%%Creator: %!PS-Adobe-2.0 dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: privacy.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o privacy.ps privacy %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.03.03:1742 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A 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/IEn 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 IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/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 A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A 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/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 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 A 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 A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 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}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 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 A 1 get A mul exch 0 get A 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/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 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/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
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2723 y(2.)49 b(A)32 b(sc)m(heme)i(for)e Fh(k)k Fk(databases)d(with)g Fh(O)s Fk(\()p Fh(n)2089 2687 y Fo(1)p Fn(=k)2201 2723 y Fk(\))g(comm)m(unication.)447 2927 y(3.)49 b(A)40 b(sc)m(heme)h(for)f Fh(O)s Fk(\(log)16 b Fh(n)p Fk(\))40 b(databases)h(with)f Fh(O)s Fk(\(log)2501 2884 y Fo(2)2557 2927 y Fh(n)17 b Fk(log)g(log)f Fh(n)p Fk(\))41 b(comm)m(unica-)572 3047 y(tion.)474 3250 y(In)51 b(this)f(pap)s(er,)56 b(w)m(e)51 b(impro)m(v)m(e)f(their)h(result,)k(constructing)50 b(a)h(proto)s(col)e (for)h Fh(k)328 3371 y Fk(databases)33 b(with)g Fh(O)s Fk(\()p Fh(n)1170 3335 y Fo(1)p Fn(=)p Fo(\(2)p Fn(k)r Fm(\000)p Fo(1\))1463 3371 y Fk(\))f(comm)m(unication.)474 3491 y Fg(Related)51 b(w)m(ork.)80 b Fk(Proto)s(cols)44 b(for)h(priv)-5 b(ate)44 b(information)d(retriev)-5 b(al)44 b(in)g([7])h(and)328 3611 y(this)e(pap)s(er)g(ha)m(v)m(e)h(used)g (ideas)f(from)f(sev)m(eral)i(related)f(problems)f(\(instance)h(hiding) 328 3732 y(and)33 b(m)m(ultipart)m(y)e(comm)m(unication)f(complexit)m (y\).)474 3852 y(Instance)40 b(hiding[1)n(,)f(5)o(,)g(6)o(])g(is)e(the) i(problem)e(of)h(obtaining)e(the)j Fh(i)2928 3816 y Fo(th)3037 3852 y Fk(bit)e(from)g(the)328 3973 y(oracle)h(so)h(that)f Fh(i)h Fk(remains)e(secret.)63 b(There)40 b(are)f(some)f(similarities)c (and)39 b(some)f(sub-)328 4093 y(stan)m(tial)26 b(di\013erences)j(b)s (et)m(w)m(een)i(instance)d(hiding)e(and)i(priv)-5 b(ate)27 b(information)e(retriev)-5 b(al)328 4213 y(\(see)34 b([7)o(])f(for)f (more)g(detailed)f(discussion\).)474 4334 y(T)-8 b(ec)m(hniques)47 b(from)42 b(instance)j(hiding)e(w)m(ere)i(relev)-5 b(an)m(t)44 b(to)g(proto)s(cols)f(for)g(priv)-5 b(ate)328 4454 y(information)29 b(retriev)-5 b(al)31 b(in)h([7].)44 b(Ho)m(w)m(ev)m(er,)35 b(they)e(are)g(not)f(used)i(in)e(this)g(pap)s(er.)474 4575 y(Multipart)m(y)f(comm)m(unication)f(complexit)m(y)h(is)g(also)g (related)g(to)g(priv)-5 b(ate)31 b(informa-)328 4695 y(tion)k(retriev)-5 b(al.)52 b(Pudlak,)37 b(R\177)-49 b(odl,)36 b(Sgall[11)n(])g(and)g(Am)m(bainis[3)n(])g(ha)m(v)m(e)h (considered)g(the)328 4815 y(problem)h(of)h(computing)f Fh(x)1376 4831 y Fo(\()p Fn(i)p Fo(+)p Fn(j)t Fo(\))21 b(mo)r(d)d Fn(n)1804 4815 y Fk(where)40 b Fh(x)g Fk(is)f(a)g(string)g (of)g Fh(n)h Fk(bits)f(and)g Fh(i;)17 b(j)46 b Fk(are)328 4936 y(in)m(tegers)33 b(in)f(the)h(follo)m(wing)d(mo)s(del:)1922 5214 y(2)p eop %%Page: 3 3 3 2 bop 474 631 a Fk(Pla)m(y)m(er)29 b(1)f(kno)m(ws)h Fh(x;)17 b(i)p Fk(,)30 b(Pla)m(y)m(er)e(2)g(kno)m(ws)i Fh(x;)17 b(j)6 b Fk(.)42 b(Eac)m(h)29 b(of)e(them)h(sends)i(one)e (message)328 751 y(to)34 b(Pla)m(y)m(er)g(3.)48 b(Pla)m(y)m(er)34 b(3)g(computes)h(the)f(result,)h(using)e(only)h(the)g(messages)h (receiv)m(ed)328 872 y(from)c(Pla)m(y)m(ers)j(1)e(and)h(2.)474 992 y(An)m(y)c(proto)s(col)d(for)h(the)h(ab)s(o)m(v)m(e)g(problem)e (can)i(b)s(e)g(easily)f(transformed)g(in)m(to)g(proto-)328 1112 y(col)d(for)g(priv)-5 b(ate)24 b(information)e(retriev)-5 b(al.)39 b(Th)m(us,)28 b(w)m(e)e(can)f(obtain)e(non)m(trivial)g(proto)s (cols)328 1233 y(for)32 b(priv)-5 b(ate)32 b(information)d(retriev)-5 b(al)31 b(with)i Fh(o)p Fk(\()p Fh(n)p Fk(\))f(comm)m(unication.)474 1353 y(Another)d(comm)m(unication)c(complexit)m(y)j(problem)f(w)m(as)i (studied)f(b)m(y)h(Babai,)f(Kim-)328 1474 y(mel)j(and)i(Lok)-5 b(am[4)o(].)43 b(It)33 b(also)e(can)i(b)s(e)g(applied)e(to)i(priv)-5 b(ate)32 b(information)d(retriev)-5 b(al.)474 1594 y(Ho)m(w)m(ev)m(er,) 28 b(all)21 b(these)k(proto)s(cols)e(are)g(less)h(e\016cien)m(t)g(than) g(the)g(proto)s(cols)e(for)h(priv)-5 b(ate)328 1714 y(information)43 b(retriev)-5 b(al)44 b(designed)i(in)f([7].)83 b(Still,)47 b(the)f(ideas)g(from)e([3,)i(4,)g(11)o(])g(\(not)328
1835 y(explicit)26 b(proto)s(cols\))h(can)i(b)s(e)f(useful)g(in)f(the)h (study)i(of)d(priv)-5 b(ate)27 b(information)e(retriev)-5 b(al.)328 1955 y(In)22 b(particular,)h(this)e(pap)s(er)i(is)e(based)i (on)f(the)g(idea)g(of)f(com)m(bining)g(t)m(w)m(o)h(proto)s(cols)f(whic) m(h)328 2076 y(app)s(eared)33 b(in)f(the)h(setting)f(of)g(m)m(ultipart) m(y)f(comm)m(unication)f(complexit)m(y[3,)i(11].)328 2408 y Fl(2)161 b(Mo)t(del)328 2627 y Fk(F)-8 b(ormally)g(,)30 b(w)m(e)35 b(view)e(the)h(database)g(as)f(a)g(string)g Fh(x)g Fk(consisting)g(of)g Fh(n)g Fk(bits.)45 b Fh(k)37 b Fk(denotes)328 2748 y(the)23 b(n)m(um)m(b)s(er)h(of)f(iden)m(tical)e (databases.)41 b(W)-8 b(e)24 b(assume)g(that)f(the)g(user)h(wishes)g (to)f(retriev)m(e)328 2868 y(a)32 b(single)g(bit)f Fh(x)887 2883 y Fn(i)949 2868 y Fk(from)g(the)i(database.)474 2989 y(W)-8 b(e)27 b(require)f(that,)i(for)d(ev)m(ery)j(database,)g (indices)e Fh(i;)17 b(j)32 b Fk(and)27 b(an)m(y)g(message)f(from)f(the) 328 3109 y(user,)k(the)e(probabilit)m(y)e(of)h(the)h(database)h (receiving)e(this)g(message)i(is)e(equal)g(when)i(the)328 3229 y(user)k(retriev)m(es)g(the)f Fh(i)1121 3193 y Fo(th)1223 3229 y Fk(bit)g(and)g(when)h(the)f(user)h(retriev)m(es)g(the)g Fh(j)2788 3193 y Fo(th)2889 3229 y Fk(bit.)43 b(This)31 b(means)328 3350 y(that)h(database)h(do)s(es)g(not)g(get)f(an)m(y)i (information)29 b(ab)s(out)j Fh(i)p Fk(.)474 3470 y(There)i(are)e(sev)m (eral)h(extensions)g(of)f(this)g(mo)s(del.)41 b([7])32 b(considered)h(sc)m(hemes)h(whic)m(h)328 3590 y(allo)m(w)k(to)h (retriev)m(e)i(blo)s(c)m(ks)f(of)f(information)e(and)j(giv)m(e)f(a)h (higher)f(degree)i(of)e(priv)-5 b(acy)328 3711 y(\(kno)m(wing)46 b Fh(k)35 b Ff(\000)d Fk(1)46 b(of)g Fh(k)j Fk(queries)e(giv)m(es)g(no) f(information)d(ab)s(out)j(the)h(bit)e(that)h(the)328 3831 y(user)i(retriev)m(es\).)90 b(Ostro)m(vsky)50 b(and)e(Shoup[9])g (ha)m(v)m(e)h(extended)h(the)e(results)g(of)f([7])328 3952 y(and)35 b(designed)g(sc)m(hemes)i(for)e(priv)-5 b(ate)34 b(information)d(storage.)51 b(Using)35 b(their)f(sc)m(hemes,) 328 4072 y(the)40 b(user)g(can)g(b)s(oth)f(read)h(and)f(write)h(to)f (the)h(database)f(without)g(rev)m(ealing)g(whic)m(h)328 4192 y(bit)c(is)g(accessed.)55 b(They)37 b(ha)m(v)m(e)h(sho)m(wn)f (that)e(an)m(y)h(proto)s(col)f(for)g(priv)-5 b(ate)35 b(information)328 4313 y(retriev)-5 b(al)33 b(can)i(b)s(e)g (transformed)f(to)g(the)h(proto)s(col)e(for)h(priv)-5 b(ate)34 b(information)e(storage)328 4433 y(with)g(a)g(sligh)m(t)g (increase)h(in)f(the)h(n)m(um)m(b)s(er)g(of)f(databases)h(and)g(comm)m (unication.)474 4553 y(Ho)m(w)m(ev)m(er,)i(in)d(this)g(pap)s(er,)h(w)m (e)h(consider)f(only)f(the)h(basic)f(one-bit)g(mo)s(del)f(of)h([7].)474 4674 y Fh(S)c Ff(\010)23 b Fh(i)33 b Fk(denotes)g Fh(S)28 b Ff([)23 b(f)p Fh(i)p Ff(g)p Fk(,)32 b(if)g Fh(i)39 b(=)-60 b Ff(2)28 b Fh(S)38 b Fk(and)33 b Fh(S)28 b Ff(\000)22 b(f)p Fh(i)p Ff(g)33 b Fk(if)e Fh(i)d Ff(2)g Fh(S)6 b Fk(.)1922 5214 y(3)p eop %%Page: 4 4 4 3 bop 328 631 a Fl(3)161 b(Result)328 850 y Fk(Consider)35 b(some)g(proto)s(col)e(for)i(priv)-5 b(ate)34 b(information)e(retriev) -5 b(al.)49 b(Do)s(es)35 b(the)g(user)h(use)328 970 y(all)21 b(bits)h(in)g(the)i(messages)g(from)d(the)j(databases?)41 b(In)23 b(some)g(proto)s(cols,)h(only)e(a)h(few)g(bits)328 1091 y(are)33 b(really)e(neccessary)-8 b(.)46 b(If)33 b(the)g(user)h(kno)m(ws)g(in)e(adv)-5 b(ance)33 b(whic)m(h)g(bits)g (are)f(necessary)-8 b(,)328 1211 y(t)m(w)m(o)28 b(proto)s(cols)e(can)i (b)s(e)g(com)m(bined,)g(obtaining)d(the)j(third)f(with)g(more)f (databases)j(and)328 1332 y(less)k(comm)m(unication.)474 1452 y(Belo)m(w,)43 b(w)m(e)e(sho)m(w)g(ho)m(w)g(to)f(com)m(bine)g(a)g
(proto)s(col)f(for)h(2)g(databases)h(and)f(a)g(pro-)328 1572 y(to)s(col)45 b(for)g Fh(k)35 b Ff(\000)c Fk(1)46 b(databases,)51 b(obtaining)44 b(a)h(proto)s(col)g(for)h Fh(k)j Fk(databases)e(with)e(less)328 1693 y(comm)m(unication.)447 1895 y(1.)k(The)33 b(user)g(in)f(the)h Fh(k)i Fk(database)e(proto)s (col)e(sim)m(ulates)g(the)i(user)g(in)e(the)i(proto)s(col)572 2016 y(for)26 b(2)g(databases.)42 b(Let)27 b Fh(x)1495 2031 y Fo(1)1561 2016 y Fk(denote)h(the)e(message)h(sen)m(t)h(to)e(the) h(1)2922 1979 y Fo(st)3008 2016 y Fk(database)g(and)572 2136 y Fh(x)627 2151 y Fo(2)699 2136 y Fk(the)33 b(message)g(sen)m(t)h (to)e(the)h(2)1782 2100 y Fo(nd)1897 2136 y Fk(database)g(in)f(the)h(2) f(database)h(proto)s(col.)572 2295 y(The)40 b(user)g(sends)h Fh(x)1315 2310 y Fo(1)1394 2295 y Fk(to)d(the)i(1)1743 2258 y Fo(st)1841 2295 y Fk(database)g(and)f Fh(x)2506 2310 y Fo(2)2585 2295 y Fk(to)f(the)i(2)2934 2258 y Fo(nd)3016 2295 y Fk(,)h Fh(:)17 b(:)g(:)p Fk(,)40 b(the)g Fh(k)3495 2258 y Fo(th)572 2415 y Fk(database.)447 2612 y(2.)49 b(Then,)32 b(the)f(user)g(computes)g(the)g(length)f(of)g(the)h(reply)f (from)g(the)h(2)3079 2576 y Fo(nd)3191 2612 y Fk(database)572 2732 y(in)37 b(the)i(2)f(database)h(proto)s(col)e(and)h(the)h(p)s (ositions)e(of)h(necessary)j(bits)d(in)f(this)572 2853 y(reply)-8 b(.)40 b(F)-8 b(urther,)25 b Fh(m)f Fk(denotes)h(the)e (length)g(and)g Fh(n)2356 2868 y Fo(1)2396 2853 y Fh(;)17 b(:)g(:)g(:)f(;)h(n)2673 2868 y Fn(i)2724 2853 y Fk(denote)24 b(the)g(p)s(ositions)572 2973 y(of)32 b(the)h(necessary)i(bits.)572 3132 y(The)d(user)h(sim)m(ulates)d(the)i(user)g(in)f(the)h(proto)s (cols)e(for)h Fh(k)23 b Ff(\000)d Fk(1)31 b(databases)i(where)572 3252 y Fh(n)630 3216 y Fo(th)630 3277 y(1)701 3252 y Fk(,)i Fh(:)17 b(:)g(:)p Fk(,)36 b Fh(n)999 3216 y Fo(th)999 3277 y Fn(i)1105 3252 y Fk(bits)e(from)g(an)h Fh(m)g Fk(bit)f(database)i(are)f(retriev)m(ed,)h(sending)g(to)e(the)572 3373 y(\()p Fh(i)26 b Fk(+)h(1\))859 3336 y Fo(st)957 3373 y Fk(database)39 b(the)g(messages)h(whic)m(h)g(are)e(sen)m(t)i(to) f(the)g Fh(i)2961 3336 y Fo(th)3071 3373 y Fk(database)h(in)572 3493 y(the)33 b(\()p Fh(k)25 b Ff(\000)e Fk(1\))32 b(database)h(proto)s (col.)447 3690 y(3.)49 b(The)30 b(1)818 3654 y Fo(st)906 3690 y Fk(database)g(sim)m(ulates)e(the)h(1)1946 3654 y Fo(st)2035 3690 y Fk(database)g(in)g(2)f(database)i(proto)s(col)e (and)572 3810 y(sends)34 b(the)f(user)h(the)f(same)f(message.)447 4007 y(4.)49 b(The)j(2)840 3971 y Fo(nd)923 4007 y Fk(,)k Fh(:)17 b(:)g(:)p Fk(,)56 b(the)c Fh(k)1445 3971 y Fo(th)1568 4007 y Fk(database)g(sim)m(ulate)e(the)i(2)2637 3971 y Fo(nd)2771 4007 y Fk(database)g(in)f(the)h(2)572 4128 y(database)42 b(proto)s(col.)68 b(Instead)42 b(of)f(sending)h(the)g (message)f(to)g(the)h(user,)j(they)572 4248 y(consider)33 b(it)e(as)i(a)f(new)i Fh(m)p Fk(-bit)d(database.)572 4407 y(F)-8 b(urther,)37 b(they)g(sim)m(ulate)e(databases)i(in)f(the)h (\()p Fh(k)28 b Ff(\000)d Fk(1\))36 b(database)g(proto)s(col)f(for)572 4527 y(the)42 b(retriev)-5 b(al)41 b(of)h(the)g Fh(n)1497 4491 y Fo(th)1497 4552 y(1)1568 4527 y Fk(,)j Fh(:)17 b(:)g(:)o Fk(,)45 b(the)d Fh(n)2061 4491 y Fo(th)2061 4552 y Fn(i)2175 4527 y Fk(bit)f(and)h(send)h(the)g(messages)g(from)572 4647 y(these)34 b(proto)s(cols)d(to)h(the)h(user.)447 4844 y(5.)49 b(The)41 b(user)h(sim)m(ulates)d(the)i(user)g(in)f(the)h (\()p Fh(k)30 b Ff(\000)e Fk(1\))41 b(database)f(proto)s(col)f(for)h (the)572 4965 y(retriev)-5 b(al)31 b(of)i(the)g Fh(n)1292 4929 y Fo(th)1292 4989 y(1)1363 4965 y Fk(,)g Fh(:)17 b(:)g(:)p Fk(,)33 b(the)g Fh(n)1824 4929 y Fo(th)1824 4989 y Fn(i)1928 4965 y Fk(bits.)44 b(Then,)34 b(kno)m(wing)f(the)h
(message)f(from)1922 5214 y(4)p eop %%Page: 5 5 5 4 bop 572 631 a Fk(the)40 b(1)796 595 y Fo(st)895 631 y Fk(database)g(and)g(all)e(the)i(necessary)j(bits)c(from)g(the)h (second)h(message,)572 751 y(the)d(user)h(sim)m(ulates)e(the)i(user)g (in)e(the)i(2)f(database)g(proto)s(col.)59 b(The)39 b(result)f(of)572 872 y(this)32 b(sim)m(ulation)e(is)i(the)h(bit)f(that)g(the)h(user)g (wishes)h(to)e(retriev)m(e.)474 1100 y(If)k(w)m(e)h(wish)f(to)f(apply)h (this)f(idea,)h(2)g(database)g(proto)s(col)f(should)g(satisfy)h (certain)328 1220 y(constrain)m(ts:)447 1424 y(1.)49 b(The)30 b(most)f(of)g(comm)m(unication)e(go)s(es)i(from)f(the)i (databases)h(to)e(the)g(user.)44 b(\(The)572 1544 y(amoun)m(t)37 b(of)h(comm)m(unication)d(from)i(the)h(user)h(to)f(databases)h (increases)g(when)572 1665 y(t)m(w)m(o)32 b(proto)s(cols)e(are)h(com)m (bined.)42 b(Hence,)33 b(if)d(it)g(is)h(already)f(large,)h(the)g(com)m (bina-)572 1785 y(tion)g(is)h(useless.\))447 1988 y(2.)49 b(Only)32 b(few)h(bits)f(from)g(the)h(messages)g(receiv)m(ed)h(b)m(y)g (the)f(user)g(are)g(necessary)-8 b(.)447 2192 y(3.)49 b(The)34 b(user)h(kno)m(ws)g(in)d(adv)-5 b(ance)35 b(whic)m(h)f(bits)f (are)g(necessary)-8 b(,)36 b(i.e.)46 b(the)34 b(p)s(ositions)572 2312 y(of)e(these)i(bits)e(do)g(not)h(dep)s(end)h(on)e(the)h (databases')h(con)m(ten)m(ts.)474 2516 y(Belo)m(w,)f(w)m(e)h(use)f(the) g(idea)f(of)g(com)m(bining)f(t)m(w)m(o)i(proto)s(cols)f(to)g(pro)m(v)m (e)328 2719 y Fg(Theorem)37 b(1)49 b Fe(L)-5 b(et)50 b Fh(k)58 b Ff(\025)d Fk(2)p Fe(.)88 b(Ther)-5 b(e)49 b(exists)g(a)g(pr)-5 b(oto)g(c)g(ol)49 b(for)h(private)f(information) 328 2839 y(r)-5 b(etrieval)35 b(with)f Fh(k)k Fe(datab)-5 b(ases)34 b(and)g Fh(O)s Fk(\()p Fh(n)1806 2803 y Fo(1)p Fn(=)p Fo(\(2)p Fn(k)r Fm(\000)p Fo(1\))2099 2839 y Fk(\))h Fe(bits)g(of)g(c)-5 b(ommunic)g(ation.)328 3043 y Fg(Pro)s(of.)43 b Fk(By)33 b(induction.)474 3163 y(The)23 b(proto)s(col)d(for)h(2)h (databases)g(w)m(as)h(constructed)h(b)m(y)e(Chor,)i(Goldreic)m(h,)f (Kushile-)328 3284 y(vitz)30 b(and)h(Sudan[7)q(].)43 b(The)31 b(proto)s(col)e(for)i Fh(k)i Fk(databases)f(is)e(obtained)g (as)h(the)g(com)m(bina-)328 3404 y(tion)g(of)i(the)g(proto)s(cols)e (for)h(2)g(databases)i(and)f(\()p Fh(k)25 b Ff(\000)d Fk(1\))33 b(databases.)474 3524 y(First,)50 b(w)m(e)d(describ)s(e)g (the)g(2)g(database)g(proto)s(col)e(that)h(w)m(e)i(use)g(to)e(obtain)f (a)i Fh(k)328 3645 y Fk(database)33 b(proto)s(col)e(from)g(a)i(\()p Fh(k)25 b Ff(\000)d Fk(1\))33 b(database)g(proto)s(col.)447 3848 y(1.)49 b(Let)35 b Fh(l)g Fk(=)d Ff(d)993 3815 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)1080 3777 y Ff(p)p 1163 3777 59 4 v 71 x Fh(n)q Ff(e)p Fk(.)51 b(The)37 b(database)e(can)h(b)s (e)f(considered)h(as)g(a)f(2)p Fh(k)27 b Ff(\000)d Fk(1)35 b(dimen-)572 3968 y(sional)e(cub)s(e)i Ff(f)p Fk(0)p Fh(;)17 b(:)g(:)g(:)f(;)h(l)25 b Ff(\000)f Fk(1)p Ff(g)1655 3932 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1823 3968 y Fk(.)50 b(Eac)m(h)36 b(p)s(osition)d Fh(i)e Ff(2)h(f)p Fk(0)p Fh(;)17 b(:)g(:)g(:)e(;)i(n)24 b Ff(\000)g Fk(1)p Ff(g)34 b Fk(in)g(the)572 4089 y(database)f(coresp)s(onds)h(to)e(some)g(p)s (osition)f(\()p Fh(i)2282 4104 y Fo(1)2322 4089 y Fh(;)17 b(:)g(:)g(:)f(;)h(i)2574 4104 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)2742 4089 y Fk(\))32 b(in)g(the)h(cub)s(e.)572 4251 y(The)e(user)h(c)m(ho)s(oses)g(indep)s(enden)m(tly)f(\(2)p Fh(k)21 b Ff(\000)d Fk(1\))31 b(random)e(subsets)k(of)d Ff(f)p Fk(0)p Fh(;)17 b(:)g(:)g(:)e(;)i(l)j Ff(\000)572 4371 y Fk(1)p Ff(g)p Fk(:)75 b Fh(S)839 4335 y Fo(1)833 4396 y(1)878 4371 y Fk(,)53 b Fh(:)17 b(:)g(:)p Fk(,)53 b Fh(S)1219 4335 y Fo(1)1213 4396 y(2)p Fn(k)r Fm(\000)p
Fo(1)1381 4371 y Fk(.)92 b(Let)49 b Fh(S)1757 4335 y Fo(2)1751 4396 y(1)1851 4371 y Fk(=)55 b Fh(S)2048 4335 y Fo(1)2042 4396 y(1)2121 4371 y Ff(\010)33 b Fh(i)2264 4386 y Fo(1)2304 4371 y Fk(,)53 b Fh(:)17 b(:)g(:)o Fk(,)53 b Fh(S)2644 4335 y Fo(2)2638 4396 y(2)p Fn(k)r Fm(\000)p Fo(1)2861 4371 y Fk(=)i Fh(S)3058 4335 y Fo(1)3052 4396 y(2)p Fn(k)r Fm(\000)p Fo(1)3254 4371 y Ff(\010)33 b Fh(i)3397 4386 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)572 4491 y Fk(where)j(\()p Fh(i)927 4506 y Fo(1)967 4491 y Fh(;)17 b(:)g(:)g(:)e(;)i(i)1218 4506 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1387 4491 y Fk(\))34 b(is)h(the)g(p)s(osition)e(of)i(the)g (required)h(bit)e(in)g(the)h(\(2)p Fh(k)27 b Ff(\000)d Fk(1\))572 4612 y(dimensional)30 b(cub)s(e.)572 4774 y(He)g(sends)i Fh(S)1043 4738 y Fo(1)1037 4798 y(1)1082 4774 y Fh(;)17 b(:)g(:)g(:)f(;)h(S)1367 4738 y Fo(1)1361 4798 y(2)p Fn(k)r Fm(\000)p Fo(1)1559 4774 y Fk(to)29 b(the)i(1)1890 4738 y Fo(st)1979 4774 y Fk(database)f(and)h Fh(S)2637 4738 y Fo(2)2631 4798 y(1)2676 4774 y Fh(;)17 b(:)g(:)g(:)f(;)h(S)2961 4738 y Fo(2)2955 4798 y(2)p Fn(k)r Fm(\000)p Fo(1)3152 4774 y Fk(to)30 b(the)g(2)3483 4738 y Fo(nd)572 4894 y Fk(database.)1922 5214 y(5)p eop %%Page: 6 6 6 5 bop 447 631 a Fk(2.)49 b(The)31 b(1)819 595 y Fo(st)909 631 y Fk(database)g(computes)g(the)f(exclusiv)m(e-or)h(of)f(the)h(bits) f(in)g(p)s(ositions)f(\()p Fh(j)3499 646 y Fo(1)3539 631 y Fk(,)572 751 y Fh(:)17 b(:)g(:)o Fk(,)29 b Fh(j)782 766 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)950 751 y Fk(\))f(suc)m(h)g(that)g Fh(j)1477 766 y Fo(1)1544 751 y Ff(2)g Fh(S)1704 715 y Fo(1)1698 776 y(1)1744 751 y Fk(,)g Fh(:)17 b(:)g(:)p Fk(,)28 b Fh(j)2009 766 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)2205 751 y Ff(2)g Fh(S)2365 715 y Fo(1)2359 776 y(2)p Fn(k)r Fm(\000)p Fo(1)2555 751 y Fk(and)f(sends)i(it)e(to)g(the)h(user.)572 910 y(The)43 b(database)g(also)f(computes)h(the)g(exclusiv)m(e-or)f(of) g(the)h(bits)f(in)g(p)s(ositions)572 1031 y(\()p Fh(j)650 1046 y Fo(1)690 1031 y Fk(,)d Fh(:)17 b(:)g(:)p Fk(,)40 b Fh(j)978 1046 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1146 1031 y Fk(\))f(suc)m(h)g(that)g Fh(j)1706 1046 y Fo(1)1783 1031 y Ff(2)f Fh(S)1953 995 y Fm(0)1947 1055 y Fo(1)1986 1031 y Fk(,)i Fh(:)17 b(:)g(:)p Fk(,)40 b Fh(j)2275 1046 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)2481 1031 y Ff(2)e Fh(S)2651 995 y Fm(0)2645 1055 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)2851 1031 y Fk(for)g(eac)m(h)h(p)s(ossible)572 1151 y Fh(S)638 1115 y Fm(0)632 1176 y Fo(1)671 1151 y Fk(,)33 b Fh(:)17 b(:)g(:)o Fk(,)33 b Fh(S)971 1115 y Fm(0)965 1176 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1166 1151 y Fk(suc)m(h)h(that)613 1349 y(\(a\))48 b Fh(S)852 1313 y Fm(0)846 1374 y Fn(j)911 1349 y Fk(=)27 b Fh(S)1080 1313 y Fo(1)1074 1374 y Fn(j)1142 1349 y Ff(\010)22 b Fh(t)33 b Fk(for)f(some)g Fh(j)i Ff(2)28 b(f)p Fk(1)p Fh(;)17 b(:)g(:)g(:)e(;)i Fk(2)p Fh(k)25 b Ff(\000)e Fk(1)p Ff(g)32 b Fk(and)h Fh(t)27 b Ff(2)i(f)p Fk(0)p Fh(;)17 b(:)g(:)g(:)e(;)i(l)24 b Ff(\000)f Fk(1)p Ff(g)p Fk(;)608 1505 y(\(b\))48 b Fh(S)852 1469 y Fm(0)846 1530 y Fn(i)903 1505 y Fk(=)28 b Fh(S)1073 1469 y Fo(1)1067 1530 y Fn(i)1144 1505 y Fk(for)33 b(all)d Fh(i)e Ff(6)p Fk(=)f Fh(j)6 b Fk(.)572 1703 y(The)41 b(exclusiv)m(e-xor)g(for)e(eac)m(h)i(p)s(ossible)f Fh(S)2192 1667 y Fm(0)2186 1728 y Fo(1)2225 1703 y Fk(,)i Fh(:)17 b(:)g(:)p Fk(,)42 b Fh(S)2544 1667 y Fm(0)2538 1728 y
Fo(2)p Fn(k)r Fm(\000)p Fo(1)2746 1703 y Fk(is)d(sen)m(t)j(to)d(the)i (user,)572 1823 y(to)s(o.)447 2021 y(3.)49 b(The)29 b(2)817 1985 y Fo(nd)928 2021 y Fk(database)f(computes)h(the)g(exclusiv)m(e-or) g(of)f(the)g(bits)g(in)g(p)s(ositions)f(\()p Fh(j)3499 2036 y Fo(1)3539 2021 y Fk(,)572 2141 y Fh(:)17 b(:)g(:)o Fk(,)29 b Fh(j)782 2156 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)950 2141 y Fk(\))f(suc)m(h)g(that)g Fh(j)1477 2156 y Fo(1)1544 2141 y Ff(2)g Fh(S)1704 2105 y Fo(2)1698 2166 y(1)1744 2141 y Fk(,)g Fh(:)17 b(:)g(:)p Fk(,)28 b Fh(j)2009 2156 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)2205 2141 y Ff(2)g Fh(S)2365 2105 y Fo(2)2359 2166 y(2)p Fn(k)r Fm(\000)p Fo(1)2555 2141 y Fk(and)f(sends)i(it)e(to)g(the)h(user.)572 2301 y(F)-8 b(urther,)35 b(the)g(2)1174 2264 y Fo(nd)1291 2301 y Fk(database)g(computes)g(the)g(exclusiv)m(e-or)g(of)f(the)h (bits)f(in)g(p)s(o-)572 2421 y(sitions)43 b(\()p Fh(j)966 2436 y Fo(1)1006 2421 y Fh(;)17 b(:)g(:)g(:)f(;)h(j)1265 2436 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1433 2421 y Fk(\))44 b(suc)m(h)i(that)e Fh(j)2010 2436 y Fo(1)2098 2421 y Ff(2)k Fh(S)2278 2385 y Fm(0)2272 2446 y Fo(1)2312 2421 y Fk(,)f Fh(:)17 b(:)g(:)p Fk(,)47 b Fh(j)2615 2436 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)2831 2421 y Ff(2)h Fh(S)3011 2385 y Fm(0)3005 2446 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)3218 2421 y Fk(for)c(eac)m(h)572 2541 y(p)s(ossible)32 b Fh(S)1005 2505 y Fm(0)999 2566 y Fo(1)1038 2541 y Fk(,)h Fh(:)17 b(:)g(:)o Fk(,)33 b Fh(S)1338 2505 y Fm(0)1332 2566 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1532 2541 y Fk(suc)m(h)i(that)613 2739 y(\(a\))48 b(F)-8 b(or)33 b(eac)m(h)i Fh(i)30 b Ff(2)g(f)p Fk(1)p Fh(;)17 b(:)g(:)g(:)e(;)i Fk(2)p Fh(k)26 b Ff(\000)d Fk(1)p Ff(g)34 b Fh(S)2084 2703 y Fm(0)2078 2764 y Fn(i)2140 2739 y Fk(is)g(equal)f(to)h Fh(S)2685 2703 y Fo(2)2679 2764 y Fn(i)2758 2739 y Fk(or)f Fh(S)2944 2703 y Fo(2)2938 2764 y Fn(i)3006 2739 y Ff(\010)24 b Fh(t)3142 2754 y Fn(i)3204 2739 y Fk(for)33 b(some)786 2859 y Fh(t)821 2874 y Fn(i)877 2859 y Ff(2)c(f)p Fk(0)p Fh(;)17 b(:)g(:)g(:)e(;)i(l)24 b Ff(\000)f Fk(1)p Ff(g)p Fk(;)608 3016 y(\(b\))48 b(There)34 b(exist)f(at)f(least)g(t)m(w)m(o)i Fh(i)28 b Ff(2)g(f)p Fk(1)p Fh(;)17 b(:)g(:)g(:)e(;)i Fk(2)p Fh(k)25 b Ff(\000)e Fk(1)p Ff(g)32 b Fk(suc)m(h)i(that)e Fh(S)3156 2980 y Fm(0)3150 3040 y Fn(i)3207 3016 y Fk(=)27 b Fh(S)3376 2980 y Fo(2)3370 3040 y Fn(i)3416 3016 y Fk(.)572 3213 y(The)41 b(exclusiv)m(e-xor)g(for)e(eac)m(h)i(p)s (ossible)f Fh(S)2192 3177 y Fm(0)2186 3238 y Fo(1)2225 3213 y Fk(,)i Fh(:)17 b(:)g(:)p Fk(,)42 b Fh(S)2544 3177 y Fm(0)2538 3238 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)2746 3213 y Fk(is)d(sen)m(t)j(to)d(the)i(user,)572 3334 y(to)s(o.)447 3532 y(4.)49 b(F)-8 b(or)42 b(eac)m(h)i(p)s(ossible)e Fh(S)1430 3495 y Fm(0)1424 3556 y Fo(1)1464 3532 y Fk(,)j Fh(:)17 b(:)g(:)p Fk(,)46 b Fh(S)1790 3495 y Fm(0)1784 3556 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1995 3532 y Fk(suc)m(h)e(that)f Fh(S)2513 3495 y Fm(0)2507 3556 y Fn(i)2579 3532 y Fk(is)g(either)f Fh(S)3040 3495 y Fo(1)3034 3556 y Fn(i)3123 3532 y Fk(or)g Fh(S)3318 3495 y Fo(2)3312 3556 y Fn(i)3358 3532 y Fk(,)j(the)572 3652 y(user)35 b(\014nds)g(the)g(exclusiv)m(e-or)f(of)g(bits)g(in)f(p)s (ositions)g(\()p Fh(j)2625 3667 y Fo(1)2665 3652 y Fh(;)17 b(:)g(:)g(:)e(;)i(j)2923 3667 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)3092 3652 y Fk(\))34 b(satisfying)572 3772 y Fh(j)612 3787 y Fo(1)679 3772 y Ff(2)28 b Fh(S)839 3736 y Fm(0)833 3797 y Fo(1)873 3772 y Fk(,)k Fh(:)17 b(:)g(:)p Fk(,)33 b Fh(j)1147 3787 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1343
3772 y Ff(2)28 b Fh(S)1503 3736 y Fm(0)1497 3797 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1665 3772 y Fk(:)613 3970 y(\(a\))48 b(If)39 b Fh(S)956 3934 y Fm(0)950 3995 y Fn(i)1017 3970 y Fk(=)f Fh(S)1197 3934 y Fo(2)1191 3995 y Fn(i)1275 3970 y Fk(for)g(at)h(most)f(one)h Fh(i)p Fk(,)h(then)g(the)f(exclusiv)m (e-or)g(is)f(one)h(of)f(the)786 4090 y(bits)33 b(sen)m(t)g(b)m(y)h(the) f(1)1533 4054 y Fo(st)1625 4090 y Fk(database.)608 4247 y(\(b\))48 b(If)39 b Fh(S)956 4211 y Fm(0)950 4271 y Fn(i)1018 4247 y Fk(=)g Fh(S)1199 4211 y Fo(2)1193 4271 y Fn(i)1277 4247 y Fk(for)f(at)h(least)g(t)m(w)m(o)g Fh(i)p Fk(,)i(then)f(the)g(exclusiv)m(e-or)f(is)f(one)i(of)e(the)786 4367 y(bits)33 b(sen)m(t)g(b)m(y)h(the)f(2)1533 4331 y Fo(nd)1648 4367 y Fk(database.)572 4565 y(The)h(user)f(computes)h (the)f(exclusiv)m(e-or)g(of)g(all)d(these)35 b(v)-5 b(alues.)44 b(It)33 b(is)f(the)i(neces-)572 4685 y(sary)f(bit)f(from)f(the)i (database.)572 4844 y(\()p Fh(S)676 4808 y Fo(2)670 4869 y Fn(j)763 4844 y Fk(=)48 b Fh(S)953 4808 y Fo(1)947 4869 y Fn(j)1022 4844 y Ff(\010)31 b Fh(i)1163 4859 y Fn(j)1200 4844 y Fk(.)79 b(Hence,)49 b Fh(i)1671 4859 y Fn(j)1753 4844 y Fk(b)s(elongs)44 b(to)g(exactly)h(one)g(of)f Fh(S)2974 4808 y Fo(1)2968 4869 y Fn(j)3057 4844 y Fk(and)h Fh(S)3325 4808 y Fo(2)3319 4869 y Fn(j)3409 4844 y Fk(and)572 4965 y Fh(i)605 4980 y Fo(1)672 4965 y Ff(2)28 b Fh(S)832 4929 y Fm(0)826 4989 y Fo(1)866 4965 y Fk(,)k Fh(:)17 b(:)g(:)p Fk(,)32 b Fh(i)1132 4980 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1329 4965 y Ff(2)c Fh(S)1489 4929 y Fm(0)1483 4989 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1683 4965 y Fk(for)k(exactly)h(one)g(c)m (hoice)g(of)f Fh(S)2814 4929 y Fm(0)2808 4989 y Fo(1)2848 4965 y Fh(;)17 b(:)g(:)g(:)e(;)i(S)3132 4929 y Fm(0)3126 4989 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)3294 4965 y Fk(.)1922 5214 y(6)p eop %%Page: 7 7 7 6 bop 572 631 a Fk(F)-8 b(or)23 b(eac)m(h)i(other)f(p)s(osition)e(\() p Fh(i)1629 595 y Fm(0)1629 656 y Fo(1)1668 631 y Fh(;)17 b(:)g(:)g(:)f(;)h(i)1920 595 y Fm(0)1920 656 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)2088 631 y Fk(\))24 b(w)m(e)h(ha)m(v)m(e)g Fh(i)2534 595 y Fm(0)2534 656 y Fo(1)2601 631 y Ff(2)j Fh(S)2761 595 y Fm(0)2755 656 y Fo(1)2795 631 y Fk(,)d Fh(:)17 b(:)g(:)p Fk(,)26 b Fh(i)3048 595 y Fm(0)3048 656 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)3244 631 y Ff(2)i Fh(S)3404 595 y Fm(0)3398 656 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)572 751 y Fk(for)k(an)g(ev)m(en)j(n)m(um)m(b)s(er)e(\(p)s(ossibly) e(zero\))i(of)f(com)m(binations)f Fh(S)2861 715 y Fm(0)2855 776 y Fo(1)2895 751 y Fh(;)17 b(:)g(:)g(:)f(;)h(S)3180 715 y Fm(0)3174 776 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)3341 751 y Fk(.)572 913 y(Hence,)41 b(the)f(exclusiv)m(e-or)f(computed)f(b)m (y)i(the)f(user)h(con)m(tains)e(the)i(bit)d(in)h(the)572 1034 y(p)s(osition)31 b(\()p Fh(i)1015 1049 y Fo(1)1054 1034 y Fh(;)17 b(:)g(:)g(:)f(;)h(i)1306 1049 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1474 1034 y Fk(\))33 b(exactly)g(once)h(and)f(an) m(y)g(other)g(bit)f(an)h(ev)m(en)h(n)m(um)m(b)s(er)572 1154 y(of)44 b(times.)80 b(It)45 b(follo)m(ws)e(that)i(this)f(exclusiv) m(e-or)h(is)g(equal)f(to)h(the)g(bit)f(in)g(the)572 1274 y(p)s(osition)31 b(\()p Fh(i)1015 1289 y Fo(1)1054 1274 y Fh(;)17 b(:)g(:)g(:)f(;)h(i)1306 1289 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1474 1274 y Fk(\),)33 b(i.e.)43 b(the)33 b(bit)e(that)i(the)g(user)g(wishes)h(to)e(retriev)m(e.\))474 1478 y Fe(The)j(amount)f(of)h(tr)-5 b(ansmitte)g(d)35 b(bits.)447 1681 y Fk(1.)49 b(Comm)m(unication)30 b(from)h(the)i(user)h
(to)e(the)h(databases.)572 1843 y(T)-8 b(o)41 b(transmit)e(a)i(set)h Fh(S)1445 1807 y Fn(i)1439 1868 y(j)1475 1843 y Fk(,)h(the)f(user)g (needs)g Fh(l)i Fk(=)2430 1810 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)2517 1772 y Ff(p)p 2600 1772 59 4 v 71 x Fh(n)e Fk(bits.)68 b(\(F)-8 b(or)40 b(eac)m(h)i Fh(x)g Ff(2)572 1963 y(f)p Fk(0)p Fh(;)17 b(:)g(:)g(:)e(;)i(l)25 b Ff(\000)e Fk(1)p Ff(g)p Fk(,)34 b(the)g(user)g(m)m(ust)g(sa)m(y)g(whether)h Fh(x)30 b Ff(2)g Fh(S)2613 1927 y Fn(i)2607 1988 y(j)2643 1963 y Fk(.\))47 b(The)34 b(user)h(transmits)572 2084 y(2)p Fh(k)27 b Ff(\000)d Fk(1)35 b(sets)i(\()p Fh(S)1182 2048 y Fo(1)1176 2108 y(1)1221 2084 y Fk(,)f Fh(:)17 b(:)g(:)p Fk(,)36 b Fh(S)1528 2048 y Fo(1)1522 2108 y(2)p Fn(k)r Fm(\000)p Fo(1)1690 2084 y Fk(\))f(to)g(the)h(1)2105 2048 y Fo(st)2200 2084 y Fk(database)g(and)f(2)p Fh(k)27 b Ff(\000)e Fk(1)35 b(sets)h(to)f(the)572 2204 y(2)621 2168 y Fo(nd)736 2204 y Fk(database.)572 2366 y(So,)54 b(the)c(total)e(amoun)m(t)h(of)g(comm)m(unication)e(in)i(this)g (direction)g(is)g(2\(2)p Fh(k)36 b Ff(\000)572 2487 y Fk(1\))686 2453 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)774 2415 y Ff(p)p 857 2415 V 72 x Fh(n)28 b Fk(=)f Fh(O)s Fk(\()1189 2453 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)1276 2415 y Ff(p)p 1359 2415 V 72 x Fh(n)q Fk(\).)447 2690 y(2.)49 b(Comm)m(unication)30 b(from)h(the)i(1)1726 2654 y Fo(st)1818 2690 y Fk(database)g(to)f(the)h (user.)572 2852 y(The)e(1)819 2816 y Fo(st)908 2852 y Fk(database)f(computes)g(the)h(exclusiv)m(e-or)f(of)f(the)h(bits)g(for) f(sev)m(eral)i(com-)572 2972 y(binations)e(of)h Fh(S)1168 2936 y Fm(0)1162 2997 y Fo(1)1201 2972 y Fh(;)17 b(:)g(:)g(:)f(;)h(S) 1486 2936 y Fm(0)1480 2997 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1679 2972 y Fk(and)30 b(sends)i(it)e(to)g(the)h(user.)44 b(The)31 b(amoun)m(t)f(of)g(bits)572 3093 y(transmitted)36 b(b)m(y)h(the)g(1)1468 3056 y Fo(st)1564 3093 y Fk(database)g(is)g (equal)f(to)h(the)g(n)m(um)m(b)s(er)g(of)f(the)h(com)m(bi-)572 3213 y(nations)32 b(of)g Fh(S)1091 3177 y Fm(0)1085 3238 y Fo(1)1124 3213 y Fh(;)17 b(:)g(:)g(:)f(;)h(S)1409 3177 y Fm(0)1403 3238 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)1571 3213 y Fk(,)33 b(i.e.)43 b(\(2)p Fh(k)24 b Ff(\000)f Fk(1\))p Fh(l)h Fk(+)e(1.)572 3375 y Fh(k)32 b Fk(is)d(a)f(constan)m(t) i(and)g Fh(l)f Fk(=)f Ff(d)1638 3342 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)1725 3304 y Ff(p)p 1808 3304 V 71 x Fh(n)q Ff(e)p Fk(.)42 b(Hence,)32 b(the)d(amoun)m(t)g(of)f(comm)m(unication) 572 3495 y(in)k(this)g(direction)f(is)h Fh(O)s Fk(\()1523 3462 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)1611 3424 y Ff(p)p 1694 3424 V 71 x Fh(n)p Fk(\),)h(to)s(o.)447 3699 y(3.)49 b(Comm)m(unication)30 b(from)h(the)i(2)1726 3662 y Fo(nd)1841 3699 y Fk(database)g(to)f(the)h(user.)572 3861 y(Similarly)f(to)k(the)h (previous)g(case,)i(the)d(amoun)m(t)g(of)g(bits)g(transmitted)g(b)m(y)h (the)572 3981 y(2)621 3945 y Fo(nd)736 3981 y Fk(database)c(is)f(equal) g(to)g(the)h(n)m(um)m(b)s(er)g(of)f(com)m(binations)f Fh(S)2911 3945 y Fm(0)2905 4006 y Fo(1)2945 3981 y Fh(;)17 b(:)g(:)g(:)f(;)h(S)3230 3945 y Fm(0)3224 4006 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)3392 3981 y Fk(.)572 4143 y(F)-8 b(or)48 b(the)h(2)996 4107 y Fo(nd)1127 4143 y Fk(database,)k(the)c (amoun)m(t)f(of)g(suc)m(h)j(com)m(binations)c(is)h(at)g(most)572 4263 y(\(2)659 4227 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)849 4263 y Ff(\000)22 b Fk(2)p Fh(k)s Fk(\))p Fh(l)1120 4227 y Fo(2)p Fn(k)r Fm(\000)p Fo(3)1316 4263 y Fk(=)28 b Fh(O)s Fk(\()p Fh(n)1594 4227 y Fo(\(2)p Fn(k)r Fm(\000)p Fo(3\))p Fn(=)p Fo(\(2)p Fn(k)r Fm(\000)p Fo(1\))2070 4263 y Fk(\))33 b(b)s(ecause:)613 4467 y(\(a\))48 b(Those)38
b Fh(i)e Fk(for)g(whic)m(h)h Fh(S)1649 4430 y Fm(0)1643 4491 y Fn(i)1706 4467 y Ff(6)p Fk(=)d Fh(S)1882 4430 y Fo(2)1876 4491 y Fn(i)1958 4467 y Fk(form)h(a)h(subset)i(of)d Ff(f)p Fk(1)p Fh(;)17 b(:)g(:)g(:)f(;)h Fk(2)p Fh(k)27 b Ff(\000)e Fk(1)p Ff(g)36 b Fk(with)786 4587 y(at)k(most)e(2)p Fh(k)30 b Ff(\000)e Fk(3)39 b(elemen)m(ts.)64 b(\(F)-8 b(or)39 b(at)g(least)g(t)m(w)m(o)h Fh(i)g Ff(2)g(f)p Fk(1)p Fh(;)17 b(:)g(:)g(:)e(;)i Fk(2)p Fh(k)30 b Ff(\000)d Fk(1)p Ff(g)p Fk(,)786 4707 y Fh(S)852 4671 y Fo(2)846 4732 y Fn(i)919 4707 y Fk(=)h Fh(S)1089 4671 y Fm(0)1083 4732 y Fn(i)1112 4707 y Fk(.\))786 4848 y(The)34 b(amoun)m(t)e(of)g (suc)m(h)i(subsets)h(is)d(2)2158 4812 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)2348 4848 y Ff(\000)22 b Fk(2)p Fh(k)s Fk(.)1922 5214 y(7)p eop %%Page: 8 8 8 7 bop 608 631 a Fk(\(b\))48 b(If)42 b(w)m(e)h(ha)m(v)m(e)g(c)m(hosen) g Fh(i)f Fk(for)f(whic)m(h)h Fh(S)2188 595 y Fo(2)2182 656 y Fn(i)2270 631 y Ff(6)p Fk(=)h Fh(S)2455 595 y Fm(0)2449 656 y Fn(i)2478 631 y Fk(,)h(it)d(remains)g(to)g(c)m(ho)s(ose)h Fh(t)3510 646 y Fn(i)3539 631 y Fk(.)786 751 y(There)34 b(are)f Fh(l)i Fk(p)s(ossible)c(v)-5 b(alues)33 b(of)f Fh(t)2098 766 y Fn(i)2159 751 y Fk(for)g(eac)m(h)h Fh(i)p Fk(.)786 887 y Fh(t)821 902 y Fn(i)891 887 y Fk(is)40 b(c)m(hosen)j(for)d(at)h(most)f(2)p Fh(k)31 b Ff(\000)d Fk(3)41 b(v)-5 b(alues)41 b(of)f Fh(i)p Fk(.)69 b(Hence,)44 b(there)e(are)f(at)786 1008 y(most)32 b Fh(l)1056 972 y Fo(2)p Fn(k)r Fm(\000)p Fo(3)1257 1008 y Fk(p)s(ossible)g(com)m (binations)f(of)h Fh(t)2364 1023 y Fn(i)2393 1008 y Fk(.)328 1201 y(So,)c(the)f(user)h(transmits)e Fh(O)s Fk(\()1421 1168 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)1508 1130 y Ff(p)p 1591 1130 59 4 v 71 x Fh(n)p Fk(\))h(bits,)h(the)f(1)2138 1165 y Fo(st)2224 1201 y Fk(database)h Fh(O)s Fk(\()2768 1168 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)2856 1130 y Ff(p)p 2939 1130 V 71 x Fh(n)p Fk(\))f(bits)f(and)h(the)328 1322 y(2)377 1285 y Fo(nd)492 1322 y Fk(database)33 b Fh(O)s Fk(\()p Fh(n)1073 1285 y Fo(\(2)p Fn(k)r Fm(\000)p Fo(3\))p Fn(=)p Fo(\(2)p Fn(k)r Fm(\000)p Fo(1\))1549 1322 y Fk(\))g(bits.)474 1442 y(F)-8 b(rom)22 b(the)i(2)929 1406 y Fo(nd)1035 1442 y Fk(database's)h(answ)m(er)g(the)f(user)h (needs)g(a)e(constan)m(t)i(amoun)m(t)e(\(2)3317 1406 y Fo(2)p Fn(k)r Fm(\000)p Fo(1)3488 1442 y Ff(\000)328 1562 y Fk(2)p Fh(k)s Fk(\))30 b(of)g(bits.)42 b(The)31 b(p)s(ositions)e(of)h(these)i(bits)e(in)f(the)i(message)g(from)e(the)i (2)3079 1526 y Fo(nd)3191 1562 y Fk(database)328 1683 y(do)h(not)h(dep)s(end)h(on)e(the)h(con)m(ten)m(ts)h(of)e(the)h (database.)474 1803 y(Hence,)43 b(w)m(e)d(can)f(com)m(bine)g(the)h (describ)s(ed)g(proto)s(col)e(with)h(a)g(\()p Fh(k)29 b Ff(\000)e Fk(1\))39 b(database)328 1923 y(proto)s(col,)31 b(using)h(the)h(metho)s(d)f(describ)s(ed)i(at)e(the)h(b)s(eginning)e (of)h(this)g(section.)474 2044 y Fe(Communic)-5 b(ation)34 b(in)g(the)h Fh(k)j Fe(datab)-5 b(ase)34 b(pr)-5 b(oto)g(c)g(ol.)447 2217 y Fk(1.)49 b(Comm)m(unication)30 b(from)h(the)i(user)h(to)e(the)h (databases.)572 2374 y(The)g(user)h(sends)g(to)e(the)h(databases:)613 2567 y(\(a\))48 b(The)36 b(information)c(from)i(the)i(2)e(database)i (proto)s(col:)47 b Fh(O)s Fk(\()2985 2534 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)3072 2496 y Ff(p)p 3155 2496 V 71 x Fh(n)p Fk(\))35 b(bits)g(to)786 2688 y(eac)m(h)f(database.)608 2839 y(\(b\))48 b(The)37 b(information)32 b(for)i(the)i(sim)m(ulations) d(of)i(the)g(\()p Fh(k)27 b Ff(\000)e Fk(1\))35 b(database)g(pro-)786
2960 y(to)s(col:)47 b Fh(O)s Fk(\()1213 2927 y Fd(2)p Fc(k)q Fb(\000)p Fd(3)1300 2888 y Ff(p)p 1383 2888 86 4 v 72 x Fh(m)p Fk(\))35 b(bits)f(where)i Fh(m)f Fk(is)f(the)h(length)g (of)f(the)h(message)g(from)786 3080 y(the)e(2)1003 3044 y Fo(nd)1118 3080 y Fk(database)g(in)f(the)h(2)f(database)h(proto)s (col.)42 b(W)-8 b(e)33 b(ha)m(v)m(e)1710 3265 y Fh(m)28 b Fk(=)g Fh(O)s Fk(\()p Fh(n)2101 3224 y Fo(\(2)p Fn(k)r Fm(\000)p Fo(3\))p Fn(=)p Fo(\(2)p Fn(k)r Fm(\000)p Fo(1\))2577 3265 y Fk(\))p Fh(:)786 3450 y Fk(Hence,)34 b Fh(O)s Fk(\()1246 3417 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)1334 3378 y Ff(p)p 1417 3378 59 4 v 72 x Fh(n)p Fk(\))e(bits)h(are)f(transmitted) g(for)g(this)g(purp)s(ose.)447 3643 y(2.)49 b(Comm)m(unication)29 b(from)i(the)h(1)1724 3607 y Fo(st)1815 3643 y Fk(database)h(to)e(the)i (user.)44 b(It)32 b(is)f(the)i(same)e(as)i(in)572 3763 y(the)g(2)f(database)h(proto)s(col,)e(i.e.)43 b Fh(O)s Fk(\()1948 3730 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)2036 3692 y Ff(p)p 2119 3692 V 71 x Fh(n)p Fk(\))32 b(bits.)447 3957 y(3.)49 b(Comm)m(unication)30 b(from)h(the)i(2)1726 3921 y Fo(nd)1809 3957 y Fk(,)f Fh(:)17 b(:)g(:)p Fk(,)32 b(the)h Fh(k)2264 3921 y Fo(th)2368 3957 y Fk(database)g(to)f(the)h (user.)572 4114 y(In)h(eac)m(h)h(sim)m(ulation)d(of)h(\()p Fh(k)26 b Ff(\000)e Fk(1\))34 b(database)g(proto)s(col,)f(these)j (databases)f(com-)572 4234 y(m)m(unicate)1071 4419 y Fh(O)s Fk(\()1214 4380 y Fd(2)p Fc(k)q Fb(\000)p Fd(3)1302 4343 y Ff(p)p 1385 4343 86 4 v 76 x Fh(m)p Fk(\))27 b(=)h Fh(O)s Fk(\()1782 4358 y Fd(2)p Fc(k)q Fb(\000)p Fd(3)1869 4325 y Ff(p)p 1952 4325 536 4 v 94 x Fh(n)2010 4390 y Fo(\(2)p Fn(k)r Fm(\000)p Fo(3\))p Fn(=)p Fo(\(2)p Fn(k)r Fm(\000)p Fo(1\))2488 4419 y Fk(\))f(=)h Fh(O)s Fk(\()2800 4380 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)2887 4343 y Ff(p)p 2970 4343 59 4 v 76 x Fh(n)p Fk(\))572 4604 y(bits.)55 b(The)37 b(amoun)m(t)f(of)h(sim)m(ulations)d(p)s(erformed)i(b)m(y)h (the)g(databases)h(is)e(equal)572 4724 y(to)31 b(the)h(amoun)m(t)f(of)f (bits)i(needed)h(b)m(y)f(the)g(user)g(from)e(the)i(2)2785 4688 y Fo(nd)2899 4724 y Fk(database's)g(mes-)572 4844 y(sage,)42 b(i.e.)63 b(constan)m(t.)j(Hence,)42 b(the)e(comm)m (unication)e(b)m(y)i(these)h(databases)f(is)572 4965 y Fh(O)s Fk(\()715 4932 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)802 4893 y Ff(p)p 885 4893 V 72 x Fh(n)p Fk(\),)33 b(to)s(o.)1922 5214 y(8)p eop %%Page: 9 9 9 8 bop 328 631 a Fk(W)-8 b(e)29 b(ha)m(v)m(e)g(constructed)h(a)e (proto)s(col)f(with)h Fh(k)j Fk(databases)e(and)g Fh(O)s Fk(\()2759 598 y Fd(2)p Fc(k)q Fb(\000)p Fd(1)2846 560 y Ff(p)p 2929 560 59 4 v 71 x Fh(n)p Fk(\))g(comm)m(unica-)328 751 y(tion)35 b(from)g(a)h(proto)s(col)e(with)i(\()p Fh(k)28 b Ff(\000)d Fk(1\))35 b(databases)i(and)g Fh(O)s Fk(\()2588 718 y Fd(2)p Fc(k)q Fb(\000)p Fd(3)2676 680 y Ff(p)p 2759 680 V 71 x Fh(n)p Fk(\))f(comm)m(unication.)328 872 y Fa(2)474 992 y Fk(Using)i(the)h(construction)f(of)g(Ostro)m(vsky) j(and)d(Shoup[9])h(and)f(the)h(proto)s(col)e(de-)328 1112 y(scrib)s(ed)28 b(ab)s(o)m(v)m(e,)h(w)m(e)f(can)g(obtain)e(a)h(sc) m(heme)i(in)e(whic)m(h)h(b)s(oth)f(reading)f(and)i(writing)e(are)328 1233 y(priv)-5 b(ate.)43 b(This)33 b(sc)m(heme)g(has)g(\()p Fh(k)25 b Fk(+)d(1\))33 b(databases)g(and)g Fh(O)s Fk(\()p Fh(n)2565 1197 y Fo(1)p Fn(=)p Fo(\(2)p Fn(k)r Fm(\000)p Fo(1\))2874 1233 y Fk(log)17 b Fh(n)p Fk(\))33 b(comm)m(uni-)328 1353 y(cation)f(complexit)m(y)f(for)h(an)m(y)i Fh(k)d Ff(\025)d Fk(2.)328 1686 y Fl(References)377 1905 y Fk([1])48
b(M.)31 b(Abadi,)f(J.)g(F)-8 b(eigen)m(baum)28 b(and)i(J.)g(Kilian,)e Fe(On)k(hiding)f(information)g(fr)-5 b(om)32 b(an)528 2025 y(or)-5 b(acle)p Fk(,)32 b(Journal)g(of)g(Computer)h(and)f(System) i(Sciences,)g(39\(1989\),)d(pp.)i(21-50)377 2229 y([2])48 b(N.)k(Adam)f(and)g(J.)h(W)-8 b(ortmann,)55 b Fe(Se)-5 b(curity)53 b(c)-5 b(ontr)g(ol)51 b(metho)-5 b(ds)52 b(for)g(statistic)-5 b(al)528 2349 y(datab)g(ases:)72 b(a)49 b(c)-5 b(omp)g(ar)g(ative)48 b(study)p Fk(,)k(A)m(CM)d (Somputing)d(Surv)m(eys,)54 b(21\(1989\),)528 2470 y(pp.)33 b(515-555)377 2673 y([3])48 b(A.)32 b(Am)m(bainis,)e Fe(Upp)-5 b(er)33 b(b)-5 b(ounds)34 b(on)f(multip)-5 b(arty)34 b(c)-5 b(ommunic)g(ation)32 b(c)-5 b(omplexity)33 b(of)528 2793 y(shifts)p Fk(,)43 b(Pro)s(ceedings)e(of)g(ST)-8 b(A)m(CS'96,)44 b(Lecture)e(Notes)f(in)f(Computer)h(Science,)528 2914 y(v)m(ol.)33 b(1047\(1996\),)d(pp.)j(631-642)377 3117 y([4])48 b(L.)35 b(Babai,)f(P)-8 b(.)35 b(Kimmel,)e(S.V.)i(Lok)-5 b(am,)34 b Fe(Simultane)-5 b(ous)36 b(messages)g(versus)h(c)-5 b(om-)528 3238 y(munic)g(ation,)p Fk(,)29 b(Pro)s(ceedings)g(of)e(ST)-8 b(A)m(CS'95,)31 b(Lecture)e(Notes)g(in)e(Computer)i(Sci-)528 3358 y(ence,)34 b(v)m(ol.)f(900\(1995\),)e(pp.)i(361-372)377 3561 y([5])48 b(R.)57 b(Bea)m(v)m(er,)64 b(J.)56 b(F)-8 b(eigen)m(baum,)61 b Fe(Hiding)c(instanc)-5 b(es)55 b(in)i(multior)-5 b(acle)56 b(queries)p Fk(,)528 3682 y(Pro)s(ceedings)d(of)f(ST)-8 b(A)m(CS'90,)58 b(Lecture)53 b(Notes)g(in)f(Computer)g(Science,)58 b(v)m(ol.)528 3802 y(415\(1990\),)31 b(pp.)i(37-48)377 4006 y([6])48 b(R.)h(Bea)m(v)m(er,)54 b(J.)48 b(F)-8 b(eigen)m(baum,)52 b(J.)d(Kilian,)g(P)-8 b(.)49 b(Roga)m(w)m(a)m(y)-8 b(,)53 b Fe(Se)-5 b(curity)50 b(with)f(low)528 4126 y(c)-5 b(ommunic)g(ation)34 b(overhe)-5 b(ad)p Fk(,)31 b(Crypto'90)377 4329 y([7])48 b(B.)38 b(Chor,)h(O.)e(Goldreic)m(h,)g(E.)h(Kushilevitz,) g(M.)g(Sudan,)h Fe(Private)g(information)528 4450 y(r)-5 b(etrieval)p Fk(,)34 b(Pro)s(ceedings)f(of)g(F)m(OCS'95,)h(pp.)f (41-50.)f(T)-8 b(o)33 b(app)s(ear)h(in)e(the)i(Journal)528 4570 y(of)f(the)g(A)m(CM.)377 4773 y([8])48 b(D.)33 b(Denning,)e Fe(Crypto)-5 b(gr)g(aphy)35 b(and)g(Data)f(Se)-5 b(curity)p Fk(,)33 b(Addison-W)-8 b(esley)g(,)33 b(1982)1922 5214 y(9)p eop %%Page: 10 10 10 9 bop 377 631 a Fk([9])48 b(R.)54 b(Ostro)m(vsky)-8 b(,)62 b(V.)54 b(Shoup,)60 b Fe(Private)55 b(information)f(stor)-5 b(age)p Fk(,)59 b(to)54 b(app)s(ear)g(in)528 751 y(STOC'97.)328 955 y([10])48 b(P)-8 b(.)39 b(Pudlak,)h(V.)e(R\177)-49 b(odl,)39 b Fe(Mo)-5 b(di\014e)g(d)39 b(r)-5 b(anks)40 b(of)g(tensors)g(and)f(the)h(size)g(of)g(cir)-5 b(cuits)p Fk(,)528 1075 y(Pro)s(ceedings)37 b(of)f(25-th)f(A)m(CM)i(STOC,)g (1993,)f(pp.)g(523-531.)f(Preliminary)e(v)m(er-)528 1196 y(sion)g(of)f([11)o(].)328 1399 y([11])48 b(P)-8 b(.)49 b(Pudlak,)j(V.)c(R\177)-49 b(odl,)51 b(J.)d(Sgall,)i Fe(Bo)-5 b(ole)g(an)48 b(cir)-5 b(cuits,)52 b(r)-5 b(anks)49 b(of)g(tensors)f(and)528 1519 y(c)-5 b(ommunic)g(ation)34 b(c)-5 b(omplexity)p Fk(,)32 b(to)g(app)s(ear)h(in)e(SIAM)j(J.)f (Computing.)328 1723 y([12])48 b(J.)33 b(D.)f(Ullman,)e Fe(Principles)k(of)h(Datab)-5 b(ase)34 b(Systems.)e Fk(1982.)1898 5214 y(10)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF