%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: TechRep.dvi %%Pages: 33 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: MSBM10 %%EndComments %DVIPSCommandLine: dvips TechRep %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1998.10.08:0918 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /lessornotequal put dup 162 /greaterornotequal put dup 163 /notlessequal put dup 164 /notgreaterequal put dup 165 /notless put dup 166 /notgreater put dup 167 /notprecedes put dup 168 /notfollows put dup 169 /lessornotdbleql put dup 170 /greaterornotdbleql put dup 173 /notlessorslnteql put dup 174 /notgreaterorslnteql put dup 175 /lessnotequal put dup 176 /greaternotequal put dup 177 /notprecedesoreql put dup 178 /notfollowsoreql put dup 179 /precedeornoteqvlnt put dup 180 /followornoteqvlnt put dup 181 /lessornotsimilar put dup 182 /greaterornotsimilar put dup 183 /notlessdblequal put dup 184 /notgreaterdblequal put dup 185 /precedenotslnteql put dup 186 /follownotslnteql put dup 187 /precedenotdbleqv put dup 188 /follownotdbleqv put dup 189 /lessnotdblequal put dup 190 /greaternotdblequal put dup 191 /notsimilar put dup 192 /notapproxequal put dup 193 /upslope put dup 194 /downslope put dup 195 /notsubsetoreql put dup 196 /epsiloninv put dup 0 /lessornotequal put dup 1 /greaterornotequal put dup 2 /notlessequal put dup 3 /notgreaterequal put dup 4 /notless put dup 5 /notgreater put dup 6 /notprecedes put dup 7 /notfollows put dup 8 /lessornotdbleql put dup 9 /greaterornotdbleql put dup 10 /notlessorslnteql put dup 11 /notgreaterorslnteql put dup 12 /lessnotequal put dup 13 /greaternotequal put dup 14 /notprecedesoreql put dup 15 /notfollowsoreql put dup 16 /precedeornoteqvlnt put dup 17 /followornoteqvlnt put dup 18 /lessornotsimilar put dup 19 /greaterornotsimilar put dup 20 /notlessdblequal put dup 21 /notgreaterdblequal put dup 22 /precedenotslnteql put dup 23 /follownotslnteql put dup 24 /precedenotdbleqv put dup 25 /follownotdbleqv put dup 26 /lessnotdblequal put dup 27 /greaternotdblequal put dup 28 /notsimilar put dup 29 /notapproxequal put dup 30 /upslope put dup 31 /downslope put dup 32 /notsubsetoreql put dup 33 /notsupersetoreql put dup 34 /notsubsetordbleql put dup 35 /notsupersetordbleql put dup 36 /subsetornotdbleql put dup 37 /supersetornotdbleql put dup 38 /subsetornoteql put dup 39 /supersetornoteql put dup 40 /subsetnoteql put dup 41 /supersetnoteql put dup 42 /notsubseteql put dup 43 /notsuperseteql put dup 44 /notparallel put dup 45 /notbar put dup 46 /notshortbar put dup 47 /notshortparallel put dup 48 /notturnstile put dup 49 /notforces put dup 50 /notsatisfies put dup 51 /notforcesextra put dup 52 /nottriangeqlright put dup 53 /nottriangeqlleft put dup 54 /nottriangleleft put dup 55 /nottriangleright put dup 56 /notarrowleft put dup 57 /notarrowright put dup 58 /notdblarrowleft put dup 59 /notdblarrowright put dup 60 /notdblarrowboth put dup 61 /notarrowboth put dup 62 /dividemultiply put dup 63 /emptyset put dup 64 /notexistential put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /hatwide put dup 92 /hatwider put dup 93 /tildewide put dup 94 /tildewider put dup 96 /Finv put dup 97 /Gmir put dup 102 /Omegainv put dup 103 /eth put dup 104 /equalorsimilar put dup 105 /beth put dup 106 /gimel put dup 107 /daleth put dup 108 /lessdot put dup 109 /greaterdot put dup 110 /multicloseleft put dup 111 /multicloseright put dup 112 /barshort put dup 113 /parallelshort put dup 114 /integerdivide put dup 115 /similar put dup 116 /approxequal put dup 117 /approxorequal put dup 118 /followsorequal put dup 119 /precedesorequal put dup 120 /archleftdown put dup 121 /archrightdown put dup 122 /Digamma put dup 123 /kappa put dup 124 /k put dup 125 /planckover2pi put dup 126 /planckover2pi1 put dup 127 /epsiloninv put dup 128 /notsubsetoreql put dup 160 /space put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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(public)e(k)m(ey)j(cryptosystems.)58 b(W)-8 b(e)37 b(also)f(giv)m(e)g (practical)118 2860 y(results)31 b(of)i(an)f(implemen)m(tation.)46 b(By)33 b(comparing)f(it)g(to)h(other)g(public)c(domain)j(pac)m(k)-5 b(ages)34 b(w)m(e)f(can)g(sa)m(y)118 2973 y(that)41 b(w)m(e)g(ha)m(v)m (e)h(the)f(fastest)h(in)m(teger)f(division)c(on)k(a)g(SP)-8 b(AR)m(C)40 b(arc)m(hitecture)i(compared)e(to)i(all)d(other)118 3085 y(in)m(teger)31 b(pac)m(k)-5 b(ages)32 b(w)m(e)f(kno)m(w)f(of.)259 3198 y(In)38 b(general,)i(in)m(tegers)e(of)h(arbitrary)d(length)i(are)g (represen)m(ted)g(in)m(ternally)f(b)m(y)g(com)m(bining)g(sev)m(eral)118 3311 y(in)m(tegers)31 b(in)m(to)g(one)g(large)g(v)m(ector,)i(just)d(as) h(\(in)f(C)g(notation\))i(a)f Fx(long)f FB(ma)m(y)h(consist)f(of)h(t)m (w)m(o)i(consecutiv)m(e)118 3424 y Fx(short)p FB(s)f(on)h(the)g (underlying)e(mac)m(hine.)49 b(All)32 b(algorithms)g(on)h(large)g(in)m (tegers)h(can)f(then)g(b)s(e)g(p)s(erformed)118 3537 y(on)39 b(these)h(v)m(ectors.)68 b(So)39 b(let)g(us)f(in)m(tro)s(duce)g (some)i(imp)s(ortan)m(t)e(terms)h(for)g(this)f(situation.)65 b(A)40 b Fw(digit)47 b FB(is)118 3650 y(the)36 b(largest)g(in)m(teger)h (unit)d(that)j(can)f(b)s(e)f(handled)f(directly)h(b)m(y)h(our)f(mac)m (hine's)h(hardw)m(are.)57 b(So,)37 b(in)e(C)118 3763 y(notation,)j(a)f(digit)e(is)g(an)h Fx(unsigned)46 b(long)g(int)p FB(.)58 b(A)m(t)37 b(the)f(time)g(of)g(this)g(writing,)g(it)f(is)h(b)s (est)f(to)i(ha)m(v)m(e)118 3876 y(a)d(32-bit)g(in)m(teger)h(in)d(mind)g (when)h(sp)s(eaking)f(of)i(a)g(digit.)50 b(No)m(w)35 b(with)d FA(\014)c Fv(\000)22 b FB(1)34 b(de\014ned)f(as)h(the)g (maximal)118 3989 y(v)-5 b(alue)38 b(of)h(a)h(digit,)g FA(\014)k FB(is)38 b(the)h Fw(b)-5 b(ase)46 b FB(of)39 b(our)f(n)m(um)m(b)s(er)g(system.)66 b(This)37 b(means)i(that)g(the)g FA(n)p FB(-digit)f(v)m(ector)118 4102 y([)p FA(a)191 4116 y Fu(n)p Ft(\000)p Fy(1)329 4102 y FA(;)15 b(a)417 4116 y Fy(1)456 4102 y FA(;)g(:)g(:)g(:)i(;)e(a)706 4116 y Fy(0)746 4102 y FB(])771 4117 y Fu(\014)847 4102 y FB(represen)m(ts)28 b(the)g FA(n)p FB(-digit)f(n)m(um)m(b)s(er)g FA(A)f FB(=)f FA(a)2289 4116 y Fu(n)p Ft(\000)p Fy(1)2426 4102 y FA(\014)2482 4069 y Fu(n)p Ft(\000)p Fy(1)2635 4102 y FB(+)16 b FA(:)f(:)g(:)i FB(+)f FA(a)2979 4116 y Fy(1)3019 4102 y FA(\014)21 b FB(+)16 b FA(a)3226 4116 y Fy(0)3293 4102 y FB(with)27 b(resp)s(ect)118 4215 y(to)k(the)g(base)f FA(\014)5 b FB(.)41 b(W)-8 b(e)32 b(will)27 b(omit)k FA(\014)k FB(in)29 b(this)g(notation)i(if)e(the)i(base)g(results)e (from)h(the)g(con)m(text.)259 4327 y FA(\014)g FB(is)23 b(mac)m(hine)h(dep)s(enden)m(t)g(and)g(has)g(gro)m(wn)g(successiv)m (ely)g(in)f(the)i(last)f(30)h(y)m(ears)g(from)f(2)3301 4294 y Fy(8)3365 4327 y FB(o)m(v)m(er)i(2)3599 4294 y Fy(16)3699 4327 y FB(to)118 4440 y(2)163 4407 y Fy(32)238 4440 y FB(,)k(and)e(will)e(b)s(e)j(2)806 4407 y Fy(64)910 4440 y FB(on)g(all)f(fast)h(mac)m(hines)f(in)g(the)h(near)g(future.)39 b(With)29 b(the)g(assumption)e(that)j FA(\014)k FB(will)118 4553 y(alw)m(a)m(ys)27 b(b)s(e)f(a)h(p)s(o)m(w)m(er)g(of)g(t)m(w)m(o,)i FA(\015)h FB(:=)1351 4485 y Fv(p)p 1427 4485 57 4 v 68 x FA(\014)i FB(is)25 b(a)i(n)m(um)m(b)s(er)f(with)f(half)h(as)g(man)m (y)h(bits)f(as)h FA(\014)k FB(and)26 b(will)e(b)s(e)i(called)118 4666 y(the)34 b Fw(halfb)-5 b(ase)p FB(.)51 b(By)33 b(a)h Fw(halfdigit)43 b FB(w)m(e)34 b(mean)f(a)h(digit)e(whose)h(most)h (signi\014can)m(t)e(half)g(is)h(zero,)i(or)e(in)f(other)118 4779 y(w)m(ords,)j(a)g(digit)e(that)i(is)e(less)h(than)g FA(\015)5 b FB(.)53 b(A)34 b(n)m(um)m(b)s(er)f FA(A)i FB(is)e(said)g(to)i Fw(normalize)-5 b(d)47 b FB(if)33 b(its)h(most)h(signi\014can)m(t)118 4892 y(digit)29 b FA(a)378 4906 y Fu(n)p Ft(\000)p Fy(1)544 4892 y FB(is)g(greater)i(or)e (equal)h(than)f FA(\014)5 b(=)p FB(2,)31 b(i.e.,)f(if)f(it)g(has)g(a)h (leading)f(1)h(in)e(its)h(binary)f(represen)m(tation.)118 5005 y(If)d FA(X)32 b FB(=)25 b([)p FA(x)484 5019 y Fu(n)p Ft(\000)p Fy(1)622 5005 y FA(;)15 b(:)g(:)g(:)h(;)f(x)875 5019 y Fy(0)915 5005 y FB(])25 b(and)g FA(Y)45 b FB(=)25 b([)p FA(y)1401 5019 y Fu(m)p Ft(\000)p Fy(1)1557 5005 y FA(;)15 b(:)g(:)g(:)i(;)e(y)1804 5019 y Fy(0)1843 5005 y FB(])26 b(w)m(e)f(denote)h(with)d([)p FA(X)r(;)15 b(Y)22 b FB(])j(the)g Fw(c)-5 b(omp)g(osition)30 b(of)e FA(X)35 b Fw(and)118 5118 y FA(Y)20 b FB(,)37 b(i.e.,)f(the)f FA(n)23 b FB(+)g FA(m)p FB(-digit)34 b(n)m(um)m(b)s(er)g([)p FA(x)1503 5132 y Fu(n)p Ft(\000)p Fy(1)1640 5118 y FA(;)15 b(:)g(:)g(:)i(;)e(x)1894 5132 y Fy(0)1934 5118 y FA(;)g(y)2019 5132 y Fu(m)p Ft(\000)p Fy(1)2176 5118 y FA(;)g(:)g(:)g(:)h(;)f(y)2422 5132 y Fy(0)2462 5118 y FB(].)55 b(Let)35 b FA(K)7 b FB(\()p FA(n)p FB(\))33 b(:=)g FA(c)3144 5132 y Fu(K)3236 5118 y Fv(\001)24 b FA(n)3340 5085 y Fy(log)12 b(3)3517 5118 y FB(denote)118 5231 y(the)31 b Fw(Kar)-5 b(atsub)g(a)35 b(multiplic)-5 b(ation)34 b(time)k FB(for)30 b(some)h(con)m(v)m(enien)m (t)g(constan)m(t)h FA(c)2711 5245 y Fu(K)2779 5231 y FB(.)1926 5841 y(1)p eop %%Page: 2 6 2 5 bop 118 399 a Fz(2)135 b(Division)46 b(with)f(remainder)118 601 y FB(In)36 b(this)g(section)h(w)m(e)g(deal)f(with)g(the)g(problem)g (to)h(\014nd)e(for)i(non-negativ)m(e)g(in)m(tegers)g FA(A)g FB(and)f FA(B)42 b FB(a)37 b(rep-)118 714 y(resen)m(tation)e FA(A)e FB(=)f FA(Q)23 b Fv(\001)g FA(B)28 b FB(+)23 b FA(R)35 b FB(with)e(the)i(\(in)m(tegral\))g(quotien)m(t)g FA(Q)e FB(=)f Fv(b)p FA(A=B)5 b Fv(c)35 b FB(and)f(remainder)f FA(R)j FB(with)118 827 y(0)26 b Fv(\024)f FA(R)h(<)f(B)5 b FB(.)118 1071 y Fs(2.1)112 b(Dividing)36 b(t)m(w)m(o)g(digits)g(b)m (y)i(one)g(digit)118 1242 y FB(The)c(w)m(ell)f(kno)m(wn)h(sc)m(ho)s(ol) g(metho)s(d)f(for)h(dividing)d(in)m(teger)j(n)m(um)m(b)s(ers)f(uses)h (as)g(basic)g(step)g(the)g(division)118 1355 y(of)j(a)h(t)m(w)m(o)h (digit)d(n)m(um)m(b)s(er)f(b)m(y)i(a)h(one)g(digit)d(n)m(um)m(b)s(er)h (to)i(mak)m(e)h(a)e(guess)g(ab)s(out)g(the)h(next)f(digit)f(of)h(the) 118 1468 y(quotien)m(t)g([5)q(].)60 b(In)36 b(elemen)m(tary)h(sc)m(ho)s 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b(to)i(compute)f(the)g(desired)f (result.)48 b(Supp)s(ose)31 b(further)h(that)i(the)f(mac)m(hine)g(has)g (no)g(built-in)118 2371 y(in)m(teger)g(division)d(instruction)h(and/or) h(that)h(w)m(e)h(w)m(an)m(t)f(to)h(write)d(p)s(ortable)h(C/C++)g(co)s (de)h(so)g(that)g(w)m(e)118 2484 y(cannot)h(use)f(mac)m(hine)g (instructions)e(\(lik)m(e)i Fx(UDIV)f FB(on)i(a)f(SP)-8 b(AR)m(C)34 b(Ultra\))f(that)h(are)g(explicitly)d(designed)118 2597 y(for)42 b(this)f(task.)76 b(Then)41 b(w)m(e)i(need)e(a)i (function)e Fx(DivTwoDigitsByOne)c FB(based)42 b(on)f(C/C++'s)h(in)m (ternal)118 2710 y Fx(unsigned)k(int)29 b FB(arithmetic.)40 b(Y)-8 b(ou)30 b(ma)m(y)h(w)m(onder)e(wh)m(y)h(w)m(e)h(b)s(egin)d(with) h(suc)m(h)h(a)g(basic)f(problem)g(but)g(as)118 2823 y(y)m(ou)i(will)d (so)s(on)i(see)h(its)e(solution)g(will)f(expand)i(to)h(a)g(fast)f (recursiv)m(e)g(algorithm)g(for)g(dividing)d(arbitrary)118 2936 y(in)m(tegers.)59 b(The)36 b(main)f(idea)h(is)g(to)h(reduce)f(the) 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FA(\014)2682 2650 y Fu(n)2729 2683 y FA(=)p FB(2)f Fv(\024)e FA(B)44 b(<)38 b(\014)3246 2650 y Fu(n)3293 2683 y Fw(.)65 b(A)n(lgorithm)118 2796 y FA(D)193 2815 y Fy(2)p Fu(n=)p Fy(1)p Fu(n)422 2796 y Fw(c)-5 b(omputes)33 b(the)g(quotient)g FA(Q)25 b FB(=)g Fv(b)p FA(A=B)5 b Fv(c)34 b Fw(with)f(r)-5 b(emainder)35 b FA(R)26 b FB(=)f FA(A)20 b Fv(\000)g FA(Q)g Fv(\001)h FA(B)5 b Fw(.)226 2974 y(1.)45 b(If)f FA(n)f Fw(is)h(o)-5 b(dd)45 b(or)f(smal)5 b(ler)45 b(than)g(some)g(c)-5 b(onvenient)44 b(c)-5 b(onstant,)48 b(c)-5 b(ompute)45 b FA(Q)e Fw(and)i FA(R)f Fw(by)g(scho)-5 b(ol)345 3087 y(division)34 b(and)f(r)-5 b(eturn.)226 3271 y(2.)45 b(Split)j FA(A)e Fw(into)h(four)g(p)-5 b(arts)49 b FA(A)i FB(=)f([)p FA(A)1669 3285 y Fy(1)1709 3271 y FA(;)15 b(:)g(:)g(:)h(;)f(A)1978 3285 y Fy(4)2018 3271 y FB(])47 b Fw(with)g FA(A)2369 3285 y Fu(i)2448 3271 y FA(<)j(\014)2625 3238 y Fu(n=)p Fy(2)2743 3271 y Fw(.)84 b(Split)47 b FA(B)k Fw(into)c(two)h(p)-5 b(arts)345 3384 y FA(B)30 b FB(=)25 b([)p 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Fv(c)34 b Fw(as)h FA(Q)1982 3878 y Fy(2)2048 3864 y FB(=)27 b Fv(b)p FB([)p FA(R)2280 3878 y Fy(1)p Fu(;)p Fy(1)2375 3864 y FA(;)15 b(R)2484 3878 y Fy(1)p Fu(;)p Fy(2)2579 3864 y FA(;)g(A)2687 3878 y Fy(4)2727 3864 y FB(])p FA(=)p FB([)p FA(B)2891 3878 y Fy(1)2931 3864 y FA(;)g(B)3040 3878 y Fy(2)3080 3864 y FB(])p Fv(c)34 b Fw(with)h(r)-5 b(emainder)345 3977 y FA(R)q Fw(,)33 b(using)f(algorithm)j FA(D)1198 3995 y Fy(3)p Fu(n=)p Fy(2)p Fu(n)1393 3977 y Fw(.)226 4161 y(5.)45 b(R)-5 b(eturn)34 b FA(Q)25 b FB(=)g([)p FA(Q)940 4175 y Fy(1)979 4161 y FA(;)15 b(Q)1091 4175 y Fy(2)1131 4161 y FB(])33 b Fw(and)h FA(R)q Fw(.)259 4339 y FB(No)m(w)f(let)f(us)g (argue)g(wh)m(y)g(algorithm)f FA(D)1654 4357 y Fy(2)p Fu(n=)p Fy(1)p Fu(n)1882 4339 y FB(is)g(correct)j(\(under)d(the)h (assumption)f(that)h FA(D)3490 4357 y Fy(3)p Fu(n=)p Fy(2)p Fu(n)3718 4339 y FB(is)118 4452 y(correct\).)118 4629 y FC(Lemma)h(1.)46 b Fw(L)-5 b(et)32 b(for)i FA(k)28 b Fv(2)d Fo(N)32 b Fw(and)i FA(B)c(>)24 b FB(0)p 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b(No)m(w)948 2074 y FA(A)21 b Fv(\000)f FB(\(1)h(+)f FA(Q)1392 2088 y Fy(1)1431 2074 y FB(\))p FA(\014)1522 2036 y Fu(k)1565 2074 y FA(B)88 b FB(=)82 b(\()p FA(A)1978 2088 y Fy(1)2039 2074 y Fv(\000)19 b FA(Q)2201 2088 y Fy(1)2241 2074 y FA(B)5 b FB(\))p FA(\014)2406 2036 y Fu(k)2469 2074 y FB(+)20 b FA(A)2628 2088 y Fu(r)2686 2074 y Fv(\000)g FA(B)5 b(\014)2907 2036 y Fu(k)1722 2212 y FB(=)82 b FA(R)1944 2226 y Fy(1)1984 2212 y FA(\014)2040 2174 y Fu(k)2103 2212 y FB(+)20 b FA(A)2262 2226 y Fu(r)2320 2212 y Fv(\000)g FA(B)5 b(\014)2541 2174 y Fu(k)1722 2350 y FA(<)82 b(B)5 b(\014)2005 2312 y Fu(k)2068 2350 y Fv(\000)20 b FA(B)5 b(\014)2289 2312 y Fu(k)1722 2487 y FB(=)82 b(0)118 2688 y(implies)22 b(that)j FA(Q)g(<)g FB(\(1)8 b(+)g FA(Q)1044 2702 y Fy(1)1084 2688 y FB(\))p FA(\014)1175 2655 y Fu(k)1219 2688 y FB(.)38 b(This)23 b(sho)m(ws)h(that)h FA(Q)2002 2702 y Fu(r)2065 2688 y FA(<)g(\014)2217 2655 y Fu(k)2260 2688 y FB(.)39 b(Moreo)m(v)m(er,)28 b(for)c FA(Q)2949 2655 y Ft(0)2949 2710 y Fu(r)3012 2688 y FB(=)h Fv(b)p FB(\()p FA(R)3252 2702 y Fy(1)3292 2688 y FA(\014)3348 2655 y Fu(k)3400 2688 y FB(+)8 b FA(A)3547 2702 y Fu(r)3585 2688 y FB(\))p FA(=B)d Fv(c)118 2801 y FB(and)30 b FA(Q)367 2768 y Ft(0)415 2801 y FB(=)25 b FA(Q)583 2815 y Fy(1)623 2801 y FA(\014)679 2768 y Fu(k)742 2801 y FB(+)20 b FA(Q)905 2768 y Ft(0)905 2823 y Fu(r)973 2801 y FB(w)m(e)31 b(ha)m(v)m(e)1100 3001 y FA(A)20 b Fv(\000)g FA(B)5 b(Q)1425 2963 y Ft(0)1531 3001 y FB(=)83 b FA(A)1753 3015 y Fy(1)1792 3001 y FA(\014)1848 2963 y Fu(k)1911 3001 y FB(+)20 b FA(A)2070 3015 y Fu(r)2129 3001 y Fv(\000)g FA(B)5 b(Q)2366 3015 y Fy(1)2404 3001 y FA(\014)2460 2963 y Fu(k)2523 3001 y Fv(\000)20 b FA(B)5 b(Q)2760 2963 y Ft(0)2760 3023 y Fu(r)1531 3139 y FB(=)83 b(\()p FA(A)1788 3153 y Fy(1)1848 3139 y Fv(\000)20 b FA(B)5 b(Q)2085 3153 y Fy(1)2124 3139 y FB(\))p FA(\014)2215 3101 y Fu(k)2278 3139 y FB(+)20 b FA(A)2437 3153 y Fu(r)2495 3139 y Fv(\000)g FA(B)5 b(Q)2732 3101 y Ft(0)2732 3161 y Fu(r)1531 3277 y FB(=)83 b FA(R)1754 3291 y Fy(1)1793 3277 y FA(\014)1849 3239 y Fu(k)1912 3277 y FB(+)20 b FA(A)2071 3291 y Fu(r)2129 3277 y Fv(\000)g FA(B)5 b(Q)2366 3239 y Ft(0)2366 3299 y Fu(r)2404 3277 y FA(;)1234 b FB(\(1\))118 3477 y(whic)m(h)29 b(is)g(a)i(n)m(um)m(b)s(er)e(from)h([0) p FA(;)15 b(B)25 b Fv(\000)20 b FB(1].)41 b(This)29 b(sho)m(ws)h(that)h FA(Q)25 b FB(=)g FA(Q)2453 3444 y Ft(0)2506 3477 y FB(and)30 b(hence)g FA(Q)3007 3491 y Fu(r)3070 3477 y FB(=)25 b FA(Q)3238 3444 y Ft(0)3238 3499 y Fu(r)3276 3477 y FB(.)41 b(Equation)29 b(1)118 3590 y(completes)i(the)f(pro)s(of)g(of)h(our)e (lemma.)p 3714 3590 4 62 v 3718 3532 55 4 v 3718 3590 V 3771 3590 4 62 v 259 3703 a(No)m(w)43 b(w)m(e)g(come)g(to)g (algorithm)e FA(D)1492 3721 y Fy(3)p Fu(n=)p Fy(2)p Fu(n)1688 3703 y FB(.)77 b(Our)41 b(goal)h(is)g(to)h(divide)d(a)i(3)p FA(n)p FB(-digit)g(n)m(um)m(b)s(er)f FA(A)h FB(b)m(y)g(a)118 3816 y(2)p FA(n)p FB(-digit)f(n)m(um)m(b)s(er)f FA(B)46 b FB(computing)41 b(an)g FA(n)p 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b(A,digit_array)k(B,)i(integer)f(n\))118 511 y({)118 624 y(\(19\))94 b(if)48 b(\(n)f(is)g(odd)g(or)g(n)h(<=)f(DIV_LIMIT\)) 118 737 y(\(20\))190 b(SchoolDivision\(A,B,Q,S\);)41 b(//)47 b(base)g(of)g(the)g(recursion)118 850 y(\(21\))94 b(else)47 b({)118 963 y(\(22\))190 b(Let)47 b([A1,A2,A3,A4]=A)c(and)k ([B1,B2]=B;)118 1076 y(\(23\))190 b([Q1,R]=DivThreeLongHalve)o(sByT)o (wo\()o(A1,A)o(2,A3)o(,B1)o(,B2,)o(n/2\))o(;)118 1189 y(\(24\))g(Let)47 b([R1,R2]=R;)118 1302 y(\(25\))190 b([Q2,S]=DivThreeLongHalve)o(sByT)o(wo\()o(R1,R)o(2,A4)o(,B1)o(,B2,)o (n/2\))o(;)118 1415 y(\(26\))g(})118 1528 y(\(27\))94 b(Return)47 b(the)f(quotient)g(Q=[Q1,Q2])f(and)i(the)g(remainder)e(S;) 118 1641 y(})259 1853 y FB(Pro)s(cedure)28 b Fx (DivThreeLongHalvesByTwo)23 b FB(has)28 b(a)h(similar)d(structure)i(as) h Fx(DivThreeHalvesByTwo)p FB(,)118 1966 y(except)i(that)g(lines)e (\(6\))i(and)f(\(7\))i(no)m(w)e(b)s(ecome)118 2154 y Fx(\(6'\))47 b([Q,C]=RecursiveDivision)o(\([A1)o(,A2])o(,B1)o(\))42 b(;)118 2341 y 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2906 y(metic)43 b(and)e(temp)s(orary)h(space)h(managemen)m (t\).)78 b Fx(DivThreeLongHalvesByTwo)36 b FB(mak)m(es)43 b(a)g(recursiv)m(e)118 3019 y(call)28 b(in)f(line)g(\(6'\))j(whic)m(h)d (tak)m(es)j FA(D)s FB(\()p FA(n=)p FB(2\).)41 b(The)28 b(bac)m(km)m(ultiplication)e FA(Q)17 b Fv(\001)f FA(B)2722 3033 y Fy(2)2790 3019 y FB(in)28 b(line)e(\(8\))k(tak)m(es)g FA(M)10 b FB(\()p FA(n=)p FB(2\).)118 3132 y(The)39 b(remaining)f (additions)f(and)i(subtractions)f(tak)m(e)j FA(O)s FB(\()p FA(n)p FB(\).)68 b(Let)40 b FA(c)2569 3146 y Fu(r)2647 3132 y FB(b)s(e)e(a)i(con)m(v)m(enien)m(t)h(constan)m(t)f(so)118 3245 y(that)29 b(the)g(o)m(v)m(erall)g(time)f(for)h(all)e(splitting)g (op)s(erations,)h(additions,)g(and)g(subtractions)f(is)h(less)g(than)g FA(c)3661 3259 y Fu(r)3700 3245 y FA(n)p FB(.)118 3357 y Fx(DivThreeLongHalvesByTwo)c FB(is)30 b(called)f(t)m(wice,)i(so)g(w)m (e)g(ha)m(v)m(e)g(the)g(recursion)734 3675 y FA(D)s FB(\()p FA(n)p FB(\))83 b Fv(\024)g FB(2)p FA(D)s FB(\()p FA(n=)p 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FD(as)g(computed)h(b)n(y)f Fk(D)2744 4862 y Fh(\000)118 5178 y FC(Lemma)33 b(4.)46 b Fw(F)-7 b(or)33 b FA(n)25 b FB(=)g(2)1006 5145 y Fu(k)1049 5178 y Fw(,)33 b(the)g(running)f(time)h(of)g FA(D)1995 5145 y Ft(\000)2086 5178 y Fw(is)g(at)g(most)h FB(3)p FA(=)p FB(2)p FA(K)7 b FB(\()p FA(n)p FB(\))21 b(+)f FA(O)s FB(\()p FA(n)15 b FB(log)i FA(n)p FB(\))p Fw(.)259 5366 y FC(Pro)s(of:)41 b FB(W)-8 b(e)29 b(denote)g(the)f(running)e (time)h(of)i(algorithm)e FA(D)2314 5333 y Ft(\000)2401 5366 y FB(for)h(an)g(instance)g(of)g(size)g FA(n)g FB(b)m(y)g FA(D)3570 5333 y Ft(\000)3629 5366 y FB(\()p FA(n)p FB(\).)118 5479 y(Similarly)-8 b(,)26 b(w)m(e)j(denote)g(the)g(running)d(time)i (of)h(algorithm)e FA(D)2233 5446 y Fy(+)2320 5479 y FB(b)m(y)i FA(D)2523 5446 y Fy(+)2582 5479 y FB(\()p FA(n)p FB(\).)40 b(Remem)m(b)s(er)28 b(that)h FA(D)3499 5446 y Fy(+)3558 5479 y FB(\()p FA(n)p FB(\))d(=)118 5592 y(2)p FA(K)7 b FB(\()p FA(n)p FB(\).)80 b(T)-8 b(o)44 b(compute)f FA(D)1083 5559 y Ft(\000)1142 5592 y FB(\()p FA(n)p FB(\),)k(w)m(e)d (call)f(once)h FA(D)1964 5559 y Fy(+)2022 5592 y FB(\()p FA(n=)p FB(2\),)k(do)c(one)f(Karatsuba)g(m)m(ultiplication)e(on)1903 5841 y(12)p eop %%Page: 13 17 13 16 bop 118 399 a FB(in)m(tegers)38 b(with)e FA(n=)p FB(2)i(digits,)g(and)e(do)i(one)f(recursiv)m(e)g(call)g(to)h FA(D)2388 366 y Ft(\000)2447 399 y FB(\()p FA(n=)p FB(2\).)63 b(All)36 b(other)i(op)s(erations)e(ha)m(v)m(e)118 511 y(cost)31 b FA(O)s FB(\()p FA(n)p FB(\).)41 b(Hence)31 b(w)m(e)g(ha)m(v)m(e)h(the)e(recursion)885 687 y FA(D)963 649 y Ft(\000)1022 687 y FB(\()p FA(n)p FB(\))83 b(=)g FA(D)1462 649 y Fy(+)1521 687 y FB(\()p FA(n=)p FB(2\))21 b(+)f FA(K)7 b FB(\()p FA(n=)p FB(2\))21 b(+)f FA(D)2337 649 y Ft(\000)2396 687 y FB(\()p FA(n=)p FB(2\))h(+)f FA(O)s FB(\()p FA(n)p FB(\))1230 825 y(=)83 b(3)p FA(K)7 b FB(\()p FA(n=)p FB(2\))22 b(+)e FA(O)s FB(\()p FA(n)15 b FB(log)h FA(n)p FB(\))k(+)g FA(D)2429 787 y Ft(\000)2488 825 y FB(\()p FA(n=)p FB(2\))h(+)f FA(O)s FB(\()p FA(n)p FB(\))1230 963 y(=)83 b FA(K)7 b FB(\()p FA(n)p FB(\))20 b(+)g FA(D)1782 925 y Ft(\000)1841 963 y FB(\()p FA(n=)p FB(2\))h(+)f FA(O)s FB(\()p FA(n)15 b FB(log)i FA(n)p FB(\))1230 1100 y Fv(\024)83 b FA(K)7 b FB(\()p FA(n)p FB(\))20 b(+)g FA(D)1782 1063 y Ft(\000)1841 1100 y FB(\()p FA(n=)p FB(2\))h(+)f FA(c)h Fv(\001)f FA(n)15 b FB(log)i FA(n;)118 1276 y FB(where)30 b(in)f(the)i(last)f(equation)g(w)m(e)h (used)e FA(K)7 b FB(\()p FA(n=)p FB(2\))27 b(=)e(1)p FA(=)p FB(3)p FA(K)7 b FB(\()p FA(n)p FB(\).)42 b(Solving)28 b(the)j(recursion)e(w)m(e)i(get)788 1538 y FA(D)866 1501 y Ft(\000)925 1538 y FB(\()p FA(n)p FB(\))83 b Fv(\024)1333 1425 y Fu(k)1287 1452 y Fr(X)1296 1647 y Fu(i)p Fy(=0)1433 1538 y FA(K)7 b FB(\()p FA(n=)p FB(2)1697 1501 y Fu(i)1726 1538 y FB(\))21 b(+)f FA(D)1951 1501 y Ft(\000)2009 1538 y FB(\(1\))i(+)2283 1425 y Fu(k)2237 1452 y Fr(X)2245 1647 y Fu(i)p Fy(=0)2383 1538 y FA(c)f Fv(\001)f FB(\()p FA(n=)p FB(2)2668 1501 y Fu(i)2697 1538 y FB(\))15 b(log)r(\()p FA(n=)p FB(2)3045 1501 y Fu(i)3074 1538 y FB(\))1133 1858 y Fv(\024)1333 1745 y Fu(k)1287 1772 y Fr(X)1296 1967 y Fu(i)p Fy(=0)1433 1858 y FA(K)7 b FB(\()p FA(n=)p FB(2)1697 1821 y Fu(i)1726 1858 y FB(\))21 b(+)f FA(O)s FB(\(1\))h(+)f(2)p FA(c)h Fv(\001)f FA(n)15 b FB(log)i FA(n)1133 2179 y FB(=)1333 2065 y Fu(k)1287 2092 y Fr(X)1296 2288 y Fu(i)p Fy(=0)1433 2179 y FA(K)7 b FB(\()p FA(n)p FB(\))p FA(=)p FB(3)1732 2141 y Fu(i)1782 2179 y FB(+)20 b FA(O)s FB(\()p FA(n)15 b FB(log)h FA(n)p FB(\))1133 2481 y Fv(\024)83 b FA(K)7 b FB(\()p FA(n)p FB(\))1542 2368 y Ft(1)1511 2395 y Fr(X)1520 2590 y Fu(i)p Fy(=0)1658 2481 y FB(1)p FA(=)p FB(3)1793 2444 y Fu(i)1842 2481 y FB(+)20 b FA(O)s FB(\()p FA(n)15 b FB(log)i FA(n)p FB(\))1133 2701 y(=)83 b(3)p FA(=)p FB(2)p FA(K)7 b FB(\()p FA(n)p FB(\))22 b(+)d FA(O)s FB(\()p FA(n)c FB(log)i FA(n)p FB(\))p FA(:)p 2264 2701 4 62 v 2268 2643 55 4 v 2268 2701 V 2322 2701 4 62 v 118 2940 a Fs(3.2)112 b(Division)36 b(without)g(remainder:)49 b(General)38 b(outline)118 3111 y FB(Because)29 b(in)e(general)h FA(D)956 3078 y Ft(\000)1043 3111 y FB(do)s(es)g(not)g(giv)m(e)g(us)g(the)g (exact)h(division)c(result,)j(further)f(correction)h(steps)g(are)118 3224 y(required)k(to)j(adjust)891 3201 y Fr(b)873 3224 y FA(Q)f FB(un)m(til)e(w)m(e)j(are)f(left)g(with)f(the)h(correct)i FA(Q)p FB(.)52 b(The)33 b(follo)m(wing)g(lemma)h(sho)m(ws)f(ho)m(w)118 3337 y(w)m(e)e(can)g(do)f(this.)40 b(Let)939 3314 y Fr(e)920 3337 y FA(Q)31 b FB(no)m(w)f(b)s(e)g(an)m(y)g(appro)m(ximate)h(quotien) m(t)f(of)h FA(A)g FB(and)e FA(B)5 b FB(.)229 3501 y(1.)46 b FA(R)q FB(\()469 3478 y Fr(e)450 3501 y FA(Q;)15 b(A;)g(B)5 b FB(\))26 b(:=)g FA(A)10 b Fv(\000)g FA(B)1178 3478 y Fr(e)1160 3501 y FA(Q)25 b FB(is)f(the)i(remainder)e(that)i(corresp)s (onds)e(to)i(an)f(appro)m(ximate)h(quotien)m(t)3726 3478 y Fr(e)3708 3501 y FA(Q)345 3614 y FB(of)j FA(A)f FB(and)g FA(B)5 b FB(.)39 b(In)28 b(the)g(sp)s(ecial)f(case)j(of)1726 3591 y Fr(e)1708 3614 y FA(Q)25 b FB(=)g Fv(b)p FA(A=B)5 b Fv(c)p FB(,)30 b(it)e(is)f(the)i(ordinary)d(division)f(remainder.)39 b(If)345 3727 y(the)31 b(division)c(op)s(erands)i FA(A)h FB(and)g FA(B)35 b FB(are)30 b(ob)m(vious)g(from)g(the)g(con)m(text,)j (w)m(e)e(write)e FA(R)q FB(\()3293 3704 y Fr(e)3275 3727 y FA(Q)p FB(\))h(for)g(short.)345 3840 y(Moreo)m(v)m(er,)j(w)m(e)e (write)f FA(R)h FB(instead)e(of)i FA(R)q FB(\()p Fv(b)p FA(A=B)5 b Fv(c)p FB(\).)229 4018 y(2.)46 b FA(\016)s FB(\()443 3995 y Fr(e)423 4018 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))35 b(:=)f Fv(b)p FA(A=B)5 b Fv(c)25 b(\000)1322 3995 y Fr(e)1304 4018 y FA(Q)p FB(.)56 b(Again,)37 b(if)d FA(A)i FB(and)f FA(B)40 b FB(are)c(ob)m(vious)f(from)g(the)h(con)m (text)h(w)m(e)f(write)345 4131 y FA(\016)s FB(\()443 4108 y Fr(e)423 4131 y FA(Q)q FB(\))31 b(for)f(short.)118 4294 y(The)g(follo)m(wing)f(lemma)h(states)h(some)g(useful)e(relations) g(b)s(et)m(w)m(een)i(these)g(quan)m(tities.)118 4458 y FC(Lemma)i(5.)46 b Fw(F)-7 b(or)33 b(any)h(inte)-5 b(gr)g(al)34 b(nonne)-5 b(gative)33 b FA(A)p Fw(,)g FA(B)5 b Fw(,)2064 4435 y Fr(e)2046 4458 y FA(Q)32 b Fw(we)h(have)226 4622 y(1.)45 b FA(\016)s FB(\()443 4599 y Fr(e)423 4622 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))27 b(=)e Fv(b)p FA(R)q FB(\()1039 4599 y Fr(e)1021 4622 y FA(Q;)15 b(A;)g(B)5 b FB(\))p FA(=B)g Fv(c)226 4800 y Fw(2.)45 b FA(R)q FB(\()p Fv(b)p FA(A=B)5 b Fv(c)p FA(;)15 b(A;)g(B)5 b FB(\))28 b(=)c FA(R)q FB(\()p FA(\016)s FB(\()1300 4777 y Fr(e)1280 4800 y FA(Q)r(;)15 b(A;)g(B)5 b FB(\))p FA(;)15 b(R)q FB(\()1775 4777 y Fr(e)1756 4800 y FA(Q)q(;)g(A;)g(B)5 b FB(\))p FA(;)15 b(B)5 b FB(\))226 4978 y Fw(3.)45 b Fv(j)p FA(R)q FB(\()494 4955 y Fr(e)475 4978 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))p Fv(j)26 b FA(<)f FB(\(1)c(+)f Fv(j)p FA(\016)s FB(\()1267 4955 y Fr(e)1247 4978 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))p Fv(j)p FB(\))p FA(B)g Fw(.)259 5141 y FC(Pro)s(of:)88 b FB(The)30 b(quan)m(tit)m(y)h FA(q)d FB(=)d FA(\016)s FB(\()1442 5118 y Fr(e)1423 5141 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))31 b(satis\014es)893 5317 y FA(R)q FB(\()p FA(q)s(;)15 b(R)q FB(\()1206 5294 y Fr(e)1187 5317 y FA(Q;)g(A;)g(B)5 b FB(\))p FA(;)15 b(B)5 b FB(\))84 b(=)f FA(R)q FB(\()2026 5294 y Fr(e)2008 5317 y FA(Q;)15 b(A;)g(B)5 b FB(\))21 b Fv(\000)f FA(\016)s FB(\()2546 5294 y Fr(e)2527 5317 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))p FA(B)1749 5455 y FB(=)83 b(\()p FA(A)21 b Fv(\000)f FA(B)2209 5432 y Fr(e)2192 5455 y FA(Q)o FB(\))h Fv(\000)f FB(\()p Fv(b)p FA(A=B)5 b Fv(c)21 b(\000)2842 5432 y Fr(e)2824 5455 y FA(Q)p FB(\))p FA(B)1749 5592 y FB(=)83 b FA(A)20 b Fv(\000)g(b)p FA(A=B)5 b Fv(c)p FA(B)1245 b FB(\(3\))1903 5841 y(13)p eop %%Page: 14 18 14 17 bop 118 399 a FB(whic)m(h)21 b(is)g(equal)g(to)i FA(R)q FB(\()p Fv(b)p FA(A=B)5 b Fv(c)p FA(;)15 b(A;)g(B)5 b FB(\))24 b(and)d(hence)h(is)f(con)m(tained)i(in)d(the)j(in)m(terv)-5 b(al)21 b([0)p FA(;)15 b(:)g(:)g(:)i(;)e(B)8 b Fv(\000)s FB(1].)39 b(Because)118 511 y Fv(b)p FA(R)q FB(\()282 488 y Fr(e)263 511 y FA(Q;)15 b(A;)g(B)5 b FB(\))p FA(=B)g Fv(c)26 b FB(is)d(the)i(only)e(in)m(teger)i(with)e(this)g(prop)s(ert)m (y)-8 b(,)25 b(the)g(\014rst)e(t)m(w)m(o)j(parts)e(of)g(our)g(lemma)g (follo)m(w.)118 624 y(The)30 b(third)f(part)h(follo)m(ws)f(b)m(y)h (using)f(Equation)h(3)h(again:)726 824 y Fv(j)p FA(R)q FB(\()875 801 y Fr(e)856 824 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))p Fv(j)84 b(\024)e(j)p FA(R)q FB(\()1597 801 y Fr(e)1578 824 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))21 b Fv(\000)f FA(\016)s FB(\()2117 801 y Fr(e)2098 824 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))p FA(B)g Fv(j)20 b FB(+)g Fv(j)p FA(\016)s FB(\()2761 801 y Fr(e)2741 824 y FA(Q)r(;)15 b(A;)g(B)5 b FB(\))p Fv(j)p FA(B)1295 962 y FB(=)82 b Fv(j)p FA(A)21 b Fv(\000)f(b)p FA(A=B)5 b Fv(c)p FA(B)g Fv(j)21 b FB(+)f Fv(j)p FA(\016)s FB(\()2253 939 y Fr(e)2234 962 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))p Fv(j)p FA(B)1295 1100 y(<)82 b FB(\(1)21 b(+)f Fv(j)p FA(\016)s FB(\()1763 1077 y Fr(e)1743 1100 y FA(Q)r(;)15 b(A;)g(B)5 b FB(\))p Fv(j)p FB(\))p FA(B)g(:)p 2330 1100 4 62 v 2334 1041 55 4 v 2334 1100 V 2387 1100 4 62 v 118 1299 a FB(Lemma)38 b(5)g(can)g(b)s(e)f(applied)e(as)j(follo)m(ws.)61 b(Supp)s(ose)36 b(that)i(w)m(e)g(w)m(an)m(t)h(to)f(compute)g(the)g(quotien)m(t)g FA(Q)f FB(=)118 1412 y Fv(b)p FA(A=B)5 b Fv(c)34 b FB(but)e(w)m(e)i (only)e(kno)m(w)h(an)g(appro)m(ximation)1923 1389 y Fr(b)1905 1412 y FA(Q)g FB(of)g FA(Q)g FB(and)f(its)h(remainder)e FA(R)q FB(\()3085 1389 y Fr(b)3067 1412 y FA(Q;)15 b(A;)g(B)5 b FB(\),)35 b(whic)m(h)c(is)118 1525 y(b)s(ounded)f(b)m(y)i(\(1)23 b(+)e Fv(j)p FA(\016)s FB(\()937 1502 y Fr(b)917 1525 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))p Fv(j)p FB(\))p FA(B)38 b FB(according)33 b(to)g(lemma.)46 b(If)32 b FA(\016)s FB(\()2463 1502 y Fr(b)2444 1525 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))33 b(is)e(not)i(to)s(o)g(large,)g(sa)m(y)g(if)e(it)118 1638 y(is)i(b)s(ounded)e(b)m(y)j(a)g(constan)m(t,)i(then)e(w)m(e)g(can) g(compute)g(the)g(quotien)m(t)g FA(\016)s FB(\()2701 1615 y Fr(b)2682 1638 y FA(Q)q(;)15 b(A;)g(B)5 b FB(\))31 b(=)g Fv(b)p FA(R)q FB(\()3309 1615 y Fr(b)3290 1638 y FA(Q;)15 b(A;)g(B)5 b FB(\))p FA(=B)g Fv(c)118 1751 y FB(b)m(y)29 b(ordinary)f(sc)m(ho)s(ol)g(division.)38 b(By)29 b(the)g(second)h(part)f(of)g(the)g(lemma,)g(the)h(remainder)d (of)i(this)f(division)118 1864 y(is)g(the)h(w)m(an)m(ted)h FA(R)q FB(\()p Fv(b)p FA(A=B)5 b Fv(c)p FA(;)15 b(A;)g(B)5 b FB(\).)42 b(If)29 b FA(B)k FB(has)c(at)h(most)f FA(n)g FB(digits,)f(the)h(sc)m(ho)s(ol)g(division)d(tak)m(es)31 b(only)d(time)118 1977 y FA(O)s FB(\()p FA(n)p FB(\))h(b)s(ecause)f (all)g(w)m(e)h(ha)m(v)m(e)h(to)f(do)g(is)e(to)j(compute)e(a)h(\(b)s (ounded\))f(n)m(um)m(b)s(er)f(of)i(bits)e(and)h(eac)m(h)i(iteration)118 2089 y(tak)m(es)i(time)e FA(O)s FB(\()p FA(n)p FB(\).)259 2202 y(With)c(this)f(lemma)h(at)i(hands)d(our)h(algorithm)f(for)h (division)e(without)h(remainder)g(has)h(the)h(follo)m(wing)118 2315 y(outline.)118 2499 y FC(Algorithm)35 b(5.)45 b(Division)28 b(without)d(remainder)g(\(Outline\):)80 b Fw(L)-5 b(et)26 b FA(A)f Fw(and)i FA(B)j Fw(b)-5 b(e)26 b(as)g(in)g(A)n(lgorithm)118 2612 y(4.)42 b(Pr)-5 b(o)g(c)g(e)g(e)g(d)35 b(in)d(thr)-5 b(e)g(e)34 b(phases.)226 2796 y(1.)45 b(Cal)5 b(l)34 b FA(D)619 2763 y Ft(\000)710 2796 y Fw(to)f(c)-5 b(ompute)34 b(an)f(appr)-5 b(oximation)1924 2773 y Fr(b)1906 2796 y FA(Q)32 b Fw(of)h(the)g(exact)g(quotient)g FA(Q)25 b FB(=)g Fv(b)p FA(A=B)5 b Fv(c)p Fw(.)226 2982 y(2.)45 b(T)-7 b(ry)34 b(to)g(c)-5 b(ompute)34 b FA(\016)s FB(\()1084 2959 y Fr(b)1065 2982 y FA(Q)q FB(\))27 b(=)e FA(Q)c Fv(\000)1498 2959 y Fr(b)1480 2982 y FA(Q)33 b Fw(in)g(line)-5 b(ar)34 b(time.)44 b(This)33 b(phase)i(may)f(fail)f(with)h(a)g(smal)5 b(l)34 b(pr)-5 b(ob)g(a-)345 3095 y(bility.)226 3281 y(3.)45 b(If)31 b(phase)i(2)e(fails,)h(c)-5 b(ompute)32 b(the)g(r)-5 b(emainder)33 b FA(R)q FB(\()2050 3258 y Fr(b)2032 3281 y FA(Q)p FB(\))25 b(=)g FA(A)17 b Fv(\000)g FA(B)k Fv(\001)2583 3258 y Fr(b)2565 3281 y FA(Q)p Fw(,)31 b(using)g(Kar)-5 b(atsub)g(a)33 b(multiplic)-5 b(a-)345 3394 y(tion.)43 b(Then)33 b(c)-5 b(ompute)33 b FA(\016)s FB(\()1263 3371 y Fr(b)1243 3394 y FA(Q)r FB(\))25 b(=)g Fv(b)p FA(R)q FB(\()1637 3371 y Fr(b)1618 3394 y FA(Q)p FB(\))p FA(=B)5 b Fv(c)34 b Fw(using)e(scho)-5 b(ol)35 b(division.)226 3580 y(4.)45 b(Set)33 b FA(Q)25 b FB(=)712 3557 y Fr(b)694 3580 y FA(Q)20 b FB(+)g FA(\016)s FB(\()975 3557 y Fr(b)955 3580 y FA(Q)q FB(\))p Fw(.)259 3763 y FB(W)-8 b(e)29 b(describ)s(e)d(the)i(simpler)e(phase)h(3)h(\014rst.)39 b(This)26 b(is)h(the)h(bad)f(case)i(but)e(as)h(w)m(e)g(will)d(see)j(it) g(will)d(o)s(ccur)118 3876 y(only)30 b(with)f(a)h(v)m(ery)h(small)e (probabilit)m(y)-8 b(.)118 4060 y FC(Lemma)33 b(6.)46 b Fw(Phase)33 b(3)g(has)g(running)g(time)g FA(K)7 b FB(\()p FA(n)p FB(\))20 b(+)g FA(O)s FB(\()p FA(n)15 b FB(log)h(log)h FA(n)p FB(\))p Fw(.)259 4244 y FC(Pro)s(of:)40 b FB(The)25 b(Karatsuba)i(m)m(ultiplication)c(in)i(phase)h(3)h(tak)m(es)g(time)g FA(K)7 b FB(\()p FA(n)p FB(\).)39 b(By)26 b(Lemma)h(5)g(w)m(e)f(kno)m (w)118 4357 y(that)40 b(in)e(fact)i FA(\016)s FB(\()724 4334 y Fr(b)705 4357 y FA(Q)q FB(\))g(=)g FA(Q)26 b Fv(\000)1177 4334 y Fr(b)1158 4357 y FA(Q)40 b FB(=)g Fv(b)p FA(R)q FB(\()1545 4334 y Fr(b)1526 4357 y FA(Q)p FB(\))p FA(=B)5 b Fv(c)40 b FB(and)f(b)m(y)g(Lemma)g(3)g(w)m(e)h(kno)m(w)f(that)h(this) e(quotien)m(t)h(is)118 4470 y(b)s(ounded)c(b)m(y)j(2)15 b(log)r(\()p FA(n)p FB(\).)61 b(Hence)39 b FA(\016)s FB(\()1394 4447 y Fr(b)1375 4470 y FA(Q)q FB(\))e(\014ts)g(in)m(to)h (one)f(digit)f(for)i(all)e(practical)h(v)-5 b(alues)36 b(of)i FA(\014)k FB(and)37 b(of)h FA(n)p FB(.)118 4582 y(Under)f(the)g(assumption)f(of)i(2)15 b(log)i FA(n)37 b(<)f(\014)5 b FB(,)40 b(w)m(e)e(can)g(compute)g Fv(b)p FA(R)q FB(\()2530 4559 y Fr(b)2512 4582 y FA(Q)p FB(\))p FA(=B)5 b Fv(c)38 b FB(in)e(linear)g(time)h(b)m(y)h(sc)m(ho)s(ol)118 4695 y(division.)f(In)26 b(general,)i FA(\016)s FB(\()1033 4672 y Fr(b)1013 4695 y FA(Q)r FB(\))f(has)g(size)g FA(O)s FB(\(log)17 b(log)g FA(n)p FB(\))27 b(and)f(the)i(division)c(tak)m(es) 29 b(time)e FA(O)s FB(\()p FA(n)15 b FB(log)h(log)h FA(n)p FB(\).)p 3714 4695 V 3718 4637 55 4 v 3718 4695 V 3771 4695 4 62 v 118 4938 a Fs(3.3)112 b(Probabilistic)34 b(quotien)m(t)j(correction:)48 b(The)38 b(algorithm)118 5110 y FB(In)26 b(this)g(section)h(w)m(e)g(explain)e(the)i(algorithm)f (that)h(implemen)m(ts)e(phase)i(2)g(of)g(Algorithm)e(5.)40 b(Henceforth)118 5223 y(w)m(e)23 b(mak)m(e)g(t)m(w)m(o)g(\(realistic\)) e(assumptions)g(on)h(out)g(mac)m(hine)g(and)f(the)h(n)m(um)m(b)s(ers)f (with)g(whic)m(h)f(w)m(e)j(compute.)118 5406 y FC(Assumption)35 b(1.)153 b Fw(1.)46 b(The)33 b(digit)f(length)i FA(\014)j Fw(is)c(at)g(le)-5 b(ast)34 b FB(2)2259 5373 y Fy(32)2334 5406 y Fw(.)226 5592 y(2.)45 b(The)33 b(numb)-5 b(er)33 b(of)g(digits)g(is)g(b)-5 b(ounde)g(d)34 b(by)e FA(n)25 b(<)g(\014)5 b(=)p FB(2)p Fw(.)1903 5841 y FB(14)p eop %%Page: 15 19 15 18 bop 259 399 a FB(Assumption)21 b(1.2)j(is)d(justi\014ed)g(b)s (ecause)i(already)f(for)g FA(n)j(>)g FB(2)2310 366 y Fy(20)2408 399 y FB(it)d(is)f(b)s(etter)i(to)h(use)e(Newton's)h(metho)s (d)118 511 y(in)d(com)m(bination)h(with)f(a)i(FFT)g(t)m(yp)s(e)g(m)m (ultiplication,)e(see)i(section)g(4.)38 b(Assumption)20 b(1.1)i(implies)d(that)j(the)118 624 y(follo)m(wing)27 b(pro)s(cedure)h(do)s(es)g(not)h(w)m(ork)g(for)g(8)g(bit)f(or)g(16)i (bit)e(systems.)40 b(Straigh)m(tforw)m(ard)28 b(mo)s(di\014cations)118 737 y(are)j(required.)259 850 y(Phase)49 b(2)f(basically)f(do)s(es)h (the)g(same)h(job)f(as)g(phase)g(3,)53 b(but)48 b(it)g(a)m(v)m(oids)g (the)h(exp)s(ensiv)m(e)e(bac)m(k{)118 963 y(m)m(ultiplication)28 b FA(B)d Fv(\001)860 940 y Fr(b)842 963 y FA(Q)p FB(.)43 b(Instead,)31 b(here)g(w)m(e)h(only)e(appro)m(ximate)h(this)f(pro)s (duct.)42 b(F)-8 b(or)31 b(tec)m(hnical)g(reasons)118 1076 y(that)i(will)c(b)s(ecome)k(clear)f(later,)h(w)m(e)f(c)m(ho)s(ose) h FA(Q)1774 1043 y Ft(0)1826 1076 y FB(:=)28 b Fw(max)12 b FB(\(0)p FA(;)2263 1053 y Fr(b)2244 1076 y FA(Q)22 b Fv(\000)f FB(2)15 b(log)j FA(n)p FB(\))31 b(for)h(the)h(bac)m(km)m (ultiplication)118 1189 y(and)38 b(appro)m(ximate)h(the)g(pro)s(duct)f FA(P)52 b FB(=)39 b FA(B)31 b Fv(\001)26 b FA(Q)1795 1156 y Ft(0)1818 1189 y FB(.)66 b(This)37 b(giv)m(es)i(us)f(upp)s(er)f (and)h(lo)m(w)m(er)h(b)s(ounds)e(for)h(the)118 1302 y(remainder)26 b FA(R)q FB(\()p FA(Q)722 1269 y Ft(0)745 1302 y FB(\))g(=)e FA(A)14 b Fv(\000)g FA(Q)1140 1269 y Ft(0)1177 1302 y Fv(\001)g FA(B)5 b FB(,)27 b(whic)m(h)f(in)g(turn)g(giv)m(e)h(us)g(upp) s(er)e(and)h(lo)m(w)m(er)i(b)s(ounds)d FA(\016)3216 1317 y Fu(l)3267 1302 y Fv(\024)g FA(\016)s FB(\()p FA(Q)3513 1269 y Ft(0)3538 1302 y FB(\))g Fv(\024)g FA(\016)3734 1317 y Fu(h)118 1415 y FB(for)35 b(the)g(desired)e(quan)m(tit)m(y)j FA(\016)s FB(\()p FA(Q)1255 1382 y Ft(0)1279 1415 y FB(\))d(=)g Fv(b)p FA(A=B)5 b Fv(c)24 b(\000)f FA(Q)1908 1382 y Ft(0)1931 1415 y FB(.)55 b(Our)34 b(appro)m(ximations)f(will)g(b)s(e)h(go)s(o)s (d)h(enough)g(to)118 1528 y(guaran)m(tee)29 b FA(\016)572 1543 y Fu(h)632 1528 y Fv(\000)14 b FA(\016)757 1543 y Fu(l)809 1528 y Fv(\024)25 b FB(1.)40 b(Phase)27 b(2)h(is)f (successful)f(if)g(and)h(only)g(if)f FA(\016)2419 1543 y Fu(h)2490 1528 y FB(=)f FA(\016)2626 1543 y Fu(l)2652 1528 y FB(.)40 b(As)27 b(w)m(e)h(will)d(pro)m(v)m(e,)k(the)f(ev)m(en)m (t)118 1641 y FA(\016)158 1656 y Fu(h)229 1641 y Fv(6)p FB(=)d FA(\016)365 1656 y Fu(l)420 1641 y FB(that)31 b(causes)f(phase)f(2)h(to)g(fail)e(is)h(v)m(ery)h(unlik)m(ely)-8 b(.)38 b(W)-8 b(e)31 b(start)f(no)m(w)f(with)g(the)g(details)g(of)h (phase)f(2.)259 1753 y(With)h(resp)s(ect)g(to)i(the)e(base)h FA(\014)5 b FB(,)30 b(the)h(n)m(um)m(b)s(ers)e FA(A)p FB(,)i FA(B)j FB(and)c FA(Q)2387 1720 y Ft(0)2441 1753 y FB(read)g(as)1065 2034 y FA(A)25 b FB(=)1254 1921 y Fy(2)p Fu(n)p Ft(\000)p Fy(1)1273 1948 y Fr(X)1281 2143 y Fu(i)p Fy(=0)1438 2034 y FA(a)1486 2048 y Fu(i)1514 2034 y FA(\014)1570 1997 y Fu(i)1598 2034 y FA(;)86 b(B)30 b FB(=)1904 1921 y Fu(n)p Ft(\000)p Fy(1)1905 1948 y Fr(X)1914 2143 y Fu(i)p Fy(=0)2052 2034 y FA(b)2091 2048 y Fu(i)2120 2034 y FA(\014)2176 1997 y Fu(i)2204 2034 y FA(;)86 b(Q)2387 1997 y Ft(0)2436 2034 y FB(=)2532 1921 y Fu(n)p Ft(\000)p Fy(1)2532 1948 y Fr(X)2541 2143 y Fu(i)p Fy(=0)2680 2034 y FA(q)2724 1997 y Ft(0)2721 2057 y Fu(i)2748 2034 y FA(\014)2804 1997 y Fu(i)118 2326 y FB(In)26 b(the)g(sc)m(ho)s(ol)h(m)m(ultiplication)c(of)k FA(B)j FB(and)c FA(Q)1673 2293 y Ft(0)1696 2326 y FB(,)i(there)e(are)h FA(n)2180 2293 y Fy(2)2245 2326 y FB(pro)s(ducts)e(of)i(digits)e FA(b)3001 2340 y Fu(i)3029 2326 y FA(q)3073 2293 y Ft(0)3070 2351 y Fu(j)3106 2326 y FB(.)40 b(W)-8 b(e)27 b(group)f(these)118 2439 y(pro)s(ducts)j(b)m(y)h(the)h(sum)e(of)i(the)f(exp)s(onen)m(ts)h FA(i)20 b FB(+)g FA(j)5 b FB(.)41 b(F)-8 b(or)32 b(0)25 b Fv(\024)g FA(k)k(<)c FB(2)p FA(n)20 b Fv(\000)g FB(1)31 b(w)m(e)f(set)1590 2649 y FA(w)1655 2664 y Fu(k)1723 2649 y FB(=)1856 2563 y Fr(X)1819 2761 y Fu(i)p Fy(+)p Fu(j)t Fy(=)p Fu(k)2039 2649 y FA(b)2078 2663 y Fu(i)2106 2649 y FA(q)2150 2612 y Ft(0)2147 2672 y Fu(j)2183 2649 y FA(\014)2239 2612 y Fu(k)2282 2649 y FA(:)118 2965 y FB(It)h(is)g(easy)g(to)h(see)g(that)g FA(P)39 b FB(=)26 b FA(Q)1234 2932 y Ft(0)1278 2965 y Fv(\001)21 b FA(B)36 b FB(is)30 b(the)h(sum)f(of)i(the)f FA(w)2198 2980 y Fu(k)2241 2965 y FB(,)g(i.e.,)h FA(P)40 b FB(=)2665 2897 y Fr(P)2761 2923 y Fy(2)p Fu(n)p Ft(\000)p Fy(2)2761 2992 y Fu(k)r Fy(=0)2948 2965 y FA(w)3013 2980 y Fu(k)3056 2965 y FB(.)j(The)31 b(next)g(lemma)118 3078 y(sho)m(ws)k(that)g(the)g (parts)f FA(w)1048 3092 y Fy(0)1088 3078 y FA(;)15 b(:)g(:)g(:)h(;)f(w) 1354 3092 y Fu(n)p Ft(\000)p Fy(4)1527 3078 y FB(only)34 b(con)m(tribute)g(a)h(small)e(amoun)m(t)j(to)f FA(P)13 b FB(,)36 b(hence)f(w)m(e)g(will)d(not)118 3191 y(compute)f(them.)40 b(Recall)30 b(that)h(w)m(e)g(w)m(an)m(t)g(to)h(p)s(erform)d(phase)h(2)g (in)f(linear)g(time.)118 3377 y FC(Lemma)k(7.)46 b Fw(F)-7 b(or)33 b(any)h FA(l)g Fw(with)f FB(0)26 b Fv(\024)f FA(l)i(<)e FB(2)p FA(l)r Fw(,)1589 3547 y Fu(l)q Ft(\000)p Fy(1)1579 3575 y Fr(X)1581 3772 y Fu(k)r Fy(=0)1726 3661 y FA(w)1791 3676 y Fu(k)1859 3661 y FA(<)g FB(2)p FA(n)20 b Fv(\001)h FA(\014)2177 3623 y Fu(l)q Fy(+1)2293 3661 y FA(:)259 3942 y FC(Pro)s(of:)423 4095 y Fu(l)q Ft(\000)p Fy(1)413 4122 y Fr(X)414 4320 y Fu(k)r Fy(=0)560 4208 y FA(w)625 4223 y Fu(k)693 4208 y Fv(\024)798 4095 y Fu(l)q Ft(\000)p Fy(1)788 4122 y Fr(X)790 4320 y Fu(k)r Fy(=0)920 4208 y FB(\()p FA(k)j FB(+)19 b(1\))p FA(\014)1252 4171 y Fu(k)r Fy(+2)1412 4208 y Fv(\024)25 b FA(l)d Fv(\001)e FA(\014)1658 4171 y Fy(2)1718 4208 y Fv(\001)1773 4095 y Fu(l)q Ft(\000)p Fy(1)1763 4122 y Fr(X)1765 4320 y Fu(k)r Fy(=0)1910 4208 y FA(\014)1966 4171 y Fu(k)2034 4208 y FB(=)25 b FA(l)d Fv(\001)2293 4147 y FA(\014)2349 4114 y Fy(2)p 2234 4188 213 4 v 2234 4271 a FA(\014)k Fv(\000)20 b FB(1)2477 4208 y Fv(\001)h FB(\()p FA(\014)2614 4171 y Fu(l)2661 4208 y Fv(\000)f FB(1\))26 b FA(<)f(l)d Fv(\001)e FB(2)p FA(\014)26 b Fv(\001)20 b FA(\014)3271 4171 y Fu(l)3298 4208 y FA(:)p 3419 4208 4 62 v 3423 4150 55 4 v 3423 4208 V 3477 4208 4 62 v 259 4490 a FB(W)-8 b(e)32 b(no)m(w)e(apply)f(the)i(previous)e(lemma)h(with)f FA(l)e FB(=)e FA(n)20 b Fv(\000)f FB(3,)31 b(whic)m(h)f(giv)m(es)g(the) h(b)s(ound)1343 4657 y Fu(n)p Ft(\000)p Fy(4)1343 4684 y Fr(X)1345 4882 y Fu(k)r Fy(=0)1491 4770 y FA(w)1556 4785 y Fu(k)1624 4770 y FA(<)25 b FB(2\()p FA(n)20 b Fv(\000)g FB(3\))p FA(\014)2102 4733 y Fu(n)p Ft(\000)p Fy(2)2266 4770 y FA(<)25 b(\014)2418 4733 y Fu(n)p Ft(\000)p Fy(1)3663 4770 y FB(\(4\))118 5051 y(b)s(ecause)i(of)g(assumption)f (1.2.)41 b(If)26 b(w)m(e)i(omit)e(in)g FA(P)40 b FB(the)27 b(terms)g FA(w)2305 5065 y Fy(0)2345 5051 y FA(;)15 b(:)g(:)g(:)h(;)f (w)2611 5065 y Fu(n)p Ft(\000)p Fy(4)2749 5051 y FB(,)28 b(w)m(e)f(get)i(an)e(appro)m(ximation)118 5164 y FA(P)189 5131 y Ft(0)241 5164 y FB(:=)366 5096 y Fr(P)462 5122 y Fy(2)p Fu(n)p Ft(\000)p Fy(2)462 5191 y Fu(k)r Fy(=)p Fu(n)p Ft(\000)p Fy(3)708 5164 y FA(w)773 5179 y Fu(k)848 5164 y FB(of)32 b FA(P)13 b FB(.)47 b(Remem)m(b)s(er)32 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Fw(is)h(less)g (likely)g(than)h(the)f(others.)41 b(In)26 b(p)-5 b(articular,)29 b(e)-5 b(ach)27 b(value)f FA(r)2271 5493 y Fu(n)p Ft(\000)p Fy(1)2433 5479 y Fv(2)f FB([0)p FA(;)15 b(:)g(:)g(:)i(;)e(b)2830 5493 y Fu(n)p Ft(\000)p Fy(1)2973 5479 y Fv(\000)5 b FB(1])27 b Fw(has)g(a)f(pr)-5 b(ob)g(ability)118 5592 y FA(p)32 b Fw(with)i FA(p)25 b(<)g FB(1)p FA(=b)690 5606 y Fu(n)p Ft(\000)p Fy(1)828 5592 y Fw(.)1903 5841 y FB(17)p eop %%Page: 18 22 18 21 bop 259 399 a FC(Pro)s(of)44 b(of)f(claim:)53 b FB(Recall)37 b(that)h(w)m(e)f(assumed)g(0)g FA(<)f(B)41 b(<)c(\014)2468 366 y Fu(n)2552 399 y FB(and)g(0)g Fv(\024)f FA(A)h(<)f(\014)3193 366 y Fu(n)3240 399 y FA(B)5 b FB(.)61 b(F)-8 b(or)38 b(ev)m(ery)118 511 y(p)s(ossible)28 b(remainder)h FA(R)q FB(,)h(0)c Fv(\024)f FA(R)h(<)f(B)34 b FB(there)d(are)g(exactly) g FA(\014)2230 478 y Fu(n)2307 511 y FB(v)-5 b(alues)30 b(of)g FA(A)h FB(that)g(generate)h(this)d FA(R)q FB(,)h(one)118 624 y(for)j(eac)m(h)i(quotien)m(t)e FA(Q)p FB(,)i(0)30 b Fv(\024)g FA(Q)g(<)g(\014)1396 591 y Fu(n)1443 624 y FB(.)50 b(Hence)34 b(ev)m(ery)g(remainder)e FA(R)i FB(is)f(equally)f(lik)m(ely)-8 b(.)48 b(F)-8 b(urthermore,)118 737 y(to)40 b(ev)m(ery)f(0)h Fv(\024)g FA(r)723 751 y Fu(n)p Ft(\000)p Fy(1)899 737 y FA(<)f(b)1048 751 y Fu(n)p Ft(\000)p Fy(1)1224 737 y FB(corresp)s(ond)f(exactly)i FA(\014)2074 704 y Fu(n)p Ft(\000)p Fy(1)2250 737 y FB(p)s(ossible)c(v) -5 b(alues)38 b(of)i FA(R)f FB(ha)m(ving)g(\014rst)f(digit)118 850 y FA(r)159 864 y Fu(n)p Ft(\000)p Fy(1)296 850 y FB(,)33 b(namely)e(all)g FA(R)d FB(=)g FA(r)1036 864 y Fu(n)p Ft(\000)p Fy(1)1173 850 y FA(\014)1229 817 y Fu(n)p Ft(\000)p Fy(1)1388 850 y FB(+)21 b FA(z)36 b FB(with)31 b(0)d Fv(\024)f FA(z)33 b(<)27 b(\014)2167 817 y Fu(n)p Ft(\000)p Fy(1)2305 850 y FB(.)45 b(Only)30 b(for)i FA(r)2783 864 y Fu(n)p Ft(\000)p Fy(1)2948 850 y FB(=)c FA(b)3086 864 y Fu(n)p Ft(\000)p Fy(1)3255 850 y FB(there)k(are)g(less)118 963 y(remainders)e(that)i(generate)h(the)f (\014rst)e(digit)h 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Fu(n)p Ft(\000)p Fy(1)1143 2796 y Fv(\025)25 b FA(r)1283 2758 y Ft(\000)1366 2796 y Fv(\025)g FA(r)1506 2758 y Fy(+)1585 2796 y Fv(\000)20 b FB(\(2)h(+)f FA(\016)s FB(\()p FA(Q)2018 2758 y Ft(0)2042 2796 y FB(\)\))26 b Fv(\025)f FA(b)2273 2810 y Fu(n)p Ft(\000)p Fy(1)2431 2796 y Fv(\000)19 b FB(\(2)i(+)f FA(\016)s FB(\()p FA(Q)2863 2758 y Ft(0)2888 2796 y FB(\)\))118 2968 y(as)31 b(desired.)39 b(Next,)31 b(if)f FA(\016)s FB(\()p FA(Q)1055 2935 y Ft(0)1079 2968 y FB(\))c(=)f FA(\016)1276 2983 y Fu(h)1346 2968 y FB(=)g FA(\016)1482 2983 y Fu(l)1529 2968 y FB(+)20 b(1)30 b(it)g(follo)m(ws)g(b)m(y)g (Claim)f(6)h(that)783 3140 y FA(r)824 3154 y Fu(n)p Ft(\000)p Fy(1)981 3140 y FB(+)20 b FA(b)1111 3154 y Fu(n)p Ft(\000)p Fy(1)1274 3140 y Fv(\024)25 b FA(r)1414 3103 y Fy(+)1497 3140 y Fv(\024)g FA(r)1637 3103 y Ft(\000)1716 3140 y FB(+)20 b(\(2)h(+)f FA(\016)s FB(\()p FA(Q)2149 3103 y Ft(0)2173 3140 y FB(\)\))26 b Fv(\024)f FA(b)2404 3154 y Fu(n)p Ft(\000)p Fy(1)2562 3140 y FB(+)19 b(\(2)i(+)f FA(\016)s FB(\()p FA(Q)2994 3103 y Ft(0)3019 3140 y FB(\)\))p FA(:)118 3313 y FB(F)-8 b(rom)31 b(this)e(w)m(e)i(get)h FA(r)861 3327 y Fu(n)p Ft(\000)p Fy(1)1023 3313 y Fv(\024)25 b FB(2)c(+)f FA(\016)s FB(\()p FA(Q)1426 3280 y Ft(0)1450 3313 y FB(\).)p 3714 3313 V 3718 3255 55 4 v 3718 3313 V 3771 3313 4 62 v 118 3474 a FC(Claim)33 b(8.)46 b Fw(Claims)34 b(7)f(and)g(3)g(pr)-5 b(ove)34 b(L)-5 b(emma)34 b(9.)259 3634 y FC(Pro)s(of)h(of)g(claim)e(and)h(lemma:)k FB(Of)30 b(all)e(1)20 b(+)e FA(b)1961 3648 y Fu(n)p Ft(\000)p Fy(1)2128 3634 y FB(p)s(ossible)27 b(v)-5 b(alues)29 b(of)h FA(r)2883 3648 y Fu(n)p Ft(\000)p Fy(1)3020 3634 y FB(,)h(only)d(2)p FA(\016)s FB(\()p FA(Q)3469 3601 y Ft(0)3494 3634 y FB(\))19 b(+)g(6)26 b Fv(\024)118 3747 y FB(2)21 b Fv(\001)g FB(66)g(+)f(6)26 b FA(<)g FB(2)645 3714 y Fy(8)715 3747 y FB(v)-5 b(alues)30 b(cause)i(phase)e(2) h(to)g(fail)f(b)m(y)g(Claim)f(7.)42 b(By)31 b(Claim)e(3,)j(the)f (probabilit)m(y)d(for)i(eac)m(h)118 3860 y(v)-5 b(alue)30 b(is)f(at)i(most)g(1)p FA(=b)907 3874 y Fu(n)p Ft(\000)p Fy(1)1045 3860 y FB(.)41 b(Hence)31 b(w)m(e)g(\014nd)e(that)1129 4033 y Fw(pr)-5 b(ob)8 b FB(\()p FA(\016)1379 4048 y Fu(l)1431 4033 y FA(<)25 b(\016)1567 4048 y Fu(h)1612 4033 y FB(\))g Fv(\024)g FB(2)1813 3995 y Fy(8)1853 4033 y FA(=b)1937 4047 y Fu(n)p Ft(\000)p Fy(1)2100 4033 y Fv(\024)g FB(2)2241 3995 y Fy(8)2281 4033 y FA(=)p FB(2)2371 3995 y Fy(31)2472 4033 y FB(=)g(2)2613 3995 y Ft(\000)p Fy(23)2743 4033 y FA(:)118 4205 y FB(This)k(concludes)g(the)i(pro)s(of) e(of)i(Lemma)g(9.)p 3714 4205 V 3718 4147 55 4 v 3718 4205 V 3771 4205 4 62 v 118 4486 a Fz(4)135 b(Other)45 b(division)h(metho)t(ds)118 4689 y FB(In)31 b(this)g(section)h(w)m(e)g (lo)s(ok)f(at)i(other)f(sub)s(quadratic)d(metho)s(ds)i(for)h(division)d (that)j(migh)m(t)f(b)s(e)h(applicable)118 4802 y(for)25 b(large)h(n)m(um)m(b)s(ers)d(in)h(practice.)40 b(Neglecting)26 b(the)f(F)-8 b(ast)27 b(F)-8 b(ourier)24 b(T)-8 b(ransformation)25 b(b)s(ecause)g(of)h(its)e(h)m(uge)118 4915 y(constan)m(t)35 b(o)m(v)m(erhead)f(w)m(e)g(consider)e(Karatsuba's)h(metho)s(d)f(as)i (the)f(fastest)h(m)m(ultiplication)c(in)i(practice)118 5028 y(for)25 b(n)m(um)m(b)s(ers)f(with)g(reasonable)h(length.)39 b(With)24 b(this)h(assumption)e(w)m(e)j(dev)m(elop)f(another)h(fast)f (algorithm)118 5141 y(for)d(division)d(b)m(y)j(com)m(bining)e(three)i (w)m(ell)f(kno)m(wn)h(metho)s(ds)f(\(Barrett's)i(metho)s(d)f(and)f(in)m (v)m(erses)h(computed)118 5254 y(b)m(y)36 b(Newtonian)h(iteration)f (com)m(bined)f(with)g(Karatsuba)i(m)m(ultiplication\))c(and)j(argue)h (that)g(recursiv)m(e)118 5367 y(division)d(is)h(still)f(sup)s(erior)g (ev)m(en)j(if)f(w)m(e)h(assume)f(that)h(an)f(actual)h(implemen)m (tation)e(of)i(the)f(com)m(bined)118 5479 y(metho)s(ds)24 b(uses)f(all)h(kno)m(wn)f(tric)m(ks)i(and)e(shortcuts.)39 b(So)24 b(to)h(sp)s(eak,)g(w)m(e)g(tak)m(e)h(the)e(b)s(est)g(ingredien) m(ts)f(kno)m(wn)118 5592 y(so)31 b(far)f(to)h(mak)m(e)g(a)g(mix)e(of)i (algorithms)e(for)h(division.)1903 5841 y(19)p eop %%Page: 20 24 20 23 bop 118 399 a Fs(4.1)112 b(Barrett's)37 b(metho)s(d)118 570 y FB(Barrett's)d(metho)s(d)f(is)f(kno)m(wn)h(in)f(applied)e (cryptograph)m(y)k(as)f(an)g(algorithm)f(for)h(mo)s(dular)e(reduction.) 118 683 y(Here)25 b(the)f(goal)h(is)e(to)i(reduce)f(a)h(n)m(um)m(b)s (er)e FA(A)h FB(mo)s(dulo)f(another)h(n)m(um)m(b)s(er)f FA(M)10 b FB(,)26 b(whic)m(h)d(means)h(that)h(only)e(the)118 796 y(remainder)32 b(of)i(the)g(quotien)m(t)g Fv(b)p FA(A=)-5 b(M)10 b Fv(c)35 b FB(is)d(imp)s(ortan)m(t)h(but)g(not)h(the)g (quotien)m(t)g(itself.)49 b(As)33 b(it)h(turns)e(out,)118 909 y(mo)s(dular)38 b(reduction)h(is)g(no)h(easier)g(than)g(division)d (with)i(remainder.)68 b(Ho)m(w)m(ev)m(er,)45 b(Barrett's)c(metho)s(d) 118 1022 y(pro\014ts)h(m)m(uc)m(h)h(from)f(the)h(fact)h(that)f(in)e (some)j(applications)c(man)m(y)j(mo)s(dular)e(reductions)h(are)h(done) 118 1135 y(mo)s(dulo)26 b(the)h(same)h(n)m(um)m(b)s(er,)e(in)g(whic)m (h)g(case)j(it)d(is)h(advisable)e(to)j(precompute)f(the)h(\(shifted\))e (in)m(v)m(erse)h(of)118 1248 y FA(M)35 b FB(up)25 b(to)h(a)g(certain)f (precision.)37 b(Let)26 b(us)f(sk)m(etc)m(h)h(Barrett's)h(algorithm)d (as)i(found)e(in)g([3)q(])h(in)f(our)h(notation.)118 1394 y FC(Algorithm)35 b(7.)45 b Fw(\(Barr)-5 b(ett)43 b(division\))e(L)-5 b(et)41 b FA(A)g Fw(and)g FA(M)51 b Fw(b)-5 b(e)40 b(nonne)-5 b(gative)42 b(inte)-5 b(gers)41 b(such)g(that)h FA(\014)3532 1361 y Fu(n)p Ft(\000)p Fy(1)3709 1394 y Fv(\024)118 1507 y FA(M)35 b(<)25 b(\014)393 1474 y Fu(n)473 1507 y Fw(and)34 b FA(A)25 b(<)g(\014)895 1474 y Fy(2)p Fu(n)978 1507 y Fw(.)41 b(The)33 b(algorithm)i(r)-5 b(eturns)33 b(the)g(quotient)g FA(Q)25 b FB(=)g Fv(b)p FA(A=)-5 b(M)10 b Fv(c)34 b Fw(with)g(r)-5 b(emainder)34 b FA(R)q Fw(.)226 1654 y(1.)45 b(Pr)-5 b(e)g(c)g(ompute)35 b(the)e(shifte)-5 b(d)33 b(inverse)g(of)g FA(M)42 b Fw(as)34 b FA(\026)25 b FB(=)g Fv(b)p FA(\014)2211 1621 y Fy(2)p Fu(n)2293 1654 y FA(=)-5 b(M)10 b Fv(c)27 b FA(<)e(\014)2650 1621 y Fu(n)p Fy(+1)2787 1654 y Fw(.)226 1825 y(2.)45 b(Set)33 b FA(A)569 1839 y Fy(1)634 1825 y FB(=)25 b Fv(b)p FA(A)c Fv(\001)f FA(\014)960 1792 y Ft(\000)p Fy(\()p Fu(n)p Ft(\000)p Fy(1\))1207 1825 y Fv(c)p Fw(.)226 1996 y(3.)45 b(Set)519 1973 y Fr(b)501 1996 y FA(Q)25 b FB(=)g Fv(b)p FA(A)802 2010 y Fy(1)862 1996 y Fv(\001)c FA(\026)f Fv(\001)g FA(\014)1084 1963 y Ft(\000)p Fy(\()p Fu(n)p Fy(+1\))1331 1996 y Fv(c)p Fw(.)226 2168 y(4.)45 b(Set)33 b FA(R)q FB(\()624 2145 y Fr(b)606 2168 y FA(Q)p FB(\))26 b(=)f FA(A)20 b Fv(\000)g FA(M)30 b Fv(\001)1196 2145 y Fr(b)1178 2168 y FA(Q)o Fw(.)226 2339 y(5.)45 b(Compute)34 b FA(\016)s FB(\()828 2316 y Fr(b)809 2339 y FA(Q)q FB(\))26 b(=)f Fv(b)p FA(R)q FB(\()1202 2316 y Fr(b)1184 2339 y FA(Q)p FB(\))p FA(=)-5 b(M)10 b Fv(c)34 b Fw(by)e(scho)-5 b(ol)35 b(division)e(and)g(the)g(over)-5 b(al)5 b(l)34 b(r)-5 b(emainder)34 b FA(R)26 b FB(=)f FA(R)q FB(\()3600 2316 y Fr(b)3581 2339 y FA(Q)p FB(\))c Fv(\000)345 2452 y FA(M)10 b(\016)s FB(\()541 2429 y Fr(b)521 2452 y FA(Q)r FB(\))p Fw(.)226 2623 y(6.)45 b(R)-5 b(eturn)34 b FA(R)f Fw(and)h FA(Q)25 b FB(=)1140 2600 y Fr(b)1122 2623 y FA(Q)20 b FB(+)g FA(\016)s FB(\()1402 2600 y Fr(b)1383 2623 y FA(Q)q FB(\))p Fw(.)259 2770 y FB(This)k(form)m(ulation)h(of)g(Barrett's)i(algorithm)e(requires)f(a) i(few)g(explanations.)38 b(First,)26 b(the)g(precompu-)118 2883 y(tation)h(of)f FA(\026)g FB(is)f(not)i(strictly)e(a)i(part)f(of)g (the)g(algorithm;)h(w)m(e)g(will)d(discuss)g(this)h(step)h(in)f(the)i (next)f(section.)118 2995 y(Next,)45 b(if)c(w)m(e)h(omitted)f(the)h(do) m(wn)m(w)m(ards-rounding)d(in)h(steps)i(1{3)h(and)d(computed)i(ev)m (ery)g(op)s(eration)118 3108 y(exactly)-8 b(,)35 b(the)e(resulting)e (quotien)m(t)i(w)m(ould)e(b)s(e)h(exact.)49 b(W)-8 b(e)34 b(omit)f(the)g(error)f(analysis)f(here;)j(details)e(can)118 3221 y(b)s(e)g(found)f(in)g([1)q(].)47 b(It)32 b(su\016ces)h(to)g(kno)m (w)f(that)h(only)f(the)g(least)h(signi\014can)m(t)e(digit)h(of)3087 3198 y Fr(b)3069 3221 y FA(Q)g FB(is)g(not)g(exact,)j(so)118 3334 y(w)m(e)c(\014nd)e(that)i FA(\016)s FB(\()728 3311 y Fr(b)709 3334 y FA(Q)q FB(\))f(is)g(computable)g(in)f(linear)g(time)h (b)m(y)g(sc)m(ho)s(ol)g(division,)e(see)j(Lemma)f(5.)259 3447 y(The)35 b(running)d(time)j(of)g(Barrett's)i(algorithm)d(dep)s (ends)f(on)i(the)g(c)m(hosen)h(m)m(ultiplication)c(metho)s(d.)118 3560 y(There)j(are)g(only)f(t)m(w)m(o)j(m)m(ultiplications)32 b(that)j(are)h(not)f(mere)h(digit{shifts,)e(namely)g(the)i(computation) 118 3673 y(of)45 b FA(A)304 3687 y Fy(1)373 3673 y Fv(\001)30 b FA(\026)44 b FB(in)f(step)h(3)h(and)f(the)g(computation)g(of)h FA(M)39 b Fv(\001)2167 3650 y Fr(b)2148 3673 y FA(Q)44 b FB(in)f(step)i(4.)83 b(All)43 b(op)s(erations)h(b)s(eside)e(the)118 3786 y(precomputation)33 b(of)h FA(\026)f FB(can)h(b)s(e)f(done)g(in)f (linear)g(time.)50 b(W)-8 b(e)35 b(conclude)d(that)i(Barrett's)h (algorithm)e(has)118 3899 y(running)26 b(time)i(2)p FA(M)10 b FB(\()p FA(n)p FB(\).)41 b(Used)28 b(with)f(Karatsuba)i(m)m (ultiplication,)d(Barrett's)k(algorithm)d(has)h(the)h(same)118 4012 y(running)c(time)j(as)g FA(D)843 4030 y Fy(2)p Fu(n=)p Fy(1)p Fu(n)1039 4012 y FB(.)40 b(In)27 b(other)h(w)m(ords,)g (Barrett's)h(algorithm)e(is)g(asymptotically)h(no)f(faster)i(than)118 4125 y(our)36 b(recursiv)m(e)f(division)e Fw(even)k(if)g(the)h(pr)-5 b(e)g(c)g(omputation)42 b(of)c(the)g(divisor's)h(inverse)e(is)h(ne)-5 b(gle)g(cte)g(d)p FB(.)58 b(This)118 4237 y(statemen)m(t)33 b(is)e(true)g(ev)m(en)h(if)e(sp)s(ecial)g(v)m(ersions)g(of)i(Karatsuba) f(m)m(ultiplication)d(are)k(used)e(that)i(are)g(\014ne-)118 4350 y(tuned)37 b(for)h(the)h(particular)d(situation)h(inside)f (Barrett's)j(algorithm.)63 b(The)38 b(k)m(ey)h(for)e(the)i(sp)s(eed-up) d(of)118 4463 y(the)30 b(m)m(ultiplication)d(in)h(step)i(3)g(is)e(that) j(here)e(w)m(e)h(only)f(need)h(the)g FA(n)18 b FB(+)g(1)30 b(most)h(signi\014can)m(t)d(digits)g(of)i(the)118 4576 y(result.)60 b(Similarly)-8 b(,)36 b(in)g(step)h(4)h(w)m(e)f(only)f (need)h(\(roughly\))g(the)g FA(n)25 b FB(+)f(1)38 b(least)f (signi\014can)m(t)f(digits)g(of)h(the)118 4689 y(result)f FA(M)f Fv(\001)570 4666 y Fr(b)551 4689 y FA(Q)p FB(,)40 b(b)s(ecause)d(the)g(higher)f FA(n)25 b FB(+)f(1)38 b(digits)e(are)i (kno)m(wn)f(to)h(b)s(e)e(equal)h(to)h(those)g(of)g FA(A)f FB(b)m(y)g(the)118 4802 y(error)d(analysis.)50 b(Unfortunately)-8 b(,)35 b(the)f(adv)-5 b(an)m(tage)36 b(w)m(e)f(get)g(from)f(using)e (these)j(sp)s(ecial)d(m)m(ultiplication)118 4915 y(routines)d(o)m(v)m (er)j(normal)d(Karatsuba)i(tends)f(to)h(zero)g(as)f FA(n)g FB(increases;)h(for)f(details)f(see)i([6)q(].)259 5028 y(It)38 b(should)d(b)s(e)i(remark)m(ed)h(that)g(in)e(cryptograph)m(y)i (Barrett's)h(algorithm)d(is)h(curren)m(tly)f(used)h(with)118 5141 y(sp)s(ecial)d(v)m(ersions)g(of)h(sc)m(ho)s(ol)g(m)m (ultiplication.)52 b(Then)34 b(the)i(n)m(um)m(b)s(er)d(of)j(digit)d(m)m (ultiplications)f(for)j Fw(b)-5 b(oth)118 5254 y FB(pro)s(ducts)36 b(in)h(Barrett's)i(algorithm)d(together)k(is)c FA(n)1948 5221 y Fy(2)2012 5254 y FB(+)25 b FA(O)s FB(\()p FA(n)p FB(\),)40 b(the)e(same)g(n)m(um)m(b)s(er)e(of)i(steps)g(that)g(are)118 5367 y(needed)27 b(for)g Fw(one)34 b FB(ordinary)26 b(sc)m(ho)s(ol)h(m) m(ultiplication)d(of)j FA(n)p FB(-digit)f(n)m(um)m(b)s(ers.)38 b(This)26 b(classical)g(implemen)m(ta-)118 5479 y(tion)e(of)g (Barrett's)h(algorithm)e(is)g(ab)s(out)h(as)g(fast)h(as)f(sc)m(ho)s(ol) g(division,)f(but)g(it)h(a)m(v)m(oids)g(division)d(op)s(erations)118 5592 y(altogether)32 b([8].)1903 5841 y(20)p eop %%Page: 21 25 21 24 bop 118 399 a Fs(4.2)112 b(In)m(v)m(erses)38 b(b)m(y)g(Newtonian) f(iteration)118 570 y FB(Giv)m(en)30 b(a)h(real)f(n)m(um)m(b)s(er)f FA(v)g Fv(2)c FB(\(1)p FA(=)p FB(2)p FA(;)15 b FB(1])33 b(and)d(an)g(appropriate)f(start)i(v)-5 b(alue)30 b(for)g FA(x)p FB(,)h(the)f(iteration)1613 774 y FA(x)25 b Fv( )g FB(2)c Fv(\001)f FA(x)g Fv(\000)g FA(v)k Fv(\001)c FA(x)2245 737 y Fy(2)118 979 y FB(con)m(v)m(erges)38 b(to)g(the)e(in)m(v)m(erse)g (1)p FA(=v)s 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y(using)f(Karatsuba)i(m)m (ultiplication)d(is)i(2)p FA(K)7 b FB(\()p FA(n)p FB(\).)57 b(This)34 b(is)h(still)e(not)j(the)g(b)s(est)g(one)g(can)g(do;)j(half) 34 b(of)i(the)118 1769 y(m)m(ultiplications)27 b(actually)j(compute)g (squares,)g(whic)m(h)f(is)g(considerably)g(simpler)e(than)j(p)s (erforming)e(or-)118 1882 y(dinary)37 b(m)m(ultiplications.)62 b(Seeking)38 b(for)g(the)h(b)s(est)f(p)s(erformance)g(one)g(can)h (implemen)m(t)e(the)i(squaring)118 1995 y(b)m(y)32 b(a)g(straigh)m (tforw)m(ard)f(v)-5 b(arian)m(t)32 b(of)g(Karatsuba)g(m)m (ultiplication,)d(whose)j(running)d(time)i(w)m(e)h(denote)h(b)m(y)118 2108 y FA(K)195 2122 y Fu(sq)266 2108 y FB(\()p FA(n)p FB(\).)57 b(One)35 b(should)f(exp)s(ect)i(1)p FA(=)p FB(2)p FA(K)7 b FB(\()p FA(n)p FB(\))35 b FA(<)f(K)1818 2122 y Fu(sq)1889 2108 y FB(\()p FA(n)p FB(\))g FA(<)g(K)7 b FB(\()p FA(n)p FB(\))35 b(in)g(practice.)57 b(There)35 b(are)h(more)g(tric)m(ks)118 2221 y(one)k(could)e(use)h(to)h(sp)s(eed)f 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1854 638 Crs 2039 697 Crs 2224 746 Crs 2409 826 Crs 2594 895 Crs 2779 984 Crs 2964 1073 Crs 3149 1182 Crs 3334 1271 Crs 3519 1380 Crs 3704 1498 Crs 3888 1627 Crs 4073 1746 Crs 4258 1894 Crs 4443 2003 Crs 4628 2132 Crs 4813 2320 Crs 4998 2498 Crs 5183 2666 Crs 5368 2844 Crs 5553 3052 Crs 5738 3200 Crs 5923 3388 Crs 6107 3635 Crs 6292 3774 Crs 6477 4041 Crs 6662 4229 Crs 6847 4437 Crs 7032 4694 Crs 1683 4599 Crs stroke grestore end showpage %%EndDocument @endspecial 653 3527 a Fp(Figure)i(5:)36 b FD(Comparing)27 b(F)-7 b(astDivision)27 b(to)g(Sc)n(ho)r(ol)g(Division)h(in)g(a)f (small)g(range)118 3868 y Fs(5.2)112 b(Hand-co)s(ded)39 b(assem)m(bler)e(routines)118 4040 y FB(Using)e(handwritten)e(assem)m (bler)i(co)s(de)h(for)f(the)g(innermost)f(lo)s(ops)h(lo)m(w)m(ers)g (the)h(running)c(times)j(signif-)118 4153 y(ican)m(tly)-8 b(,)46 b(but)c(also)h(mo)m(v)m(es)h(up)e(the)h(break-ev)m(en)h(p)s(oin) m(t)d(of)i Fx(FastDivision)d FB(with)h(resp)s(ect)i(to)h(sc)m(ho)s(ol) 118 4266 y(division)34 b(to)j(ab)s(out)f(64)i(digits.)58 b(In)36 b(this)f(con)m(text)k(it)d(is)f(more)i(useful)e(to)i(compare)g Fx(FastDivision)d FB(to)118 4378 y(other)26 b(in)m(teger)g(pac)m(k)-5 b(ages.)41 b(These)26 b(other)g(pac)m(k)-5 b(ages)28 b(w)m(ere)e(CLN)f([12)q(]\),)j(libI)23 b([11)q(],)28 b(and)d(LIP)g([13)q(])h(a)m(v)-5 b(ailable)118 4491 y(to)39 b(us)f(in)e(the)j(LiDIA)e(library)f([15)q(].)65 b(Figure)37 b(10)i(sho)m(ws)f(that)h(for)f(n)m(um)m(b)s(ers)f(less)g(than)h(100)i (mac)m(hine)118 4604 y(digits)30 b(there)i(is)f(no)h(signi\014can)m(t)e (di\013erence)i(in)e(the)i(running)d(times)i(compared)h(to)g(CLN)g (\(whic)m(h)f(is)f(the)118 4717 y(fastest)k(public)c(domain)h(library)f (for)j(in)m(teger)g(arithmetic)e(b)s(eside)g(ours)h(at)i(the)e(time)h (of)f(this)g(writing\).)118 4830 y(These)27 b(\014gures)g(sho)m(w)h (the)g(a)m(v)m(erage)i(running)25 b(time)i(p)s(er)g(division.)36 b(Ho)m(w)m(ev)m(er,)31 b(Figure)c(11)h(sho)m(ws)g(that)g(w)m(e)118 4943 y(are)i(m)m(uc)m(h)g(faster)g(as)g(the)f(digits)g(b)s(ecome)h (larger.)40 b(W)-8 b(e)31 b(exp)s(ect)f(that)g(w)m(e)g(can)g(lo)m(w)m (er)g(our)f(running)e(times)118 5056 y(ev)m(en)k(more)g(if)e(w)m(e)i (ha)m(v)m(e)g(more)g(time)f(to)h(\014ne-tune)f(our)g(assem)m(bler)g(co) s(de.)118 5299 y Fs(5.3)112 b(F)-9 b(ast)38 b(mo)s(dular)f(m)m (ultiplication)118 5471 y FB(The)29 b(e\013ectiv)m(eness)i(of)f(public) d(k)m(ey)k(cryptosystems)f(dep)s(ends)e(hea)m(vily)h(on)g(the)h(abilit) m(y)e(to)j(quic)m(kly)d(ev)-5 b(al-)118 5584 y(uate)28 b(expressions)e(of)h(the)h(form)e FA(x)1305 5551 y Fu(e)1367 5584 y Fw(mo)-5 b(d)37 b FA(m)p FB(,)27 b(where)g FA(x)g FB(is)f(the)i Fw(message)34 b FB(to)28 b(b)s(e)f(encrypted)g(or)g (decrypted)1903 5841 y(22)p eop %%Page: 23 27 23 26 bop 768 2289 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 2834 @rwi @setspecial %%BeginDocument: BigRange.eps /gnudict 120 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth 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rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 476 420 M 63 0 V 6493 0 R -63 0 V 392 420 M (0) Rshow 476 825 M 63 0 V 6493 0 R -63 0 V 392 825 M (5) Rshow 476 1229 M 63 0 V 6493 0 R -63 0 V -6577 0 R (10) Rshow 476 1634 M 63 0 V 6493 0 R -63 0 V -6577 0 R (15) Rshow 476 2039 M 63 0 V 6493 0 R -63 0 V -6577 0 R (20) Rshow 476 2444 M 63 0 V 6493 0 R -63 0 V -6577 0 R (25) Rshow 476 2848 M 63 0 V 6493 0 R -63 0 V -6577 0 R (30) Rshow 476 3253 M 63 0 V 6493 0 R -63 0 V -6577 0 R (35) Rshow 476 3658 M 63 0 V 6493 0 R -63 0 V -6577 0 R (40) Rshow 476 4063 M 63 0 V 6493 0 R -63 0 V -6577 0 R (45) Rshow 476 4467 M 63 0 V 6493 0 R -63 0 V -6577 0 R (50) Rshow 476 4872 M 63 0 V 6493 0 R -63 0 V -6577 0 R (55) Rshow 840 420 M 0 63 V 0 4389 R 0 -63 V 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1043 Pls 2297 1053 Pls 2370 1062 Pls 2443 1076 Pls 2516 1092 Pls 2588 1090 Pls 2661 1194 Pls 2734 1215 Pls 2807 1227 Pls 2880 1231 Pls 2953 1242 Pls 3026 1241 Pls 3098 1268 Pls 3171 1288 Pls 3244 1304 Pls 3317 1305 Pls 3390 1319 Pls 3463 1326 Pls 3535 1328 Pls 3608 1351 Pls 3681 1351 Pls 3754 1360 Pls 3827 1534 Pls 3900 1581 Pls 3973 1556 Pls 4045 1578 Pls 4118 1615 Pls 4191 1624 Pls 4264 1641 Pls 4337 1637 Pls 4410 1675 Pls 4482 1653 Pls 4555 1723 Pls 4628 1698 Pls 4701 1698 Pls 4774 1722 Pls 4847 1712 Pls 4920 1719 Pls 4992 1935 Pls 5065 1961 Pls 5138 1995 Pls 5211 2006 Pls 5284 2017 Pls 5357 2055 Pls 5429 2079 Pls 5502 2076 Pls 5575 2066 Pls 5648 2087 Pls 5721 2099 Pls 5794 2113 Pls 5866 2121 Pls 5939 2137 Pls 6012 2142 Pls 6085 2151 Pls 6158 2206 Pls 6231 2221 Pls 6304 2238 Pls 6376 2270 Pls 6449 2280 Pls 6522 2295 Pls 6595 2304 Pls 6668 2338 Pls 6741 2326 Pls 6813 2342 Pls 6886 2345 Pls 6959 2363 Pls 7032 2373 Pls 1599 4739 Pls 1.000 UP 1.000 UL LT1 1316 4599 M (SchoolDiv) Rshow 476 770 Crs 549 785 Crs 622 809 Crs 695 831 Crs 767 853 Crs 840 876 Crs 913 899 Crs 986 923 Crs 1059 943 Crs 1132 967 Crs 1204 992 Crs 1277 1028 Crs 1350 1046 Crs 1423 1069 Crs 1496 1102 Crs 1569 1127 Crs 1642 1160 Crs 1714 1191 Crs 1787 1239 Crs 1860 1263 Crs 1933 1272 Crs 2006 1303 Crs 2079 1330 Crs 2151 1357 Crs 2224 1398 Crs 2297 1425 Crs 2370 1461 Crs 2443 1488 Crs 2516 1546 Crs 2588 1561 Crs 2661 1616 Crs 2734 1669 Crs 2807 1689 Crs 2880 1717 Crs 2953 1772 Crs 3026 1774 Crs 3098 1820 Crs 3171 1867 Crs 3244 1912 Crs 3317 1937 Crs 3390 1995 Crs 3463 2021 Crs 3535 2072 Crs 3608 2121 Crs 3681 2172 Crs 3754 2211 Crs 3827 2264 Crs 3900 2359 Crs 3973 2347 Crs 4045 2392 Crs 4118 2426 Crs 4191 2472 Crs 4264 2530 Crs 4337 2562 Crs 4410 2661 Crs 4482 2660 Crs 4555 2747 Crs 4628 2791 Crs 4701 2826 Crs 4774 2897 Crs 4847 2954 Crs 4920 2964 Crs 4992 3018 Crs 5065 3081 Crs 5138 3137 Crs 5211 3197 Crs 5284 3248 Crs 5357 3315 Crs 5429 3386 Crs 5502 3412 Crs 5575 3457 Crs 5648 3509 Crs 5721 3569 Crs 5794 3640 Crs 5866 3693 Crs 5939 3764 Crs 6012 3803 Crs 6085 3883 Crs 6158 3943 Crs 6231 4000 Crs 6304 4060 Crs 6376 4130 Crs 6449 4254 Crs 6522 4284 Crs 6595 4324 Crs 6668 4413 Crs 6741 4453 Crs 6813 4515 Crs 6886 4579 Crs 6959 4653 Crs 7032 4716 Crs 1599 4599 Crs stroke grestore end showpage %%EndDocument @endspecial 692 2572 a Fp(Figure)32 b(6:)k FD(Comparing)26 b(F)-7 b(astDivision)27 b(to)h(Sc)n(ho)r(ol)f(Division)g(in)h(a)f(big)h (range)768 4965 y @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 2834 @rwi @setspecial %%BeginDocument: Fast2Kara.eps /gnudict 120 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 476 420 M 63 0 V 6493 0 R -63 0 V 392 420 M (0) Rshow 476 1162 M 63 0 V 6493 0 R -63 0 V -6577 0 R (5) Rshow 476 1904 M 63 0 V 6493 0 R -63 0 V -6577 0 R (10) Rshow 476 2646 M 63 0 V 6493 0 R -63 0 V -6577 0 R (15) Rshow 476 3388 M 63 0 V 6493 0 R -63 0 V -6577 0 R (20) Rshow 476 4130 M 63 0 V 6493 0 R -63 0 V -6577 0 R (25) Rshow 476 4872 M 63 0 V 6493 0 R -63 0 V -6577 0 R (30) Rshow 788 420 M 0 63 V 0 4389 R 0 -63 V 788 280 M (80) Cshow 1413 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (100) Cshow 2037 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (120) Cshow 2661 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (140) Cshow 3286 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (160) Cshow 3910 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (180) Cshow 4534 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (200) Cshow 5159 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (220) Cshow 5783 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (240) Cshow 6408 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (260) Cshow 7032 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (280) Cshow 1.000 UL LTb 476 420 M 6556 0 V 0 4452 V -6556 0 V 476 420 L 0 2646 M currentpoint gsave translate 90 rotate 0 0 M (sec) Cshow grestore 3754 140 M (Digits) Cshow 1.000 UP 1.000 UL LT0 1232 4739 M (KaraMult) Rshow 476 693 Pls 538 703 Pls 601 706 Pls 663 711 Pls 726 715 Pls 788 715 Pls 851 798 Pls 913 798 Pls 976 816 Pls 1038 824 Pls 1100 831 Pls 1163 830 Pls 1225 830 Pls 1288 828 Pls 1350 930 Pls 1413 954 Pls 1475 962 Pls 1537 954 Pls 1600 969 Pls 1662 985 Pls 1725 960 Pls 1787 972 Pls 1850 991 Pls 1912 996 Pls 1975 997 Pls 2037 1006 Pls 2099 1017 Pls 2162 1015 Pls 2224 1021 Pls 2287 1023 Pls 2349 1229 Pls 2412 1245 Pls 2474 1251 Pls 2536 1238 Pls 2599 1238 Pls 2661 1236 Pls 2724 1267 Pls 2786 1269 Pls 2849 1284 Pls 2911 1288 Pls 2974 1288 Pls 3036 1302 Pls 3098 1306 Pls 3161 1296 Pls 3223 1312 Pls 3286 1312 Pls 3348 1567 Pls 3411 1566 Pls 3473 1561 Pls 3535 1567 Pls 3598 1612 Pls 3660 1604 Pls 3723 1621 Pls 3785 1622 Pls 3848 1629 Pls 3910 1619 Pls 3973 1635 Pls 4035 1662 Pls 4097 1638 Pls 4160 1656 Pls 4222 1670 Pls 4285 1656 Pls 4347 1940 Pls 4410 1934 Pls 4472 2011 Pls 4534 2015 Pls 4597 1999 Pls 4659 2038 Pls 4722 2041 Pls 4784 2041 Pls 4847 2015 Pls 4909 2043 Pls 4972 2035 Pls 5034 2043 Pls 5096 2051 Pls 5159 2070 Pls 5221 2054 Pls 5284 2063 Pls 5346 2141 Pls 5409 2140 Pls 5471 2152 Pls 5533 2171 Pls 5596 2186 Pls 5658 2173 Pls 5721 2170 Pls 5783 2156 Pls 5846 2195 Pls 5908 2179 Pls 5971 2187 Pls 6033 2184 Pls 6095 2204 Pls 6158 2198 Pls 6220 2213 Pls 6283 2239 Pls 6345 2872 Pls 6408 2823 Pls 6470 2858 Pls 6532 2878 Pls 1515 4739 Pls 1.000 UP 1.000 UL LT1 1232 4599 M (FastDiv) Rshow 476 933 Crs 538 953 Crs 601 942 Crs 663 956 Crs 726 978 Crs 788 987 Crs 851 1085 Crs 913 1098 Crs 976 1137 Crs 1038 1146 Crs 1100 1175 Crs 1163 1187 Crs 1225 1189 Crs 1288 1205 Crs 1350 1339 Crs 1413 1382 Crs 1475 1391 Crs 1537 1423 Crs 1600 1438 Crs 1662 1451 Crs 1725 1444 Crs 1787 1466 Crs 1850 1527 Crs 1912 1557 Crs 1975 1563 Crs 2037 1580 Crs 2099 1597 Crs 2162 1622 Crs 2224 1652 Crs 2287 1649 Crs 2349 1839 Crs 2412 1877 Crs 2474 1900 Crs 2536 1907 Crs 2599 1928 Crs 2661 1925 Crs 2724 1975 Crs 2786 2011 Crs 2849 2041 Crs 2911 2042 Crs 2974 2069 Crs 3036 2081 Crs 3098 2085 Crs 3161 2127 Crs 3223 2127 Crs 3286 2143 Crs 3348 2462 Crs 3411 2548 Crs 3473 2502 Crs 3535 2542 Crs 3598 2610 Crs 3660 2628 Crs 3723 2658 Crs 3785 2652 Crs 3848 2722 Crs 3910 2680 Crs 3973 2809 Crs 4035 2763 Crs 4097 2763 Crs 4160 2808 Crs 4222 2788 Crs 4285 2802 Crs 4347 3198 Crs 4410 3246 Crs 4472 3308 Crs 4534 3327 Crs 4597 3348 Crs 4659 3418 Crs 4722 3462 Crs 4784 3456 Crs 4847 3438 Crs 4909 3476 Crs 4972 3498 Crs 5034 3525 Crs 5096 3539 Crs 5159 3568 Crs 5221 3576 Crs 5284 3594 Crs 5346 3695 Crs 5409 3722 Crs 5471 3753 Crs 5533 3811 Crs 5596 3830 Crs 5658 3858 Crs 5721 3873 Crs 5783 3936 Crs 5846 3915 Crs 5908 3943 Crs 5971 3949 Crs 6033 3982 Crs 6095 4001 Crs 6158 4008 Crs 6220 4051 Crs 6283 4108 Crs 6345 4630 Crs 6408 4655 Crs 6470 4792 Crs 1515 4599 Crs stroke grestore end showpage %%EndDocument @endspecial 750 5248 a Fp(Figure)j(7:)36 b FD(Comparing)26 b(F)-7 b(astDivison)27 b(with)h(Karatsuba)e(m)n(ultiplication)1903 5841 y FB(23)p eop %%Page: 24 28 24 27 bop 768 2289 a @beginspecial 50 @llx 50 @lly 446 @urx 327 @ury 2834 @rwi @setspecial %%BeginDocument: Surfacetest.eps /gnudict 120 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 3960 5277 M (Varying the sizes of A and B) Cshow 1.000 UL LT2 7122 4920 M ("Surfacetest.dat" using 2:1:3) Rshow 7206 4920 M 399 0 V 3432 3856 M 74 46 V 74 27 V 74 46 V 73 26 V 74 405 V 74 -34 V 74 -14 V 74 27 V 74 6 V 73 -54 V 74 86 V 74 6 V 74 6 V 74 26 V 73 -14 V 74 -33 V 74 -54 V 74 -14 V 74 -34 V 73 -93 V 74 66 V 74 -34 V 74 -14 V 74 -94 V 73 66 V 74 -113 V 74 -34 V 74 -54 V 74 -133 V 73 -34 V 74 26 V 74 -134 V 74 26 V 74 -73 V 73 -54 V 74 -34 V 74 -233 V 74 -74 V 74 -74 V 73 -54 V 74 -113 V 74 -74 V 74 -94 V 74 -73 V 73 -194 V 3390 3793 M 73 66 V 74 46 V 74 46 V 74 6 V 74 345 V 73 -14 V 74 7 V 74 6 V 74 26 V 74 -54 V 74 106 V 73 -34 V 74 66 V 74 -54 V 74 7 V 74 -54 V 73 6 V 74 -34 V 74 -54 V 74 -53 V 74 66 V 73 -74 V 74 -54 V 74 -53 V 74 26 V 74 -74 V 73 -14 V 74 -94 V 74 -113 V 74 -34 V 74 -54 V 73 -54 V 74 26 V 74 -73 V 74 -54 V 74 -193 V 73 -74 V 74 -74 V 74 -74 V 74 -93 V 74 -74 V 73 -94 V 74 -93 V 74 -94 V 74 -94 V 3347 3769 M 74 66 V 74 26 V 73 26 V 74 46 V 74 325 V 74 -13 V 74 26 V 73 -34 V 74 26 V 74 6 V 74 26 V 74 -14 V 73 66 V 74 -14 V 74 7 V 74 -74 V 74 -74 V 74 26 V 73 -53 V 74 -34 V 74 6 V 74 -14 V 74 -74 V 73 -53 V 74 6 V 74 -14 V 74 -94 V 74 -93 V 73 -54 V 74 -54 V 74 -54 V 74 -53 V 74 -14 V 73 -114 V 74 -133 V 74 -74 V 74 -54 V 74 -114 V 73 -53 V 74 -74 V 74 -94 V 74 -94 V 74 -173 V 73 -14 V 74 -14 V 3304 3745 M 74 66 V 74 26 V 74 6 V 74 26 V 73 325 V 74 7 V 74 -14 V 74 26 V 74 26 V 73 -34 V 74 26 V 74 6 V 74 66 V 74 -53 V 73 6 V 74 -54 V 74 -14 V 74 -54 V 74 -33 V 74 -54 V 73 6 V 74 -14 V 74 -54 V 74 -93 V 74 26 V 73 -114 V 74 -14 V 74 -33 V 74 -94 V 74 -54 V 73 -34 V 74 -73 V 74 -34 V 74 -154 V 74 -93 V 73 -54 V 74 -74 V 74 -93 V 74 -74 V 74 -74 V 73 -94 V 74 -173 V 74 -14 V 74 -14 V 74 -14 V 3262 3701 M 74 46 V 73 26 V 74 46 V 74 6 V 74 346 V 74 -54 V 73 26 V 74 -14 V 74 26 V 74 -54 V 74 106 V 73 -53 V 74 46 V 74 26 V 74 -34 V 74 -14 V 73 -114 V 74 7 V 74 -34 V 74 -74 V 74 46 V 74 -34 V 73 -93 V 74 -74 V 74 -14 V 74 -94 V 74 66 V 73 -73 V 74 -94 V 74 -54 V 74 -33 V 74 -54 V 73 -194 V 74 -73 V 74 -54 V 74 -74 V 74 -54 V 73 -113 V 74 -74 V 74 -74 V 74 -93 V 74 -94 V 73 -14 V 74 -14 V 74 -14 V 3219 3657 M 74 66 V 74 26 V 74 6 V 73 26 V 74 326 V 74 -14 V 74 -34 V 74 6 V 73 26 V 74 -34 V 74 66 V 74 7 V 74 66 V 73 -34 V 74 -14 V 74 -154 V 74 47 V 74 -34 V 73 -14 V 74 -74 V 74 6 V 74 6 V 74 -153 V 73 -74 V 74 -14 V 74 -73 V 74 66 V 74 -34 V 74 -94 V 73 -74 V 74 -93 V 74 -114 V 74 -94 V 74 -53 V 73 -74 V 74 -74 V 74 -73 V 74 -94 V 74 -74 V 73 -74 V 74 -113 V 74 -14 V 74 -14 V 74 -14 V 73 6 V 3177 3633 M 73 26 V 74 46 V 74 6 V 74 46 V 74 306 V 73 -14 V 74 -14 V 74 6 V 74 -14 V 74 -13 V 73 86 V 74 -74 V 74 106 V 74 -14 V 74 -94 V 73 -54 V 74 7 V 74 -14 V 74 -74 V 74 -14 V 73 -14 V 74 -73 V 74 -54 V 74 -94 V 74 -34 V 73 -33 V 74 -14 V 74 -54 V 74 -14 V 74 -73 V 74 -194 V 73 -54 V 74 -73 V 74 -74 V 74 -54 V 74 -94 V 73 -73 V 74 -74 V 74 -174 V 74 -13 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 74 -13 V 3134 3589 M 74 46 V 74 26 V 73 26 V 74 26 V 74 266 V 74 6 V 74 26 V 73 -34 V 74 6 V 74 -13 V 74 26 V 74 26 V 73 46 V 74 -114 V 74 6 V 74 -53 V 74 26 V 73 -14 V 74 -54 V 74 -34 V 74 -73 V 74 -94 V 73 -54 V 74 6 V 74 -53 V 74 -74 V 74 -14 V 73 -14 V 74 -74 V 74 -213 V 74 -14 V 74 -93 V 74 -74 V 73 -34 V 74 -94 V 74 -93 V 74 -74 V 74 -74 V 73 -93 V 74 -14 V 74 6 V 74 -34 V 74 -14 V 73 -14 V 74 7 V 3091 3545 M 74 26 V 74 26 V 74 66 V 74 6 V 73 266 V 74 -14 V 74 -14 V 74 6 V 74 7 V 73 -14 V 74 26 V 74 6 V 74 66 V 74 -94 V 73 -33 V 74 -54 V 74 6 V 74 6 V 74 -34 V 73 -53 V 74 -114 V 74 -54 V 74 -14 V 74 -73 V 73 -34 V 74 -54 V 74 6 V 74 -34 V 74 -193 V currentpoint stroke M 73 -54 V 74 -74 V 74 -53 V 74 -74 V 74 -94 V 73 -73 V 74 -54 V 74 -94 V 74 -94 V 74 -13 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 74 6 V 74 -33 V 3049 3521 M 74 26 V 73 26 V 74 26 V 74 26 V 74 266 V 74 26 V 73 -14 V 74 -53 V 74 26 V 74 -14 V 74 26 V 73 6 V 74 -14 V 74 -14 V 74 -33 V 74 -34 V 73 26 V 74 -54 V 74 -54 V 74 -93 V 74 -54 V 73 -54 V 74 -54 V 74 -33 V 74 6 V 74 -74 V 73 6 V 74 -213 V 74 -34 V 74 -74 V 74 -73 V 73 -74 V 74 -74 V 74 -74 V 74 -53 V 74 -174 V 73 -14 V 74 -14 V 74 -13 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 6 V 74 -33 V 3006 3477 M 74 46 V 74 26 V 74 6 V 73 27 V 74 265 V 74 -34 V 74 46 V 74 -93 V 73 66 V 74 -34 V 74 46 V 74 -34 V 74 6 V 73 6 V 74 -73 V 74 -14 V 74 -34 V 74 -14 V 73 -33 V 74 -114 V 74 -54 V 74 -54 V 74 7 V 73 -74 V 74 -14 V 74 -74 V 74 -133 V 74 -74 V 73 -54 V 74 -54 V 74 -73 V 74 -74 V 74 -74 V 73 -73 V 74 -154 V 74 -14 V 74 -14 V 74 6 V 73 -33 V 74 -14 V 74 -14 V 74 6 V 74 -34 V 74 -13 V 73 -14 V 2964 3453 M 73 6 V 74 46 V 74 7 V 74 46 V 74 245 V 73 -34 V 74 7 V 74 -54 V 74 26 V 74 -34 V 73 46 V 74 -14 V 74 -13 V 74 6 V 74 -34 V 73 -54 V 74 -14 V 74 -93 V 74 -34 V 74 -54 V 73 -34 V 74 -53 V 74 6 V 74 -74 V 74 -34 V 73 -173 V 74 -34 V 74 -74 V 74 -54 V 74 -53 V 73 -74 V 74 -74 V 74 -74 V 74 -153 V 74 -14 V 73 -14 V 74 -14 V 74 -13 V 74 -14 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 74 -13 V 74 -14 V 2921 3409 M 74 26 V 74 26 V 73 7 V 74 6 V 74 245 V 74 26 V 74 -33 V 73 6 V 74 -34 V 74 -34 V 74 66 V 74 -14 V 73 7 V 74 -54 V 74 -34 V 74 -34 V 74 -74 V 73 -53 V 74 -14 V 74 -54 V 74 -54 V 74 -33 V 73 -14 V 74 -34 V 74 -174 V 74 -53 V 74 -54 V 73 -74 V 74 -53 V 74 -54 V 74 -74 V 74 -74 V 73 -153 V 74 -14 V 74 6 V 74 -34 V 74 -14 V 73 -13 V 74 6 V 74 -34 V 74 -14 V 74 -14 V 73 6 V 74 -33 V 74 -14 V 2878 3385 M 74 6 V 74 27 V 74 26 V 74 6 V 73 225 V 74 -13 V 74 -14 V 74 6 V 74 6 V 73 -34 V 74 46 V 74 -33 V 74 -14 V 74 -14 V 73 -34 V 74 -14 V 74 -153 V 74 -34 V 74 6 V 73 -54 V 74 -53 V 74 -54 V 74 6 V 74 -154 V 73 -73 V 74 -54 V 74 -54 V 74 -54 V 74 -53 V 73 -94 V 74 -74 V 74 -133 V 74 6 V 74 -34 V 73 -14 V 74 -14 V 74 6 V 74 -33 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -13 V 73 -14 V 2836 3341 M 74 26 V 73 7 V 74 26 V 74 6 V 74 246 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 74 -54 V 74 7 V 73 6 V 74 6 V 74 26 V 74 -54 V 74 -54 V 73 -113 V 74 -34 V 74 -14 V 74 -54 V 74 -53 V 73 -34 V 74 -134 V 74 -53 V 74 -54 V 74 -54 V 73 -74 V 74 -73 V 74 -54 V 74 -74 V 74 -74 V 73 -73 V 74 -14 V 74 -14 V 74 -14 V 74 6 V 73 -33 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 7 V 74 -34 V 74 -14 V 2793 3297 M 74 47 V 74 -14 V 74 26 V 73 6 V 74 246 V 74 -74 V 74 46 V 74 -54 V 73 26 V 74 -53 V 74 6 V 74 6 V 74 26 V 73 -134 V 74 66 V 74 -93 V 74 -74 V 74 -34 V 73 -14 V 74 -53 V 74 -34 V 74 -154 V 74 -54 V 73 -53 V 74 -54 V 74 -54 V 74 -54 V 74 -73 V 73 -54 V 74 -94 V 74 -53 V 74 -34 V 74 -14 V 73 -14 V 74 6 V 74 -34 V 74 -13 V 74 -14 V 73 6 V 74 -34 V 74 -14 V 74 -14 V 74 7 V 73 -34 V 74 -14 V 2751 3274 M 73 26 V 74 6 V 74 6 V 74 46 V 74 166 V 73 -14 V 74 26 V 74 -54 V 74 6 V 74 7 V 73 -34 V 74 -34 V 74 -34 V 74 -14 V 74 6 V 73 -93 V 74 -74 V 74 -34 V 74 6 V 74 -73 V 73 -134 V 74 -34 V 74 -53 V 74 -54 V 74 -74 V 73 -34 V 74 -73 V 74 -74 V 74 -134 V 74 -14 V 73 -13 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -13 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -14 V 74 -13 V 74 6 V 74 -34 V 2708 3250 M 74 6 V 74 26 V 73 6 V 74 6 V 74 206 V 74 -14 V 74 -34 V 73 -14 V 74 -13 V 74 6 V 74 6 V 74 -34 V 73 -94 V 74 6 V 74 -13 V currentpoint stroke M 74 -74 V 74 -54 V 73 -34 V 74 -13 V 74 -154 V 74 -54 V 74 -54 V 73 -53 V 74 -54 V 74 -54 V 74 -74 V 74 -73 V 73 -54 V 74 -74 V 74 -14 V 74 -13 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -13 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -13 V 73 -14 V 74 -14 V 74 -14 V 2665 3186 M 74 6 V 74 26 V 74 26 V 74 -14 V 73 206 V 74 -14 V 74 -14 V 74 -34 V 74 -33 V 73 -14 V 74 26 V 74 -54 V 74 -34 V 74 -53 V 73 26 V 74 -54 V 74 -74 V 74 -53 V 74 -114 V 73 -34 V 74 -54 V 74 -53 V 74 -54 V 74 -54 V 73 -54 V 74 -73 V 74 -74 V 74 -74 V 74 -14 V 73 -13 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -13 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -14 V 74 -13 V 74 -14 V 74 -14 V 73 -14 V 2623 3142 M 73 46 V 74 -14 V 74 6 V 74 -14 V 74 206 V 74 6 V 73 -14 V 74 -33 V 74 -14 V 74 -74 V 74 26 V 73 -34 V 74 -33 V 74 -54 V 74 26 V 74 -54 V 73 -74 V 74 -133 V 74 -54 V 74 -34 V 74 -34 V 73 -73 V 74 -34 V 74 -74 V 74 -54 V 74 -133 V 73 -14 V 74 -14 V 74 -14 V 74 -13 V 74 -14 V 73 6 V 74 -34 V 74 -14 V 74 -14 V 74 -13 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 73 -13 V 74 -14 V 74 -14 V 74 -14 V 2580 3118 M 74 26 V 74 6 V 73 -14 V 74 6 V 74 206 V 74 -54 V 74 6 V 74 -33 V 73 -54 V 74 6 V 74 6 V 74 -14 V 74 -93 V 73 -34 V 74 6 V 74 -54 V 74 -153 V 74 -14 V 73 -54 V 74 -54 V 74 -33 V 74 -74 V 74 -54 V 73 -54 V 74 -133 V 74 6 V 74 -34 V 74 -14 V 73 -14 V 74 -13 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -14 V 74 -13 V 74 -14 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 74 -13 V 74 -14 V 73 -14 V 74 -14 V 2537 3074 M 74 6 V 74 26 V 74 26 V 74 -14 V 73 186 V 74 -34 V 74 -13 V 74 -34 V 74 -54 V 74 -14 V 73 26 V 74 -73 V 74 -54 V 74 -14 V 74 -14 V 73 -153 V 74 -34 V 74 -54 V 74 -34 V 74 -73 V 73 -34 V 74 -54 V 74 -54 V 74 -73 V 74 -74 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -13 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -13 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 73 -13 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 2495 3050 M 74 26 V 73 -14 V 74 6 V 74 -13 V 74 185 V 74 -33 V 73 -54 V 74 -14 V 74 -14 V 74 -34 V 74 6 V 73 -33 V 74 -54 V 74 -34 V 74 -114 V 74 -53 V 74 -34 V 73 -54 V 74 -34 V 74 -53 V 74 -74 V 74 -34 V 73 -74 V 74 -73 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 6 V 74 -33 V 74 -14 V 74 -14 V 73 6 V 74 -34 V 74 -13 V 74 -14 V 74 6 V 73 -34 V 74 -14 V 74 -14 V 74 -13 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 2452 3006 M 74 26 V 74 -14 V 74 6 V 73 -13 V 74 185 V 74 -53 V 74 -14 V 74 -34 V 73 -34 V 74 -14 V 74 -13 V 74 -54 V 74 -14 V 73 -154 V 74 -13 V 74 -54 V 74 -34 V 74 -34 V 74 -73 V 73 -34 V 74 -54 V 74 -74 V 74 -53 V 74 -34 V 73 -14 V 74 -14 V 74 6 V 74 -34 V 74 -13 V 73 -14 V 74 6 V 74 -34 V 74 -14 V 74 -14 V 73 7 V 74 -34 V 74 -14 V 74 -14 V 74 6 V 73 -34 V 74 -13 V 74 -14 V 74 6 V 74 -34 V 73 -14 V 2410 2962 M 73 6 V 74 26 V 74 -13 V 74 6 V 74 146 V 73 -14 V 74 -14 V 74 -74 V 74 6 V 74 -73 V 73 6 V 74 -34 V 74 -134 V 74 -33 V 74 -34 V 73 -34 V 74 -54 V 74 -54 V 74 -53 V 74 -54 V 74 -94 V 73 -34 V 74 -13 V 74 -14 V 74 6 V 74 -34 V 73 -14 V 74 -14 V 74 7 V 74 -34 V 74 -14 V 73 -14 V 74 6 V 74 -34 V 74 -13 V 74 -14 V 73 6 V 74 -34 V 74 -14 V 74 -14 V 74 7 V 73 -34 V 74 -14 V 74 -14 V 74 6 V 2367 2938 M 74 26 V 74 -34 V 73 27 V 74 6 V 74 126 V 74 -34 V 74 -34 V 73 -34 V 74 -14 V 74 -33 V 74 -14 V 74 -134 V 73 -54 V 74 -33 V 74 -34 V 74 -54 V 74 -14 V 73 -73 V 74 -34 V 74 -74 V 74 -74 V 74 -14 V 73 -13 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 73 -13 V 74 6 V 74 -34 V 74 -14 V 74 -14 V 73 -13 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -13 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -14 V 2324 2914 M 74 26 V currentpoint stroke M 74 -33 V 74 -14 V 74 6 V 73 126 V 74 -54 V 74 6 V 74 -34 V 74 -53 V 73 -14 V 74 -114 V 74 -34 V 74 -53 V 74 -14 V 73 -54 V 74 -34 V 74 -54 V 74 -53 V 74 -114 V 73 -14 V 74 -14 V 74 -13 V 74 6 V 74 -34 V 73 -14 V 74 -14 V 74 6 V 74 -33 V 74 -14 V 74 -14 V 73 6 V 74 -34 V 74 -14 V 74 -13 V 74 6 V 73 -34 V 74 -14 V 74 -14 V 74 6 V 74 -33 V 73 -14 V 74 -14 V 74 6 V 74 -34 V 74 -14 V 2282 2850 M 74 6 V 73 7 V 74 -14 V 74 -14 V 74 126 V 74 -34 V 73 -14 V 74 -34 V 74 -13 V 74 -114 V 74 -54 V 73 -14 V 74 -53 V 74 -34 V 74 -34 V 74 -54 V 73 -53 V 74 -54 V 74 -74 V 74 -14 V 74 -14 V 73 7 V 74 -34 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -13 V 74 6 V 74 -34 V 74 -14 V 73 -14 V 74 6 V 74 -33 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -14 V 74 -13 V 74 -14 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 2239 2826 M 74 -13 V 74 6 V 74 6 V 73 -14 V 74 86 V 74 -14 V 74 26 V 74 -73 V 73 -134 V 74 -34 V 74 -14 V 74 -34 V 74 -73 V 73 -14 V 74 -34 V 74 -74 V 74 -73 V 74 -34 V 73 -14 V 74 -14 V 74 -14 V 74 -13 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -13 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 73 -13 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -13 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 73 -13 V 2197 2763 M 73 26 V 74 6 V 74 -14 V 74 -34 V 74 106 V 73 -34 V 74 -14 V 74 -133 V 74 -14 V 74 -34 V 73 -34 V 74 -33 V 74 -54 V 74 -34 V 74 -34 V 73 -113 V 74 -14 V 74 6 V 74 -34 V 74 -14 V 73 -14 V 74 -13 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -13 V 74 -14 V 74 -14 V 74 -14 V 73 -14 V 74 -14 V 74 -13 V 74 -14 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 74 -13 V 74 -14 V 74 6 V 73 -34 V 74 -14 V 74 -14 V 74 -13 V 2154 2739 M 74 6 V 74 -14 V 73 -14 V 74 6 V 74 106 V 74 -34 V 74 -153 V 73 -14 V 74 -14 V 74 -34 V 74 -34 V 74 -33 V 73 -34 V 74 -74 V 74 -54 V 74 -53 V 74 6 V 73 -34 V 74 -14 V 74 -14 V 74 6 V 74 -33 V 73 -14 V 74 -14 V 74 6 V 74 -34 V 74 -13 V 73 -14 V 74 6 V 74 -34 V 74 -14 V 74 -14 V 73 7 V 74 -34 V 74 -14 V 74 -14 V 74 6 V 73 -34 V 74 -13 V 74 -14 V 74 6 V 74 -34 V 73 -14 V 74 -14 V 74 7 V 2111 2715 M 74 6 V 74 -14 V 74 -14 V 74 -14 V 73 86 V 74 -113 V 74 -54 V 74 -14 V 74 -34 V 73 -34 V 74 -53 V 74 -34 V 74 -34 V 74 -34 V 73 -73 V 74 6 V 74 -34 V 74 -14 V 74 -14 V 73 6 V 74 -13 V 74 -34 V 74 -14 V 74 -14 V 73 6 V 74 -34 V 74 -13 V 74 -14 V 74 6 V 73 -34 V 74 -14 V 74 6 V 74 -33 V 74 -14 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -13 V 73 -14 V 74 -14 V 74 -14 V 74 -14 V 74 -14 V 73 -13 V 2069 2671 M 74 6 V 73 -14 V 74 -14 V 74 -14 V 74 -33 V 74 -14 V 73 -14 V 74 -34 V 74 -34 V 74 -53 V 74 -14 V 73 -34 V 74 -74 V 74 -54 V 74 -13 V 74 -14 V 73 -14 V 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4080 M 399 0 V 6443 3073 M -93 -9 V -27 -1 V -87 -7 V -48 0 V -51 -1 V -65 -9 V -65 -9 V -51 -1 V -51 -1 V -78 -7 V -39 -3 V -102 -2 V -106 -4 V -35 -2 V -92 -14 V -102 -3 V -39 -3 V -78 -7 V -51 -1 V -51 -1 V -102 -2 V -116 -10 V -103 -2 V -51 -1 V -58 -5 V -58 -5 V -51 -1 V -102 -3 V -117 -10 V -95 2 V -58 -5 V -58 -5 V -95 2 V -110 -6 V -19 -2 V -92 -6 V -33 -2 V -84 -8 V -33 -3 V -83 -7 V -42 -1 V -62 -2 V -67 -6 V -50 -4 V -73 -5 V -39 -3 V -78 -7 V -36 -2 V -105 -3 V -36 7 V -80 -17 V -37 7 V -37 6 V -67 -19 V -70 -3 V -53 26 V -74 14 V 22 28 V 21 28 V 22 28 V 41 24 V 41 24 V 35 26 V -74 13 V -31 38 V -31 38 V 18 28 V 67 20 V -52 26 V 6443 1875 M -93 -8 V -27 -2 V -87 -7 V -48 1 V -51 -1 V -65 -9 V -65 -9 V -51 -1 V -51 -2 V -78 -6 V -39 -4 V -102 -2 V -106 -4 V -35 -1 V -92 -15 V -102 -2 V -39 -4 V -78 -6 V -51 -1 V -51 -1 V -102 -3 V -116 -10 V -103 -2 V -51 -1 V -58 -5 V -58 -5 V -51 -1 V -102 -2 V -117 -10 V -95 1 V -58 -5 V -58 -5 V -95 2 V -110 -6 V -19 -2 V -92 -5 V -33 -3 V -84 -7 V -33 -3 V -83 -7 V -42 -2 V -62 -2 V -67 -6 V -50 -4 V -73 -4 V -39 -4 V -78 -6 V -36 -3 V -105 -3 V -36 7 V -80 -17 V -37 7 V -37 7 V -67 -20 V -70 -2 V -53 25 V -74 14 V 22 28 V 21 28 V 22 28 V 41 24 V 41 24 V 35 26 V -74 14 V -31 37 V -31 38 V 18 29 V 67 19 V -52 26 V LT0 7122 3940 M ( 0.1) Rshow 7206 3940 M 399 0 V 6655 2833 M -70 -2 V -59 -6 V -58 -5 V -58 -5 V -58 -5 V -59 -5 V -58 -5 V -70 -3 V -39 -3 V -78 -7 V -38 -3 V -78 -7 V -35 -1 V -106 -4 V -116 -10 V -117 -11 V -116 -10 V -117 -10 V -116 -10 V -106 -4 V -29 -3 V -87 -7 V -48 1 V -58 -5 V -46 -8 V -78 -6 V -38 -4 V -71 -3 V -70 -2 V -46 -8 V -59 -5 V -51 -1 V -77 -7 V -39 -3 V -102 -2 V -117 -10 V -116 -10 V -102 -3 V -39 -3 V -78 -7 V -51 -1 V -79 -17 V -74 14 V -117 -10 V -116 -10 V -88 6 V -78 -7 V -45 -7 V -96 1 V -116 -10 V -95 2 V -95 2 V -74 14 V -116 -10 V -74 14 V -117 -11 V -73 14 V -74 14 V -74 14 V -31 38 V 6655 1835 M -70 -3 V -59 -5 V -58 -5 V -58 -5 V -58 -5 V -59 -5 V -58 -5 V -70 -3 V -39 -3 V -78 -7 V -38 -3 V -78 -7 V -35 -1 V -106 -4 V -116 -11 V -117 -10 V -116 -10 V -117 -10 V -116 -10 V -106 -4 V -29 -3 V -87 -7 V -48 1 V -58 -5 V -46 -8 V -78 -7 V -38 -3 V -71 -3 V -70 -2 V -46 -8 V -59 -5 V -51 -1 V -77 -7 V -39 -3 V -102 -2 V -117 -10 V -116 -10 V -102 -3 V -39 -3 V -78 -7 V -51 -1 V -79 -17 V -74 14 V -117 -10 V -116 -10 V -88 6 V -78 -7 V -45 -8 V -96 2 V -116 -10 V -95 2 V -95 2 V -74 14 V -116 -10 V -74 13 V -117 -10 V -73 14 V -74 14 V -74 14 V -31 38 V 1.000 UL LTb 6870 1827 M 4739 629 L 1049 1321 M 4739 629 L 1049 1321 M 3180 2518 L 6870 1827 M 3180 2518 L 1049 1321 M 0 2395 V 1.000 UL LTa 1049 1321 M 3180 2518 L 1.000 UL LTb 1049 1321 M 54 30 V 975 1252 M (0) Cshow 3180 2518 M -55 -31 V 1.000 UL LTa 1418 1252 M 3549 2449 L 1.000 UL LTb 1418 1252 M 54 30 V -128 -99 R (10) Cshow 3549 2449 M -55 -31 V 1.000 UL LTa 1787 1182 M 3918 2380 L 1.000 UL LTb 1787 1182 M 54 30 V -128 -99 R (20) Cshow 3918 2380 M -55 -31 V 1.000 UL LTa 2156 1113 M 4287 2311 L 1.000 UL LTb 2156 1113 M 54 30 V -128 -99 R (30) Cshow 4287 2311 M -55 -31 V 1.000 UL LTa 2525 1044 M 4656 2242 L 1.000 UL LTb 2525 1044 M 54 30 V 2451 975 M (40) Cshow 4656 2242 M -55 -31 V 1.000 UL LTa 2894 975 M 5025 2173 L 1.000 UL LTb 2894 975 M 54 30 V 2820 906 M (50) Cshow 5025 2173 M -55 -31 V 1.000 UL LTa 3263 906 M 5394 2103 L 1.000 UL LTb 3263 906 M 54 30 V 3189 837 M (60) Cshow 5394 2103 M -55 -31 V 1.000 UL LTa 3632 837 M 5763 2034 L 1.000 UL LTb 3632 837 M 54 30 V 3558 768 M (70) Cshow 5763 2034 M -55 -31 V 1.000 UL LTa 4001 767 M 6132 1965 L 1.000 UL LTb 4001 767 M 54 30 V 3927 698 M (80) Cshow 6132 1965 M -55 -31 V 1.000 UL LTa 4370 698 M 6501 1896 L 1.000 UL LTb 4370 698 M 54 30 V 4296 629 M (90) Cshow 6501 1896 M -55 -31 V 1.000 UL LTa 4739 629 M 6870 1827 L 1.000 UL LTb 4739 629 M 54 30 V 4665 560 M (100) Cshow 6870 1827 M -55 -31 V 2362 676 M (B) Cshow 1.000 UL LTa 4739 629 M 1049 1321 L 1.000 UL LTb 4739 629 M -62 11 V 144 -37 R (0) Lshow 1049 1321 M 61 -12 V 1.000 UL LTa 4952 749 M 1262 1440 L 1.000 UL LTb 4952 749 M -62 11 V 144 -37 R (10) Lshow 1262 1440 M 61 -12 V 1.000 UL LTa 5165 869 M 1475 1560 L 1.000 UL LTb 5165 869 M -62 11 V 144 -37 R (20) Lshow 1475 1560 M 61 -12 V 1.000 UL LTa 5378 988 M 1688 1680 L 1.000 UL LTb 5378 988 M -62 11 V 144 -37 R (30) Lshow 1688 1680 M 61 -12 V 1.000 UL LTa 5592 1108 M 1901 1800 L 1.000 UL LTb 5592 1108 M -62 11 V 144 -37 R (40) Lshow 1901 1800 M 61 -12 V 1.000 UL LTa 5805 1228 M 2114 1920 L 1.000 UL LTb 5805 1228 M -62 11 V 144 -37 R (50) Lshow 2114 1920 M 61 -12 V 1.000 UL LTa 6018 1348 M 2327 2039 L 1.000 UL LTb 6018 1348 M -62 11 V 144 -37 R (60) Lshow 2327 2039 M 61 -12 V 1.000 UL LTa 6231 1468 M 2541 2159 L 1.000 UL LTb 6231 1468 M -62 11 V 144 -37 R (70) Lshow 2541 2159 M 61 -12 V 1.000 UL LTa 6444 1587 M 2754 2279 L 1.000 UL LTb 6444 1587 M -62 11 V 144 -37 R (80) Lshow 2754 2279 M 61 -12 V 1.000 UL LTa 6657 1707 M 2967 2399 L 1.000 UL LTb 6657 1707 M -62 11 V 144 -37 R (90) Lshow 2967 2399 M 61 -12 V 1.000 UL LTa 6870 1827 M 3180 2518 L 1.000 UL LTb 6870 1827 M -62 11 V 144 -37 R (100) Lshow 3180 2518 M 61 -12 V 6727 1055 M (A) Cshow 1049 2119 M 63 0 V -189 0 R (0) Rshow 1049 2319 M 63 0 V -189 0 R (0.1) Rshow 1049 2518 M 63 0 V -189 0 R (0.2) Rshow 1049 2718 M 63 0 V -189 0 R (0.3) Rshow 1049 2918 M 63 0 V -189 0 R (0.4) Rshow 1049 3117 M 63 0 V -189 0 R (0.5) Rshow 1049 3317 M 63 0 V -189 0 R (0.6) Rshow 1049 3516 M 63 0 V -189 0 R (0.7) Rshow 1049 3716 M 63 0 V -189 0 R (0.8) Rshow stroke grestore end showpage %%EndDocument @endspecial 1107 2571 a Fp(Figure)31 b(8:)36 b FD(V)-7 b(arying)27 b(the)h(input)g(sizes)f(of)h(A)g(and)f(B)768 4965 y @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 2834 @rwi @setspecial %%BeginDocument: RatioFast2Kara.eps /gnudict 120 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 560 420 M 63 0 V 6409 0 R -63 0 V 476 420 M (1.4) Rshow 560 977 M 63 0 V 6409 0 R -63 0 V 476 977 M (1.5) Rshow 560 1533 M 63 0 V 6409 0 R -63 0 V -6493 0 R (1.6) Rshow 560 2090 M 63 0 V 6409 0 R -63 0 V -6493 0 R (1.7) Rshow 560 2646 M 63 0 V 6409 0 R -63 0 V -6493 0 R (1.8) Rshow 560 3203 M 63 0 V 6409 0 R -63 0 V -6493 0 R (1.9) Rshow 560 3759 M 63 0 V 6409 0 R -63 0 V -6493 0 R (2) Rshow 560 4316 M 63 0 V 6409 0 R -63 0 V -6493 0 R (2.1) Rshow 560 4872 M 63 0 V 6409 0 R -63 0 V -6493 0 R (2.2) Rshow 560 420 M 0 63 V 0 4389 R 0 -63 V 560 280 M (0) Cshow 1639 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (50) Cshow 2717 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (100) Cshow 3796 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (150) Cshow 4875 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (200) Cshow 5953 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (250) Cshow 7032 420 M 0 63 V 0 4389 R 0 -63 V 0 -4529 R (300) Cshow 1.000 UL LTb 560 420 M 6472 0 V 0 4452 V -6472 0 V 560 420 L 0 2646 M currentpoint gsave translate 90 rotate 0 0 M (Ratio) Cshow grestore 3796 140 M (Digits) Cshow 1.000 UL LT0 560 -7370 M 1.000 UL LT0 560 3759 M 6472 0 V -122 -33 R 122 33 V -122 32 V 3.000 UL LT2 560 3759 M 6472 0 V -122 -33 R 122 33 V -122 32 V 1.000 UP 1.000 UL LT0 6381 4739 M ("RA.dat") Rshow 6465 4739 M 399 0 V 733 976 M 43 696 V 43 -443 V 862 791 L 43 1229 V 43 -538 V 43 1309 V 44 -887 V 43 1316 V 43 -1548 V 43 1544 V 43 414 V 43 1097 V 43 -2763 V 44 470 V 43 79 V 43 909 V 43 -842 V 43 -58 V 43 381 V 44 -150 V 43 -420 V 43 267 V 43 239 V 43 202 V 43 718 V 43 0 V 44 422 V 43 373 V 43 -1864 V 43 101 V 43 303 V 43 -5 V 43 -310 V 44 100 V 43 265 V 43 168 V 43 -906 V 43 196 V 43 94 V 43 -62 V 44 221 V 43 198 V 43 21 V 43 260 V 43 -691 V 43 3 V 43 -46 V 44 479 V 43 -132 V 43 -167 V 43 398 V 43 -3 V 43 237 V 44 204 V 43 29 V 43 1 V 43 -39 V 43 263 V 43 164 V 43 -56 V 44 -1587 V 43 67 V 43 79 V 43 212 V 43 141 V 43 -1 V 43 -46 V 44 215 V 43 13 V 43 -45 V 43 172 V 43 -86 V 43 -24 V 43 388 V 44 -199 V 43 102 V 43 -844 V 43 431 V 43 -184 V 43 142 V 43 -66 V 44 148 V 43 -3 V 43 -41 V 43 257 V 43 -101 V 43 451 V 43 -441 V 44 204 V 43 46 V 43 -201 V 43 174 V 43 -549 V 43 215 V 44 -286 V 43 39 V 43 178 V 43 -6 V 43 134 V 43 -20 V 43 103 V 44 -56 V 43 135 V 43 33 V 43 2 V 43 -30 V 43 137 V 43 2 V 44 -165 V 43 96 V 43 26 V 43 67 V 43 -30 V 43 171 V 43 66 V 44 284 V 43 -310 V 43 191 V 43 -38 V 43 122 V 43 -61 V 43 60 V 44 41 V 43 7 V 43 -1723 V 43 253 V 43 168 V 43 16 V 44 196 V 43 10 V 43 -12 V 43 51 V 733 976 Pls 776 1672 Pls 819 1229 Pls 862 791 Pls 905 2020 Pls 948 1482 Pls 991 2791 Pls 1035 1904 Pls 1078 3220 Pls 1121 1672 Pls 1164 3216 Pls 1207 3630 Pls 1250 4727 Pls 1293 1964 Pls 1337 2434 Pls 1380 2513 Pls 1423 3422 Pls 1466 2580 Pls 1509 2522 Pls 1552 2903 Pls 1596 2753 Pls 1639 2333 Pls 1682 2600 Pls 1725 2839 Pls 1768 3041 Pls 1811 3759 Pls 1854 3759 Pls 1898 4181 Pls 1941 4554 Pls 1984 2690 Pls 2027 2791 Pls 2070 3094 Pls 2113 3089 Pls 2156 2779 Pls 2200 2879 Pls 2243 3144 Pls 2286 3312 Pls 2329 2406 Pls 2372 2602 Pls 2415 2696 Pls 2458 2634 Pls 2502 2855 Pls 2545 3053 Pls 2588 3074 Pls 2631 3334 Pls 2674 2643 Pls 2717 2646 Pls 2760 2600 Pls 2804 3079 Pls 2847 2947 Pls 2890 2780 Pls 2933 3178 Pls 2976 3175 Pls 3019 3412 Pls 3063 3616 Pls 3106 3645 Pls 3149 3646 Pls 3192 3607 Pls 3235 3870 Pls 3278 4034 Pls 3321 3978 Pls 3365 2391 Pls 3408 2458 Pls 3451 2537 Pls 3494 2749 Pls 3537 2890 Pls 3580 2889 Pls 3623 2843 Pls 3667 3058 Pls 3710 3071 Pls 3753 3026 Pls 3796 3198 Pls 3839 3112 Pls 3882 3088 Pls 3925 3476 Pls 3969 3277 Pls 4012 3379 Pls 4055 2535 Pls 4098 2966 Pls 4141 2782 Pls 4184 2924 Pls 4227 2858 Pls 4271 3006 Pls 4314 3003 Pls 4357 2962 Pls 4400 3219 Pls 4443 3118 Pls 4486 3569 Pls 4529 3128 Pls 4573 3332 Pls 4616 3378 Pls 4659 3177 Pls 4702 3351 Pls 4745 2802 Pls 4788 3017 Pls 4832 2731 Pls 4875 2770 Pls 4918 2948 Pls 4961 2942 Pls 5004 3076 Pls 5047 3056 Pls 5090 3159 Pls 5134 3103 Pls 5177 3238 Pls 5220 3271 Pls 5263 3273 Pls 5306 3243 Pls 5349 3380 Pls 5392 3382 Pls 5436 3217 Pls 5479 3313 Pls 5522 3339 Pls 5565 3406 Pls 5608 3376 Pls 5651 3547 Pls 5694 3613 Pls 5738 3897 Pls 5781 3587 Pls 5824 3778 Pls 5867 3740 Pls 5910 3862 Pls 5953 3801 Pls 5996 3861 Pls 6040 3902 Pls 6083 3909 Pls 6126 2186 Pls 6169 2439 Pls 6212 2607 Pls 6255 2623 Pls 6299 2819 Pls 6342 2829 Pls 6385 2817 Pls 6428 2868 Pls 6664 4739 Pls stroke grestore end showpage %%EndDocument @endspecial 455 5247 a Fp(Figure)k(9:)36 b FD(The)28 b(ratio)f(b)r(et)n(w)n(een)g(F)-7 b(astDiv)28 b(and)f(Karatsuba)f(m)n (ultiplication)i(is)f(less)g(than)h(2)1903 5841 y FB(24)p eop %%Page: 25 29 25 28 bop 768 2289 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 2834 @rwi @setspecial %%BeginDocument: allsmall.eps /gnudict 120 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 252 280 M 63 0 V 6717 0 R -63 0 V 168 280 M (0) Rshow 252 1045 M 63 0 V 6717 0 R -63 0 V -6801 0 R (1) Rshow 252 1811 M 63 0 V 6717 0 R -63 0 V -6801 0 R (2) Rshow 252 2576 M 63 0 V 6717 0 R -63 0 V -6801 0 R (3) Rshow 252 3341 M 63 0 V 6717 0 R -63 0 V -6801 0 R (4) Rshow 252 4107 M 63 0 V 6717 0 R -63 0 V -6801 0 R (5) Rshow 252 4872 M 63 0 V 6717 0 R -63 0 V -6801 0 R (6) Rshow 252 280 M 0 63 V 0 4529 R 0 -63 V 252 140 M (0) Cshow 1382 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (20) Cshow 2512 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (40) Cshow 3642 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (60) Cshow 4772 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (80) Cshow 5902 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (100) Cshow 7032 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (120) Cshow 1.000 UL LTb 252 280 M 6780 0 V 0 4592 V -6780 0 V 252 280 L 1.000 UP 1.000 UL LT0 1176 4739 M ("LIP.dat") Rshow 1260 4739 M 399 0 V 704 302 M 113 13 V 113 9 V 113 12 V 113 11 V 226 29 V 226 44 V 226 47 V 226 48 V 452 126 V 452 147 V 452 142 V 452 211 V 904 479 V 904 653 V 904 575 V 452 376 V 704 302 Pls 817 315 Pls 930 324 Pls 1043 336 Pls 1156 347 Pls 1382 376 Pls 1608 420 Pls 1834 467 Pls 2060 515 Pls 2512 641 Pls 2964 788 Pls 3416 930 Pls 3868 1141 Pls 4772 1620 Pls 5676 2273 Pls 6580 2848 Pls 1459 4739 Pls 1.000 UP 1.000 UL LT1 1176 4599 M (CLN) Rshow 1260 4599 M 399 0 V 704 314 M 113 9 V 113 8 V 113 14 V 113 12 V 226 25 V 226 34 V 226 31 V 226 45 V 452 91 V 452 113 V 452 145 V 452 154 V 904 356 V 904 442 V 904 513 V 452 281 V 704 314 Crs 817 323 Crs 930 331 Crs 1043 345 Crs 1156 357 Crs 1382 382 Crs 1608 416 Crs 1834 447 Crs 2060 492 Crs 2512 583 Crs 2964 696 Crs 3416 841 Crs 3868 995 Crs 4772 1351 Crs 5676 1793 Crs 6580 2306 Crs 1459 4599 Crs 1.000 UP 1.000 UL LT2 1176 4459 M (FastDiv) Rshow 1260 4459 M 399 0 V 704 321 M 113 10 V 113 12 V 113 12 V 113 15 V 226 35 V 226 44 V 226 39 V 226 50 V 452 111 V 452 142 V 452 139 V 452 204 V 904 295 V 904 360 V 904 496 V 452 166 V 704 321 Star 817 331 Star 930 343 Star 1043 355 Star 1156 370 Star 1382 405 Star 1608 449 Star 1834 488 Star 2060 538 Star 2512 649 Star 2964 791 Star 3416 930 Star 3868 1134 Star 4772 1429 Star 5676 1789 Star 6580 2285 Star 1459 4459 Star 1.000 UP 1.000 UL LT3 1176 4319 M ("libI.dat") Rshow 1260 4319 M 399 0 V 704 314 M 113 13 V 113 18 V 113 20 V 113 24 V 226 52 V 226 67 V 226 72 V 226 88 V 452 202 V 452 255 V 452 292 V 452 328 V 904 821 V 904 985 V 904 1132 V 125 189 V 704 314 Box 817 327 Box 930 345 Box 1043 365 Box 1156 389 Box 1382 441 Box 1608 508 Box 1834 580 Box 2060 668 Box 2512 870 Box 2964 1125 Box 3416 1417 Box 3868 1745 Box 4772 2566 Box 5676 3551 Box 6580 4683 Box 1459 4319 Box stroke grestore end showpage %%EndDocument @endspecial 643 2572 a Fp(Figure)32 b(10:)k FD(Comparing)26 b(F)-7 b(astDivision)27 b(to)h(other)f(pac)n(k)-5 b(ages)25 b(in)j(a)f(small)h(range)768 4965 y @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 2834 @rwi @setspecial %%BeginDocument: allbig.eps /gnudict 120 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 420 280 M 63 0 V 6549 0 R -63 0 V 336 280 M (0) Rshow 420 854 M 63 0 V 6549 0 R -63 0 V 336 854 M (50) Rshow 420 1428 M 63 0 V 6549 0 R -63 0 V -6633 0 R (100) Rshow 420 2002 M 63 0 V 6549 0 R -63 0 V -6633 0 R (150) Rshow 420 2576 M 63 0 V 6549 0 R -63 0 V -6633 0 R (200) Rshow 420 3150 M 63 0 V 6549 0 R -63 0 V -6633 0 R (250) Rshow 420 3724 M 63 0 V 6549 0 R -63 0 V -6633 0 R (300) Rshow 420 4298 M 63 0 V 6549 0 R -63 0 V -6633 0 R (350) Rshow 420 4872 M 63 0 V 6549 0 R -63 0 V -6633 0 R (400) Rshow 420 280 M 0 63 V 0 4529 R 0 -63 V 420 140 M (100) Cshow 1155 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (200) Cshow 1889 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (300) Cshow 2624 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (400) Cshow 3359 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (500) Cshow 4093 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (600) Cshow 4828 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (700) Cshow 5563 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (800) Cshow 6297 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (900) Cshow 7032 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (1000) Cshow 1.000 UL LTb 420 280 M 6612 0 V 0 4592 V -6612 0 V 420 280 L 1.000 UP 1.000 UL LT0 1092 4739 M (LIP) Rshow 1176 4739 M 399 0 V 420 312 M 88 7 V 118 11 V 235 29 V 235 32 V 235 38 V 235 49 V 470 106 V 470 136 V 471 161 V 470 176 V 940 499 V 941 473 V 940 631 V 764 684 V 508 319 Pls 626 330 Pls 861 359 Pls 1096 391 Pls 1331 429 Pls 1566 478 Pls 2036 584 Pls 2506 720 Pls 2977 881 Pls 3447 1057 Pls 4387 1556 Pls 5328 2029 Pls 6268 2660 Pls 1375 4739 Pls 1.000 UP 1.000 UL LT1 1092 4599 M (CLN) Rshow 1176 4599 M 399 0 V 420 305 M 88 5 V 118 9 V 235 21 V 235 25 V 235 29 V 235 38 V 470 78 V 470 113 V 471 102 V 470 136 V 940 282 V 941 300 V 940 361 V 764 236 V 508 310 Crs 626 319 Crs 861 340 Crs 1096 365 Crs 1331 394 Crs 1566 432 Crs 2036 510 Crs 2506 623 Crs 2977 725 Crs 3447 861 Crs 4387 1143 Crs 5328 1443 Crs 6268 1804 Crs 1375 4599 Crs 1.000 UP 1.000 UL LT2 1092 4459 M (FastDiv) Rshow 1176 4459 M 399 0 V 420 304 M 88 6 V 118 5 V 235 16 V 235 19 V 235 21 V 235 15 V 470 53 V 470 50 V 471 68 V 470 41 V 940 145 V 941 163 V 940 237 V 764 89 V 508 310 Star 626 315 Star 861 331 Star 1096 350 Star 1331 371 Star 1566 386 Star 2036 439 Star 2506 489 Star 2977 557 Star 3447 598 Star 4387 743 Star 5328 906 Star 6268 1143 Star 1375 4459 Star 1.000 UP 1.000 UL LT3 1092 4319 M (libI) Rshow 1176 4319 M 399 0 V 420 333 M 88 13 V 118 21 V 235 48 V 235 59 V 235 67 V 235 84 V 470 186 V 470 257 V 471 285 V 470 295 V 940 747 V 941 1039 V 940 1202 V 204 236 V 508 346 Box 626 367 Box 861 415 Box 1096 474 Box 1331 541 Box 1566 625 Box 2036 811 Box 2506 1068 Box 2977 1353 Box 3447 1648 Box 4387 2395 Box 5328 3434 Box 6268 4636 Box 1375 4319 Box stroke grestore end showpage %%EndDocument @endspecial 683 5248 a Fp(Figure)j(11:)36 b FD(Comparing)26 b(F)-7 b(astDivision)28 b(to)f(other)g(pac)n(k)-5 b(ages)26 b(in)i(a)f(big)g(range)1903 5841 y FB(25)p eop %%Page: 26 30 26 29 bop 118 399 a FB(and)33 b(the)h(pair)e(\()p FA(e;)15 b(m)p FB(\))35 b(is)e(the)h Fw(key)p FB(.)50 b(The)33 b(basis)g(of)h(these)g Fw(mo)-5 b(dular)38 b(exp)-5 b(onentiation)43 b FB(algorithms)33 b(is)f(the)118 511 y(computation)e(of)h(the)g (expression)e FA(x)20 b Fv(\001)g FA(y)28 b Fw(mo)-5 b(d)37 b FA(m)p FB(,)30 b(where)g FA(x)p FB(,)g FA(y)s FB(,)h(and)f FA(m)g FB(are)g FA(n)p FB(-digit)f(n)m(um)m(b)s(ers.)259 624 y(There)k(are)h(three)g(metho)s(ds)e(in)h(use)g(for)g(this)f Fw(mo)-5 b(dular)38 b(multiplic)-5 b(ation)p FB(,)37 b(ordinary)32 b(sc)m(ho)s(ol)h(metho)s(d)118 737 y(\(whic)m(h)k(m)m (ultiplies)d FA(x)j FB(and)g FA(y)k FB(as)d(in)m(tegers)f(and)g(p)s (erforms)f(an)i(in)m(teger)g(division)c(afterw)m(ards\),)40 b(Mon)m(t-)118 850 y(gomery)d(m)m(ultiplication,)d(and)h(Barrett)i (reduction)e(\(see)i([8])f(and)g(section)g(4\).)57 b(They)36 b(all)e(need)i FA(O)s FB(\()p FA(n)3705 817 y Fy(2)3744 850 y FB(\))118 963 y(op)s(erations)h(on)h(single)f(digits)f(and)h(p)s (erform)g(equally)g(w)m(ell)f(in)h(concrete)i(implemen)m(tations)e ([8].)64 b(The)118 1076 y(exact)38 b(n)m(um)m(b)s(ers)d(of)i(steps)f (of)h(the)g(sc)m(ho)s(ol)f(metho)s(d)g(is)g(2)p FA(n)2168 1043 y Fy(2)2231 1076 y FB(+)24 b FA(O)s FB(\()p FA(n)p FB(\).)60 b(It)36 b(is)g(easy)h(to)g(write)f(a)h(function)118 1189 y Fx(FastModMult)21 b FB(that)j(p)s(erforms)e(this)h(task)h (asymptotically)g(faster)g(b)m(y)f(computing)g FA(x)7 b Fv(\001)g FA(y)27 b FB(with)c(Karatsuba)118 1302 y(m)m(ultiplication) k(and)j(reducing)f(mo)s(dulo)g FA(m)h FB(b)m(y)g Fx(FastDivision)d FB(afterw)m(ards.)118 1489 y FC(Theorem)34 b(3.)46 b Fw(The)41 b(expr)-5 b(ession)42 b FA(x)26 b Fv(\001)h FA(y)43 b Fw(mo)-5 b(d)51 b FA(m)40 b Fw(wher)-5 b(e)42 b FA(x)p Fw(,)h FA(y)s Fw(,)g(and)e FA(m)g Fw(ar)-5 b(e)42 b FA(n)p Fw(-digit)e(numb)-5 b(ers)41 b(c)-5 b(an)42 b(b)-5 b(e)118 1602 y(c)g(ompute)g(d)35 b(in)d FB(3)p FA(K)7 b FB(\()p FA(n)p FB(\))21 b(+)f FA(O)s FB(\()p FA(n)15 b FB(log)h FA(n)p FB(\))33 b Fw(steps.)259 1790 y FB(Figure)d(12)i(sho)m(ws)e(exp)s(erimen)m(tal)g(running)d(times)k (measured)f(with)f(p)s(ortable)h(C/C++)f(co)s(de.)42 b(The)118 1903 y(break-ev)m(en)c(p)s(oin)m(t)d(is)h(ab)s(out)g(30)h (digits)e(or)i(960)h(bits,)f(so)g(w)m(e)g(are)g(w)m(ell)e(in)g(the)i (cryptographic)f(range.)118 2016 y(This)28 b(can)h(b)s(e)g(used)f(to)i (substan)m(tially)e(sp)s(eedup)f(public)g(k)m(ey)j(cryptosystems)f(in)f (the)i(near)f(future)f(when)118 2129 y(secure)j(cryptosystems)f(are)h (assumed)f(to)h(ha)m(v)m(e)g(1024)i(bits)c(as)h(their)g(minimal)d(k)m (ey)32 b(length.)768 3900 y @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 2834 @rwi @setspecial %%BeginDocument: ModMult.eps /gnudict 120 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 728 420 M 31 0 V 6273 0 R -31 0 V 644 420 M (0) Rshow 728 1056 M 31 0 V 6273 0 R -31 0 V -6357 0 R (0.002) Rshow 728 1692 M 31 0 V 6273 0 R -31 0 V -6357 0 R (0.004) Rshow 728 2328 M 31 0 V 6273 0 R -31 0 V -6357 0 R (0.006) Rshow 728 2964 M 31 0 V 6273 0 R -31 0 V -6357 0 R (0.008) Rshow 728 3600 M 31 0 V 6273 0 R -31 0 V -6357 0 R (0.01) Rshow 728 4236 M 31 0 V 6273 0 R -31 0 V -6357 0 R (0.012) Rshow 728 4872 M 31 0 V 6273 0 R -31 0 V -6357 0 R (0.014) Rshow 728 420 M 0 31 V 0 4421 R 0 -31 V 728 280 M (0) Cshow 1043 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (10) Cshow 1358 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (20) Cshow 1674 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (30) Cshow 1989 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (40) Cshow 2304 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (50) Cshow 2619 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (60) Cshow 2934 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (70) Cshow 3250 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (80) Cshow 3565 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (90) Cshow 3880 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (100) Cshow 4195 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (110) Cshow 4510 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (120) Cshow 4826 420 M 0 31 V 0 4421 R 0 -31 V 0 -4561 R (130) Cshow 5141 420 M 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b(Implemen)m(ting)38 b(the)h(Riv)m(est,)k (Shamir,)d(and)f(Adleman)f(public)f(k)m(ey)k(encryption)305 714 y(algorithm)j(on)h(a)h(standard)e(digital)g(signal)g(pro)s(cessor.) 85 b(In)44 b Fw(A)-5 b(dvanc)g(es)47 b(in)f(Cryptolo)-5 b(gy,)53 b(Pr)-5 b(o)g(c.)305 827 y(Crypto)34 b(86,)f(LNCS)g(263)p FB(,)f(pages)f(311{323.)164 1015 y([2])46 b(C.)28 b(Burnik)m(el,)g(R.)g (Fleisc)m(her,)h(K.)f(Mehlhorn,)g(and)g(S.)g(Sc)m(hirra.)36 b(A)29 b(strong)f(and)g(easily)g(computable)305 1128 y(separation)34 b(b)s(ound)f(for)h(arithmetic)g(expressions)f(in)m(v)m (olving)g(square)i(ro)s(ots.)53 b Fw(Pr)-5 b(o)g(c)g(e)g(e)g(dings)38 b(of)f(the)305 1241 y(8th)d(A)n(CM-SIAM)c(Symp)-5 b(osium)34 b(on)f(Discr)-5 b(ete)33 b(A)n(lgorithms)p FB(,)f(1997.)164 1428 y([3])46 b(A.J.)28 b(Menezes,)i(P)-8 b(.C.)28 b(v.)h(Oorsc)m(hot,) g(and)e(S.A.)h(V)-8 b(anstone.)38 b Fw(Handb)-5 b(o)g(ok)32 b(of)f(Applie)-5 b(d)31 b(Crypto)-5 b(gr)g(aphy)p FB(.)305 1541 y(CR)m(C)30 b(Press,)g(1996.)164 1729 y([4])46 b(T.)34 b(Jeb)s(elean.)50 b(Practical)34 b(in)m(teger)h(division)c(with)i (Karatsuba)g(complexit)m(y)-8 b(.)52 b(In)33 b Fw(Pr)-5 b(o)g(c.)37 b(Symp.)g(on)305 1842 y(Symb)-5 b(olic)34 b(and)f(A)n(lgebr)-5 b(aic)33 b(Computation,)h(ISSA)n(C)p FB(,)d(pages)f(339{341,)k(1997.)164 2029 y([5])46 b(D.)31 b(Kn)m(uth:)40 b(The)30 b(Art)g(Of)g(Computer)f(Programming,)h(V)-8 b(ol.)31 b(2,)g(3rd)f(Edition,)f(Section)h(4.3)164 2217 y([6])46 b(W.)40 b(Krandic)m(k)f(and)g(J.R.)h(Johnson.)68 b(E\016cien)m(t)40 b(m)m(ultiprecision)d(\015oating)j(p)s(oin)m(t)e(m)m (ultiplication)305 2330 y(with)33 b(exact)i(rounding.)49 b(T)-8 b(ec)m(hnical)34 b(Rep)s(ort)f(93-76,)k(RISC-Linz,)d(Johannes)f (Kepler)g(Univ)m(ersit)m(y)-8 b(,)305 2443 y(1993.)164 2630 y([7])46 b(K.)30 b(Mehlhorn,)f(S.)h(N\177)-45 b(aher:)41 b(LED)m(A,)32 b(A)e(Platform)g(for)g(Com)m(binatorial)f(and)h (Geometric)h(Comput-)305 2743 y(ing,)f(Cam)m(bridge)f(Univ)m(ersit)m(y) h(Press,)g(1998)164 2931 y([8])46 b(A.)g(Bosselaers,)j(R.)d(Go)m(v)-5 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b(Heidelb)s(erg)f(1991)118 4133 y([12])47 b(B.)j(Haible:)78 b(CLN,)49 b(a)g(class)h(library)c(for)k(n)m(um)m(b)s(ers.)e(Av)-5 b(ailable)48 b(via)h(anon)m(ymous)g(FTP)g(from)305 4246 y Fx(ftp://ftp.santafe.edu/pu)o(b/gn)o(u/cl)o(n.t)o(ar.g)o(z)24 b FB(\(1996\))118 4434 y([13])47 b(A.)31 b(Lenstra:)40 b(lip:)f(long)30 b(in)m(teger)h(pac)m(k)-5 b(age,)32 b(Bellcore)f(1989)118 4621 y([14])47 b(Th.)d(P)m(apanik)m(olaou:)70 b(En)m(t)m(wurf)44 b(und)f(En)m(t)m(wic)m(klung)h(einer)g(ob)5 b(jektorien)m(tierten)45 b(Bibliothek)e(f)s(\177)-48 b(ur)305 4734 y(algorithmisc)m(he)29 b(Zahlen)m(theorie,)h(PhD)g (Thesis,)f(Univ)m(ersit\177)-45 b(at)30 b(des)g(Saarlandes,)g(1997)118 4922 y([15])47 b(I.)35 b(Biehl,)h(J.)f(Buc)m(hmann,)i(Th.)e(P)m(apanik) m(olaou,)i(LiDIA:)e(a)g(library)f(for)h(computational)g(n)m(um)m(b)s (er)305 5035 y(theory,)c(Univ)m(ersit\177)-45 b(at)30 b(des)g(Saarlandes,)f(SFB)i(124)h(rep)s(ort,)e(1995)1903 5841 y(27)p eop %%Page: 28 32 28 31 bop 0 630 1386 4 v 2285 202 100 7 v 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b(Simpler)22 b(and)i(F)-6 b(aster)24 b(Static)h(A)n(C)2994 1767 y Fg(0)3052 1790 y Fy(Dictionaries)0 1902 y(MPI-I-97-2-012)169 b(L.)23 b(Bac)n(hmair,)f(H.)h(Ganzinger,)h(A.)f(V)-6 b(oronk)n(o)n(v)178 b(Elimination)22 b(of)h(Equalit)n(y)h(via)g(T)-6 b(ransformation)23 b(with)2079 1981 y(Ordering)g(Constrain)n(ts)0 2092 y(MPI-I-97-2-011)169 b(L.)23 b(Bac)n(hmair,)f(H.)h(Ganzinger)607 b(Strict)24 b(Basic)f(Sup)r(erp)r(osition)i(and)f(Chaining)0 2204 y(MPI-I-97-2-010)169 b(S.)23 b(V)-6 b(orob)n(y)n(o)n(v,)24 b(A.)f(V)-6 b(oronk)n(o)n(v)643 b(Complexit)n(y)23 b(of)g(Nonrecursiv)n (e)h(Logic)g(Programs)e(with)2079 2283 y(Complex)h(V)-6 b(alues)0 2398 y(MPI-I-97-2-009)169 b(A.)23 b(Bo)r(c)n(kma)n(yr,)g(F.)g (Eisen)n(brand)558 b(On)23 b(the)i(Ch)n(v\023)-35 b(atal)24 b(Rank)h(of)e(P)n(olytop)r(es)i(in)e(the)i(0/1)f(Cub)r(e)0 2510 y(MPI-I-97-2-008)169 b(A.)23 b(Bo)r(c)n(kma)n(yr,)g(T.)g(Kasp)r (er)679 b(A)23 b(Unifying)g(F)-6 b(ramew)n(ork)23 b(for)g(In)n(teger)i (and)f(Finite)g(Domain)2079 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y(MPI-I-97-1-023)169 b(S.A.)23 b(Leonardi,)g(A.P)-6 b(.)23 b(Marc)n(hetti-Spaccamela)133 b(Randomized)24 b(on-line)f(call)g(con)n(trol)h(revisited)0 4546 y(MPI-I-97-1-022)169 b(E.)23 b(Althaus,)g(K.)g(Mehlhorn)672 b(Maxim)n(um)21 b(Net)n(w)n(ork)j(Flo)n(w)g(with)f(Floating)i(P)n(oin)n (t)2079 4625 y(Arithmetic)0 4736 y(MPI-I-97-1-021)169 b(J.F.)23 b(Sib)r(eyn)1089 b(F)-6 b(rom)22 b(P)n(arallel)h(to)h (External)g(List)g(Ranking)0 4848 y(MPI-I-97-1-020)169 b(G.S.)23 b(Bro)r(dal)1075 b(Finger)23 b(Searc)n(h)i(T)-6 b(rees)23 b(with)h(Constan)n(t)h(Insertion)g(Time)0 4960 y(MPI-I-97-1-019)169 b(D.)23 b(Alb)r(erts,)g(C.)g(Gut)n(w)n(enger,)i(P) -6 b(.)23 b(Mutzel,)638 5039 y(S.)g(N\177)-35 b(aher)2079 4960 y(A)n(GD-Library:)30 b(A)23 b(Library)g(of)g(Algorithms)f(for)h (Graph)2079 5039 y(Dra)n(wing)0 5150 y(MPI-I-97-1-018)169 b(R.)23 b(Fleisc)n(her)1074 b(On)23 b(the)i(Bahncard)f(Problem)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF