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b(F)o(or)0 1169 y(e)o(xample,)17 b(suppose)g(node)g Ft(a)e FB(is)g(sending)i(messages)f(to)g(node)h Ft(b)p FB(.)k(The)16 b(ad-)0 1260 y(v)o(ersary)i(sees)g(the)f(same)h (number)h(of)e(messages)i(from)e(node)i Ft(a)e FB(whether)0 1352 y(it)28 b(were)h(con)m(v)o(ersing)h(with)e Ft(b)g FB(or)h(not,)i(and)e(all)f(of)h(the)g(messages)g(are)0 1443 y(sent)18 b(to)f(the)h(same)g(broadcast)h(address.)24 b(Similarly)-5 b(,)16 b(whether)i Ft(a)g FB(talks)f(to)0 1534 y Ft(b)24 b FB(or)g(not,)h Ft(b)f FB(recei)n(v)o(es)g(the)h(same)f (number)h(of)f(messages)h(from)f Ft(a)g FB(o)o(v)o(er)0 1626 y(an)o(y)h(suitably)g(lar)o(ge)g(interv)n(al.)41 b(Note)24 b(that)h(the)g(adv)o(ersary)h(is)e(not)h(able)0 1717 y(to)h Fy(tr)o(ace)f FB(a)h(message)h(from)e(the)h(sender)h(to)e (a)h(recei)n(v)o(er)g(or)f(vice-v)o(ersa)0 1808 y(because)19 b(of)f(the)g(hop-by-hop)j(encryption,)e(and)f(thus,)h(e)n(v)o(en)f(if)g (one)g(end)0 1900 y(of)i(a)g(communication)i(colludes)f(with)e(the)h (adv)o(ersary)-5 b(,)22 b(the)e(anon)o(ymity)0 1991 y(of)f(the)g(other) g(party)g(is)g(not)g(compromised.)83 2085 y(This)f(nai)n(v)o(e)h (solution)f(does)h(not)g(scale)f(due)h(to)f(its)g(broadcast)h(nature.)0 2176 y(As)i(the)g(number)h(of)f(people)h(in)f(the)h(channel)g (increases,)g(the)f(a)o(v)n(ailable)0 2267 y(bandwidth)c(for)g(an)o(y)f (useful)h(communication)h(decreases)f(linearly)-5 b(,)16 b(and)0 2359 y(end-to-end)26 b(reliability)e(decreases)i(e)o (xponentially)-5 b(.)42 b(It)24 b(is)g(possible)h(to)0 2450 y(increase)19 b(the)g(bandwidth)i(utilization)d(and)i(reliability) e(by)i(limiting)e(the)0 2541 y(number)g(of)f(people)h(in)e(a)h (broadcast)h(group,)g(b)o(ut)f(then)g(tw)o(o)g(parties)g(who)0 2633 y(w)o(ant)i(to)g(communicate)h(may)g(end)f(up)h(in)e(dif)n(ferent) h(groups.)83 2727 y FA(P)143 2695 y Fz(5)200 2727 y FB(is)j(based)i (upon)g(this)e(basic)i(broadcast)f(channel)h(principle;)h(we)0 2818 y(scale)d(the)h(system)f(by)h(creating)g(a)f(hierarchy)h(of)f (broadcast)h(channels.)0 2909 y(Clearly)-5 b(,)17 b(an)o(y)g (broadcast-based)h(system,)g(including)f FA(P)1473 2877 y Fz(5)1536 2909 y FB(will)f(not)h(pro-)0 3001 y(vide)k(high)g (bandwidth)h(ef)n(\002cienc)o(y)-5 b(,)21 b(both)g(in)g(terms)f(of)h (ho)n(w)g(man)o(y)g(bits)0 3092 y(it)f(tak)o(es)i(a)f(sender)o (\226recei)n(v)o(er)h(pair)f(to)g(e)o(xchange)i(a)e(bit)g(of)g (information,)0 3183 y(and)h(ho)n(w)g(man)o(y)g(e)o(xtra)g(bits)f(the)g (netw)o(ork)i(carries)e(to)g(carry)h(one)g(bit)f(of)0 3274 y(useful)26 b(information.)45 b FA(P)684 3243 y Fz(5)744 3274 y FB(allo)n(ws)26 b(users)g(to)g(choose)h(ho)n(w)g Fy(inef)o(\002cient)0 3366 y FB(the)21 b(communication)i(is,)f(and)g (pro)o(vides)g(a)f(scalable)h(control)g(structure)0 3457 y(for)c(securely)h(and)g(anon)o(ymously)i(connecting)f(users)e(in)h (dif)n(ferent)f(log-)0 3548 y(ical)h(broadcast)h(groups.)k(W)-6 b(e)18 b(present)i(an)f(o)o(v)o(ervie)n(w)h(of)e FA(P)1550 3517 y Fz(5)1603 3548 y FB(ne)o(xt.)0 3786 y Fv(1.2)99 b(Solution)25 b(o)o(v)o(er)o(view)0 3933 y FA(P)60 3901 y Fz(5)129 3933 y FB(scales)34 b(the)h(nai)n(v)o(e)g(solution)g(by)h (creating)f(a)f(broadcast)i(hierar)o(-)0 4024 y(chy)-5 b(.)53 b(Dif)n(ferent)28 b(le)n(v)o(els)h(of)g(the)f(hierarchy)i(pro)o (vide)f(dif)n(ferent)g(le)n(v)o(els)0 4115 y(of)22 b(anon)o(ymity)h (\(and)f(unlinkability\),)h(at)f(the)g(cost)g(of)f(communication)0 4207 y(bandwidth)26 b(and)g(reliability)-5 b(.)41 b(Users)25 b(of)g(the)g(system)g(locally)g(select)g(a)0 4298 y(le)n(v)o(el)j(of)g (anon)o(ymity)i(and)f(communication)g(ef)n(\002cienc)o(y)g(and)g(can)f (lo-)0 4389 y(cally)g(map)h(themselv)o(es)g(to)f(a)g(le)n(v)o(el)g (which)h(pro)o(vides)g(requisite)g(per)o(-)0 4481 y(formance.)59 b(At)30 b(an)o(y)i(time,)g(it)e(is)h(possible)g(for)g(indi)n(vidual)g (users)g(in)0 4572 y FA(P)60 4540 y Fz(5)117 4572 y FB(to)23 b(decrease)h(anon)o(ymity)g(by)f(choosing)h(a)f(more)g(communication)0 4663 y(ef)n(\002cient)31 b(channel.)60 b(\(Unfortunately)-5 b(,)35 b(it)30 b(is)h(not)g(possible)h(to)f(re)o(gain)0 4755 y(stronger)c(anon)o(ymity\).)47 b(Ob)o(viously)-5 b(,)29 b(it)c(is)h(possible)h(to)f(choose)i(a)e(set)0 4846 y(of)17 b(parameters)h(that)g(is)f(not)h(supported)g(by)g(the)g (system)g(\(e.g.,)f(mutually)0 4937 y(incompatible)j(le)n(v)o(els)f(of) g(bandwidth)h(utilization)f(and)g(anon)o(ymity\).)83 5031 y(Chaum)41 b(introduced)g(sender)g(anon)o(ymity)g(in)f([1],)k(and) d(sender)o(-)0 5122 y(recei)n(v)o(er)j(anon)o(ymity)g(\(as)f(the)g (dining)h(cryptographers)h(problem\))0 5214 y(in)26 b([2].)43 b(The)25 b(solution)i(presented)f(in)g([2])f(pro)o(vides)i(sender)m(,)h (recei)n(v)o(er)0 5305 y(anon)o(ymity)j(and)f(unlinkability)-5 b(.)57 b(Ho)n(we)n(v)o(er)m(,)32 b(it)d(does)i(not)f(pro)o(vide)g(a)0 5396 y(bandwidth)37 b(vs.)73 b(communication)37 b(ef)n(\002cienc)o(y)f (trade-of)n(f,)k(and)c(one)0 5488 y(\223con)m(v)o(ersation\224)21 b(can)f(be)f(only)h(sustained)g(at)f(an)o(y)h(one)f(time.)24 b(The)19 b(clos-)1988 451 y(est)28 b(prior)g(w)o(ork)h(to)e FA(P)2598 420 y Fz(5)2660 451 y FB(is)h([5].)50 b(This)27 b(system)i(also)f(allo)n(ws)g(a)f(trade-)1988 543 y(of)n(f)e(between)g (bandwidth)h(and)f(communication)i(ef)n(\002cienc)o(y)-5 b(,)26 b(b)o(ut)e(lik)o(e)1988 634 y(in)g([2],)g(in)g(this)f(system)h (only)g(one)h(sender)o(-recei)n(v)o(er)f(pair)g(may)g(simul-)1988 725 y(taneously)35 b(communicate)f(in)f(this)g(system.)65 b(Thus,)37 b(these)c(systems)1988 817 y(cannot)25 b(be)f(used)h(to)e (implement)h(lar)o(ge)g(anon)o(ymous)i(communication)1988 908 y(groups.)e(A)18 b(number)h(of)e(recent)i(protocols)f([8)q(,)f(4])h (also)g(pro)o(vide)g(anon)o(y-)1988 999 y(mous)28 b(communication)h(o)o (v)o(er)f(the)f(Internet.)49 b(These)27 b(protocols)h(ha)o(v)o(e)1988 1091 y(the)23 b(same)h(underlying)g(systems)g(assumptions)g(as)f FA(P)3422 1059 y Fz(5)3457 1091 y FB(;)h(ho)n(we)n(v)o(er)m(,)h(un-) 1988 1182 y(lik)o(e)16 b FA(P)2175 1150 y Fz(5)2225 1182 y FB(these)g(protocols)g(cannot)h(withstand)f(an)g(all-po)n(werful)g (passi)n(v)o(e)1988 1273 y(adv)o(ersary)-5 b(.)1988 1525 y Fv(1.3)100 b(Roadmap)1988 1676 y FB(The)17 b(rest)f(of)h(this)f (paper)h(is)f(structured)h(as)g(follo)n(ws:)22 b(we)16 b(discuss)h(related)1988 1767 y(w)o(ork)27 b(in)g(Section)f(2.)46 b(W)-6 b(e)26 b(describe)i(the)e FA(P)3181 1736 y Fz(5)3242 1767 y FB(algorithm)h(in)f(detail)h(in)1988 1859 y(Section)22 b(3,)h(and)g(present)f(a)g(set)g(of)g(analytic)g(bounds)i(on)e (performance)1988 1950 y(in)f(Section)f(5.)27 b(In)20 b(Section)g(6,)h(we)f(analyze)h(results)f(from)g(pack)o(et-le)n(v)o(el) 1988 2041 y FA(P)2048 2010 y Fz(5)2097 2041 y FB(simulator)l(.)i(W)-6 b(e)14 b(discuss)i(future)f(w)o(ork)g(and)h(conclude)g(in)f(Section)g (7.)1988 2335 y FC(2)120 b(Related)30 b(W)-9 b(ork)1988 2513 y FB(W)j(e)16 b(discuss)g(prior)g(w)o(ork)h(related)f(to)g FA(P)3016 2481 y Fz(5)3050 2513 y FB(.)22 b(W)-6 b(e)15 b(be)o(gin)i(with)e(a)h(discussion)1988 2604 y(of)23 b(the)f(Dining)g(Cryptographers)i(problem,)g(and)f(discuss)f(some)h (other)1988 2695 y(systems)d(that)e(pro)o(vide)i(anon)o(ymity)g(o)o(v)o (er)g(the)f(Internet.)1988 2947 y Fv(2.1)100 b(Dining)24 b(Cryptographers)i(and)g(Mixes)1988 3098 y Fs(Dining)21 b(Cryptographers)73 b FB(The)18 b(Dining)i(Cryptographers)g(\(DC-)1988 3189 y(net\))g(protocol)g([2])f(pro)o(vides)i(sender)f(anon)o(ymity)g (under)h(an)e(adv)o(ersary)1988 3281 y(model)24 b(similar)e(to)h FA(P)2569 3249 y Fz(5)2603 3281 y FB(.)35 b(DC-Net)23 b(assumes)g(a)g(public)h(k)o(e)o(y)g(infrastruc-)1988 3372 y(ture,)30 b(and)e(users)g(send)h(encrypted)g(broadcasts)g(to)e (the)h(entire)g(group,)1988 3463 y(thus)33 b(achie)n(ving)h(recei)n(v)o (er)f(anon)o(ymity)-5 b(.)65 b(Ho)n(we)n(v)o(er)m(,)36 b(unlik)o(e)d FA(P)3730 3432 y Fz(5)3765 3463 y FB(,)i(all)1988 3555 y(members)30 b(of)f(the)h(group)g(are)f(made)h(a)o(w)o(are)f(of)g Fy(when)h FB(a)f(message)h(is)1988 3646 y(sent,)22 b(so)f(Dining)g (Cryptographers)h(does)g(not)f(ha)o(v)o(e)g(the)g(same)h(le)n(v)o(el)e (of)1988 3737 y(sender)o(-recei)n(v)o(er)k(anon)o(ymity)-5 b(.)35 b(Also,)23 b(in)f(DC-net,)h(only)g(one)g(user)g(can)1988 3829 y(send)g(at)f(a)g(time,)g(so)g(it)f(tak)o(es)h(additional)h (bandwidth)g(to)f(handle)h(colli-)1988 3920 y(sions)f(and)g(contention) h([10)q(].)30 b(Lastly)-5 b(,)21 b(a)h(DC-net)f(participant)h(\002x)o (es)f(its)1988 4011 y(anon)o(ymity)d(vs.)23 b(bandwidth)17 b(trade)g(of)n(f)f(when)h(joining)g(the)g(system,)g(and)1988 4103 y(there)j(are)f(no)g(pro)o(visions)h(to)f(rescale)g(that)g(trade)g (of)n(f)g(when)h(others)f(join)1988 4194 y(the)g(system.)1988 4421 y Fs(Mixes)75 b FB(A)20 b Fy(mix)g FB(is)f(a)h(process)h(that)e (pro)o(vides)i(anon)o(ymity)g(via)f(pack)o(et)1988 4512 y(re-shuf)n(\003ing.)47 b(Mix)o(es)27 b(were)f(introduced)i(by)f(Chaum) g(in)f([1)q(].)45 b(Mix)o(es)1988 4603 y(w)o(ork)31 b(best)g(in)f (series,)i(and)f(need)g(a)g(constant)g(amount)g(of)f(traf)n(\002c)g(to) 1988 4695 y(a)o(v)o(oid)17 b(delay)h(while)f(preserving)h(anon)o(ymity) -5 b(.)24 b FA(P)3277 4663 y Fz(5)3328 4695 y FB(does)18 b(both)g(by)g(creat-)1988 4786 y(ing)23 b(a)f(hierarchy)g(of)g(mix)o (es,)h(and)g(the)f(constant)h(stream)e(of)i(signal)f(and)1988 4877 y(noise)e(pack)o(ets)g(serv)o(e)f(to)g(k)o(eep)h(the)f(mix)o(es)g (operational.)1988 5129 y Fv(2.2)100 b(Recent)128 b(Inter)o(net-based)h (Anonymous)2213 5245 y(Communications)25 b(W)-7 b(ork)1988 5396 y FB(W)h(e)20 b(describe)h(four)g(anon)o(ymity)h(protocols)f(that) f(can)h(be)g(implemented)1988 5488 y(o)o(v)o(er)f(the)f(Internet)1926 5737 y Fu(2)p eop %%Page: 3 3 3 2 bop 0 451 a Fs(Xor)m(-T)-6 b(r)o(ees)73 b FB(Lik)o(e)67 b FA(P)691 420 y Fz(5)725 451 y FB(,)78 b(Xor)o(-T)m(rees)67 b([5])g(pro)o(vides)g(sender)o(-,)0 543 y(recei)n(v)o(er)o(-,)23 b(and)g(sender)o(-recei)n(v)o(er)g(anon)o(ymity)-5 b(.)35 b(Ho)n(we)n(v)o(er)m(,)23 b(unlik)o(e)g FA(P)1852 511 y Fz(5)1887 543 y FB(,)0 634 y(Xor)o(-T)m(rees)h(do)h(not)f(admit)h(a)f 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(ascertain)g(each)g(other')l(s)0 3962 y(public)j(k)o(e)o(ys)h(using)g (an)f(out-of-band)h(mechanism.)83 4055 y(Assume)f Ft(N)27 b FB(indi)n(viduals)759 4023 y Fp(1)807 4055 y FB(wish)18 b(to)h(form)f(an)h(anon)o(ymous)h(commu-)0 4146 y(nication)25 b(system)f(using)h FA(P)742 4115 y Fz(5)776 4146 y FB(.)39 b(Assume)24 b(each)h(of)f(these)h FA(P)1583 4115 y Fz(5)1641 4146 y FB(users)f(\(or)0 4238 y(group)g(members\))f(ha)o(v)o(e)g (public)h(k)o(e)o(ys)f Ft(K)1111 4246 y Fz(0)1146 4238 y Ft(;)13 b(:)g(:)g(:)26 b(K)1360 4246 y Fo(N)5 b Fx(\000)p Fz(1)1496 4238 y FB(.)35 b FA(P)1610 4206 y Fz(5)1667 4238 y FB(will)22 b(use)0 4329 y(these)d Ft(N)28 b FB(public)19 b(k)o(e)o(ys,)h(called)f Fy(communication)h(k)o(e)n(ys)g FB(to)f(create)g(a)g Fy(lo)o(g-)0 4420 y(ical)g(br)m(oadcast)h(hier)o (ar)m(c)o(hy)p FB(.)0 4653 y Fv(3.1)99 b(The)26 b FF(P)492 4617 y Fn(5)557 4653 y Fv(logical)e(br)n(oadcast)h(hierar)n(ch)o(y)0 4798 y FB(The)g FA(P)201 4766 y Fz(5)261 4798 y FB(logical)h(broadcast) g(hierarchy)g(is)f(a)h(binary)g(tree)f(\()p FA(L)p FB(\))g(which)0 4889 y(is)18 b(constructed)i(using)f(the)g(public)g(k)o(e)o(ys)g Ft(K)1144 4897 y Fz(0)1179 4889 y Ft(;)13 b(:)g(:)g(:)g(;)h(K)1415 4897 y Fo(N)5 b Fx(\000)p Fz(1)1551 4889 y FB(.)22 b(Each)d(node)0 4980 y(of)28 b FA(L)g FB(consists)g(of)g(a)g(bitstring)g(of)g(a)g (speci\002ed)h(length.)51 b(W)-6 b(e)28 b(present)0 5072 y(the)18 b(algorithm)h(assuming)g(each)g(node)g(of)f FA(L)g FB(contains)h(both)g(a)f(bitstring)0 5163 y(and)29 b(a)g(bitmask.)53 b(The)28 b(bitmask)h(speci\002es)g(ho)n(w)g(man)o(y)h (of)e(the)h(most)0 5254 y(signi\002cant)h(bits)g(in)g(the)g(bitstring)g (are)g(v)n(alid.)56 b(Though)32 b(not)e(strictly)p 0 5330 763 4 v 90 5385 a Fp(1)120 5409 y Fw(It)17 b(is)h(entirely)i (possible)f(that)f(the)h Fm(N)24 b Fw(k)o(e)o(ys)18 b(belong)h(to)f Fm(n)f Fw(dif)n(ferent)j(indi)n(vid-)0 5488 y(uals,)d(such)h(that)g Fm(n)h(<)h(N)7 b Fw(.)20 b(W)-5 b(e)16 b(discuss)i(this)g(issue)f(in)g (Section)i(3.5.)1988 439 y FB(necessary)-5 b(,)20 b(the)e(addition)h (of)f(the)g(bitmask)g(will)f(signi\002cantly)i(ease)f(our)1988 530 y(e)o(xposition.)32 b(W)-6 b(e)21 b(use)g(the)h(notation)g(\()p Ft(b)p FB(/)p Ft(m)p FB(\))e(to)h(represent)h(the)g(contents)1988 621 y(of)c(a)f(group,)h(where)f Ft(b)g FB(is)g(the)g(bitstring,)g(and)h Ft(m)f FB(is)g(the)g(number)h(of)f(v)n(alid)1988 712 y(bits.)2071 804 y(The)22 b(root)g(of)g FA(L)f FB(consists)h(of)g(the)g (null)g(bitstring)g(and)g(a)g(zero)g(length)1988 895 y(mask.)j(W)-6 b(e)18 b(represent)i(the)f(root)g(with)g(the)g(label)g Fl(\()p Ft(?=)p Fl(0\))p FB(.)25 b(The)19 b(left)f(child)1988 986 y(of)g(the)g(root)f(contains)i(the)f(group)g Fl(\(0)p Ft(=)p Fl(1\))h FB(and)f(the)g(right)f(child)h(is)f Fl(\(1)p Ft(=)p Fl(1\))p FB(.)1988 1078 y(The)28 b(rest)g(of)g(the)g(tree)f(is)h (constructed)h(as)f(sho)n(wn)g(in)g(Figure)g(1.)50 b(F)o(or)1988 1169 y(e)o(xample,)32 b(note)c(that)g(group)i Fl(\(0)p Ft(=)p Fl(1\))f FB(represents)g(the)f(bitstring)g Fl(0)h FB(and)1988 1260 y(the)19 b(group)h Fl(\(00)p Ft(=)p Fl(2\))g FB(represents)g(the)f(bitstring)g Fl(00)p FB(.)2071 1352 y(Each)25 b(group)h(in)f FA(L)f FB(corresponds)j(to)e(a)f (broadcast)i(channel)g(in)f FA(P)3841 1320 y Fz(5)3875 1352 y FB(.)1988 1443 y(A)c(message)g(sent)g(to)g(a)g(group)g(is)g (\(unreliably\))g(forw)o(arded)h(to)e(a)h(subset)1988 1534 y(of)g(all)e(members)i(of)f(the)h(system.)27 b(Suppose)21 b(user)g Ft(A)e FB(sends)i(a)g(message)1988 1626 y(to)j(group)h Fl(\()p Ft(b=m)p Fl(\))p FB(.)37 b(This)24 b(message)h(will)d(be)i (forw)o(arded)h(to)f(user)g Ft(B)k FB(in)1988 1717 y(group)21 b Fl(\()p Ft(b)2245 1685 y Fx(0)2267 1717 y Ft(=m)2373 1685 y Fx(0)2396 1717 y Fl(\))e FB(if)g(and)i(only)f(if)g(the)f Ft(k)j FB(most)e(signi\002cant)g(bits)g(of)f Ft(b)h FB(and)1988 1808 y Ft(b)2021 1777 y Fx(0)2064 1808 y FB(are)g(the)h(same,)g(where)g Ft(k)h FB(is)e(de\002ned)h(to)g(be)g Fl(min)o FA(f)p Ft(m;)13 b(m)3561 1777 y Fx(0)3583 1808 y FA(g)p FB(.)28 b(W)-6 b(e)20 b(call)1988 1900 y(this)f(common)h(pre\002x)f(testing)g (the)g(\223min-common-pre\002x)i(check\224.)2071 1991 y(Thus,)c(a)e(message)i(sent)e(to)h(a)g(group)g Fl(\()p Ft(b=m)p Fl(\))f FB(is)h(sent)f(to)h(three)g(distinct)1988 2082 y(re)o(gions)k(of)f(the)g FA(L)f FB(tree:)2075 2203 y FA(\017)41 b Fk(Local:)24 b FB(A)19 b(message)h(sent)f(on)g Fl(\()p Ft(b=m)p Fl(\))g FB(is)f(broadcast)i(to)f(all)f(mem-)2154 2294 y(bers)i(of)f(the)f Fl(\()p Ft(b=m)p Fl(\))h FB(group.)2075 2415 y FA(\017)41 b Fk(P)o(ath)29 b(to)h(r)o(oot:)45 b FB(F)o(or)29 b(each)h Ft(m)2991 2383 y Fx(0)3055 2415 y Ft(<)41 b(m)p FB(,)31 b(this)f(message)g(is)f(also)2154 2506 y(broadcast)f(to)f(all)f(members)i(of)f(the)g(group)h Fl(\()p Ft(b)3425 2518 y Fo(m)3479 2504 y Fj(0)3506 2506 y Ft(=m)3612 2475 y Fx(0)3634 2506 y Fl(\))p FB(,)g(where)2154 2598 y Ft(b)2187 2610 y Fo(m)2241 2596 y Fj(0)2287 2598 y FB(denotes)20 b(the)f Ft(m)p FB('-bit)f(pre\002x)h(of)g Ft(b)p FB(.)2075 2718 y FA(\017)41 b Fk(Subtr)o(ee:)21 b FB(Lastly)-5 b(,)17 b(for)f(all)h Ft(m)2930 2687 y Fx(00)2991 2718 y Ft(>)k(m)p FB(,)c(this)f(message)i(is)e(also)h(sent) 2154 2810 y(to)k(all)g(groups)h Fl(\()p Ft(b)p FA(j)d Ft(?)g(=m)2819 2778 y Fx(00)2859 2810 y Fl(\))p FB(,)i(where)g Ft(b)p FA(j)e Ft(?)g(=k)24 b FB(is)c(an)o(y)i(bitstring)e(of)2154 2901 y(length)g Ft(k)h FB(that)e(be)o(gins)g(with)g(the)g(string)g Ft(b)p FB(.)2071 3022 y(F)o(or)e(e)o(xample,)h(an)o(y)f(message)h(sent) f(to)g(the)g(root)g Fl(\()p Ft(?=)p Fl(0\))h FB(is)e(forw)o(arded)1988 3113 y(to)27 b(all)g(members)h(of)f(all)g(groups.)49 b(An)o(y)27 b(message)h(sent)g(to)f Fl(\(0101)p Ft(=)p Fl(4\))1988 3204 y FB(is)d(sent)g(to)f(members)h(of)g(the)g Fl(0101)h FB(group)g(\(local\).)37 b(This)23 b(message)i(is)1988 3296 y(also)e(sent)g(to)f(all)g(members)h(of)g(the)f Fl(\(010)p Ft(=)p Fl(3\))i FB(group;)h(all)d(members)h(of)1988 3387 y(the)e Fl(\(01)p Ft(=)p Fl(2\))g FB(and)g Fl(\(0)p Ft(=)p Fl(1\))g FB(groups,)g(and)g(the)f(all)g(members)g(of)h(the)f Fl(\()p Ft(?=)p Fl(0\))1988 3478 y FB(group.)58 b(Further)m(,)32 b(this)d(message)i(is)f(sent)g(to)f(e)n(v)o(ery)i(member)g(of)e(the) 1988 3570 y Fl(\(0101)23 b Ft(?)f(=k)r Fl(\))j FB(groups,)h(where)f Ft(k)34 b(>)e Fl(4)p FB(,)26 b(and)f Fl(0101)p Ft(?)h FB(is)f(an)o(y)g(bitstring)1988 3661 y(that)30 b(starts)f(with)g Fl(0101)p FB(.)57 b(In)30 b(general,)j(when)d(a)g(message)g(is)g(sent)g (to)1988 3752 y(group)g Fl(\()p Ft(b=m)p Fl(\))p FB(,)f(members)g(of)f (groups)h Fl(\()p Ft(b)3124 3721 y Fx(0)3146 3752 y Ft(=m)3252 3721 y Fx(0)3275 3752 y Fl(\))f FB(are)g(forw)o(arded)h(this)1988 3844 y(message)c(if)e(and)h(only)g(if)f(the)g(nodes)i(of)e FA(L)g FB(corresponding)j(to)d Fl(\()p Ft(b=m)p Fl(\))1988 3935 y FB(and)33 b Fl(\()p Ft(b)2191 3903 y Fx(0)2214 3935 y Ft(=m)2320 3903 y Fx(0)2342 3935 y Fl(\))f FB(ha)o(v)o(e)g(an)h (ancestor)o(-descendant)i(relationship.)63 b(Note)1988 4026 y(that)25 b(these)f(broadcast)i(groups)f(should)g(be)g (implemented)g(as)g(peer)o(-to-)1988 4118 y(peer)f(unicast)f(trees)g (in)g(the)g(underlying)h(netw)o(ork)g(\(and)g(not)f(multicast)1988 4209 y(trees\).)41 b(These)25 b(channels)g(may)h(lose)f(messages)g(and) h(require)f(no)g(par)o(-)1988 4300 y(ticular)e(consistenc)o(y)-5 b(,)26 b(reliability)-5 b(,)23 b(or)g(quality-of-service)i(guarantees.) 1988 4392 y(W)-6 b(e)26 b(describe)h(the)f(precise)h(netw)o(orking)g (and)g(systems)g(requirements)1988 4483 y(of)19 b FA(P)2129 4451 y Fz(5)2182 4483 y FB(and)h(underlying)g(protocols)g(in)f(Section) g(4.3.)2071 4574 y(Each)i(user)g(in)f(the)h(system)f(joins)h(a)f(set)g (of)h(such)g(broadcast)h(groups.)1988 4666 y(In)i(general,)i (communication)g(ef)n(\002cienc)o(y)e(increases)g(as)g(the)h(groups')l (s)1988 4757 y(mask)h(size)f(increases;)j(ho)n(we)n(v)o(er)m(,)f(as)e (we)g(shall)g(see,)h(this)f(increase)h(in)1988 4848 y(ef)n(\002cienc)o (y)16 b(comes)h(at)e(an)h(e)o(xpense)h(of)f(reduced)h(anon)o(ymity)-5 b(.)23 b(The)16 b(depth)1988 4940 y(of)g(the)f FA(L)g FB(tree)g(is)g(de\002ned)h(at)f(run-time)h(and)g(depends)h(on)f(the)f (number)h(of)1988 5031 y(people)i(in)f(the)g(system)g(\()p Ft(N)8 b FB(\),)16 b(and)i(on)f(the)g(security)g(parameters)g(chosen) 1988 5122 y(by)k(indi)n(vidual)g(users.)27 b(Ho)n(we)n(v)o(er)m(,)20 b(we)g(need)h(to)f(\002x)g(a)g(maximum)h(depth)1988 5214 y FA(L)2041 5223 y Fo(d)2095 5214 y FB(of)c(this)h(tree:)k(choosing)d (this)f(parameter)m(,)g(a-priori,)f(is)g(not)h(dif)n(\002cult,)1988 5305 y(since)28 b(such)h(a)f(system)g(can)g(accommodate)h (approximately)h Fl(2)3701 5273 y Fx(L)3744 5285 y Fi(d)3806 5305 y FA(\001)25 b Ft(k)1988 5396 y FB(users,)d(where)g Ft(k)h FB(is)e(the)g(least)g(number)i(of)e(people)h(in)f(an)o(y)h (channel.)31 b(In)1988 5488 y(our)20 b(implementation,)f(we)g(ha)o(v)o (e)g(chosen)h FA(L)3155 5497 y Fo(d)3210 5488 y FB(to)f(be)g(32.)1926 5737 y Fu(3)p eop %%Page: 4 4 4 3 bop 1894 457 a Fh(?)p Fs(/0)1812 565 y Fg(\036\035)1812 300 y(\037\034)1064 789 y Fs(0/1)982 897 y Fg(\036\035)982 632 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FB(,)f(it)g(can)g(determine)h Ft(b)997 3934 y Fo(B)1050 3926 y FB(;)g(ho)n(we)n(v)o(er)m(,)h Ft(A)e FB(does)h(not)f(kno)n(w)0 4017 y(the)k(v)n(alue)h(of)g Ft(m)460 4025 y Fo(B)512 4017 y FB(.)45 b(Ev)o(en)27 b(without)f(an)o(y)h(kno)n(wledge)h(of)e(the)h(masks,)0 4108 y Ft(A)g FB(and)g Ft(B)k FB(can)d(be)o(gin)g(to)f(communicate)h (using)g(the)f Fl(\()p Ft(?=)p Fl(0\))h FB(channel.)0 4200 y(Unfortunately)-5 b(,)30 b(this)d(communication)h(channel)h(can)e (be)g(quite)h(lossy)0 4291 y(since)e(messages)g(ha)o(v)o(e)g(a)f (higher)h(probability)g(of)g(getting)g(lost)f(in)g(the)0 4382 y(channels)f(\223higher\224)g(up)g(in)f(the)g FA(L)g FB(tree)g(and)h Fl(\()p Ft(?=)p Fl(0\))g FB(is)e(the)i(most)f(lossy)0 4474 y(communication)e(channel)f(of)f(all.)83 4574 y(The)27 b(communication)i(ef)n(\002cienc)o(y)e(can)h(be)f(impro)o(v)o(ed)h(as)f (follo)n(ws.)0 4666 y(Instead)19 b(of)g(sending)h(messages)g(through)g Fl(\()p Ft(?=)p Fl(0\))p FB(,)f Ft(A)f FB(could)h(try)g(to)f(send)0 4757 y(messages)k(on)f(to)g(some)g(group)h Fl(\()p Ft(b)913 4765 y Fo(B)965 4757 y Ft(=m)p Fl(\))p FB(,)f Ft(m)j(>)h Fl(0)p FB(.)k Ft(B)24 b FB(w)o(ould)e(recei)n(v)o(e)0 4848 y(these)k(messages,)i(and)f Fy(may)f FB(reply)g(back)g(to)g Ft(A)p FB(.)43 b(Ho)n(we)n(v)o(er)m(,)28 b(in)d(doing)0 4940 y(so,)g Ft(B)i FB(sacri\002ces)c(some)h(anon)o(ymity)h(since)f Ft(A)f FB(can)h(no)n(w)g(map)g Ft(B)j FB(to)c(a)0 5031 y(smaller)g(set)h(of)g(users.)37 b(\()p Ft(A)23 b FB(maps)h Ft(B)k FB(to)23 b(a)h(smaller)f(set)h(using)g(a)g(\223Dif-)0 5122 y(ference)29 b(Attack\224)g(described)h(in)e(Section)g(3.5\).)53 b(In)28 b(general,)j Ft(B)i FB(can)0 5214 y(choose)21 b(to)e(communicate)j(back)e(to)g Ft(A)f FB(using)h(an)o(y)g(length)g (mask;)h(ho)n(w-)0 5305 y(e)n(v)o(er)m(,)e(longer)h(masks)g(trade-of)n (f)g(anon)o(ymity)h(for)e(communication)i(ef)n(\002-)0 5396 y(cienc)o(y)-5 b(.)39 b Ft(B)27 b FB(can)d(selecti)n(v)o(ely)h (\223trust\224)e(some)i(users)f(and)g(re)n(v)o(eal)g(longer)0 5488 y(masks)17 b(for)f(these)h(trusted)f(users,)h(b)o(ut)f(in)g (general,)h(the)g(anon)o(ymity)g(of)g Ft(B)1988 2279 y FB(is)j(bounded)i(by)e(the)h(longest)f(mask)h(that)e(it)h(has)g(re)n (v)o(ealed)h(to)f(an)o(y)g(other)1988 2370 y(member)l(.)j(Lastly)-5 b(,)17 b(note)f(that)h(the)f(communication)j(ef)n(\002cienc)o(y)d(is)h (upper)1988 2462 y(bounded)26 b(by)d(the)h(smaller)f(of)g(the)g(tw)o(o) h(masks)g(re)n(v)o(ealed)g(by)f(either)g(of)1988 2553 y(the)c(participants)h(in)e(a)h(communication.)2071 2648 y(Suppose)31 b Ft(A)e FB(and)h Ft(B)k FB(ha)o(v)o(e)29 b(agreed)i(upon)g(the)f(length)g(of)f(a)h(mask)1988 2740 y(\(say)24 b Ft(m)p FB(\),)f(and)h(wish)f(to)f(communicate.)37 b(This)23 b(means)h Ft(A)e FB(is)h(joined)h(to)1988 2831 y(some)17 b(group)f Fl(\()p Ft(b)2414 2839 y Fo(A)2464 2831 y Ft(=m)2570 2839 y Fo(A)2621 2831 y Fl(\))f FB(and)h Ft(B)k FB(is)15 b(joined)h(to)g(some)g(group)h Fl(\()p Ft(b)3634 2839 y Fo(B)3686 2831 y Ft(=m)3792 2839 y Fo(B)3845 2831 y Fl(\))p FB(,)1988 2922 y(where)24 b(both)h Ft(m)2419 2930 y Fo(A)2492 2922 y FB(and)f Ft(m)2691 2930 y Fo(B)2767 2922 y FB(are)g(greater)g(than)g(or)g(equal)g(to)f Ft(m)p FB(.)37 b(The)o(y)1988 3014 y(can)16 b(no)n(w)g(communicate)h(by)f (sending)h(messages)f(to)f(a)h(lo)n(west)f(common)1988 3105 y(ancestor)25 b(channel)g(that)e(has)h(both)g Fl(\()p Ft(b)2994 3113 y Fo(A)3044 3105 y Ft(=m)3150 3113 y Fo(A)3201 3105 y Fl(\))f FB(and)h Fl(\()p Ft(b)3448 3113 y Fo(B)3501 3105 y Ft(=m)3607 3113 y Fo(B)3659 3105 y Fl(\))g FB(as)f(de-)1988 3196 y(scendants.)32 b(Ho)n(we)n(v)o(er)m(,)23 b(if)e Ft(A)f FB(and)i Ft(B)j FB(only)d(join)g(one)g(group)g(each,)h(and)1988 3288 y(the)e(public)g(k)o(e)o(ys)h(are)f(uniformly)g(hashed)h(to)f (channels)h(on)f(the)g FA(L)f FB(tree,)1988 3379 y(then)k(there)f(is)f (approximately)i Fl(50\045)g FB(probability)f(that)g Fl(\()p Ft(?=)p Fl(0\))g FB(will)f(be)1988 3470 y(the)i(only)h(channel) g(that)f Ft(A)f FB(and)h Ft(B)k FB(w)o(ould)c(ha)o(v)o(e)g(in)g (common!)40 b(Simi-)1988 3562 y(larly)-5 b(,)17 b(for)f(an)o(y)g(gi)n (v)o(en)h(node,)h(an)e(e)o(xponentially)h(high)g(number)g(of)f(nodes) 1988 3653 y(w)o(ould)29 b(be)f(\223f)o(arther\224)g(a)o(w)o(ay)g(on)g (the)g(logical)f(broadcast)i(tree,)g(and)g(in)1988 3744 y(general,)17 b(the)e(communication)i(on)f(the)f(system)g(will)g(not)g (be)h(v)o(ery)f(ef)n(fec-)1988 3836 y(ti)n(v)o(e.)32 b(Once)22 b(again,)g(we)g(are)g(reduced)h(to)e(using)i(the)e(inef)n (\002cient)h(global)1988 3927 y(broadcast)e(channel)g(for)f(most)g (communications.)2071 4022 y(Our)e(solution)f(to)h(this)e(inef)n (\002cient)i(routing)f(is)g(as)g(follo)n(ws:)22 b(each)17 b(user)1988 4114 y(joins)29 b(a)f(small)g(number)h(of)f(groups)i(on)f (the)f(logical)g(tree.)51 b(F)o(or)28 b(each)1988 4205 y(joined)35 b(group,)k(users)34 b(generate)h(another)g(public-pri)n(v)n (ate)h(k)o(e)o(y)f(pair)m(,)1988 4296 y(called)18 b Fy(r)m(outing)f(k)o (e)n(y)p FB(.)23 b(These)18 b(routing)g(k)o(e)o(ys)g(are)f(generated)h (locally)-5 b(,)18 b(and)1988 4388 y(do)k(not)f(require)g(an)o(y)g (global)g(coordination.)31 b(In)20 b(f)o(act,)h(it)f(should)i(not)f(be) 1988 4479 y(possible)d(to)e(map)h(a)g(user')l(s)g(routing)g(k)o(e)o(y)h (to)e(their)h(communication)h(k)o(e)o(y)-5 b(,)1988 4570 y(otherwise,)23 b(the)f(user')l(s)g(anon)o(ymity)h(can)f(be)g (compromised)i(\(using)e(an)1988 4662 y(\223Intersection)e(Attack\224,) f(Section)g(3.5\).)2071 4757 y(When)h(user)f Ft(A)f FB(joins)h(a)g (group)h Ft(c)p FB(,)f(it)f(periodically)i(sends)g(a)f(message)1988 4848 y(to)28 b(the)f(channel)i(listing)e(other)h(channels)h(that)e(it)g (is)g(joined)h(to.)49 b(This)1988 4940 y(message)24 b(serv)o(es)g(as)f (a)g(routing)g(adv)o(ertisement,)i(and)f(will)e(be)h(used)h(to)1988 5031 y(ef)n(\002ciently)i(send)g(messages)g(along)g(the)g(lo)n(wer)f (le)n(v)o(els)g(of)h(the)f FA(L)g FB(tree.)1988 5122 y(In)h(general,)i(the)e(adv)o(ertisements)g(from)g(a)g(node)h(contains) f(the)g(set)f(of)1988 5214 y(channels)j(it)d(can)h(directly)g(reach,)i (the)e(set)g(of)g(channels)h(it)f(can)g(reach)1988 5305 y(using)d(one)f(other)g(node,)h(and)g(so)e(on.)32 b(In)22 b(ef)n(fect,)g(these)g Ft(t)f FB(routing)i(k)o(e)o(ys)1988 5396 y(generate)g(\223lateral\224)f(edges)h(in)f(the)g(tree.)32 b(In)22 b(Section)g(5,)h(we)f(sho)n(w)h(that)1988 5488 y(typically)-5 b(,)27 b(each)f(user)g(needs)g(to)f(join)g(only)h(a)f (fe)n(w)g(groups)h(\()p FA(\024)33 b Fl(3)p FB(\))26 b(for)1926 5737 y Fu(4)p eop %%Page: 5 5 5 4 bop 0 451 a FB(an)o(y)19 b(tw)o(o)f(users)h(in)f FA(P)562 420 y Fz(5)614 451 y FB(to)g(ha)o(v)o(e)h(short)f(paths)h(\()p FA(\024)i Fl(2)d FB(channel)h(crossings\))0 543 y(between)h(them)f (with)f(high)i(probability)-5 b(.)83 641 y(W)f(e)28 b(note)h(that)f(a)h (user)f(joins)h(a)f(set)h(of)f(groups)i(only)f(when)g(it)f(en-)0 732 y(ters)23 b(the)g(system,)h(and)g(should)g(not)g(change)g(the)g (set)e(of)i(channels)g(it)f(is)0 823 y(part)i(of.)42 b(Otherwise,)26 b(once)g(again,)g(yet)f(another)h(intersection)g (attack)0 915 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2059 y(same)h(size)f(and)h(is)f(encrypted)i Fy(per)o(-hop)p FB(,)g(and)f(thus)f(it)g(is)g(not)h(possible)g(to)0 2151 y(map)23 b(an)g(outgoing)i(message)e(\(pack)o(et\))h(to)f(a)g (speci\002c)f(pack)o(et)i(that)f(the)0 2242 y(node)f(recei)n(v)o(ed)g (in)e(the)h(past.)29 b(Ho)n(we)n(v)o(er)m(,)22 b(a)f(passi)n(v)o(e)g (observ)o(er)h(can)f(still)0 2333 y(mount)26 b(an)f(easy)h(statistical) e(attack)i(and)g(trace)f(a)g(communication)i(by)0 2425 y Fy(corr)m(elating)e FB(a)g(pack)o(et)g(stream)f(from)g(a)g (communicating)i(source)f(to)f(a)0 2516 y(sink.)83 2614 y(Thus,)k(we)e(add)h(the)f(notion)h(of)g(noise)f(to)h(the)f(system.)45 b(The)27 b(noise)0 2705 y(pack)o(ets)38 b(should)g(be)g(added)g(such)g (that)f(a)g(passi)n(v)o(e)h(correlation)g(at-)0 2797 y(tack)24 b(becomes)h(infeasible.)38 b(There)24 b(are)g(man)o(y)g (possible)h(good)g(noise-)0 2888 y(generation)18 b(algorithms,)g(and)f (we)g(use)h(the)f(follo)n(wing)g(simple)h(scheme.)83 2986 y(Each)28 b FA(P)320 2954 y Fz(5)383 2986 y FB(user)m(,)i(at)e (all)f(times,)j(generates)f(\002x)o(ed)g(amount)g(of)f(traf-)0 3078 y(\002c)c(destined)i(to)e(channel)i(chosen)g(uniformly)f(at)f (random.)42 b(A)24 b(pack)o(et)0 3169 y(transmitted)19 b(from)g(a)g(node)h(is)e(one)i(of)f(the)g(follo)n(wing:)86 3314 y FA(\017)42 b FB(A)29 b(pack)o(et)i(\(noise)e(or)h(signal\))f (that)g(w)o(as)h(recei)n(v)o(ed)g(from)g(some)166 3405 y(incoming)21 b(interf)o(ace)g(that)f(this)g(node)h(is)f(forw)o(arding) h(onto)g(some)166 3496 y(other)26 b(channel\(s\).)46 b(\(The)25 b(precise)i(forw)o(arding)g(rule)f(for)f FA(P)1795 3465 y Fz(5)1856 3496 y FB(is)166 3588 y(described)20 b(in)f(Section)g(4\).)86 3726 y FA(\017)42 b FB(A)19 b(signal)g(pack)o(et)h(that)f(has)g(been)h(locally)f(generated.)86 3864 y FA(\017)42 b FB(A)19 b(noise)g(pack)o(et)h(that)f(has)g(been)h (locally)f(generated.)83 4009 y(Note)j(that)f(to)h(an)g(e)o(xternal)g (observ)o(er)m(,)h(there)f(is)f(no)h(discernible)g(dif-)0 4100 y(ference)35 b(between)g(these)g(three)f(scenarios.)70 b(In)35 b(general,)j(only)d(the)0 4192 y(source)24 b(and)h(destination) f(of)g(a)f(communication)i(can)g(distinguish)f(be-)0 4283 y(tween)19 b(noise)g(and)g(signal)f(pack)o(ets.)24 b(The)o(y)19 b(are)g(treated)f(with)g(equal)h(dis-)0 4374 y(dain)g(at)g(all)f(other)i(nodes.)0 4609 y Fs(Message)41 b(Dr)o(opping)f(Algorithms)74 b FB(In)37 b(an)o(y)g(communication)0 4701 y(system)25 b(without)f(e)o(xplicit)g(feedback,)i(e.g.)39 b(our)25 b(channel)g(broadcasts,)0 4792 y(message)e(queues)g(may)g(b)o (uild)f(at)g(slo)n(w)g(nodes)h(or)f(at)g(nodes)h(with)f(high)0 4883 y(de)o(gree.)53 b(In)29 b FA(P)420 4851 y Fz(5)454 4883 y FB(,)i(members)e(may)h(simply)e(drop)i(an)o(y)f(message)h(the)o (y)0 4975 y(do)i(not)g(ha)o(v)o(e)g(the)g(bandwidth)h(or)e(processing)i (capacity)g(to)e(handle.)0 5066 y(The)19 b(global)g(properties)h(of)f (the)g(system)g(depend)i(upon)f(ho)n(w)f(messages)0 5157 y(are)24 b(dropped.)38 b(W)-6 b(e)23 b(ha)o(v)o(e)h(considered)h(tw)o (o)f(dif)n(ferent)g(dropping)h(algo-)0 5248 y(rithms:)p 0 5330 763 4 v 90 5385 a Fp(2)120 5409 y Fw(Clearly)l(,)g(these)f(are)f (one)g(hop)g(\223transport)i(le)n(v)o(el\224)g(peerings,)g(and)e(not)g (one)0 5488 y(\223physical)c(hop\224)f(peerings.)2075 439 y FA(\017)41 b FB(Uniform)21 b(drop:)28 b(This)20 b(is)h(the)f(simplest)h(scheme)h(in)e(which)h(mes-)2154 530 y(sages)k(from)g(the)f(input)h(queue)g(are)f(dropped)i(with)e (equal)h(prob-)2154 621 y(ability)j(until)f(the)h(input)g(queue)h(size) f(is)f(belo)n(w)h(the)g(maximum)2154 712 y(threshold.)2075 832 y FA(\017)41 b FB(Non-uniform)30 b(drop:)44 b(In)29 b(this)g(scheme,)j(messages)e(which)g(are)2154 923 y(destined)19 b(to)g(a)f(channel)h(higher)g(up)g(in)f FA(L)g FB(are)g(dropped)i (preferen-)2154 1014 y(tially)-5 b(.)2154 1120 y(W)f(e)42 b(ha)o(v)o(e)g(e)o(xperimented)i(with)d(se)n(v)o(eral)i(v)n(ariations)g (of)f(this)2154 1211 y(scheme;)35 b(the)29 b(speci\002c)f(scheme)i (which)f(we)g(use)g(for)g(our)g(sim-)2154 1302 y(ulations)e(drops)g (pack)o(ets)h(destined)f(for)f(higher)h(nodes)g(with)f(an)2154 1393 y(e)o(xponentially)21 b(higher)e(probability)-5 b(.)2154 1499 y(If)25 b(most)h(of)f(the)g(end-to-end)i(paths)f(in)f(a)h FA(P)3337 1467 y Fz(5)3396 1499 y FB(netw)o(ork)g(can)g(use)2154 1590 y(\223lateral\224)d(edges,)h(i.e.)34 b(between)24 b(channels)g(at)e(the)h(same)g(logical)2154 1681 y(height,)k(then)e (this)f(scheme)i(pro)o(vides)f(lo)n(wer)g(drop)g(rate.)40 b(Ho)n(w-)2154 1773 y(e)n(v)o(er)25 b(an)o(y)f(communication)i(that)e (must)g(use)g(\223higher\224)h(channels)2154 1864 y(ha)o(v)o(e)20 b(proportionately)g(high)f(drops.)1988 2087 y Fv(3.4)100 b(Anonymity)24 b(Analysis)1988 2229 y FB(Assume)18 b(node)h Ft(a)e FB(has)h(joined)g(the)g FA(L)f FB(at)g(node)i Fl(\()p Ft(b=m)p Fl(\))p FB(.)j(Let)16 b Ft(E)t FA(f)p Fl(\()p Ft(b=m)p Fl(\))p FA(g)1988 2320 y FB(be)22 b(the)f(set)g(of)h (members)g(who)f(recei)n(v)o(e)h(a)g(broadcast)g(message)g(sent)g(to) 1988 2412 y(channel)30 b Fl(\()p Ft(b=m)p Fl(\))p FB(.)51 b(\(Recall)28 b(that)g(this)g(set)g(includes)i(all)d(members)i(of)1988 2503 y(group)24 b Fl(\()p Ft(b=m)p Fl(\))p FB(,)f(all)f(members)i(of)f (the)f(groups)i(belo)n(w)g Fl(\()p Ft(b=m)p Fl(\))p FB(,)f(and)g(all) 1988 2594 y(members)d(of)f(groups)h(on)f(the)g(path)h(to)f(the)f(root)i (of)e FA(L)p FB(\).)1988 2755 y Fk(Claim)h(3.1)42 b Fy(The)15 b(anonymity)g(of)g(a)g(node)g(communicating)h(using)g(c)o(han-)1988 2847 y(nel)25 b Fl(\()p Ft(b=m)p Fl(\))f Fy(is)g(equivalent)h(to)g(the) f(set)g(of)h(member)o(s)g(who)g(ar)m(e)g(part)g(of)1988 2938 y Ft(E)t FA(f)p Fl(\()p Ft(b=m)p Fl(\))p FA(g)p Fy(.)1988 3121 y Fk(Pr)o(oof)o(.)2071 3212 y FB(W)-6 b(e)48 b(consider)g(the)g(sender)o(-,)55 b(recei)n(v)o(er)o(-,)f(and)49 b(sender)o(-recei)n(v)o(er)1988 3304 y(anon)o(ymity)21 b(cases)e(separately)-5 b(.)2075 3437 y FA(\017)41 b FB(Sender)20 b(Anon)o(ymity)2154 3542 y(Sender)26 b(anon)o(ymity)h(is)e (the)h(size)g(of)f(the)h(set)g(of)f(the)h(nodes)h(that)2154 3633 y Fy(could)33 b FB(ha)o(v)o(e)d(sent)g(a)g(particular)f(pack)o(et) i(to)f(a)g(gi)n(v)o(en)g(host)g(\(say)2154 3725 y Ft(B)t FB(\).)c(A)19 b(recei)n(v)o(er)i(who)f(can)g(only)h(monitor)f(their)g (o)n(wn)g(links)g(can-)2154 3816 y(not)e(determine)g(the)g(source)g(of) f(a)h(pack)o(et)g(since)g(this)f(information)2154 3907 y(is)27 b(ne)n(v)o(er)h(included)h(in)e(the)h(pack)o(et.)49 b(A)27 b(recei)n(v)o(er)m(,)i(in)f(collusion)2154 3999 y(with)e(a)f(all-po)n(werful)h(passi)n(v)o(e)g(adv)o(ersary)-5 b(,)29 b(ho)n(we)n(v)o(er)m(,)f(can)e(enu-)2154 4090 y(merate)j(the)g(set)f(of)h(nodes)h(who)f Fy(could)i FB(ha)o(v)o(e)e(sent)g(the)f(pack)o(et)2154 4181 y(by)d(computing)h (the)e(closure)h(of)f(the)h(set)f(of)g(nodes)h(who)g(ha)o(v)o(e)g(a) 2154 4273 y(causal)e(relationship)g(with)f Ft(B)t FB(.)32 b(\(In)22 b(this)g(case,)h(nodes)g Ft(X)28 b FB(and)23 b Ft(Y)2154 4364 y FB(are)e(causally)f(related)g(if)g Ft(X)26 b FB(sent)20 b(a)g(pack)o(et)h(to)f Ft(Y)d FB(\).)26 b(This)20 b(closure)2154 4455 y(w)o(ould)25 b(be)e(computed)i(o)o(v)o (er)f(some)g(\002nite)f(time)g(windo)n(w)h(on)f(the)2154 4546 y(order)d(of)f(the)g(end-to-end)h(latenc)o(y)f(in)g(the)g(system.) 2154 4652 y(Ho)n(we)n(v)o(er)m(,)28 b(in)e FA(P)2610 4620 y Fz(5)2645 4652 y FB(,)h(a)e(user)i(connected)g(to)f Fl(\()p Ft(b=m)p Fl(\))f FB(sends)i(pack-)2154 4743 y(ets)32 b(at)f(a)g(constant)h(rate)f(\(signal)g(or)h(noise\),)i(and)e(these)g (pack-)2154 4834 y(ets)d(are)f(recei)n(v)o(ed)i(by)f(all)f(users)h(in)f Ft(E)t FA(f)p Fl(\()p Ft(b=m)p Fl(\))p FA(g)p FB(.)52 b(Thus,)31 b(there)2154 4926 y(is)23 b(a)f(causal)i(relationship)f (between)h(e)n(v)o(ery)f(user)g(in)g(a)f(broadcast)2154 5017 y(group.)h(Ho)n(we)n(v)o(er)m(,)17 b(inter)o(-channel)f(routers)f (transmit)g(pack)o(ets)h(be-)2154 5108 y(tween)k(channels,)h(and)f(o)o (v)o(er)g(time)g(e)n(v)o(ery)g(node)h(in)e(the)h(system)g(is)2154 5200 y(causally)g(related)f(to)g(e)n(v)o(ery)g(other)h(node)g(in)e(the) h(system.)2154 5305 y(Suppose)38 b(a)e(malicious)h(recei)n(v)o(er)f (tries)g(to)g(e)o(xpose)i(a)e(sender)l(.)2154 5396 y(It)30 b(can,)j(at)d(best)g(\(assuming)h(there)f(are)g(no)h(other)f(cross)h (chan-)2154 5488 y(nel)26 b(pack)o(ets\),)j(causally)d(relate)g(pack)o (ets)h(to)f(its)f Fy(own)h FB(broadcast)1926 5737 y Fu(5)p eop %%Page: 6 6 6 5 bop 166 439 a FB(group.)57 b(Further)m(,)33 b(if)c(the)h(recei)n(v) o(er)g(is)g(able)g(to)g(determine)g(and)166 530 y(compromise)g(the)f (router)g(node,)j(then)d(the)g(sender)h(anon)o(ymity)166 621 y(becomes)20 b(the)f(ef)n(fecti)n(v)o(e)g(broadcast)h(group)g(of)f (the)g(sender)l(.)166 729 y(In)g(case)g(the)g(recei)n(v)o(er)g(cannot)h (compromise)g(the)f(channel)h(router)166 820 y(node,)k(the)e(sender')l (s)g(anon)o(ymity)i(is)e(the)g(size)g(of)g(the)g(entire)g(sys-)166 912 y(tem,)17 b(e)n(v)o(en)h(in)g(the)f(presence)i(of)e(an)g(all-po)n (werful)h(passi)n(v)o(e)g(adv)o(er)o(-)166 1003 y(sary)-5 b(.)86 1127 y FA(\017)42 b FB(Recei)n(v)o(er)19 b(Anon)o(ymity)166 1235 y(When)29 b Ft(A)f FB(sends)i(a)e(pack)o(et)i(to)f Ft(B)j FB(at)c(channel)i Fl(\()p Ft(b)1521 1243 y Fo(B)1574 1235 y Ft(=m)1680 1243 y Fo(B)1732 1235 y Fl(\))p FB(,)h(e)n(v-)166 1327 y(ery)25 b(member)g(of)g Ft(E)t FA(f)p Fl(\()p Ft(b)804 1335 y Fo(B)857 1327 y Ft(=m)963 1335 y Fo(B)1015 1327 y Fl(\))p FA(g)g FB(recei)n(v)o(es)g(the)g(pack)o(et.)41 b(From)166 1418 y(the)18 b(perspecti)n(v)o(e)i(of)e(an)g(e)o(xternal)h (observ)o(er)m(,)g(the)f(beha)o(vior)h(of)f(the)166 1509 y(system)k(is)f(e)o(xactly)h(the)g(same)g(whether)g Ft(B)j FB(recei)n(v)o(es)d(the)g(pack)o(et)166 1601 y(or)28 b(not.)49 b(Thus,)29 b Ft(B)t FB(')l(s)e(recei)n(v)o(er)h(anon)o(ymity) h(is)e(e)o(xactly)h(equi)n(v)n(a-)166 1692 y(lent)22 b(to)g(the)f(set)h(of)g(all)f(users)h(who)h(recei)n(v)o(e)f(the)g(pack) o(et,)h(namely)166 1783 y Ft(E)t FA(f)p Ft(b)298 1791 y Fo(B)351 1783 y Ft(=m)457 1791 y Fo(B)510 1783 y FA(g)p FB(.)86 1908 y FA(\017)42 b FB(Sender)o(-recei)n(v)o(er)19 b(anon)o(ymity)166 2016 y(Since)k(all)g(nodes)h(in)g(the)f(system)h (send)g(at)f(a)g(constant)i(rate,)e(and)166 2107 y(all)f(pack)o(ets)h (are)f(pair)o(-wise)f(encrypted)i(between)g(each)g(hop,)g(we)166 2198 y(claim)d(that)h(it)f(is)g(impossible)h(for)g(a)f(passi)n(v)o(e)h (observ)o(er)h(to)e(distin-)166 2290 y(guish)e(noise)g(from)f(signal)g (pack)o(ets.)24 b(Since)17 b(the)g(observ)o(er)h(cannot)166 2381 y(distinguish)24 b(signal)f(pack)o(ets,)i(it)e(cannot)h(discern)f (if)g(or)g(when)h Ft(A)166 2472 y FB(communicates,)32 b(and)d(thus,)h(it)e(cannot)h(determine)g(when)g Ft(A)f FB(is)166 2564 y(communicating)21 b(with)d(an)o(y)i(other)f(node)h Ft(B)t FB(.)1849 2705 y Fc(2)83 2846 y FB(Assume)c(that)f(the)g(rate)g (at)f(which)i(some)g(user)f Ft(A)f FB(sends)i(pack)o(ets)g(does)0 2937 y(not)27 b(change)h(when)f(it)f(is)g(sending)h(signal)g(v)o(ersus) g(noise)g(pack)o(ets.)47 b(In)0 3028 y(this)30 b(case,)j(the)d(distrib) o(ution)g(of)g(pack)o(ets,)j(whether)e(the)o(y)f(are)g(signal)0 3120 y(pack)o(ets)21 b(or)g(noise,)g(does)g(not)f(af)n(fect)h(the)f (security)h(of)f(the)h(node.)28 b(Thus,)0 3211 y(a)20 b(nice)h(property)g(of)f(our)g(system)h(is)f(that)g(the)g(anon)o(ymity) h(of)f(an)o(y)h(node)0 3302 y Ft(A)d FB(depends)j(only)e(upon)h(the)f (length)g(of)g(the)g(mask)h(that)e Ft(A)g FB(is)h(willing)f(to)0 3394 y(respond)i(to.)0 3619 y Fv(3.5)99 b(Attacks)0 3761 y FB(In)18 b(this)g(section,)g(we)g(outline)h(a)f(number)h(of)f (attacks)h(that)f(a)g(system)g(lik)o(e)0 3852 y FA(P)60 3820 y Fz(5)115 3852 y FB(must)j(guard)g(against,)g(and)h(sho)n(w)f (why)g FA(P)1236 3820 y Fz(5)1291 3852 y FB(is)f(in)m(vulnerable)i(to)f (all)0 3944 y(these)e(attacks.)86 4085 y FA(\017)42 b Fk(Corr)o(elation)25 b(Attack:)34 b FB(W)-6 b(e)24 b(ha)o(v)o(e)h (already)g(alluded)h(to)e(this)h(at-)166 4176 y(tack)j(in)f(which)h(a)f (passi)n(v)o(e)i(observ)o(er)f(is)f(able)h(to)f(\(statistically\))166 4267 y(track)17 b(signal)h(pack)o(ets)g(from)f(a)g(source)h(to)f(a)g (destination,)h(thus)f(vi-)166 4359 y(olating)22 b(sender)o(-recei)n(v) o(er)g(anon)o(ymity)-5 b(.)33 b(In)22 b FA(P)1380 4327 y Fz(5)1448 4359 y FB(the)f(noise)h(pack-)166 4450 y(ets)d(thw)o(art)g (this)f(attack.)86 4574 y FA(\017)42 b Fk(Intersection)21 b(Attack:)27 b FB(If)21 b(an)g(adv)o(ersary)h(kno)n(ws)g(that)f(a)g (user)h(is)166 4666 y(tw)o(o)c(dif)n(ferent)g(sets)g Ft(U)26 b FB(and)19 b Ft(V)e FB(,)g(then)i(the)f(anon)o(ymity)h(of)f (the)g(user)166 4757 y(is)23 b(reduced)i(to)f Ft(U)53 b FA(\\)44 b Ft(V)17 b FB(.)38 b(If)23 b(users)h(are)g(uniformly)g (distrib)o(uted)166 4848 y(across)16 b(such)g(sets)f(that)g(can)h(be)g (intersected,)g(then)f(the)h(anon)o(ymity)166 4940 y(for)22 b(an)o(y)g(user)g(in)g(these)g(reduces)g(e)o(xponentially)h(with)f(the) g(num-)166 5031 y(ber)g(of)f(intersecting)h(sets.)31 b(F)o(or)21 b(e)o(xample,)i(suppose)g(users)f(com-)166 5122 y(municate)17 b(using)h(both)f(their)f(routing)h(k)o(e)o(ys)h(and) f(their)f(communi-)166 5214 y(cation)22 b(k)o(e)o(y)-5 b(.)34 b(W)m(ith)21 b(each)h(k)o(e)o(y)-5 b(,)24 b(there)e(is)f(a)h (corresponding)i(set)e(of)166 5305 y(users)e(who)h(may)g(o)n(wn)f(that) g(k)o(e)o(y)-5 b(.)28 b(This)20 b(leads)g(to)g(an)h(intersection)166 5396 y(attack.)36 b(This)22 b(is)h(the)g(reason)h(a)f(user)g (communicates)i(with)e(only)166 5488 y(one)18 b(k)o(e)o(y)g(in)g FA(P)550 5456 y Fz(5)584 5488 y FB(,)f(and)h(routing)h(k)o(e)o(ys)f (cannot)g(be)g(mapped)h(back)f(to)2154 439 y(the)23 b(communication)g (k)o(e)o(ys.)34 b(Note)22 b(that)g(this)g(is)f(also)i(the)f(reason)2154 530 y(users)e(cannot)g(increase)f(their)g(anon)o(ymity)h(be)o(yond)h (the)e(smallest)2154 621 y(set)g(the)o(y)g(ha)o(v)o(e)g(e)n(v)o(er)h (been)g(mapped)g(to.)2075 743 y FA(\017)41 b Fk(Differ)o(ence)27 b(Attack:)38 b FB(If)27 b(an)g(adv)o(ersary)h(can)f(map)h(the)e(user)i (to)2154 835 y(some)c(set)f(and)h(can)f(assert)g(that)g(the)g(user)h (is)e(not)i(in)f(some)g(other)2154 926 y(set,)16 b(then)g(it)e(can)i 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b(this)g(case,)h(a)f(\(set)g(of\))g(malicious)g(users)h(\(the)2154 2647 y(mob\))g(collude)g(to)f(try)g(to)g(e)o(xpose)h(some)g(user)f Ft(A)p FB(.)41 b(These)25 b(users)2154 2739 y(can)k(all)e(join)h(the)h (same)f(channel)h(as)f Ft(A)p FB(,)h(and)g(reduce)g(the)f(ef)n(\002-) 2154 2830 y(cienc)o(y)e(of)f(the)g(channel.)42 b(This)24 b(can)i(cause)f Ft(A)f FB(to)h(e)o(xpose)h(more)2154 2921 y(bits)k(of)g(its)f(mask)h(or)f(cause)i(other)f(le)o(gitimate)f (users)h(to)g(lea)o(v)o(e)2154 3013 y(the)35 b(channel.)72 b(In)34 b(either)h(case,)j(the)d(mob)g(has)g(reduced)h Ft(A)p FB(')l(s)2154 3104 y(anon)o(ymity)21 b(to)d(the)h(set)g(of)g (remaining)h(le)o(gitimate)e(users.)2154 3211 y(This)23 b(is)f(a)h(dif)n(\002cult)f(attack)h(to)f(handle)i(in)f(a)f(system)h (which)g(pro-)2154 3302 y(vides)17 b(anon)o(ymity)-5 b(,)17 b(since)f(it)g(is)f(\(hopefully\))i(not)f(possible)g(to)g(map) 2154 3393 y(public)23 b(k)o(e)o(ys)h(back)f(to)f(indi)n(viduals.)34 b(In)23 b FA(P)3291 3362 y Fz(5)3325 3393 y 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y(the)k(channel)h(that)e(it)g(responds)i (on.)31 b(Ho)n(we)n(v)o(er)m(,)22 b(an)g(all-po)n(werful)2154 4505 y Fy(active)j FB(attack)o(er)f(can)h(mount)g(this)f(attack)g(from) g(enough)i(dif)n(fer)o(-)2154 4596 y(ent)21 b(ASs)f(to)h(e)o(xpose)h (an)o(y)f(user)l(.)29 b(Thus)21 b FA(P)3242 4564 y Fz(5)3297 4596 y FB(is)f(susceptible)i(to)e(ad-)2154 4687 y(v)o(ersaries)i(who)f (can)h(acti)n(v)o(ely)f(manipulate)h(\(e.g.)29 b(by)21 b(generating)2154 4779 y(pack)o(ets)f(from)f(arbitrary)g(ASs\))f(lar)o (ge)h(parts)g(of)g(the)g(netw)o(ork.)1988 5045 y FC(4)120 b(Details)1988 5214 y FB(In)26 b(this)f(section,)j(we)d(present)i (details)e(from)h(our)g(implementation)g(of)1988 5305 y FA(P)2048 5273 y Fz(5)2083 5305 y FB(.)37 b(W)-6 b(e)24 b(ha)o(v)o(e)g(implemented)h FA(P)2897 5273 y Fz(5)2955 5305 y FB(in)e(a)h(pack)o(et)h(le)n(v)o(el)f(simulator)m(,)h(b)o(ut) 1988 5396 y(the)20 b(details)g(from)g(our)g(implementation)h(w)o(ould)f (be)h(useful)f(in)g(a)f(\223real\224)1988 5488 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y(chunk)k(at)e(end)h(of)f(the)h(pack)o(et)g(with)f(random)h(bits)f (to)h(increment)166 4930 y(the)c(pack)o(et)h(length)f(to)g(the)g(\002x) o(ed)g(system)h(size.)0 5086 y(If)d(the)g(\223forw)o(ard\224)h(bit)e (is)h(not)g(set,)g(then)h(this)e(signal)i(pack)o(et)g(is)e(deli)n(v)o (ered)0 5177 y(locally)-5 b(.)0 5396 y Fs(F)n(orwarding)20 b(within)i(a)g(channel)74 b FB(Since)20 b(each)h(logical)f(channel)0 5488 y(is)29 b(a)h(tree,)h(each)f(node)h(can)f(use)g(the)g(follo)n (wing)g(simple)f(forw)o(arding)1988 444 y(algorithm)e(to)g(forw)o(ard)g (a)f(pack)o(et)i Ft(p)d FB(sent)i(to)g(some)g(arbitrary)f Fl(\()p Ft(b=m)p Fl(\))1988 535 y FB(channel)20 b(on)g FA(L)p FB(.)2154 665 y(F)o(orw)o(ard)g Ft(p)e FB(to)h(a)g(peer)h(on)g (channel)g Ft(c)f FB(if)n(f)g(only)h(if)e Ft(p)h FB(did)g(not)2154 756 y(come)i(in)e(on)h Ft(c)g FB(and)g(if)f Ft(c)h FB(passes)g(the)g (min-common-pre\002x)2154 847 y(check)g(with)f(respect)g(to)g Fl(\()p Ft(b=m)p Fl(\))p FB(.)2071 977 y(Note)24 b(that)f(it)g(is)g (important)g(that)g(the)h(output)g(order)g(of)f(the)g(pack)o(ets)1988 1068 y(not)18 b(be)g(determined)g(by)g(the)f(input)h(order)m(,)f(else)h (it)e(becomes)j(possible)f(to)1988 1160 y(correlate)g(pack)o(ets)h (across)e(successi)n(v)o(e)i(nodes)g(and)f(trace)f(communica-)1988 1251 y(tion)22 b(between)h(tw)o(o)e(parties.)32 b(In)22 b(other)g(w)o(ords,)h(each)f(node)h(should)g(act)1988 1342 y(lik)o(e)c(a)g Fy(mix)g FB([1].)1988 1565 y Fv(4.1)100 b(Member)26 b(Security)g(and)f(J)o(oin)f(Pr)n(ocedur)n(e)1988 1707 y FB(Analogous)29 b(to)e(the)h(de\002nition)f(in)h([8],)h(we)e (de\002ne)h(anon)o(ymity)g(for)g(a)1988 1798 y(user)19 b Ft(A)e FB(as)h(the)g(set)g Ft(S)k FB(of)c(users)h(in)f(the)g(group)h (who)f(are)h(\223indistinguish-)1988 1889 y(able\224)i(from)g(that)g Ft(A)p FB(,)f(i.e.,)g(no)h(other)g(user)g(or)f(a)h(passi)n(v)o(e)g(adv) o(ersary)h(can)1988 1981 y(resolv)o(e)e(messages)g(from)f Ft(A)f FB(to)h(a)f(granularity)i(\002ner)f(than)g Ft(S)t FB(.)2071 2072 y(W)-6 b(e)28 b(assume)g(that)g(each)h(user)f Ft(u)g FB(requires)g(a)g(minimal)f(acceptable)1988 2163 y(le)n(v)o(el)18 b(of)f(anon)o(ymity)-5 b(,)19 b(i.e.)j(each)c(user)f (requires)h(their)f(corresponding)j Ft(S)1988 2254 y FB(set)h(to)g(be)g(of)g(a)g(minimum)g(size.)30 b(W)-6 b(e)20 b(call)h(this)f(minimum)i(set)f(size)f(the)1988 2346 y Fy(security)d(par)o(ameter)p FB(,)g(and)g(denote)g(it)e(with)h Ft( )3181 2354 y Fo(u)3222 2346 y FB(.)22 b(Each)16 b(user)h Ft(u)f FB(may)g(also)1988 2437 y(de\002ne)26 b(a)g(maximum)g(required)g (le)n(v)o(el)f(of)h(security)f(\(i.e.)42 b(a)26 b(maximum)1988 2528 y(size)c(of)g(the)g(corresponding)i Ft(S)h FB(set\))d(since)g (this)f(pro)o(vides)i(a)f(bound)h(on)1988 2620 y(the)h(communication)h (inef)n(\002cienc)o(y)-5 b(.)38 b(W)-6 b(e)23 b(call)g(this)g(the)h Fy(ef)o(\002ciency)g(pa-)1988 2711 y(r)o(ameter)p FB(,)f(and)f(denote)h (it)e(with)h Ft(\021)2871 2719 y Fo(u)2912 2711 y FB(.)31 b(User)22 b Ft(A)f FB(joins)h(group)h Fl(\()p Ft(b=m)p Fl(\))p FB(,)f(s.t.)1988 2802 y Ft( )2038 2810 y Fo(A)2129 2802 y FA(\024)39 b(j)p Ft(E)t FA(f)p Fl(\()p Ft(b=m)p Fl(\))q FA(gj)h(\024)g Ft(\021)2785 2810 y Fo(A)2854 2802 y FB(in)19 b(the)g(follo)n(wing)g(manner:)2075 2932 y FA(\017)41 b Ft(A)27 b FB(initially)f(joins)h Fl(\()p Ft(?=)p Fl(0\))p FB(.)48 b(If)27 b Ft( )3038 2940 y Fo(A)3151 2932 y FA(\024)64 b(j)p Ft(E)t FA(f)p Ft(?=)p Fl(0\))q FA(gj)g(\024)g Ft(\021)3825 2940 y Fo(A)3875 2932 y FB(,)2154 3023 y(we')l(re)19 b(done.)2154 3128 y(If)27 b Ft( )2281 3136 y Fo(A)2368 3128 y Ft(>)36 b FA(j)p Ft(E)t FA(f)p Ft(?=)p Fl(0\))q FA(gj)p FB(,)29 b Ft(A)e FB(the)g(entire)f(system)i (does)f(not)g(ha)o(v)o(e)2154 3219 y(enough)k(members)e(to)f(pro)o (vide)i(the)e(requisite)h(anon)o(ymity)-5 b(.)53 b Ft(A)2154 3310 y FB(forw)o(ards)22 b(pack)o(ets)f(for)g(other)f(nodes)i(and)f (sends)h(noise)f(pack)o(ets,)2154 3402 y(b)o(ut)e(does)h(not)f (directly)g(communicate)h(with)f(other)g(nodes.)2154 3506 y(If)c FA(j)p Ft(E)t FA(f)p Ft(?=)p Fl(0)r FA(gj)22 b Ft(>)f(\021)2655 3514 y Fo(A)2706 3506 y FB(,)15 b(then)h(this)f (group)h(has)g(too)f(man)o(y)h(members,)2154 3597 y(and)k Ft(A)e FB(joins)h(the)g(appropriate)h(channel)g(of)f(the)g(form)g Fl(\()p Ft(b=)p Fl(1\))p FB(.)2075 3714 y FA(\017)41 b Ft(A)26 b FB(repeats)g(this)g(procedure)i(until)e(it)f(\002nds)h(a)g Fl(\()p Ft(b=m)p Fl(\))g FB(such)h(that)2154 3805 y Ft( )2204 3813 y Fo(A)2295 3805 y FA(\024)39 b(j)p Ft(E)t FA(f)p Fl(\()p Ft(b=m)p Fl(\))q FA(gj)h(\024)g Ft(\021)2951 3813 y Fo(A)3002 3805 y FB(.)2071 3935 y(Clearly)-5 b(,)29 b(it)e(is)g(possible)h(to)g(choose)g(incompatible)h(v)n(alues)f Ft( )i FB(and)1988 4026 y Ft(\021)40 b FB(such)e(that)f Ft(\021)58 b(>)e FA(j)p Ft(E)t FA(f)p Fl(\()p Ft(b=m)p Fl(\))p FA(g)37 b FB(and)h Ft( )58 b(>)e FA(j)p Ft(E)t FA(f)p Fl(\()p Ft(b=m)17 b Fl(+)g(1\))p FA(gj)p FB(.)1988 4118 y(In)34 b(this)f(case,)k(the)c(user)h(can)g(either)f(change)i (their)e(security)g(or)h(ef-)1988 4209 y(\002cienc)o(y)c(parameter)g (or)f(w)o(ait)g(in)g(channel)h Fl(\()p Ft(b=m)25 b Fl(+)g(1\))p FB(,)31 b(until)e Ft( )44 b FA(\024)1988 4300 y(j)p Ft(E)t FA(f)p Fl(\()p Ft(b=m)18 b Fl(+)f(1\))p FA(gj)p FB(.)2071 4392 y(In)28 b(our)h(simulations,)h(each)e(user)h(can)f(determine)h (the)f(number)g(of)1988 4483 y(people)21 b(in)e(a)g(group)i(by)f (consulting)g(an)g(\223oracle\224)g(which)f(maintains)h(an)1988 4574 y(up)h(to)f(date)h(list)e(of)h(channel)h(memberships.)29 b(In)20 b(implementation,)h(this)1988 4666 y(information)27 b(can)f(be)h(maintained)g(in)e(a)h(secure)h(distrib)o(uted)f(manner)m (,)1988 4757 y(either)k(by)g(the)f(underlying)i(application-layer)g (multicast)e(primiti)n(v)o(e,)1988 4848 y(or)19 b(at)g(a)g(well-kno)n (wn)h(centralized)f(\223topology)i(serv)o(er\224.)2071 4940 y(The)k(\223topology)i(serv)o(er\224)e(construct)g(is)g(needed)h (if)e(it)h(is)f(not)h(possi-)1988 5031 y(ble)19 b(to)g(infer)f (approximate)i(group)g(sizes.)j(The)18 b(topology)j(serv)o(er)d(k)o (eeps)1988 5122 y(pairs)25 b(of)f(the)h(form)f FA(h)p Fl(\()p Ft(b=m)p Fl(\))32 b(:)g(IP)18 b(address)p FA(i)q FB(.)39 b(It)24 b(is)g(possible)h(for)f(the)1988 5214 y(topology)e(serv)o(er)e(to)g(e)o(xpose)h(a)f(user)h(by)f(pro)o(viding) i(f)o(alse)e(information)1988 5305 y(\(reporting)31 b(a)g(group)g(is)f (lar)o(ge)g(when)h(it)f(is)g(in)g(f)o(act)g(not\).)58 b(F)o(or)29 b(e)o(xtra)1988 5396 y(security)-5 b(,)25 b(the)f(topology)h(information)f(can)g(be)g(replicated)g(at)f Ft(l)h FB(dif)n(fer)o(-)1988 5488 y(ent)d(topology)g(serv)o(ers.)27 b(A)20 b(user)h(w)o(ould)g(only)g(consider)g(the)f(minimum)1926 5737 y Fu(7)p eop %%Page: 8 8 8 7 bop 0 439 a FB(group)21 b(size)e(reported)h(by)g(all)f(topology)i (serv)o(ers;)f(this)f(w)o(ay)-5 b(,)20 b(a)g(user)g(can)0 530 y(withstand)c(up)h(to)f Ft(l)8 b FA(\000)f Fl(1)15 b FB(colluding)j(malicious)e(topology)i(serv)o(ers.)k(Sim-)0 621 y(ilar)16 b(techniques)j(can)e(be)g(used)h(to)f(handle)h(malicious) f(topology)i(serv)o(ers)0 712 y(who)24 b(return)f(a)g(v)n(alue)h 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y(violated,)19 b Ft(A)f FB(can)i(migrate)f(to)g (group)h Fl(\()p Ft(b=m)c Fl(+)h(1\))p FB(.)83 1817 y(Unfortunately)-5 b(,)24 b Ft(A)e FB(does)h Fy(not)h FB(re-gain)e(an)o(y)h(security)g(by) g(migrating)0 1908 y(\223up\224)f FA(L)f FB(\(i.e.)29 b(by)21 b(decreasing)h Ft(m)p FB(\))f(since)g(an)h(intersection)f (attack)g(\002x)o(es)0 2000 y Ft(S)47 2008 y Fo(A)120 2000 y FB(to)i(the)g(size)f(of)h(the)g(smallest)g(channel)h(that)f Ft(A)f FB(w)o(as)h(part)g(of)g(since)0 2091 y(it)j(initially)f(joined.) 46 b(Thus,)28 b(in)f(common)g(use,)i(we)d(assume)h(users)g(do)0 2182 y(not)e(lea)o(v)o(e)g(once)g(the)o(y)h(join)e(and)i(if)e(a)h (group)h(becomes)g(too)f(small,)h(all)0 2274 y(remaining)18 b(users)g(ha)o(v)o(e)g(to)g(recreate)g(a)f(ne)n(w)h(hierarchy)-5 b(.)24 b(Note)17 b(that)h(the)o(y)0 2365 y(must)29 b(then)g(use)g(a)g (ne)n(w)g(communication)i(k)o(e)o(y)-5 b(,)31 b(since)f(a)e(passi)n(v)o (e)i(ob-)0 2456 y(serv)o(er)18 b(along)h(with)e(a)h(colluding)h(recei)n 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4581 y Fz(5)871 4612 y FB(requires)20 b(state,)e (processing,)i(and)g(link)0 4704 y(bandwidth)g(at)f(each)g(host.)24 b(W)-6 b(e)18 b(discuss)i(these)f(requirements)h(in)f(turn:)86 4848 y FA(\017)42 b FB(Suppose)28 b(member)f Ft(A)f FB(communicates)i (using)g(channel)g Fl(\()p Ft(b=m)p Fl(\))166 4940 y FB(at)17 b(depth)h Ft(d)f FB(in)g FA(L)p FB(.)23 b(In)17 b(order)h(to)f(communicate)i(within)e(the)g(group,)166 5031 y Ft(A)29 b FB(may)h(ha)o(v)o(e)f(to)g(maintain)h(ne)o(xt)g(hop)g (information)g(about)g Fl(2)1869 4999 y Fo(d)166 5122 y FB(groups.)i(F)o(or)21 b(small)g Ft(d)g FB(\(e.g.)30 b Ft(d)c(<)g Fl(16)p FB(\),)d(the)e(state)g(requirements)166 5214 y(are)29 b(minimal.)54 b(Note)29 b(that)g(unlik)o(e)g(an)h(IP)e (router)m(,)k Ft(A)c FB(does)i(not)166 5305 y(ha)o(v)o(e)22 b(to)f(search)h(its)f(\223routing)i(table\224)e(for)h(e)n(v)o(ery)g (pack)o(et;)i(it)d(only)166 5396 y(searches)f(this)e(table)h(when)g(it) f(initiates)g(a)h(ne)n(w)g(data)g(connection.)166 5488 y(Thus,)g(this)f(table)h(can)h(be)f(maintained)h(in)f(secondary)h (storage.)2154 439 y(If)d Ft(d)f FB(is)h(lar)o(ge)f(\(e.g.)23 b Ft(d)e FA(\025)g Fl(24)p FB(\),)c(it)f(may)i(not)f(be)g(feasible)g (for)g(a)f(node)2154 531 y(to)j(maintain)g(information)h(about)g(all)e Fl(2)3199 499 y Fo(d)3254 531 y FB(peer)i(channels.)k(In)19 b(this)2154 622 y(case,)29 b Ft(A)d FB(could)i(maintain)f(a)f(small)h (cache)g(of)g(adv)o(ertisements,)2154 713 y(and)21 b(if)f(a)h(ne)n(w)f (communication)i(requires)f(a)g(channel)g(that)f(is)g(not)2154 805 y(in)25 b(the)g(cache,)i Ft(A)d FB(w)o(ould)i(ha)o(v)o(e)f(to)g(w)o (ait)f(until)h(it)f(hears)i(another)2154 896 y(adv)o(ertisement)20 b(for)f(that)g(channel.)2075 1014 y FA(\017)41 b FB(There)28 b(is)g(a)g(single)g(public-k)o(e)o(y)i(decryption)f(for)f(e)n(v)o(ery)g (pack)o(et)2154 1105 y(that)22 b(member)g Ft(A)g FB(recei)n(v)o(es)g(.) 31 b(Further)m(,)22 b Ft(A)f FB(has)h(to)g(encrypt)h(e)n(v)o(ery)2154 1197 y(signal)18 b(pack)o(et)g(during)f(communication.)25 b(Ho)n(we)n(v)o(er)m(,)17 b(in)g(general,)2154 1288 y Ft(A)24 b FB(does)i(not)f Fy(have)g FB(to)g(encrypt)g(noise)g(pack)o (ets;)k(it)24 b(is)g(only)h(nec-)2154 1379 y(essary)h(that)f(the)h(adv) o(ersary)g(not)g(be)f(able)h(to)f(distinguish)h(noise)2154 1471 y(pack)o(ets)g(from)e(signal)g(pack)o(ets.)40 b(Thus,)26 b(it)d(is)h(feasible)g(for)g Ft(A)g FB(to)2154 1562 y(generate)d(noise) g(pack)o(ets)g(using)g(a)f(good)h(local)f(random)h(number)2154 1653 y(source.)2075 1771 y FA(\017)41 b FB(The)f(broadcast)h(nature)g (of)e FA(P)3024 1740 y Fz(5)3098 1771 y FB(requires)i(indi)n(vidual)f (group)2154 1863 y(members)49 b(to)f(\(potentially\))g(de)n(v)o(ote)g (more)g(bandwidth)i(for)2154 1954 y(communication)42 b(than)f(pure)f(unicast)g(\(or)g(systems)h(such)f(as)2154 2045 y(Cro)n(wds)16 b([8)q(]\).)21 b(Ho)n(we)n(v)o(er)m(,)c(the)f(e)o (xtra)f(bandwidth)i(directly)f(results)2154 2137 y(in)29 b(enhanced)i(anon)o(ymity)-5 b(,)32 b(and,)f(ob)o(viously)-5 b(,)33 b(indi)n(vidual)c(users)2154 2228 y(may)19 b(choose)g(a)f(more)h (bandwidth)g(ef)n(\002cient)f(channel)h(if)f(the)o(y)h(are)2154 2319 y(willing)d(to)g(sacri\002ce)g(some)h(anon)o(ymity)-5 b(.)24 b(W)-6 b(e)16 b(analyze)h(the)f(actual)2154 2411 y(bandwidth)24 b(usage)g(and)f(ho)n(w)g(it)f(scales)h(with)f(number)i (of)e(group)2154 2502 y(members)e(and)g(dif)n(ferent)f(security)g (parameters)g(in)g(Section)g(6.)1988 2767 y FC(5)120 b(The)30 b(Random)g(Channels)h(Model)1988 2935 y FB(In)22 b(this)f(section,)h(we)g(present)g(an)g(analysis)g(of)f(the)h(paths)g (in)f(a)h FA(P)3722 2903 y Fz(5)3778 2935 y FB(net-)1988 3026 y(w)o(ork.)54 b(Let)28 b Ft(n)h FB(be)g(the)g(number)h(of)f(users) g(in)f(the)h(system,)j(and)d(sup-)1988 3118 y(pose)23 b(there)g(are)f Ft(k)i FB(channels)g(a)o(v)n(ailable)e(in)g(total.)33 b(F)o(or)22 b(some)h(inte)o(ger)f Ft(t)1988 3209 y FB(\(which)16 b(is)g(typically)g(small\227at)f(most)g Fl(5)p FB(\),)h(each)h(user)f Ft(u)f FB(independently)1988 3300 y(chooses)j Ft(t)e FB(random)h(channels)h(without)e(replacement:)23 b(we)16 b(will)g(denote)1988 3392 y(this)29 b(random)g(set)g(of)f Ft(t)h FB(channels)g(by)h Ft(S)t Fl(\()p Ft(u)p Fl(\))p FB(.)52 b(T)-6 b(w)o(o)28 b(central)h(parame-)1988 3483 y(ters)f(for)g(us)g(will)g(be:)41 b(\(i\))28 b Ft(d)p FB(,)h(the)g(maximum)g(\223hop-count\224)h(between)1988 3574 y(an)o(y)18 b Fl(2)g FB(users,)g(which)g(is)f(the)h(maximum)g (communication)h(distance)f(be-)1988 3666 y(tween)28 b(an)o(y)g(tw)o(o)g(users,)i(and)e(\(ii\))f Ft(L)2975 3674 y Fo(min)3122 3666 y FB(and)h Ft(L)3309 3674 y Fo(max)3436 3666 y FB(,)h(the)f(minimum)1988 3757 y(and)19 b(maximum)g(load)f (\(number)h(of)f(users\))g(on)g(an)o(y)h(channel.)24 b(Note)18 b(that)1988 3848 y Ft(L)2040 3856 y Fo(min)2188 3848 y FB(and)29 b Ft(L)2376 3856 y Fo(max)2531 3848 y FB(are)g(dif)n(ferent)f(than)h(the)g(security)g(parameters)f(as)1988 3940 y(the)o(y)h(count)h(only)f(the)g(set)f(of)h(users)g(local)g(to)f (a)h(channel,)j(while)c(the)1988 4031 y(security)h(parameters)f(count)h (the)f(set)g(of)g(users)h(in)f(all)f(channels)j(that)1988 4122 y(pro)o(vide)24 b(anon)o(ymity)f(for)f(a)g(gi)n(v)o(en)h(user)l(.) 34 b(W)-6 b(e)21 b(ne)o(xt)i(discuss)g(these)f(tw)o(o)1988 4214 y(parameters.)1988 4348 y Fk(The)47 b(parameter)i Ft(d)p Fk(.)109 b FB(W)-6 b(e)47 b(say)h(that)f(there)h(is)f(a)h(path)g Ft(u)76 b Fl(=)1988 4440 y Ft(u)2032 4448 y Fz(0)2067 4440 y Ft(;)13 b(u)2145 4448 y Fz(1)2180 4440 y Ft(;)g(u)2258 4448 y Fz(2)2293 4440 y Ft(;)g(:)g(:)g(:)h(;)f(u)2508 4449 y Fo(`)2567 4440 y Fl(=)28 b Ft(v)e FB(between)d(tw)o(o)g(users)g Ft(u)g FB(and)h Ft(v)s FB(,)f(if)f(for)h(each)1988 4531 y Ft(i)p FB(,)h(users)g Ft(u)2278 4539 y Fo(i)2327 4531 y FB(and)g Ft(u)2502 4539 y Fo(i)p Fz(+1)2629 4531 y FB(choose)g(a)f(channel)i(in)e(common.)37 b(The)23 b Fy(length)1988 4622 y FB(of)18 b(such)h(a)f(path)h(is)e(de\002ned)i(to) f(be)h Ft(`)p FB(,)e(and)i(the)f(minimum)g(length)h(of)f(an)o(y)1988 4714 y(path)e(between)h Ft(u)e FB(and)h Ft(v)i FB(is)e(the)f(distance)h (or)g(hop-count)h(between)f Ft(u)g FB(and)1988 4805 y Ft(v)s FB(;)25 b(if)e(there)g(is)g(no)h(such)g(path,)g(then)g(this)f (hop-count)i(is)e(de\002ned)h(to)f(be)1988 4896 y FA(1)p FB(.)39 b(W)-6 b(e)24 b(are)h(interested)f(in)h(bounding)h Ft(d)p FB(,)f(the)g(maximum)g(hop-count)1988 4988 y(between)20 b(an)o(y)f(tw)o(o)g(users.)1988 5122 y Fk(The)24 b(load)h(parameters)h Ft(L)2741 5130 y Fo(min)2885 5122 y Fk(and)e Ft(L)3082 5130 y Fo(max)3209 5122 y Fk(.)40 b FB(De\002ne)25 b(the)g(load)g(on)h (a)1988 5214 y(channel)h(to)e(be)h(the)f(number)h(of)f(users)h(who)g (chose)g(it)f(among)h(their)f Ft(t)1988 5305 y FB(random)d(channels;)f (let)f Ft(L)2686 5313 y Fo(min)2825 5305 y FB(and)h Ft(L)3005 5313 y Fo(max)3152 5305 y FB(respecti)n(v)o(ely)g(be)g(the)f(mini-)1988 5396 y(mum)c(and)h(maximum)f(loads)g(on)g(an)o(y)g(channel.)23 b(A)16 b(lo)n(wer)f(bound)i(on)f(the)1988 5488 y(former)j(is)g(needed)h (to)f(guarantee)h(the)f(security)h(of)f(the)g(system,)g(and)g(an)1926 5737 y Fu(8)p eop %%Page: 9 9 9 8 bop 0 439 a FB(upper)19 b(bound)h(on)e(the)h(latter)e(is)h (required)h(to)f(sho)n(w)h(that)f(the)g(bandwidth)0 530 y(o)o(v)o(erhead)i(is)f(not)g(signi\002cant.)83 672 y(Gi)n(v)o(en)c (the)g(abo)o(v)o(e)g(discussion,)h(our)f(basic)g(goals)g(will)f(be)h (as)f(follo)n(ws.)0 764 y(Clearly)-5 b(,)16 b(we)h(simultaneously)g(w)o (ant)g(\223small\224)f Ft(d)p FB(,)g(a)g(v)n(alue)h(of)g Ft(L)1658 772 y Fo(min)1793 764 y FB(that)0 855 y(is)25 b(\223not)i(too)f(small\224,)h(and)f Ft(L)777 863 y Fo(max)930 855 y FB(being)h(\223not)f(too)g(high\224.)45 b(It)25 b(is)h(easy)0 946 y(to)j(see)g(that)f(the)h(e)o(xpected)h(load)f(on)h (an)o(y)f(gi)n(v)o(en)g(channel)h(is)f(e)o(xactly)0 1038 y Ft(t)21 b FA(\001)g Ft(n=k)r FB(.)40 b(Thus,)26 b Ft(n=k)h FB(is)c(a)i(natural)f(parameter)h(to)f(study)h(our)g(system)0 1129 y(with.)36 b(So,)23 b(we)h(will)e(consider)i(scenarios)g(where)g Ft(k)h 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FB(denote)h(the)f (desirable)h(e)n(v)o(ent)g(that)f(\223\(i\))f(all)h(the)g(channel)0 1860 y(loads)c(are)g(within)g Fl(1)e FA(\006)f Ft(\017)j FB(of)g(the)g(e)o(xpected)h(v)n(alue)f Ft(t\025)p FB(,)f(and)i(\(ii\))e Ft(d)j FA(\024)g Ft(s)p FB(\224.)0 1951 y(W)-6 b(e)17 b(deri)n(v)o(e)h(the)f(follo)n(wing)h(suf)n(\002cient)f(conditions)i (for)e FA(A)1517 1959 y Fo(s)1568 1951 y FB(to)g(hold)h(\(for)0 2043 y Ft(s)j Fl(=)g(2)p Ft(;)14 b Fl(3)p FB(\))19 b(with)f(a)h (probability)h(of)f(at)f(least)h Fl(1)e FA(\000)g Fl(10)1360 2011 y Fx(\000)p Fz(4)1443 2043 y FB(:)68 2168 y(1.)42 b Ft(s)21 b Fl(=)g(2)p FB(:)j(tw)o(o)19 b(suf)n(\002cient)f(conditions) j(are)179 2311 y(\(P1\))41 b Ft(t)21 b Fl(=)g(3)p FB(,)e Ft(\025)i FA(\025)g Fl(100)p FB(,)e(and)h Ft(K)27 b FA(\024)21 b Fl(100)p FB(;)e(or)179 2436 y(\(P2\))41 b Ft(t)21 b Fl(=)g(4)p FB(,)e Ft(\025)i FA(\025)g Fl(80)p FB(,)e(and)g Ft(K)27 b FA(\024)21 b Fl(300)p FB(.)68 2578 y(2.)42 b Ft(s)21 b Fl(=)g(3)p FB(:)j(tw)o(o)19 b(suf)n(\002cient)f(conditions) j(are)179 2721 y(\(P3\))41 b Ft(t)21 b Fl(=)g(3)p FB(,)e Ft(\025)i FA(\025)g Fl(100)p FB(,)e(and)h Ft(K)27 b FA(\024)21 b Fl(150)p FB(;)e(or)179 2846 y(\(P4\))41 b Ft(t)21 b Fl(=)g(4)p FB(,)e Ft(\025)i FA(\025)g Fl(80)p FB(,)e(and)g Ft(K)27 b FA(\024)21 b Fl(700)p FB(.)0 2989 y(W)-6 b(e)20 b(no)n(w)h(pro)o(v)o(e)g(that)f(these)g(conditions)i(are)e(indeed)h (suf)n(\002cient,)g(in)f(the)0 3080 y(rest)f(of)f(this)h(section.)0 3308 y Fv(5.1)99 b(Analysis)24 b(A)n(ppr)n(oach)0 3450 y FB(In)36 b(our)h(analysis,)j(we)c(will)g(frequently)h(use)f(the)g (union)h(bound)h(or)0 3541 y(Boole')l(s)19 b(inequality:)24 b Fl(Pr[)p Ft(E)750 3549 y Fz(1)802 3541 y FA(_)17 b Ft(E)927 3549 y Fz(2)978 3541 y FA(_)g Ft(:)c(:)h(:)j FA(_)g Ft(E)1278 3549 y Fo(s)1311 3541 y Fl(])22 b FA(\024)1435 3481 y Fb(P)1522 3502 y Fo(s)1522 3568 y(i)p Fz(=1)1639 3541 y Fl(Pr[)p Ft(E)1799 3549 y Fo(i)1825 3541 y Fl(])p FB(.)83 3633 y(The)32 b(parameters)g Ft(L)642 3641 y Fo(min)793 3633 y 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Fz(2)738 4394 y Ft(=)p Fl(2)h FA(\000)f Ft(\017)940 4358 y Fz(3)975 4394 y Ft(=)p Fl(6\))g FA(\001)h Fl(\()p Ft(n=k)r Fl(\)\))265 4510 y FA(\024)83 b Fl(exp)o(\()p FA(\000)p Ft(t\025\017)719 4474 y Fz(2)753 4510 y Ft(=)p Fl(2\))18 b(+)f(exp)o(\()p FA(\000)p Ft(t\025)p Fl(\()p Ft(\017)1295 4474 y Fz(2)1329 4510 y Ft(=)p Fl(2)h FA(\000)e Ft(\017)1530 4474 y Fz(3)1565 4510 y Ft(=)p Fl(6\)\))p Ft(:)83 4674 y FB(No)n(w)-5 b(,)19 b(a)g(simple)g(application)g(of)g (the)g(union)h(bound)h(yields)132 4929 y Fl(Pr)q([\()p Ft(L)318 4937 y Fo(min)459 4929 y Ft(<)g Fl(\(1)c FA(\000)g Ft(\017)p Fl(\))g FA(\001)g Ft(tn=k)r Fl(\))1016 4852 y Fb(_)1108 4929 y Fl(\()p Ft(L)1190 4937 y Fo(max)1338 4929 y Ft(>)k Fl(\(1)d(+)f Ft(\017)p Fl(\))g FA(\001)g Ft(tn=k)r Fl(\)])215 5122 y FA(\024)83 b Ft(K)23 b FA(\001)17 b Fl(exp)o(\()p FA(\000)p Ft(t\025)774 5073 y(\017)805 5042 y Fz(2)p 773 5105 66 4 v 787 5172 a Fl(2)849 5122 y(\))36 b(+)17 b Ft(K)22 b FA(\001)17 b Fl(exp\()p FA(\000)p Ft(t\025)p Fl(\()1438 5073 y Ft(\017)1469 5042 y Fz(2)p 1437 5105 V 1450 5172 a Fl(2)1529 5122 y FA(\000)1616 5073 y Ft(\017)1647 5042 y Fz(3)p 1616 5105 V 1630 5172 a Fl(6)1692 5122 y(\)\)])45 b FB(\(1\))83 5305 y(W)-6 b(e)21 b(ne)o(xt)g(turn)g(to)g(bounding)i Ft(d)p FB(.)30 b(W)-6 b(e)20 b(cannot)i(directly)f(dra)o(w)g(on)h(the)0 5396 y(rich)k(random)h(graphs)f(literature,)h(since)f(we)g(are)g(w)o (orking)g(here)h(with)0 5488 y(a)d(certain)g(model)g(of)g(random)h (hyper)o(graphs)g(with)f(possibly)g(repeated)1988 439 y(hyperedges.)32 b(W)-6 b(e)21 b(ne)o(xt)h(study)g(the)f(tw)o(o)g (requirements)i(of)e(most)g(inter)o(-)1988 530 y(est:)38 b Ft(d)d FA(\024)f Fl(2)27 b FB(and)f Ft(d)35 b FA(\024)g Fl(3)p FB(.)45 b(As)26 b(can)g(be)h(e)o(xpected,)i(the)d(second)h(case) 1988 621 y(in)m(v)o(olv)o(es)22 b(more)g(w)o(ork)g(than)h(the)e (\002rst.)30 b(Our)22 b(basic)g(plan)g(is)f(as)h(follo)n(ws.)1988 712 y(Lemma)17 b(5.1)g(gi)n(v)o(es)h(an)f(upper)o(-bound)i(on)e Fl(Pr[)p Ft(d)22 b(>)f Fl(2])p FB(,)c(and)g(Lemma)g(5.2)1988 804 y(gi)n(v)o(es)22 b(an)f(upper)o(-bound)h(on)g Fl(Pr[)p Ft(d)j(>)f Fl(3])p FB(.)29 b(Then,)22 b(letting)p 3515 743 48 4 v 20 w FA(E)27 b FB(denote)21 b(the)1988 895 y(complement)g(of)e(e)n(v)o(ent)h FA(E)7 b FB(,)18 b(we)i(see)f(by)h (the)f(union)i(bound)g(that)e Fl(Pr[)p 3776 834 96 4 v FA(A)3837 903 y Fz(2)3872 895 y Fl(])1988 986 y FB(is)25 b(upper)o(-bounded)j(by)e(the)f(sum)h(of)f(\(1\))h(and)g(the)f (probability)h(bound)1988 1078 y(gi)n(v)o(en)i(by)g(Lemma)f(5.1.)48 b(Similarly)-5 b(,)28 b Fl(Pr)q([)p 3120 1017 V FA(A)3181 1086 y Fz(3)3216 1078 y Fl(])f FB(is)g(upper)o(-bounded)j(by)1988 1169 y(the)20 b(sum)h(of)f(\(1\))f(and)i(the)f(bound)h(gi)n(v)o(en)g (by)g(Lemma)e(5.2.)27 b(W)-6 b(e)19 b(shall)h(do)1988 1260 y(this)f(\223putting)h(together\224)f(in)g(Section)g(5.4.)1988 1519 y Fv(5.2)100 b(The)26 b(Requir)n(ement)h Fa(d)g FF(\024)h FH(2)1988 1672 y FB(Here,)g(we)e(w)o(ant)h(suf)n(\002cient)f (conditions)i(for)e(\223)p Ft(d)35 b FA(\024)g Fl(2)p FB(\224)27 b(to)f(hold)h(with)1988 1763 y(high)d(probability)-5 b(.)36 b(In)23 b(other)g(w)o(ords,)h(we)f(w)o(ant)g(to)g(sho)n(w)g (that)g(for)g(an)o(y)1988 1855 y(tw)o(o)30 b(users,)j(there)d(is)f(a)h (path)g(of)g(length)g(at)f(most)h Fl(2)g FB(between)h(them.)1988 1946 y(T)-6 b(o)28 b(do)h(so,)h(we)e(\002x)g(distinct)g(users)g Ft(u)g FB(and)h Ft(v)s FB(,)h(and)e(upper)o(-bound)i(the)1988 2037 y(probability)g Ft(p)f FB(that)g(there)g(is)g(no)g(path)h(of)f (length)g Fl(2)h FB(between)g(them;)1988 2129 y(then,)e(by)d(the)h (union)g(bound,)j(the)c(probability)h(of)g Ft(d)33 b(>)h Fl(2)25 b FB(is)g(at)g(most)1988 2155 y Fb(\000)2026 2186 y Fo(n)2030 2245 y Fz(2)2065 2155 y Fb(\001)2122 2220 y FA(\001)19 b Ft(p)p FB(,)i(since)h(there)f(are)g(only)2852 2155 y Fb(\000)2890 2186 y Fo(n)2894 2245 y Fz(2)2929 2155 y Fb(\001)2988 2220 y FB(choices)h(for)f(the)h(unordered)h(pair) 1988 2311 y Fl(\()p Ft(u;)14 b(v)s Fl(\))p FB(.)22 b(Thus,)17 b(we)g(need)h(to)g(sho)n(w)g(that)f Ft(p)f FB(is)h(ne)o(gligible)h(in)f (comparison)1988 2403 y(with)i Fl(\()2170 2337 y Fb(\000)2208 2369 y Fo(n)2212 2428 y Fz(2)2246 2337 y Fb(\001)2284 2403 y Fl(\))2314 2371 y Fx(\000)p Fz(1)2397 2403 y FB(,)f(which)h(we)g (proceed)h(to)f(do)g(no)n(w)-5 b(.)2071 2500 y(Our)20 b(plan)f(is)g(to)h(condition)g(on)g(the)f(v)n(alues)h(of)g Ft(S)t Fl(\()p Ft(u)p Fl(\))f FB(and)h Ft(S)t Fl(\()p Ft(v)s Fl(\))p FB(.)k(F)o(or)1988 2592 y(each)18 b(such)g(choice,)g(we) f(will)g(upper)o(-bound)i(the)e(probability)h(that)f(there)1988 2683 y(is)26 b(no)h(user)g(\(among)g(the)g(remaining)g Fl(\()p Ft(n)c FA(\000)f Fl(2\))p FB(\))k(who)h(chose)g(a)g(chan-)1988 2774 y(nel)j(that)f(intersects)g(both)g Ft(S)t Fl(\()p Ft(u)p Fl(\))g FB(and)h Ft(S)t Fl(\()p Ft(v)s Fl(\))p FB(.)54 b(Then,)31 b(the)f(maximum)1988 2866 y(such)21 b(probability)f(is)f(an)h(upper)o(-bound)i(on)e Ft(p)p FB(.)25 b(Fix)19 b Ft(S)t Fl(\()p Ft(u)p Fl(\))h FB(and)g Ft(S)t Fl(\()p Ft(v)s Fl(\))p FB(.)25 b(If)1988 2957 y Ft(S)t Fl(\()p Ft(u)p Fl(\))d FA(\\)f Ft(S)t Fl(\()p Ft(v)s Fl(\))32 b FA(6)p Fl(=)g FA(;)p FB(,)26 b(then)f Ft(u)f FB(and)i Ft(v)h FB(are)d(at)h(distance)g Fl(1)p FB(;)i(so)e(suppose)1988 3048 y Ft(S)t Fl(\()p Ft(u)p Fl(\))18 b FA(\\)f Ft(S)t Fl(\()p Ft(v)s Fl(\))k(=)g FA(;)p FB(.)i(In)c(particular)m(,)g(we)g(may)g(assume)h(that)e Ft(k)24 b FA(\025)d Fl(2)p Ft(t)p FB(.)2071 3146 y(Consider)28 b(an)o(y)g(other)g(user)f Ft(w)r FB(.)48 b(What)27 b(is)g(the)h (probability)g(of)f Ft(w)r FB(')l(s)1988 3237 y(random)32 b(choice)f Ft(S)t Fl(\()p Ft(w)r Fl(\))f FB(intersecting)h Ft(S)t Fl(\()p Ft(u)p Fl(\))f FB(and)i Ft(S)t Fl(\()p Ft(v)s Fl(\))p FB(?)57 b(The)31 b(total)1988 3329 y(number)26 b(of)f(possible)h(choices)g(for)e Ft(w)j FB(is)3115 3264 y Fb(\000)3153 3295 y Fo(k)3158 3354 y(t)3187 3264 y Fb(\001)3225 3329 y FB(.)40 b(The)25 b(number)h(of)f(pos-)1988 3420 y(sible)g(intersection)g(patterns)g(can)g(be)g(counted)h(as)f (follo)n(ws.)41 b(Suppose)1988 3511 y FA(j)p Ft(S)t Fl(\()p Ft(w)r Fl(\))23 b FA(\\)f Ft(S)t Fl(\()p Ft(u)p Fl(\))p FA(j)34 b Fl(=)f Ft(i)25 b FB(and)h FA(j)p Ft(S)t Fl(\()p Ft(w)r Fl(\))d FA(\\)f Ft(S)t Fl(\()p Ft(v)s Fl(\))p FA(j)33 b Fl(=)h Ft(j)t FB(,)27 b(where)e Ft(i;)14 b(j)38 b FA(\025)33 b Fl(1)1988 3603 y FB(and)20 b Ft(i)d Fl(+)f Ft(j)26 b FA(\024)21 b Ft(t)p FB(.)h(The)d(remaining)h Ft(t)c FA(\000)g Fl(\()p Ft(i)h Fl(+)f Ft(j)t Fl(\))j FB(elements)g(of)g Ft(S)t Fl(\()p Ft(w)r Fl(\))g FB(are)1988 3694 y(selected)h(at)e(random)i(from)f(outside)h Ft(S)t Fl(\()p Ft(u)p Fl(\))d FA([)g Ft(S)t Fl(\()p Ft(v)s Fl(\))p FB(.)23 b(Thus,)1988 3896 y Fl(Pr)q([\()p Ft(S)t Fl(\()p Ft(w)r Fl(\))16 b FA(\\)h Ft(S)t Fl(\()p Ft(u)p Fl(\))k(=)h FA(;)p Fl(\))2713 3819 y Fb(_)2805 3896 y Fl(\()p Ft(S)t Fl(\()p Ft(w)r Fl(\))16 b FA(\\)h Ft(S)t Fl(\()p Ft(v)s Fl(\))k(=)g FA(;)p Fl(\)])h(=)f(1)c FA(\000)f Ft(f)8 b Fl(\()p Ft(k)r(;)14 b(t)p Fl(\))p Ft(;)1988 4093 y FB(where)2311 4328 y Ft(f)8 b Fl(\()p Ft(k)r(;)13 b(t)p Fl(\))21 b(=)2632 4200 y Fb(P)2720 4288 y Fo(i;j)s Fx(\025)p Fz(1;)e Fo(i)p Fz(+)p Fo(j)s Fx(\024)p Fo(t)3090 4195 y Fb(\000)3128 4227 y Fo(t)3128 4286 y(i)3151 4195 y Fb(\001\000)3230 4227 y Fo(t)3228 4286 y(j)3256 4195 y Fb(\001\000)3349 4227 y Fo(k)q Fx(\000)p Fz(2)p Fo(t)3332 4286 y(t)p Fx(\000)p Fo(i)p Fx(\000)p Fo(j)3502 4195 y Fb(\001)p 2632 4311 909 4 v 3031 4324 a(\000)3069 4356 y Fo(k)3074 4415 y(t)3103 4324 y Fb(\001)3550 4328 y Ft(:)1988 4570 y FB(Thus,)25 b(since)g(dif)n(ferent)f(users)g Ft(w)h FB(mak)o(e)g(their)f(random)g(choices)h(inde-)1988 4661 y(pendently)-5 b(,)30 b(we)c(get)h(that)f Ft(p)35 b FA(\024)h Fl(\(1)23 b FA(\000)f Ft(f)8 b Fl(\()p Ft(k)r(;)14 b(t)p Fl(\)\))3277 4629 y Fo(n)p Fx(\000)p Fz(2)3433 4661 y FA(\024)35 b Fl(exp)o(\()p FA(\000)p Fl(\()p Ft(n)23 b FA(\000)1988 4752 y Fl(2\))p Ft(f)8 b Fl(\()p Ft(k)r(;)14 b(t)p Fl(\)\))p FB(.)23 b(Thus,)18 b(as)h(discussed)h(abo)o(v)o(e,)g(a) f(union)h(bound)g(yields)2248 4934 y Fl(Pr[)p Ft(d)i(>)f Fl(2])h FA(\024)f Fl(\()p Ft(n)2732 4898 y Fz(2)2767 4934 y Ft(=)p Fl(2\))c FA(\001)h Fl(exp)o(\()p FA(\000)p Fl(\()p Ft(n)f FA(\000)g Fl(2\))p Ft(f)8 b Fl(\()p Ft(k)r(;)13 b(t)p Fl(\)\))p Ft(:)173 b FB(\(2\))2071 5121 y(In)20 b(order)f(to)g(see)g(what)g(this)g(bound)i(says)e(for)g(v)n(arious)h (concrete)g(v)n(al-)1988 5213 y(ues)g(of)f(our)g(parameters)g Ft(K)q(;)14 b(\025;)e(t)p FB(,)19 b(we)f(de)n(v)o(elop:)1988 5396 y Fk(Lemma)i(5.1)42 b Fy(Suppose)31 b Ft(\025)40 b FA(\025)h Fl(2)p Ft(K)q(=)p Fl(\()p Ft(t)3051 5364 y Fz(3)3086 5396 y Fl(\()p Ft(t)24 b FA(\000)g Fl(1\)\))29 b Fy(and)h Ft(K)46 b FA(\025)41 b Fl(2)3770 5364 y Fo(t)p Fz(+2)3875 5396 y Fy(.)1988 5488 y(Then,)19 b Fl(Pr[)p Ft(d)j(>)f Fl(2])h FA(\024)f Fl(\(\()p Ft(K)5 b(e\025)p Fl(\))2824 5456 y Fz(2)2858 5488 y Ft(=)p Fl(2\))18 b FA(\001)f Fl(exp\()p FA(\000)p Fl(0)p Ft(:)p Fl(8)p Ft(t)3353 5456 y Fz(3)3388 5488 y Fl(\()p Ft(t)f FA(\000)h Fl(1\))p Ft(\025=K)5 b Fl(\))p Fy(.)1926 5737 y Fu(9)p eop %%Page: 10 10 10 9 bop 0 456 a Fk(Pr)o(oof)o(.)86 b(\(Sk)o(etch.\))42 b FB(Since)26 b Ft(f)8 b Fl(\()p Ft(k)r(;)13 b(t)p Fl(\))34 b FA(\025)g Ft(t)1165 424 y Fz(2)1200 391 y Fb(\000)1238 422 y Fo(k)q Fx(\000)p Fz(2)p Fo(t)1255 481 y(t)p Fx(\000)p Fz(2)1374 391 y Fb(\001)1412 456 y Ft(=)1450 391 y Fb(\000)1488 422 y Fo(k)1493 481 y(t)1522 391 y Fb(\001)1560 456 y FB(,)27 b(bound)g(\(2\))0 547 y(sho)n(ws)20 b(that)86 767 y Fl(Pr[)p Ft(d)i(>)f Fl(2])h FA(\024)f Fl(\()p Ft(n)570 732 y Fz(2)605 767 y Ft(e)641 732 y Fz(2)675 767 y Ft(=)p Fl(2\))d FA(\001)f Fl(exp)967 652 y Fb(\022)1028 767 y FA(\000)p Ft(nt)1162 732 y Fz(2)1197 652 y Fb(\022)1258 719 y Ft(k)i FA(\000)e Fl(2)p Ft(t)1279 818 y(t)g FA(\000)g Fl(2)1460 652 y Fb(\023)1521 767 y Ft(=)1559 652 y Fb(\022)1621 719 y Ft(k)1628 818 y(t)1663 652 y Fb(\023\023)1798 767 y Ft(:)0 1000 y FB(W)-6 b(e)26 b(can)h(no)n(w)g(do)f(a)h(calculation)g (to)f(sho)n(w)h(that)f(subject)h(to)f(our)h(con-)0 1091 y(straints)22 b(\(which)g(includes)g(the)h(constraint)f(that)g Ft(k)29 b FA(\025)e Fl(2)p Ft(t)p FB(\),)22 b(this)g(bound)0 1183 y(is)17 b(maximized)h(when)g Ft(k)24 b Fl(=)d Ft(K)h FB(and)c Ft(n)k Fl(=)f Ft(\025K)5 b FB(.)22 b(Further)17 b(simpli\002cation)0 1274 y(then)i(leads)h(to)e(the)h(bound)i(of)e(the) g(lemma.)1849 1373 y Fc(2)0 1699 y Fv(5.3)99 b(The)26 b(Requir)n(ement)h Fa(d)h FF(\024)g FH(3)0 1854 y FB(W)-6 b(e)27 b(no)n(w)i(adopt)g(a)f(dif)n(ferent)g(approach)h(to)f(get)g(an)h (upper)o(-bound)h(on)0 1945 y Fl(Pr[)p Ft(d)23 b(>)f Fl(4])p FB(.)j(Let)19 b FA(C)k FB(denote)e(our)f(set)f(of)g Ft(k)j FB(channels.)k(F)o(or)19 b(a)g(set)g Ft(A)j FA(\022)h(C)0 2037 y FB(with)d FA(j)p Ft(A)p FA(j)26 b Fl(=)e Ft(t)p FB(,)d(de\002ne)g Ft(Y)c Fl(\()p Ft(A)p Fl(\))j FB(to)g(be)h(the)g(set) g(of)g(all)f Ft(a)25 b FA(2)g Fl(\()p FA(C)d(\000)c Ft(A)p Fl(\))i FB(for)0 2128 y(which)f(the)g(follo)n(wing)h(holds:)382 2330 y FA(9)p Ft(x)g Fl(:)41 b(\()p Ft(a)21 b FA(2)g Ft(S)t Fl(\()p Ft(x)p Fl(\)\))912 2253 y Fb(^)1004 2330 y Fl(\()p Ft(S)t Fl(\()p Ft(x)p Fl(\))c FA(\\)g Ft(A)k FA(6)p Fl(=)g FA(;)p Fl(\))p Ft(:)0 2534 y FB(In)c(other)h(w)o(ords,)g Ft(Y)e Fl(\()p Ft(A)p Fl(\))h FB(is)f(the)i(set)f(of)g(channels)h(that) f(lie)g(outside)h(of)f Ft(A)p FB(,)0 2626 y(b)o(ut)i(which)h(lie)e(in)i (some)f(set)g Ft(S)t Fl(\()p Ft(x)p Fl(\))g FB(that)g(intersects)g Ft(A)p FB(.)24 b(Thus,)19 b(we)g(need)0 2717 y(to)i(sho)n(w)h(that)f (with)g(high)g(probability)-5 b(,)23 b(at)e(least)f(one)i(of)f(the)h (follo)n(wing)0 2808 y(four)e(conditions)i(holds)e(for)g(each)h(pair)f (of)g(users)h Ft(u)f FB(and)g Ft(v)s FB(:)26 b(\(i\))19 b Ft(S)t Fl(\()p Ft(u)p Fl(\))f FA(\\)0 2900 y Ft(S)t Fl(\()p Ft(v)s Fl(\))j FA(6)p Fl(=)g FA(;)p FB(;)d(\(ii\))e Ft(S)t Fl(\()p Ft(u)p Fl(\))10 b FA(\\)g Ft(Y)17 b Fl(\()p Ft(S)t Fl(\()p Ft(v)s Fl(\)\))k FA(6)p Fl(=)g FA(;)p FB(;)d(\(iii\))e Ft(Y)h Fl(\()p Ft(S)t Fl(\()p Ft(u)p Fl(\)\))10 b FA(\\)g Ft(S)t Fl(\()p Ft(v)s Fl(\))21 b FA(6)p Fl(=)g FA(;)p FB(;)0 2991 y(or)28 b(\(i)n(v\))f Ft(Y)17 b Fl(\()p Ft(S)t Fl(\()p Ft(u)p Fl(\)\))24 b FA(\\)g Ft(Y)16 b Fl(\()p Ft(S)t Fl(\()p Ft(v)s 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Fl(2\))p Fy(.)0 4502 y Fk(Pr)o(oof)o(.)46 b(\(Sk)o(etch.\))22 b FB(A)c(union)i(bound)h (using)e(\(3\))g(yields)10 4686 y Fl(Pr)q([)p Ft(d)i(>)g Fl(3])84 b FA(\024)f Fl(Pr[)p FA(9)p Ft(A)21 b Fl(:)40 b FA(j)p Ft(Y)17 b Fl(\()p Ft(A)p Fl(\))p FA(j)k(\024)g Ft(s)p Fl(])399 4856 y FA(\024)542 4741 y Fb(\022)603 4808 y Ft(k)610 4907 y(t)645 4741 y Fb(\023)723 4856 y FA(\001)762 4741 y Fb(\022)834 4808 y Ft(k)f FA(\000)c Ft(t)823 4907 y FA(d)p Ft(k)r(=)p Fl(2)p FA(e)1010 4741 y Fb(\023)1088 4856 y FA(\001)h Fl(\(exp\()p FA(\000)p Ft(t)1392 4820 y Fz(2)1426 4856 y Ft(=k)r Fl(\))g(+)g(2)1668 4820 y Fx(\000)p Fo(t)1744 4856 y Fl(\))1774 4820 y Fo(n)p Fx(\000)p Fz(1)399 5041 y FA(\024)83 b Fl(2)580 5005 y Fo(k)635 5041 y FA(\001)18 b Ft(k)716 5005 y Fo(t)761 5041 y FA(\001)f Fl(\(exp)o(\()p FA(\000)p Ft(t)1064 5005 y Fz(2)1098 5041 y Ft(=k)r Fl(\))h(+)f(2)1341 5005 y Fx(\000)p Fo(t)1417 5041 y Fl(\))1447 5005 y Fo(n)p Fx(\000)p Fz(1)1568 5041 y Ft(:)0 5224 y FB(A)g(calculation)h(sho)n(ws) g(that)f(subject)g(to)g(our)h(constraints,)g(this)f(bound)i(is)0 5316 y(maximized)k(when)h Ft(k)30 b Fl(=)f Ft(K)e FB(and)d Ft(n)k Fl(=)h Ft(\025K)5 b FB(.)34 b(Further)22 b(simpli\002cation)0 5407 y(completes)e(the)f(proof.)1237 b Fc(2)1988 456 y Fv(5.4)100 b(Putting)25 b(It)g(T)-9 b(ogether)1988 603 y FB(No)n(w)k(,)23 b(as)g(described)g(at)f(the)h(end)g(of)f (Section)g(5.1,)h(we)f(just)g(do)h(routine)1988 694 y(calculations)e (to)g(v)o(erify)f(the)g(follo)n(wing.)28 b(First,)19 b(if)h Ft(s)k Fl(=)g(2)c FB(and)h(an)o(y)g(one)1988 785 y(of)j(\(P1\))e(and)i(\(P2\))f(holds,)h(then)g(the)f(sum)g(of)h(\(1\))f (and)h(the)f(probability)1988 877 y(bound)e(gi)n(v)o(en)f(by)g(Lemma)g (5.1)f(is)g(at)g(most)h Fl(10)3213 845 y Fx(\000)p Fz(4)3296 877 y FB(.)k(Similarly)-5 b(,)19 b(if)f Ft(s)k Fl(=)g(3)1988 968 y FB(and)f(an)o(y)g(one)g(of)g(\(P3\))e(and)i(\(P4\))f(holds,)h (then)g(the)f(sum)h(of)f(\(1\))g(and)h(the)1988 1059 y(probability)27 b(bound)g(gi)n(v)o(en)f(by)g(Lemma)g(5.2)g(is)f(at)g (most)h Fl(10)3616 1028 y Fx(\000)p Fz(4)3699 1059 y FB(.)43 b(This)1988 1151 y(concludes)22 b(our)f(proof)g(sk)o(etch)g (about)g(these)g(suf)n(\002cient)f(conditions)i(for)1988 1242 y FA(A)2049 1250 y Fz(2)2103 1242 y FB(and)e FA(A)2291 1250 y Fz(3)2344 1242 y FB(respecti)n(v)o(ely)g(to)f(hold)g(with)g (high)g(probability)-5 b(.)1988 1522 y FC(6)120 b(Simulation)31 b(Results)1988 1695 y FB(In)21 b(this)f(section,)h(we)f(present)h (results)g(from)f(a)g(pack)o(et-le)n(v)o(el)i FA(P)3669 1664 y Fz(5)3724 1695 y FB(simu-)1988 1787 y(lator)l(.)30 b(Our)21 b(simulator)h(is)f(written)f(in)i(C,)e(and)i(can)g(simulate)f (the)h(entire)1988 1878 y FA(P)2048 1846 y Fz(5)2109 1878 y FB(protocol)27 b(with)f(thousands)i(of)f(participants.)46 b(W)-6 b(e)26 b(designed)i(and)1988 1969 y(implemented)20 b(\002)n(v)o(e)f(basic)g(e)o(xperiments:)2075 2102 y FA(\017)41 b FB(Measure)17 b(system)e(performance)h(as)g(the)f(number)h (of)f(participants)2154 2193 y(increase;)39 b(speci\002cally)-5 b(,)35 b(we)c(measure)i(the)e(end-to-end)j(band-)2154 2284 y(width,)15 b(latenc)o(y)-5 b(,)16 b(and)f(pack)o(et)h(drop)f (rates)f(as)h(the)f(number)i(of)e(users)2154 2376 y(in)19 b(the)g(system)g(is)g(increased)2075 2505 y FA(\017)41 b FB(Measure)18 b(the)g(ef)n(fect)f(of)g(the)g(security)g(and)h(ef)n (\002cienc)o(y)f(parameter)2154 2597 y(on)j(communication)g(ef)n (\002cienc)o(y)2075 2726 y FA(\017)41 b FB(Estimate)17 b(the)g(amount)g(of)g(time)g(it)f(tak)o(es)i FA(P)3298 2695 y Fz(5)3349 2726 y FB(systems)f(of)g(a)g(gi)n(v)o(en)2154 2818 y(number)25 b(of)e(participants)g(to)g(con)m(v)o(er)o(ge)i(\(i.e.) 35 b(ho)n(w)24 b(long)f(does)h(it)2154 2909 y(tak)o(e)18 b(a)f(user)h(to)f(\002nd)h(a)f(channel)h(that)f(satis\002es)g(their)g (security)h(and)2154 3000 y(ef)n(\002cienc)o(y)i(constraints\))2075 3130 y FA(\017)41 b FB(Measure)21 b(the)e(ef)n(fects)h(of)f(dif)n (ferent)h(noise)g(generation)h(rates)e(and)2154 3221 y(queuing)i(disciplines)2075 3351 y FA(\017)41 b FB(Measure)29 b(ho)n(w)f(the)g(system)g(beha)o(v)o(es)g(when)h(increasing)f(num-)2154 3442 y(bers)20 b(of)f(nodes)h(engage)f(in)g(end-to-end)i(communication) 1988 3656 y Fs(Simulation)i(Methodology)73 b FB(F)o(or)20 b(each)i(e)o(xperiment,)g(we)f(gener)o(-)1988 3747 y(ated)28 b(a)e(random)i(physical)g(topology)-5 b(.)48 b(W)-6 b(e)27 b(did)g(not)g(model)g(dif)n(ferent)1988 3839 y(propagation)18 b(delays)f(between)h(pairs)e(of)h(nodes;)h(instead,)f(we)f(assumed)1988 3930 y(unit)23 b(propagation)g(delay)g(between)g(an)o(y)g(tw)o(o)f (nodes.)35 b(Since)22 b(all)f(inter)o(-)1988 4021 y(node)31 b(latencies)f(are)g(the)g(same,)i(our)e(simulation)g(proceeds)h(using)g (a)1988 4113 y(synchronous)e(clock.)45 b(At)25 b(e)n(v)o(ery)i(tick,)g (all)f(pack)o(ets)h(sent)f(from)g(e)n(v)o(ery)1988 4204 y(node)20 b(is)f(recei)n(v)o(ed)h(at)e(their)h(destination.)2071 4298 y(W)-6 b(e)21 b(assume)g(an)g(unbounded)j(input)d(queue)h(length)f (and)h(a)e(bounded)1988 4389 y(output)30 b(queue.)52 b(All)28 b(pack)o(ets)h(recei)n(v)o(ed)g(at)g(a)f(node)h(at)f(a)h(gi)n (v)o(en)g(time)1988 4480 y(step)19 b(are)g(processed.)24 b(Some)19 b(of)f(these)h(pack)o(ets)h(may)f(be)g(queued)h(at)e(ap-)1988 4572 y(propriate)k(output)f(queues,)h(and)f(a)g(subset)g(of)g(them)g (may)g(be)g(deli)n(v)o(ered)1988 4663 y(locally)-5 b(.)24 b(Ne)o(xt,)18 b(each)h(node)g(generates)g(a)f(set)g(of)h(outgoing)g (pack)o(ets)h(and)1988 4754 y(enqueues)25 b(these)e(on)g(the)g(output)g (queues.)36 b(W)-6 b(e)22 b(then)h(impose)h(the)e(out-)1988 4846 y(put)e(queue)h(limit)d(and)i(according)h(to)e(the)h(queuing)g (discipline,)g(discard)1988 4937 y(pack)o(ets)f(if)d(an)o(y)i(output)g (queue)g(is)f(lar)o(ger)g(than)h(its)e(maximum)i(speci\002ed)1988 5028 y(size.)30 b(Note)22 b(that)f(during)h(the)f(discard)h(phase,)h (the)e(node)h(does)g(not)g(dis-)1988 5120 y(criminate)c(whether)f(it)g (is)g(dropping)h(its)f(o)n(wn)h(pack)o(ets)g(or)f(pack)o(ets)h(from) 1988 5211 y(some)k(other)g(node.)32 b(All)21 b(remaining)i(pack)o(ets)f (at)g(an)f(output)i(queue)g(are)1988 5302 y(deli)n(v)o(ered)d(in)f(the) g(ne)o(xt)g(time)g(step)g(to)g(the)f(ne)o(xt)i(hop)f(node.)2071 5396 y(Since)25 b(all)f(pack)o(ets)i(queued)h(at)d(a)h(node)h(are)f (deli)n(v)o(ered)h(at)e(the)h(ne)o(xt)1988 5488 y(time)20 b(step,)h(the)f(output)h(queue)g(size)f(serv)o(es)h(as)f(a)g(measure)h (of)g(both)f(the)1905 5737 y Fu(10)p eop %%Page: 11 11 11 10 bop 0 439 a FB(processing)28 b(and)f(bandwidth)g(requirements)h (at)e(a)g(node.)46 b(W)-6 b(e)26 b(instru-)0 530 y(mented)18 b(the)f(simulator)g(to)f(record)i(the)f(end-to-end)h(latencies)f (\(number)0 621 y(of)27 b(simulator)h(ticks\),)g(drop)g(rates)f(and)h (bandwidth,)j(end-to-end)d(hop)0 712 y(counts,)19 b(number)g(of)f (channel)i(crossings,)f(and)g(con)m(v)o(er)o(gence)h(times.)i(In)0 804 y(the)d(rest)g(of)g(this)g(section,)g(we)g(report)g(results)g(from) g(indi)n(vidual)h(e)o(xperi-)0 895 y(ments.)0 1124 y Fv(6.1)99 b(Scalability)-150 1297 y 16577003 11438130 3289088 3289088 26970521 19866091 startTexFig -150 1297 a %%BeginDocument: data/dropRate-1.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: dropRate-1.eps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Fri Nov 9 13:19:16 2001 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -66 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 userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 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 %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 200 scalefont setfont 1.000 UL LTb 1.000 UL LTa 900 682 M 5960 0 V 1.000 UL LTb 900 682 M 63 0 V 5897 0 R -63 0 V 780 682 M (0) Rshow 1.000 UL LTa 900 1506 M 5960 0 V 1.000 UL LTb 900 1506 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.1) Rshow 1.000 UL LTa 900 2329 M 5960 0 V 1.000 UL LTb 900 2329 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.2) Rshow 1.000 UL LTa 900 3153 M 5960 0 V 1.000 UL LTb 900 3153 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.3) Rshow 1.000 UL LTa 900 3976 M 5960 0 V 1.000 UL LTb 900 3976 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.4) Rshow 1.000 UL LTa 900 4800 M 5960 0 V 1.000 UL LTb 900 4800 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.5) Rshow 1.000 UL LTa 900 600 M 0 4200 V 1.000 UL LTb 900 600 M 0 63 V 0 4137 R 0 -63 V 900 400 M (10) Cshow 1498 600 M 0 31 V 0 4169 R 0 -31 V 1848 600 M 0 31 V 0 4169 R 0 -31 V 2096 600 M 0 31 V 0 4169 R 0 -31 V 2289 600 M 0 31 V 0 4169 R 0 -31 V 2446 600 M 0 31 V 0 4169 R 0 -31 V 2579 600 M 0 31 V 0 4169 R 0 -31 V 2694 600 M 0 31 V 0 4169 R 0 -31 V 2796 600 M 0 31 V 0 4169 R 0 -31 V 1.000 UL LTa 2887 600 M 0 3737 V 0 400 R 0 63 V 1.000 UL LTb 2887 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (100) Cshow 3485 600 M 0 31 V 0 4169 R 0 -31 V 3835 600 M 0 31 V 0 4169 R 0 -31 V 4083 600 M 0 31 V 0 4169 R 0 -31 V 4275 600 M 0 31 V 0 4169 R 0 -31 V 4433 600 M 0 31 V 0 4169 R 0 -31 V 4566 600 M 0 31 V 0 4169 R 0 -31 V 4681 600 M 0 31 V 0 4169 R 0 -31 V 4782 600 M 0 31 V 0 4169 R 0 -31 V 1.000 UL LTa 4873 600 M 0 4200 V 1.000 UL LTb 4873 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (1000) Cshow 5471 600 M 0 31 V 0 4169 R 0 -31 V 5821 600 M 0 31 V 0 4169 R 0 -31 V 6069 600 M 0 31 V 0 4169 R 0 -31 V 6262 600 M 0 31 V 0 4169 R 0 -31 V 6419 600 M 0 31 V 0 4169 R 0 -31 V 6552 600 M 0 31 V 0 4169 R 0 -31 V 6667 600 M 0 31 V 0 4169 R 0 -31 V 6769 600 M 0 31 V 0 4169 R 0 -31 V 1.000 UL LTa 6860 600 M 0 4200 V 1.000 UL LTb 6860 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (10000) Cshow 1.000 UL LTb 900 600 M 5960 0 V 0 4200 V -5960 0 V 900 600 L 200 2700 M currentpoint gsave translate 90 rotate 0 0 M (Drop Rate) Cshow grestore 3880 100 M (Group Size) Cshow 2.000 UP 1.000 UL LT0 3660 4637 M (Sending rate: 1/S) Rshow 3780 4637 M 543 0 V 1904 682 M 598 0 V 598 0 V 598 16 V 598 133 V 4894 722 L 598 84 V 598 44 V 1904 682 Pls 2502 682 Pls 3100 682 Pls 3698 698 Pls 4296 831 Pls 4894 722 Pls 5492 806 Pls 6090 850 Pls 4051 4637 Pls 2.000 UP 1.000 UL LT1 3660 4437 M (Sending rate: log S/S) Rshow 3780 4437 M 543 0 V 1904 801 M 598 171 V 598 698 V 598 441 V 598 1033 V 598 359 V 598 52 V 598 523 V 1904 801 Crs 2502 972 Crs 3100 1670 Crs 3698 2111 Crs 4296 3144 Crs 4894 3503 Crs 5492 3555 Crs 6090 4078 Crs 4051 4437 Crs stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument endTexFig 247 2928 a Fu(Figure)20 b(2:)25 b(Loss)c(rate)f(vs.)25 b(number)18 b(of)i(users)-150 3036 y 16577003 11438130 3289088 3289088 26970521 19866091 startTexFig -150 3036 a %%BeginDocument: data/rtC-1.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: rtC-1.eps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Thu Nov 8 23:33:16 2001 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -66 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 userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 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 %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 200 scalefont setfont 1.000 UL LTb 1.000 UL LTa 700 600 M 6160 0 V 1.000 UL LTb 700 600 M 63 0 V 6097 0 R -63 0 V 580 600 M (0) Rshow 1.000 UL LTa 700 1246 M 6160 0 V 1.000 UL LTb 700 1246 M 63 0 V 6097 0 R -63 0 V -6217 0 R (0.2) Rshow 1.000 UL LTa 700 1892 M 6160 0 V 1.000 UL LTb 700 1892 M 63 0 V 6097 0 R -63 0 V -6217 0 R (0.4) Rshow 1.000 UL LTa 700 2538 M 6160 0 V 1.000 UL LTb 700 2538 M 63 0 V 6097 0 R -63 0 V -6217 0 R (0.6) Rshow 1.000 UL LTa 700 3185 M 6160 0 V 1.000 UL LTb 700 3185 M 63 0 V 6097 0 R -63 0 V -6217 0 R (0.8) Rshow 1.000 UL LTa 700 3831 M 6160 0 V 1.000 UL LTb 700 3831 M 63 0 V 6097 0 R -63 0 V -6217 0 R (1) Rshow 1.000 UL LTa 700 4477 M 6160 0 V 1.000 UL LTb 700 4477 M 63 0 V 6097 0 R -63 0 V -6217 0 R (1.2) Rshow 1.000 UL LTa 700 600 M 0 4200 V 1.000 UL LTb 700 600 M 0 63 V 0 4137 R 0 -63 V 700 400 M (10) Cshow 1318 600 M 0 31 V 0 4169 R 0 -31 V 1680 600 M 0 31 V 0 4169 R 0 -31 V 1936 600 M 0 31 V 0 4169 R 0 -31 V 2135 600 M 0 31 V 0 4169 R 0 -31 V 2298 600 M 0 31 V 0 4169 R 0 -31 V 2435 600 M 0 31 V 0 4169 R 0 -31 V 2554 600 M 0 31 V 0 4169 R 0 -31 V 2659 600 M 0 31 V 0 4169 R 0 -31 V 1.000 UL LTa 2753 600 M 0 3937 V 0 200 R 0 63 V 1.000 UL LTb 2753 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (100) Cshow 3371 600 M 0 31 V 0 4169 R 0 -31 V 3733 600 M 0 31 V 0 4169 R 0 -31 V 3990 600 M 0 31 V 0 4169 R 0 -31 V 4189 600 M 0 31 V 0 4169 R 0 -31 V 4351 600 M 0 31 V 0 4169 R 0 -31 V 4489 600 M 0 31 V 0 4169 R 0 -31 V 4608 600 M 0 31 V 0 4169 R 0 -31 V 4713 600 M 0 31 V 0 4169 R 0 -31 V 1.000 UL LTa 4807 600 M 0 4200 V 1.000 UL LTb 4807 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (1000) Cshow 5425 600 M 0 31 V 0 4169 R 0 -31 V 5786 600 M 0 31 V 0 4169 R 0 -31 V 6043 600 M 0 31 V 0 4169 R 0 -31 V 6242 600 M 0 31 V 0 4169 R 0 -31 V 6404 600 M 0 31 V 0 4169 R 0 -31 V 6542 600 M 0 31 V 0 4169 R 0 -31 V 6661 600 M 0 31 V 0 4169 R 0 -31 V 6766 600 M 0 31 V 0 4169 R 0 -31 V 1.000 UL LTa 6860 600 M 0 4200 V 1.000 UL LTb 6860 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (10000) Cshow 1.000 UL LTb 700 600 M 6160 0 V 0 4200 V -6160 0 V 700 600 L 3780 100 M (Group Size) Cshow 1.000 UP 1.000 UL LT0 3580 4637 M (Avg. number of channels) Rshow 3700 4637 M 543 0 V 1737 600 M 618 0 V 618 0 V 619 2117 V 618 111 V 618 0 V 618 223 V 618 -223 V 618 223 V 1737 600 Pls 2355 600 Pls 2973 600 Pls 3592 2717 Pls 4210 2828 Pls 4828 2828 Pls 5446 3051 Pls 6064 2828 Pls 6682 3051 Pls 3971 4637 Pls stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument endTexFig 261 4668 a Fu(Figure)g(3:)25 b(A)-6 b(v)o(erage)19 b(number)f(of)i (channels)83 4940 y FB(In)e(Figure)h(2,)f(we)g(plot)h(the)f(end-to-end) i(loss)f(rate)f(as)g(the)h(number)g(of)0 5031 y(users)i(in)g(the)g (system)g(is)g(increased.)30 b(In)21 b(each)h(case,)f(there)g(is)g (only)g(one)0 5122 y(pair)26 b(of)g(communicating)i(nodes;)j(all)25 b(other)i(nodes)g(only)g(send)g(noise)0 5214 y(pack)o(ets.)53 b(F)o(or)29 b(each)g(group)h(size,)g(we)f(chose)h(ten)e(dif)n(ferent)h (random)0 5305 y(seeds)e(and)f(created)h(ten)f(dif)n(ferent)g (topologies.)45 b(F)o(or)25 b(each)i(topology)-5 b(,)0 5396 y(we)28 b(chose)h(three)f(dif)n(ferent)g(sender)o(-recei)n(v)o(er) h(pairs.)51 b(Each)28 b(point)h(on)0 5488 y(Figure)d(2)h(is)f(an)h(a)o (v)o(erage)f(of)h(all)f(these)h(runs)f(\(24)h(for)g(each)g(topology) 1988 439 y(size\).)i(Unless)21 b(otherwise)g(noted,)g(we)g(choose)h (the)f(v)n(alues)g(of)g(security)1988 530 y(parameters)29 b(as)g Ft( )43 b Fl(=)c(100)p FB(,)32 b(and)d Ft(\021)43 b Fl(=)c(300)p FB(,)32 b(and)d(implement)g(non-)1988 621 y(uniform)17 b(queuing.)23 b(All)15 b(nodes)i(in)e(the)h(system)g (were)g(connected)h(to)e(tw)o(o)1988 712 y(channels,)22 b(i.e.)k(the)o(y)21 b(had)g(one)g(communication)h(k)o(e)o(y)f(and)g (one)g(routing)1988 804 y(k)o(e)o(y)-5 b(.)2071 896 y(There)16 b(are)f(tw)o(o)g(dif)n(ferent)h(curv)o(es,)g(each)g(corresponding)h(to) e(tw)o(o)h(dif-)1988 987 y(ferent)i(sending)i(rates.)i(In)d(the)f (\223Sending)h(Rate=1/S\224)e(case,)i(each)f(node)1988 1079 y(generates)f(a)f(pack)o(et)h(at)f(each)g(time)g(stop)g(with)g (probability)g Fl(1)p Ft(=S)t FB(,)h(where)1988 1170 y Ft(S)25 b FB(is)19 b(the)i(size)f(of)g(its)g(current)h(broadcast)g (group.)28 b(Analogously)-5 b(,)22 b(in)e(the)1988 1261 y(\223Sending)j(Rate=)p Fl(log)15 b Ft(S=S)t FB(\224)22 b(case,)h(each)g(node)f(generates)h(a)f(pack)o(et)h(at)1988 1353 y(each)29 b(time)g(stop)f(with)g(probability)i Fl(log)15 b Ft(S=S)t FB(.)52 b(F)o(or)27 b(both)i(cases,)i(the)1988 1444 y(queue)j(sizes)e(at)g(each)h(node)g(were)f(v)o(ery)h(small:)49 b Fl(max)o FA(f)p Fl(10)p Ft(;)15 b Fl(log)f Ft(S)t FA(g)p FB(.)1988 1535 y(From)24 b(the)g(plot,)h(it)e(is)h(clear)g(that)g(the)g Fl(1)p Ft(=S)29 b FB(sending)c(rate)f(can)g(be)g(sus-)1988 1627 y(tained)19 b(in)g(the)g(system,)f(and)i(almost)e(no)h(signal)g (\(or)g(noise\))g(pack)o(ets)h(are)1988 1718 y(lost.)j(Ho)n(we)n(v)o (er)m(,)18 b(as)g(the)f(sending)i(rate)f(is)f(increased,)h(the)g (queues)h(in)e(the)1988 1809 y(system)j(are)f(saturated,)g(and)g(drop)h (rates)f(increase)g(with)g(group)h(size.)2071 1902 y(In)h(Figure)g(3,)h (we)f(plot)g(the)g(a)o(v)o(erage)g(number)h(of)f(channels)i(that)d(the) 1988 1993 y(signal)k(pack)o(ets)h(ha)o(v)o(e)f(to)f(cross)h(in)g(order) g(to)g(reach)g(their)f(destination.)1988 2084 y(Note)28 b(that)f(in)h(this)f(case,)j(each)e(user)g(only)g(connects)h(to)e Fl(2)h FB(channels.)1988 2176 y(As)d(predicted)g(by)g(the)g(analysis)g (in)g(Section)g(5,)g(the)g(a)o(v)o(erage)g(channel)1988 2267 y(le)n(v)o(el)20 b(hop)h(count)g(is)f(v)o(ery)h(small,)e(and)i(is) f(less)g(than)g(1)g(for)g(all)g(our)g(runs.)1988 2358 y(The)e(a)o(v)o(erage)g(end-to-end)h(hop)g(count)f(in)g(these)g(runs)g (were)f 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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 %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 200 scalefont setfont 1.000 UL LTb 1.000 UL LTa 900 600 M 5960 0 V 1.000 UL LTb 900 600 M 63 0 V 5897 0 R -63 0 V 780 600 M (0) Rshow 1.000 UL LTa 900 1020 M 377 0 V 5463 0 R 120 0 V 1.000 UL LTb 900 1020 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.1) Rshow 1.000 UL LTa 900 1440 M 377 0 V 5463 0 R 120 0 V 1.000 UL LTb 900 1440 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.2) Rshow 1.000 UL LTa 900 1860 M 5960 0 V 1.000 UL LTb 900 1860 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.3) Rshow 1.000 UL LTa 900 2280 M 5960 0 V 1.000 UL LTb 900 2280 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.4) Rshow 1.000 UL LTa 900 2700 M 5960 0 V 1.000 UL LTb 900 2700 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.5) Rshow 1.000 UL LTa 900 3120 M 5960 0 V 1.000 UL LTb 900 3120 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.6) Rshow 1.000 UL LTa 900 3540 M 5960 0 V 1.000 UL LTb 900 3540 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.7) Rshow 1.000 UL LTa 900 3960 M 5960 0 V 1.000 UL LTb 900 3960 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.8) Rshow 1.000 UL LTa 900 4380 M 5960 0 V 1.000 UL LTb 900 4380 M 63 0 V 5897 0 R -63 0 V -6017 0 R (0.9) Rshow 1.000 UL LTa 900 4800 M 5960 0 V 1.000 UL LTb 900 4800 M 63 0 V 5897 0 R -63 0 V -6017 0 R (1) Rshow 1.000 UL LTa 900 600 M 0 4200 V 1.000 UL LTb 900 600 M 0 63 V 0 4137 R 0 -63 V 900 400 M (1) Cshow 1797 600 M 0 31 V 0 4169 R 0 -31 V 2322 600 M 0 31 V 0 4169 R 0 -31 V 2694 600 M 0 31 V 0 4169 R 0 -31 V 2983 600 M 0 31 V 0 4169 R 0 -31 V 3219 600 M 0 31 V 0 4169 R 0 -31 V 3418 600 M 0 31 V 0 4169 R 0 -31 V 3591 600 M 0 31 V 0 4169 R 0 -31 V 3744 600 M 0 31 V 0 4169 R 0 -31 V 1.000 UL LTa 3880 600 M 0 63 V 0 800 R 0 3337 V 1.000 UL LTb 3880 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (10) Cshow 4777 600 M 0 31 V 0 4169 R 0 -31 V 5302 600 M 0 31 V 0 4169 R 0 -31 V 5674 600 M 0 31 V 0 4169 R 0 -31 V 5963 600 M 0 31 V 0 4169 R 0 -31 V 6199 600 M 0 31 V 0 4169 R 0 -31 V 6398 600 M 0 31 V 0 4169 R 0 -31 V 6571 600 M 0 31 V 0 4169 R 0 -31 V 6724 600 M 0 31 V 0 4169 R 0 -31 V 1.000 UL LTa 6860 600 M 0 4200 V 1.000 UL LTb 6860 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (100) Cshow 1.000 UL LTb 900 600 M 5960 0 V 0 4200 V -5960 0 V 900 600 L 200 2700 M currentpoint gsave translate 90 rotate 0 0 M (Drop Rate) Cshow grestore 3880 100 M (Sending Rate Multiplier) Cshow 2.000 UP 1.000 UL LT0 5957 1363 M (Uniform Queuing, Base rate: 1) Rshow 6077 1363 M 543 0 V 900 680 M 2983 1823 L 435 602 V 462 684 V 525 638 V 372 98 V 289 212 V 236 195 V 661 329 V 525 103 V 372 -15 V 900 680 Pls 2983 1823 Pls 3418 2425 Pls 3880 3109 Pls 4405 3747 Pls 4777 3845 Pls 5066 4057 Pls 5302 4252 Pls 5963 4581 Pls 6488 4684 Pls 6860 4669 Pls 6348 1363 Pls 2.000 UP 1.000 UL LT1 5957 1163 M (Non-uniform Queuing, Base rate 1) Rshow 6077 1163 M 543 0 V 900 640 M 2983 1866 L 435 613 V 462 564 V 525 694 V 372 86 V 289 193 V 236 56 V 661 384 V 525 148 V 372 -37 V 900 640 Crs 2983 1866 Crs 3418 2479 Crs 3880 3043 Crs 4405 3737 Crs 4777 3823 Crs 5066 4016 Crs 5302 4072 Crs 5963 4456 Crs 6488 4604 Crs 6860 4567 Crs 6348 1163 Crs 2.000 UP 1.000 UL LT2 5957 963 M (Uniform Queuing, Base rate: log S/S) Rshow 6077 963 M 543 0 V 900 1931 M 2983 4028 L 435 350 V 462 134 V 525 20 V 372 105 V 289 85 V 236 -2 V 661 -9 V 525 7 V 372 -12 V 900 1931 Star 2983 4028 Star 3418 4378 Star 3880 4512 Star 4405 4532 Star 4777 4637 Star 5066 4722 Star 5302 4720 Star 5963 4711 Star 6488 4718 Star 6860 4706 Star 6348 963 Star 2.000 UP 1.000 UL LT3 5957 763 M (Non-uniform Queuing,Base rate log S/S) Rshow 6077 763 M 543 0 V 900 1931 M 2983 4012 L 435 402 V 462 77 V 525 146 V 372 -8 V 289 114 V 236 -58 V 661 -20 V 525 67 V 372 -44 V 900 1931 Box 2983 4012 Box 3418 4414 Box 3880 4491 Box 4405 4637 Box 4777 4629 Box 5066 4743 Box 5302 4685 Box 5963 4665 Box 6488 4732 Box 6860 4688 Box 6348 763 Box stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument endTexFig 2301 2019 a Fu(Figure)20 b(4:)25 b(Loss)c(rate)f(vs.)25 b(sending)19 b(rate)1838 2127 y 16577003 11438130 3289088 3289088 26970521 19866091 startTexFig 1838 2127 a %%BeginDocument: data/senders.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: senders.eps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Fri Nov 9 16:56:11 2001 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -66 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 userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 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 %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 200 scalefont setfont 1.000 UL LTb 1.000 UL LTa 1020 600 M 5840 0 V 1.000 UL LTb 1020 600 M 63 0 V 5777 0 R -63 0 V 900 600 M (0) Rshow 1.000 UL LTa 1020 1067 M 5840 0 V 1.000 UL LTb 1020 1067 M 63 0 V 5777 0 R -63 0 V -5897 0 R (0.05) Rshow 1.000 UL LTa 1020 1533 M 5840 0 V 1.000 UL LTb 1020 1533 M 63 0 V 5777 0 R -63 0 V -5897 0 R (0.1) Rshow 1.000 UL LTa 1020 2000 M 5840 0 V 1.000 UL LTb 1020 2000 M 63 0 V 5777 0 R -63 0 V -5897 0 R (0.15) Rshow 1.000 UL LTa 1020 2467 M 5840 0 V 1.000 UL LTb 1020 2467 M 63 0 V 5777 0 R -63 0 V -5897 0 R (0.2) Rshow 1.000 UL LTa 1020 2933 M 5840 0 V 1.000 UL LTb 1020 2933 M 63 0 V 5777 0 R -63 0 V -5897 0 R (0.25) Rshow 1.000 UL LTa 1020 3400 M 5840 0 V 1.000 UL LTb 1020 3400 M 63 0 V 5777 0 R -63 0 V -5897 0 R (0.3) Rshow 1.000 UL LTa 1020 3867 M 5840 0 V 1.000 UL LTb 1020 3867 M 63 0 V 5777 0 R -63 0 V -5897 0 R (0.35) Rshow 1.000 UL LTa 1020 4333 M 5840 0 V 1.000 UL LTb 1020 4333 M 63 0 V 5777 0 R -63 0 V -5897 0 R (0.4) Rshow 1.000 UL LTa 1020 4800 M 5840 0 V 1.000 UL LTb 1020 4800 M 63 0 V 5777 0 R -63 0 V -5897 0 R (0.45) Rshow 1.000 UL LTa 1020 600 M 0 4200 V 1.000 UL LTb 1020 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (0) Cshow 1.000 UL LTa 2161 600 M 0 3737 V 0 400 R 0 63 V 1.000 UL LTb 2161 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (50) Cshow 1.000 UL LTa 3301 600 M 0 3737 V 0 400 R 0 63 V 1.000 UL LTb 3301 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (100) Cshow 1.000 UL LTa 4442 600 M 0 3737 V 0 400 R 0 63 V 1.000 UL LTb 4442 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (150) Cshow 1.000 UL LTa 5583 600 M 0 4200 V 1.000 UL LTb 5583 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (200) Cshow 1.000 UL LTa 6723 600 M 0 4200 V 1.000 UL LTb 6723 600 M 0 63 V 0 4137 R 0 -63 V 0 -4337 R (250) Cshow 1.000 UL LTb 1020 600 M 5840 0 V 0 4200 V -5840 0 V 0 -4200 V 200 2700 M currentpoint gsave translate 90 rotate 0 0 M (Drop Rate) Cshow grestore 3940 100 M (Number of simultaneous sender-receiver pairs) Cshow 1.200 UP 1.000 UL LT0 3660 4637 M (Sending rate: 1/S) Rshow 3780 4637 M 543 0 V 1248 789 M 228 -17 V 685 10 V 1140 61 V 3422 11 V 137 50 V 1248 789 Pls 1476 772 Pls 2161 782 Pls 3301 843 Pls 6723 854 Pls 4051 4637 Pls 1.200 UP 1.000 UL LT1 3660 4437 M (Sending rate: log S/S) Rshow 3780 4437 M 543 0 V 1248 3488 M 228 390 V 685 -122 V 1140 694 V 6723 3889 L 137 45 V 1248 3488 Crs 1476 3878 Crs 2161 3756 Crs 3301 4450 Crs 6723 3889 Crs 4051 4437 Crs stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument endTexFig 2196 3759 a Fu(Figure)h(5:)25 b(Loss)c(rate)f(vs.)25 b(number)19 b(of)h(senders)1988 4083 y FB(passi)n(v)o(e)f(observ)o(er)f (when)h(more)f(people)g(in)g(the)f(system)h(communicate.)1988 4175 y(W)-6 b(e)28 b(ha)o(v)o(e)h(also)g(e)o(xperimented)h(with)e(dif)n (ferent)h(v)n(alues)g(of)g Ft( )i FB(and)e Ft(\021)s FB(.)1988 4266 y(As)20 b(e)o(xpected,)h(the)f(drop)h(rate)e(increases)i (as)f(users)g(choose)h(higher)f(v)n(al-)1988 4357 y(ues)k(of)f(the)g (security)g(parameters,)i(since)e(the)o(y)g(are)g(mapped)i(to)e(lar)o (ger)1988 4448 y(broadcast)d(groups.)1988 4735 y FC(7)120 b(Conclusions)1988 4911 y FB(W)-6 b(e)19 b(di)n(vide)g(our)h (conclusions)g(for)f FA(P)2964 4879 y Fz(5)3017 4911 y FB(in)g(to)g(tw)o(o)g(parts.)1988 5156 y Fv(7.1)100 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