/.0 b hJLKNPHYcKNPX/]`\[\^]6Z^Yh_Uo{P\dfc{Z^]`Y)Kh]`\^zNiD/]`o

daYj/.0

b j '

.*

&

) b

c

5

+-,

"T%*

$ , "]T

$ ' .

. ) f

KNPHokP N21%C&M .31AX

;

&M\.31

8

X4

b

/]`odaYj65

b

@YKvdfc/]`\[\^]6iH1¤P!gNikP!Z[Ouh\^Z^bZ[c{\[jc{KNPX/]`\^\[] Z[YN_ikZ^OQuN\^Pt:dfbci{P`Poda\ cZ[OPiH

(12)

c ±

Y m

% .

. o %st"="$8qsr$ " %{ # 3tW%&

c &M.X @ &M. X

.

. . U u"u"u"o

%& , '^# D*x\

u m

%

c uIv

&o%& wF8%

sx\ . c

& %&i% &M\.%*OX

>

@L&M\.*

&

X5u

XZY

G µ a µ ¶ /N j ± r ±

Pc+fj3¥l+P;c¤]-YN]aYG7PHo{]-Z[Yc{PH_`Po{iZcK c &M\#X @ &M\3AX§sGgNuNu+]`ikP;cKvdfc)DP

daYc c]b]`OuhghcPc{KNP Z^Y`Po{i{PQ]aB3UOQ]GzNgh\^]=vlj¯gNikZ^YN_&daY¡P·1c{PHYNzNPz aPHo{ikZ^]`Y ]a

cKNPpgNb\^Z[zNPedaY¡df\^_`]`okZ[c{KNOlvdaikPHz¡]`Y;c{KNPQVkf zNZ[1Z^ikZ^]`Y¨xJLKNPP·GPHbHgGcZ^]aY¯]a cKNP Vgf

ghbH\^Z[zNPedfYEda\^_a]`o{ZcKNO _aZ[`PiUcD]¯Z^YcP_`PHoki daYNz E i{gNb|KEcKNdfcI$ E93 c +

KNPHokP K MNYXdaYNz 4 _`bz M+fj3X9Bok]`Oi{gNb|K do{P\dfc{Z^]`Y¨vZ[c¢Z[itPHdaikj&c])b|KNPb|©

cKvdfc b ¨sGgNuhux]ìkPYN]6 cKvdfctcKNPZ[Y`PHoki{PE5]a 3!OQ]1zNgN\^]&zh]GPi¢P·GZîyce9Bo{]Ò

cKNP o{P\dfc{Z^]`Y !$ E93%2chN¤P ©YN]6 cKNdfc¢j

E E

c

OQ]Gz4+

JLKNPHokP/]òkPaZ^Y ]òkzNPHo)c{]®]alhc|daZ[Y ELZcZî)i{g bZ^PYcc{]Ib]ÒuhghcP;c{KNP¡Z^Y`Po{i{P ]a

I

N ³ f g ]a

9BZ^_`gho{PD ? ¤P¤]`lhc|dfZ^YSc{KNP¤Z[Y`PHoki{P]f OQ]1zNgN\^]IcNfJLKNZ^i"_`Z[aPHi«dfYNzXcKvdrcidrcZ^iyjj

!$Nc? b ¤\[Pedaok\[jZ^ic{KNP Z^Y`Po{ikPS]a cQO]1zNgN\[]v

s1Z^YNbP&OUgN\[c{Z^uN\[Z^bedrcZ^]aY5 ?PLc]`Y iQZ[c{PHodrcZ^]aY-daYNz-zNZ[1Z[i{Z^]aYE]`Y-Y1ghOlxPo{iQ]a?i{Z87P

K MONXbedaYlxP§uxPok/]`okOPzZ[YQc{Z^OQPK MAMONXX_`Z`PYcD] N+ lNZ[c Z[Yc{PH_`Po{idaYNz36c{KNP

dazNzNZcZ^]aYvda\bH]`iyc!c{]bH]`OQuNghc{PcKNPZ[YaPHoki{PQ]a 3!O]1zNgN\[]=_`Z`PY¡c{KNP ]`ghc{uNghc ]a¤daY

P·1cPYNzNPHzpgNbH\[Z^zNPHdaY&da\^_`]ao{Z[c{KNOlvdai{Pz¥]aY&c{KNP!Vkf zNZGZ[i{Z[]`YZ^iK;MdAMNYXwX

EGFIH HY32 -W#4 VH O .W-i4+7F

PYN]6 zNPHikbHokZ^l+PcKNPokPHbgNo{ikZ[`PQda\[_`]`okZ[cKhO lvdai{Pz¡]`Y¯cKNPQVgf zNZ[1Z[i{Z^]aY5AJLKNZ^iXzNP

i{bHokZ^uhc{Z^]`Yb\^]`ikPH\[jo{PHikPHOlh\^PHicKhPS]`YhP!]f1<[BFB\> @ c?ghi{PHi§c¤]o{]`gGcZ^YhPHi\jNc{KNPSda\^_`]ao{Z[c{KNO

³ f g ³ ax¶ dfYNz c{KNP da\^_`]ao{Z[c{KNO I v ³

fhg

³ a+¶ ?V Z`PHY cD] YN]`Y17HPo{]IZ^YcP

_`PHoki=.dfYNz .

Z[cK

c &M\.X @ &M\.

XDc{KNP da\^_`]ao{Z[c{KNO ³ f g ³ a+¶ ]`ghcy

uNghci?cKNP Vgf okPHOdaZ^YhzNPHokiI.* dfYNz .*

&

]fpcKNPVkf okPHOdaZ^YhzNPHo?i{PHmgNPYNbHP ]aM\. f . X

cKvdfc¢idfc{Z^ik j &M.*OX

>

FM\.X c=@ &M\.*

&

X@ cXda\^ik])]`ghcuhghci¢c{KNPbH]`oko{Pi{u+]`YNzNZ[YN_)O«d

co{Z· c

! " " ,

"T%*$ , "]T

$

hJLKhPHY)¤PXzhPHi{bo{Z[lxP¢c{KNPXda\[_`]`o{ZcKNO I v ³

fhg

³ a+¶

i{gNbbHPHiki{Z`PUbHda\^\[i¤c] cKhPSdf\^_`]`okZ[c{KNO# ³ f g ³ ax¶

(13)

B6

G

Y\[

³ f g ³ ax¶

lNZ[c{i]a)«daYNz3tZ[iPH\[Z^OQZ^YvdrcPHz«ljzN]`Z^Yh_ d okPHbgNo{ikZ[`P bHda\^\x]`Y)cKNP \^]6LFM#X ec]fc{KNP

lNZ[c{i]a? daYhz 36JLKhPHY cKNP;cD]-bH]`oko{PHikux]aYNzNZ^Yh_Eo{PHOdaZ[YNzNPHoki M f 3 X]a cKNP Vgf

Z i{Z[YN_`\^P«ikcPu-]acKNP&VgfpgNbH\[Z^zNPHdaY-da\^_a]`o{ZcKNO Z[iuxPok/]`okOPzE]`Y M fj3 XKNZ^b|K

_`Z[aPHid YNPo{PHOdaZ[YNzNPHo uvdfZ^o M3 f .FXpJLKNPHY c{KNPHokPZ[i«d¡i{PbH]`YNz o{PHbgNo{ikZ[aP;beda\[\L]`Y

dauNuNok]6·GZ[O«dfc{PH\j FM\X c ]a cKNP¯\[Pedaiyci{Z[_`YNZ[£NbedaYc)lNZci)]a;M\3 f .FX¤JLKNP¡ikZ87HP¯]f!c{KNP

Z^YNuNgGciX]a cKNZ[iXi{PbH]`YNz o{PbHgNoki{Z[aPbHda\^\"Z^iXikZ^OQZ^\dfo§c{]c{KNP ]`YNP ]a cKNP£vokikc okPHbHgho{i{Z`P

beda\[\ a9BZ[Yvda\^\j`Hc{KNP¤b]`o{okPHikux]`YhzNZ^YN_to{PHOdaZ[YNzNPHokiyM

f X]avc{KNPLVgf okPHOdaZ^YNzhPHoi{PHmgNPYNbHP

U

fdfYNzSc{KNPD]aghcuNgGcBOdfcokZ·

Z^ipbHda\^bgN\drcPHz/ok]`O

U

daYNzc{KNP¢VgfEmgN]acZ[PHYc ]a M fj3 X9BZ^_agNo{P§XZ^\[\^gNiycodrcPHi

cKNP P·GPbHghc{Z^]`Y&]acKNZ[ida\^_`]ao{Z[c{KNO&

§]acP?cKvdfcZ^Y«c{KNP¢zNPi{bHokZ^uhc{Z^]`Y)]fcKNP da\[_`]`okZ[cKhO

³ f g ³ a+¶

Z^Y 9BZ^_`gho{P IGNd

o{]`ghc{Z^YNP!Vgf Z[iLgNi{Pz5AJLKNZîZ[id ikZÔQuN\^PtO]1zNZ[£Nbedfc{Z^]`Y]f"c{KNP zNZ[1ZîkZ^]`Y&VgfjN_aZ[`PY

daYNz3¢daiZ^YhuNghcLZ[c{K c 2&M#X @ &M3XvZ[c]`ghc{uNghc{ic{KNPHZ[o§Vkf mgN]ac{Z^PYc T1vdaYNz +

U Y[Z]\_^

Z[5.Z^ic{KNPHZ[o§Vkf okPHOdaZ^YNzhPHoH

G YX I N ³ f g ³ ax¶

JLKNP!da\^_`]ao{Z[c{KNO

I

v ³ f g ³ a+¶ gNi{Pi?ikP`Poda\c{Z^OQPHi¤c{KNP!da\^_`]ao{Z[c{KNO# ³ f g ³ a+¶

JLKhPtOdaZ^YQdazhfdaYcda_`Pc]6dfo{zNi cKNP?]acKNPoDmgvdfi{Z \^Z[YNPedaoc{ZÔQPda\^_a]ò{ZcKNOQipZîpcKNdfc

Z[ dOdfcokZ·}Z^itokPc{gNo{YNPz¡lj¯do{PHbgNo{ikZ[aPbeda\[\B]apc{KNP da\^_a]`o{ZcKNO ³ fhg ³ a+¶

cKNPi{PHmgNPYNbHP)]fLmgN]acZ[PHYciBJLKNPgNYNzNPo{\jGZ[YN_okPedaik]`Y Z[i!c{KvdfcUc{KNP«okPHOdaZ^YhzNPHokiUdaokP

i{KN]`oycPYNPHzljQcKNP§\^PHdaikcDi{Z[_`YNZ£vbedaYcDlNZ[c{iH1daYNzi{Z[YNbHP§cKNP?bedaoko{Z^PiD_a]UZ^Y cKhPtzNZ[o{PHbcZ[]`Y

]acKhP«OQ]ìkcSi{Z[_`YNZ[£NbedaYcSlNZ[c{iHcKNPi{P«cD]¯uNKNPYN]ÒQPHYvd&zN];YN]fcUZ[Yc{PHoy/PHo{Pf 9v]ò cKNdfc

o{Pedfi{]`Y5 cKhP«da\[_`]òkZ[cKhO Zî!daiSi{ZÔQuN\[Pdfi!c{KNPªXYghcK1ysGb|KN¬aYNKvda_`Pda\^_`]ao{Z[c{KNOZ[Y c{KNP

bedaikP!]fBu+]`\[j1YN]aOZ^da\^i

.W-i-NHY3+*IHZW W . +7FIH `HY32-NW4 VH O6.W-/4+7F

@Y&c{Kvdfc§i{PbcZ[]`Y5NDPSikKN]6 cKNdfccKNP!da\[_`]`o{ZcKNO

I

v ³ f g ³ a+¶ ]a 9"Z^_`gNokPkE Z[iLbH]`o

(14)

GB Division

Recursive Call Truncation

R1 x

Truncation

Recursive Call

R2 x

n 3n/4 n/2 n/4 0

c,d a’,b’

b’,r a,b

c2,d2 b’0,r0

c1,d1 a0,b0

R1

R2

(15)

B R

P[

uw

B *4"!$#

tu5!tG !vA

9H7 +

5- 4+

#.-

)

svvxxA¢uwv,!vxwx{|6u}z uwv\,

vxx{tYu}v¢ Wt¡\O

¢u5

7 < `

#

\uwv\

7 < w

7 < u}x\x¥

vAtx¡Ax

x|6uwv\Ax{t}F\x

x|6u}v¢AxVtx¡\xvAxw +

C!,#.-

¢u5Vtu}t

4+ -

+

5-$?

< 4+ - u}v¢

9H7 4+

-

+

-

7

+

5-$?

<

6 m +

#B- 1

AAxv,x¡Av

92!

9

!A"!,#

;

7

?

<

@ 7:

< .

!

#

7:#

< .

#"

9 ! *!$#"

<

§ ! ! ! 7

6'9;>=9$

+ !$# -

D

#

7 <

$`

# .

>

m +

#- . 1

\x{v,x{¡v$

!

!,

!$#

+

!*-

7

+

!,#

-

% 7 4+

# -

&

7 G

+'4+

# - . -

¨

\ !+"-$

& 7 # < ('

. # 7

< )' .

*

9 # ! *

< ('

! ! ! 7

6'9;>=9$

+#

! -

6 7 < )'

$`

+' #

. >

; ¤ x{¡Av,

. . !

#-

*.

+/

! !

J K#04K

swA|

6'9;>=9$

[ u}t$

B *4"!$#

tu5!tG !vA

9H7 +

5- +

#.-

) *

& 7

+

"!$#.-

m

#07 9

Ax{¡Av

6 !

!,

!$#

7

6'9;>=9$

+

"!$#.-

; ¤

x{¡v & 7 J 6 "B#9; = 9$ +

!$#

-

J K#1CK

s¥ww\|

J 6"B#9;>=9$