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%!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: display.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips display.dvi %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2001.02.26:0010 %%BeginProcSet: texc.pro /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N 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Fu(is)35 b(the)i(message)g(c)m(hosen)f(in)0 5280 y(Step)e(1,)h(and)e Ft(I)41 b Fu(is)33 b(the)i(random)e(set)h(c)m (hosen)h(in)d(Step)i(2.)52 b(W)-8 b(e)35 b(sa)m(y)g(the)f(ev)m(en)m(t)h Fn(B)i Fu(o)s(ccurs)c(when)g(the)i(algorithm)d Ft(B)2002 5529 y Fu(8)p eop %%Page: 9 9 9 8 bop 0 63 a Fu(wins)29 b(the)h(signature)g(forgery)g(game)i(b)m(y)e (outputting)f(a)i(forgery)g Ft(S)k Fu(on)30 b Ft(M)41 b Fu(in)29 b(the)h(deriv)m(ed)g(signature)f(sc)m(heme.)42 b(By)0 176 y(assumption)37 b(w)m(e)i(kno)m(w)f(that)h(Pr)o([)p Fn(B)s Fu(])g Ft(>)f(\017)25 b Fu(+)1637 140 y Fr(1)p 1607 155 94 4 v 1607 207 a(2)1642 188 y Fm(m)1711 176 y Fu(.)65 b(No)m(w,)41 b(when)c(b)s(oth)h(ev)m(en)m(ts)h Fn(A)f Fu(and)g Fn(B)j Fu(o)s(ccur,)f(w)m(e)f(deduce)f(the)0 289 y(follo)m(wing:)66 507 y(\(1\))46 b(Since)30 b Ft(S)35 b Fu(is)29 b(a)i(signature)f(forgery)g(for)h Ft(M)40 b Fu(w)m(e)31 b(ha)m(v)m(e)g(that)1416 711 y Fn(8)p Ft(i)25 b Fn(2)g Ft(I)32 b Fu(:)117 b Fi(mac-ver)q Fu(\()p Ft(M)5 b(;)15 b(S;)g Fh(rk)2486 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Fi(mac-gen)r Fu(,)g Fi(mac-ver)q Fu(\))g(is)f(\()p Ft(t)p Fu(,)i Ft(\017)p Fu(,)f Ft(q)s Fu(\)-secure)g(against)h(existen) m(tial)e(forgery)-8 b(.)41 b(Let)0 5237 y Ft(H)h Fu(b)s(e)35 b(the)g(hash)f(function)g(SHA-1)51 b(:)g Fn(f)p Fu(0)p Ft(;)15 b Fu(1)p Fn(g)1595 5204 y Fg(\003)1662 5237 y Fn(!)25 b(f)p Fu(0)p Ft(;)15 b Fu(1)p Fn(g)1998 5204 y Fr(160)2110 5237 y Fu(,)37 b(with)d(securit)m(y)h Ft(\017)2765 5251 y Fs(H)2865 5237 y Fn(\031)3012 5202 y Fr(1)p 2979 5217 101 4 v 2979 5271 a(2)3014 5252 y Fo(80)3089 5237 y Fu(.)55 b(Then)35 b(taking)g Ft(m)e Fu(=)g(240)2002 5529 y(9)p eop %%Page: 10 10 10 9 bop 0 63 a Fu(results)33 b(in)g(a)i(su\016cien)m(tly)d(secure)j (signature)e(sc)m(heme.)53 b(Hence,)37 b(as)d(so)s(on)g(as)h(the)f(MMA) m(C)h(length)f(is)f(sligh)m(tly)g(less)0 176 y(than)d(the)h(n)m(um)m(b) s(er)e(of)h(receiv)m(ers,)h Ft(n)p Fu(,)f(w)m(e)h(obtain)f(an)g (existen)m(tially)f(secure)i(signature)f(sc)m(heme.)0 343 y Fj(Pr)-5 b(o)g(of)43 b(of)f(The)-5 b(or)g(em)43 b(4.1.)233 b Fu(W)-8 b(e)41 b(will)d(establish)g(the)i(con)m(trap)s (ositiv)m(e.)70 b(Supp)s(ose)38 b(w)m(e)j(ha)m(v)m(e)g(a)g(forger)f Ft(B)k Fu(that)0 455 y(pro)s(duces)38 b(successful)f(existen)m(tial)i (forgeries)f(for)h(the)g(deriv)m(ed)f(signature)g(sc)m(heme)i(\()p Fi(sk)m(ey-gen)r Fu(,)h Fi(sig-gen)r Fu(,)g Fi(sig-ver)p Fu(\).)0 568 y(W)-8 b(e)40 b(build)35 b(a)k(MMA)m(C)h(forger)f Ft(A)f Fu(for)h(\()p Fi(k)m(ey-gen)s Fu(,)h Fi(mac-gen)r Fu(,)h Fi(mac-ver)q Fu(\).)66 b(Recall)38 b(that)h(w)m(e)g(mo)s(del)e (securit)m(y)i(as)g(the)0 681 y(probabilit)m(y)25 b(of)j(winning)d(a)j (game)g(against)g(a)g(certain)g(c)m(hallenger.)39 b(W)-8 b(e)29 b(describ)s(e)d(ho)m(w)i(the)g(algorithm)f Ft(A)g Fu(in)m(teracts)0 794 y(with)i(the)i(c)m(hallenger)f(in)f(this)g(game,) j(using)c Ft(B)35 b Fu(as)c(a)g(subroutine.)0 1007 y Fh(Step)k(1:)45 b Fu(The)28 b(algorithm)f Ft(A)h Fu(c)m(ho)s(oses)h(a)g (random)e(subset)g Ft(I)33 b Fu(=)25 b Fn(f)p Ft(i)2307 1021 y Fr(1)2347 1007 y Ft(;)15 b(:)g(:)g(:)h(;)f(i)2579 1021 y Fs(w)2637 1007 y Fn(g)25 b(\022)g(f)p Fu(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(n)p Fn(g)28 b Fu(and)g(sends)f(this)g(to)i(the)227 1120 y(c)m(hallenger,)i(whic)m(h)f(resp)s(onds)f(with)g(\()p Fh(rk)1660 1134 y Fs(i)1684 1143 y Fo(1)1723 1120 y Ft(;)15 b(:)g(:)g(:)h(;)f Fh(rk)2023 1134 y Fs(i)2047 1142 y Fm(w)2101 1120 y Fu(\))31 b(for)g(some)g(\()p Fh(sk)q Ft(;)15 b Fh(rk)2806 1134 y Fr(1)2845 1120 y Ft(;)g(:)g(:)g(:)i(;)e Fh(rk)3145 1134 y Fs(n)3192 1120 y Fu(\))32 b(generated)f(randomly)227 1233 y(b)m(y)g Fi(k)m(ey-gen)r Fu(.)0 1419 y Fh(Step)k(2:)45 b Fu(Algorithm)29 b Ft(A)i Fu(sets)g Ft(K)1157 1434 y Fr(pub)1304 1419 y Fu(=)25 b(\()p Fh(rk)1534 1433 y Fs(i)1558 1442 y Fo(1)1596 1419 y Ft(;)15 b(:)g(:)g(:)i(;)e Fh(rk)1897 1433 y Fs(i)1921 1441 y Fm(w)1975 1419 y Fu(\))30 b(and)g(sends)g Ft(K)2538 1434 y Fr(pub)2690 1419 y Fu(to)h Ft(B)5 b Fu(.)0 1605 y Fh(Step)35 b(3:)45 b Fu(F)-8 b(or)32 b(eac)m(h)g(query)e Ft(M)1076 1619 y Fs(i)1135 1605 y Fu(made)h(b)m(y)g Ft(B)5 b Fu(,)30 b(algorithm)g Ft(A)h Fu(sends)f(the)h(query)f Ft(H)7 b Fu(\()p Ft(M)3008 1619 y Fs(i)3037 1605 y Fu(\))31 b(to)h(the)f(c)m(hallenger.)41 b(Algo-)227 1718 y(rithm)29 b Ft(A)i Fu(then)f(passes)g(the)g(resp)s(onse)g Ft(T)1637 1732 y Fs(i)1695 1718 y Fu(bac)m(k)i(to)f Ft(B)5 b Fu(.)0 1903 y Fh(Step)35 b(4:)45 b Fu(After)40 b(at)g(most)g Ft(q)i Fu(queries,)e Ft(B)k Fu(outputs)39 b(a)h(message)g Ft(M)49 b Fu(and)39 b(a)h(candidate)f(signature)f(forgery)i Ft(S)k Fu(for)227 2016 y Ft(M)10 b Fu(.)51 b(If)34 b Ft(H)7 b Fu(\()p Ft(M)702 2030 y Fs(i)731 2016 y Fu(\))31 b(=)g Ft(H)7 b Fu(\()p Ft(M)j Fu(\))34 b(for)g(some)g Ft(i)d Fn(2)g(f)p Fu(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(q)s Fn(g)p Fu(,)35 b(the)f(algorithm)f Ft(A)h Fu(ab)s(orts)g(the)g(forgery) g(attempt,)i(as)227 2129 y(a)e(collision)c(in)i Ft(H)40 b Fu(has)32 b(b)s(een)h(found.)47 b(Otherwise,)32 b(the)h(algorithm)f Ft(A)h Fu(outputs)g(the)g(pair)f(\()p Ft(H)7 b Fu(\()p Ft(M)j Fu(\))p Ft(;)15 b(S)5 b Fu(\))34 b(as)g(its)227 2242 y(candidate)c(MMA)m(C)i(forgery)-8 b(.)141 2455 y(W)g(e)38 b(claim)d(that)j Ft(A)e Fu(wins)f(the)i(existen)m(tial)f (forgery)h(game)g(for)g(MMA)m(Cs)g(with)e(probabilit)m(y)f(at)k(least)e Ft(\017)p Fu(.)59 b(The)0 2568 y(pro)s(of)33 b(uses)f(the)i(follo)m (wing)d(concept:)48 b(w)m(e)34 b(sa)m(y)g(that)f(a)h(subset)f(of)g (users)g Ft(I)2599 2535 y Fg(0)2652 2568 y Fn(\022)d(f)p Fu(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(n)p Fn(g)34 b Fu(is)e Fj(b)-5 b(ad)35 b Fu(if)d(there)h(is)g(some)0 2681 y Ft(H)76 2695 y Fs(m)167 2681 y Fn(2)25 b(f)p Fu(0)p Ft(;)15 b Fu(1)p Fn(g)473 2648 y Fs(h)551 2681 y Fu(and)29 b(some)i(tag)h Ft(T)38 b Fn(2)25 b(f)p Fu(0)p Ft(;)15 b Fu(1)p Fn(g)1509 2648 y Fs(n)p Fg(\000)p Fs(m)1705 2681 y Fu(suc)m(h)30 b(that)1183 2880 y Fn(8)p Ft(i)25 b Fn(2)g Ft(I)1423 2847 y Fg(0)1472 2880 y Fu(:)g Fi(mac-ver)q Fu(\()p Ft(H)1933 2894 y Fs(m)2000 2880 y Ft(;)15 b(T)8 b(;)15 b Fh(rk)2239 2894 y Fs(i)2267 2880 y Fu(\))26 b(=)f Fi(`y)m(es')q Ft(;)107 b Fu(and)1311 3017 y Fn(8)p Ft(j)30 b Fn(62)25 b Ft(I)1562 2984 y Fg(0)1610 3017 y Fu(:)h Fi(mac-ver)q Fu(\()p Ft(H)2072 3031 y Fs(m)2138 3017 y Ft(;)15 b(T)8 b(;)15 b Fh(rk)2378 3031 y Fs(j)2415 3017 y Fu(\))25 b(=)g Fi(`no')q Ft(:)0 3216 y Fu(That)36 b(is,)h Ft(I)404 3183 y Fg(0)463 3216 y Fu(is)f(bad)f(when)g Ft(I)1033 3183 y Fg(0)1093 3216 y Fu(is)g(precisely)g(the)h(subset)f(of)i(receiv) m(er)f(k)m(eys)h(for)f(whic)m(h)f(some)h(tag)i Ft(T)49 b Fu(v)m(eri\014es)35 b(as)i(a)0 3329 y(v)-5 b(alid)29 b(tag)i(for)f(some)h Ft(H)819 3343 y Fs(m)916 3329 y Fu(in)e(the)h(range)h(of)g(the)f(hash)g(function)f Ft(H)7 b Fu(.)0 3472 y Fh(Lemma)33 b(4.2)46 b Fj(When)32 b Ft(I)40 b Fj(is)33 b(a)g(r)-5 b(andom)35 b(subset)d(of)h Fn(f)p Fu(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(n)p Fn(g)33 b Fj(we)f(have)h(that:) 1606 3709 y Fu(Pr)o([)p Ft(I)40 b Fj(is)33 b(b)-5 b(ad)10 b Fu(])26 b Fn(\024)2283 3647 y Fu(1)p 2202 3688 208 4 v 2202 3771 a(2)2247 3745 y Fs(m)p Fg(\000)p Fs(h)2419 3709 y Ft(:)0 3965 y Fh(Pro)s(of)139 b Fu(W)-8 b(e)43 b(use)f(the)g(b)s(ound)e(of)i(Lemma)g(3.2)h(on)f(the)g(probabilit)m(y)e (that)i(a)g Fj(p)-5 b(air)44 b Fu(\()p Ft(H)3184 3979 y Fs(m)3250 3965 y Ft(;)15 b(I)3337 3932 y Fg(0)3361 3965 y Fu(\))42 b(is)f(bad,)k(for)d(an)m(y)0 4078 y Ft(H)76 4092 y Fs(m)167 4078 y Fn(2)25 b(f)p Fu(0)p Ft(;)15 b Fu(1)p Fn(g)473 4045 y Fs(h)520 4078 y Fu(.)41 b(W)-8 b(e)32 b(obtain)d(the)i(follo)m(wing:)574 4420 y(Pr)454 4485 y Fs(I)5 b Fg(\022f)p Fr(1)p Fs(;:::)n(;n)p Fg(g)791 4420 y Fu([)p Ft(I)38 b Fu(is)29 b(bad])83 b(=)203 b(Pr)1394 4485 y Fs(I)5 b Fg(\022f)p Fr(1)p Fs(;:::)n(;n)p Fg(g)1731 4420 y Fu([)p Fn(9)p Ft(H)1883 4434 y Fs(m)1974 4420 y Fn(2)25 b(f)p Fu(0)p Ft(;)15 b Fu(1)p Fn(g)2280 4382 y Fs(h)2418 4420 y Fu(s.t.)92 b(\()p Ft(H)2742 4434 y Fs(m)2808 4420 y Ft(;)15 b(I)7 b Fu(\))31 b(is)f(bad)o(])1240 4675 y Fn(\024)1510 4589 y Fl(X)1394 4794 y Fs(H)1452 4802 y Fm(m)1510 4794 y Fg(2f)p Fr(0)p Fs(;)p Fr(1)p Fg(g)1717 4775 y Fm(h)1893 4675 y Fu(Pr)1773 4740 y Fs(I)5 b Fg(\022f)p Fr(1)p Fs(;:::)o(;n)p Fg(g)2111 4675 y Fu([\()p Ft(H)2247 4689 y Fs(m)2314 4675 y Ft(;)15 b(I)7 b Fu(\))31 b(is)e(bad])c Fn(\024)g Fu(2)2896 4638 y Fs(h)2956 4547 y Fl(\022)3067 4614 y Fu(1)p 3033 4654 113 4 v 3033 4738 a(2)3078 4711 y Fs(m)3155 4547 y Fl(\023)3248 4675 y Fu(=)3435 4614 y(1)p 3353 4654 208 4 v 3353 4738 a(2)3398 4711 y Fs(m)p Fg(\000)p Fs(h)3571 4675 y Ft(;)0 4977 y Fu(as)31 b(desired.)p 4011 4977 40 40 v 0 5167 a(W)-8 b(e)37 b(can)e(no)m(w)g(complete)h(the)g(pro)s(of)e(Theorem)h(4.1.)57 b(Supp)s(ose)34 b(there)h(is)g(a)g(forger)h Ft(B)k Fu(for)35 b(the)g(deriv)m(ed)g(signature)0 5280 y(sc)m(heme)43 b(\()p Fi(sk)m(ey-gen)s Fu(,)j Fi(sig-gen)q Fu(,)g Fi(sig-ver)q Fu(\))c(that)i(runs)d(in)g(time)h Ft(t)p Fu(,)k(mak)m(es)e Ft(q)h Fu(queries,)g(and)d(pro)s(duces)f(a)i(successful)1980 5529 y(10)p eop %%Page: 11 11 11 10 bop 0 79 a Fu(existen)m(tial)27 b(forgery)g(with)f(probabilit)m (y)f(at)j(least)f Ft(\017)14 b Fu(+)1937 43 y Fr(1)p 1865 58 179 4 v 1865 115 a(2)1900 96 y Fm(m)p Fq(\000)p Fm(h)2067 79 y Fu(+)g Ft(\017)2189 93 y Fs(H)2256 79 y Fu(.)39 b(W)-8 b(e)29 b(claim)d(algorithm)g Ft(A)h Fu(describ)s(ed)e(ab)s(o)m(v)m(e)k(wins)0 192 y(the)i(existen)m(tial)e (forgery)i(game)g(for)g(the)f(MMA)m(C)h(\()p Fi(k)m(ey-gen)s Fu(,)g Fi(mac-gen)q Fu(,)g Fi(mac-ver)q Fu(\))f(with)f(probabilit)m(y)f (at)j(least)g Ft(\017)p Fu(.)141 334 y(W)-8 b(e)31 b(sa)m(y)f(the)g(ev) m(en)m(t)h Fn(A)e Fu(o)s(ccurs)g(when)f(the)i(set)g Ft(I)36 b Fu(c)m(hosen)30 b(in)f(Step)g(1)h(of)f(algorithm)g Ft(A)g Fu(is)g Fh(not)g Fu(bad.)40 b(W)-8 b(e)31 b(sa)m(y)f(the)0 447 y(ev)m(en)m(t)38 b Fn(B)g Fu(o)s(ccurs)e(when)g(the)g(algorithm)f Ft(A)i Fu(do)s(es)f Fh(not)g Fu(ab)s(ort)g(in)f(Step)h(4.)59 b(Finally)-8 b(,)36 b(w)m(e)h(sa)m(y)g(the)f(ev)m(en)m(t)i Fn(C)j Fu(o)s(ccurs)0 560 y(when)26 b(the)g(algorithm)g Ft(B)31 b Fu(wins)25 b(the)h(existen)m(tial)g(forgery)h(game)h(b)m(y)e (outputting)g(a)h(forgery)f Ft(S)32 b Fu(on)26 b Ft(M)37 b Fu(in)25 b(the)i(deriv)m(ed)0 673 y(signature)j(sc)m(heme.)41 b(By)31 b(assumption)e(w)m(e)h(kno)m(w)h(that)g(Pr)o([)p Fn(C)5 b Fu(])26 b Fn(\025)f Ft(\017)20 b Fu(+)2485 637 y Fr(1)p 2413 652 V 2413 709 a(2)2448 690 y Fm(m)p Fq(\000)p Fm(h)2621 673 y Fu(+)g Ft(\017)2749 687 y Fs(H)2816 673 y Fu(.)141 816 y(No)m(w,)31 b(when)f(ev)m(en)m(ts)i Fn(A)p Fu(,)e Fn(B)s Fu(,)g(and)g Fn(C)35 b Fu(hold,)29 b(w)m(e)i(deduce)f (the)h(follo)m(wing:)378 951 y(\(1\))156 b Fn(8)p Ft(i)26 b Fn(2)e Ft(I)33 b Fu(:)115 b Fi(mac-ver)q Fu(\()p Ft(H)7 b Fu(\()p Ft(M)j Fu(\))p Ft(;)15 b(S;)g Fh(rk)1877 965 y Fs(i)1905 951 y Fu(\))26 b(=)f Fi(`y)m(es')151 b Fu(\()p Ft(S)36 b Fu(is)29 b(a)i(signature)e(forgery)i(for)f Ft(M)10 b Fu(\),)378 1064 y(\(2\))156 b Fn(8)p Ft(i)26 b Fn(2)e Ft(I)33 b Fu(:)115 b Ft(H)7 b Fu(\()p Ft(M)j Fu(\))26 b Fn(6)p Fu(=)f Ft(H)7 b Fu(\()p Ft(M)1634 1078 y Fs(i)1662 1064 y Fu(\))680 b(\()p Ft(A)31 b Fu(do)s(es)f(not)g(ab)s (ort\),)378 1177 y(\(3\))156 b Fn(9)p Ft(j)31 b Fn(62)25 b Ft(I)32 b Fu(:)104 b Fi(mac-ver)q Fu(\()p Ft(H)7 b Fu(\()p Ft(M)j Fu(\))p Ft(;)15 b(S;)g Fh(rk)1877 1191 y Fs(j)1913 1177 y Fu(\))26 b(=)f Fi(`y)m(es')143 b Fu(\(b)m(y)31 b(\(1\))g(and)f(the)g(fact)i(that)2453 1290 y Ft(I)37 b Fu(is)29 b(not)i(bad\).)141 1425 y(But)25 b(the)g(second)g(and)f (third)f(conditions)g(are)i(precisely)e(what)i(is)f(needed)g(for)g Ft(A)h Fu(to)h(win)d(the)i(existen)m(tial)f(forgery)0 1538 y(game)31 b(against)g(a)g(MMA)m(C.)g(So,)g(b)m(y)f(Lemma)h(4.2)g (and)f(the)g(fact)i(that)f Ft(H)37 b Fu(is)29 b(collision-resistan)m (t:)527 1742 y(Pr)o([)p Ft(A)i Fu(wins)e(MMA)m(C)i(forgery)f(game)q(]) 84 b Fn(\025)e Fu(Pr[)p Fn(C)26 b(^)19 b(A)h(^)g(B)s Fu(])1908 1880 y Fn(\025)82 b Fu(Pr[)p Fn(C)5 b Fu(])21 b Fn(\000)f Fu(Pr)o([)p Fn(:A)p Fu(])g Fn(\000)g Fu(Pr)o([)p Fn(:B)s Fu(])1908 2068 y Fn(\025)2061 1940 y Fl(\022)2128 2068 y Ft(\017)g Fu(+)2367 2006 y(1)p 2286 2047 208 4 v 2286 2130 a(2)2331 2104 y Fs(m)p Fg(\000)p Fs(h)2524 2068 y Fu(+)g Ft(\017)2652 2082 y Fs(H)2719 1940 y Fl(\023)2806 2068 y Fn(\000)2988 2006 y Fu(1)p 2907 2047 V 2907 2130 a(2)2952 2104 y Fs(m)p Fg(\000)p Fs(h)3145 2068 y Fn(\000)g Ft(\017)3273 2082 y Fs(H)3365 2068 y Fu(=)25 b Ft(\017:)0 2317 y Fu(This)30 b(probabilit)m(y)f(is)h(tak)m(en)j(o)m(v)m(er)g(the)f (random)e(coin)i(\015ips)d(of)j(the)g(c)m(hallenger)f(and)g(of)g(the)h (algorithms)f Ft(A)g Fu(and)g Ft(B)5 b Fu(.)0 2430 y(Th)m(us,)30 b(the)g(theorem)h(follo)m(ws.)p 4011 2430 40 40 v 141 2573 a(Note)39 b(that)g(the)f(construction)g(of)g(the)g(signature)f(sc) m(heme)i(ab)s(o)m(v)m(e)g(made)f(use)g(of)g(a)g(collision)e(resistan)m (t)i(hash)0 2686 y(function.)56 b(The)35 b(pro)s(of)h(can)g(b)s(e)f (easily)g(mo)s(di\014ed)f(to)j(only)e(use)g(one)h(w)m(a)m(y)h(univ)m (ersal)d(hashing)h(\(O)m(WUHF\).)j(Since)0 2799 y(O)m(WUHF's)32 b(can)e(b)s(e)g(constructed)g(from)g(one-w)m(a)m(y)i(functions,)d (there)i(is)e(no)h(need)g(to)h(rely)f(on)g(collision)e(resistance.)0 3115 y Fv(5)135 b(Coalitions)47 b(of)e(Limited)h(Size)0 3348 y Fu(A)34 b(MMA)m(C)i(\()p Fi(k)m(ey-gen)r Fu(,)g Fi(mac-gen)q Fu(,)g Fi(mac-ver)q Fu(\))e(is)g(said)f(to)i(b)s(e)e(\()p Ft(t)p Fj(,)38 b Ft(\017)p Fj(,)e Ft(q)s Fj(,)h Ft(\024)p Fu(\))p Fj(-se)-5 b(cur)g(e)37 b(against)g(sele)-5 b(ctive)36 b(for)-5 b(gery)37 b(under)0 3461 y(an)30 b(adaptive)h(chosen)f (message)f(attack)f Fu(if)e(ev)m(ery)h Ft(t)p Fu(-time)g(probabilistic) c(algorithm)j Ft(A)h Fu(wins)e(the)i(game)h(in)d(Section)i(2)0 3574 y(\(depicted)40 b(in)e(Fig.)j(1\))f(with)f(probabilit)m(y)e(less)j (than)g Ft(\017)p Fu(,)i(where)e(the)g(coalition)f Ft(I)47 b Fu(is)39 b(sub)5 b(ject)40 b(to)h(the)f(constrain)m(t)0 3687 y Fn(j)p Ft(I)7 b Fn(j)27 b(\024)f Ft(\024)p Fu(.)43 b(Similarly)-8 b(,)28 b(\()p Ft(t)p Fu(,)k Ft(\017)p Fu(,)f Ft(q)s Fu(,)g Ft(\024)p Fu(\)-securit)m(y)h(against)g(existen)m (tial)e(forgery)h(is)f(de\014ned)g(as)h(\()p Ft(t)p Fu(,)h Ft(\017)p Fu(,)f Ft(q)s Fu(\)-securit)m(y)h(against)0 3800 y(existen)m(tial)h(forgery)g(where)g(the)g(coalition)f(size)h Fn(j)p Ft(I)7 b Fn(j)34 b Fu(is)e(limited)f(b)m(y)i Ft(\024)p Fu(.)49 b(Note)35 b(that)e(for)g Ft(\024)d Fu(=)g Ft(n)p Fu(,)k(these)f(notions)g(are)0 3913 y(exactly)40 b(the)g(same)g(as)f (those)h(de\014ned)f(in)f(Section)h(2;)45 b(when)38 b Ft(\024)j(<)f(n)p Fu(,)i(the)d(securit)m(y)h(requiremen)m(ts)e(are)i (strictly)0 4025 y(w)m(eak)m(er.)141 4168 y(W)-8 b(e)32 b(sho)m(w)e(in)f(this)g(section)i(that)g(a)g(\()p Ft(t)p Fu(,)f Ft(\017)p Fu(,)h Ft(q)s Fu(,)f Ft(\024)p Fu(\)-secure)i(MMA)m(C) f(with)e(output)h(length)g(less)f(than)1772 4442 y(log)1889 4464 y Fr(2)1989 4328 y Fs(\024)1944 4356 y Fl(X)1952 4551 y Fs(i)p Fr(=0)2090 4314 y Fl(\022)2157 4381 y Ft(n)2169 4504 y(i)2212 4314 y Fl(\023)0 4723 y Fu(giv)m(es)i(rise)e(to)i(a)g (signature)f(sc)m(heme)h(of)f(nearly)g(equiv)-5 b(alen)m(t)30 b(securit)m(y)-8 b(.)141 4866 y(Let)24 b(\()p Fi(k)m(ey-gen)s Fu(,)h Fi(mac-gen)q Fu(,)g Fi(mac-ver)q Fu(\))f(b)s(e)e(a)i(MMA)m(C)g (that)g(is)f(\()p Ft(t)p Fu(,)i Ft(\017)p Fu(,)g Ft(q)s Fu(,)g Ft(\024)p Fu(\)-secure)f(against)g(selectiv)m(e)g(forgery)f (under)0 4979 y(an)31 b(adaptiv)m(e)h(c)m(hosen)f(message)i(attac)m(k.) 45 b(De\014ne)32 b(the)f(deriv)m(ed)f(signature)h(sc)m(heme)h(\()p Fi(sk)m(ey-gen)r Fu(,)g Fi(sig-gen)q Fu(,)g Fi(sig-ver)p Fu(\))g(as)0 5092 y(in)27 b(Section)i(3,)g(with)f(the)h(mo)s (di\014cation)e(that)i Fi(sk)m(ey-gen)r Fu(\()p Ft(s;)15 b(n)p Fu(\))29 b(pic)m(ks)f(a)h(random)f(subset)g Ft(I)k Fn(\022)25 b(f)p Fu(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(n)p Fn(g)29 b Fu(sub)5 b(ject)28 b(to)0 5205 y(the)j(constrain)m(t)f Fn(j)p Ft(I)7 b Fn(j)26 b(\024)f Ft(\024)p Fu(.)1980 5529 y(11)p eop %%Page: 12 12 12 11 bop 141 68 a Fu(Supp)s(ose)29 b(the)h(length)g(of)h(the)f(output) g(of)h Fi(mac-gen)q Fu(\()p Ft(M)10 b Fu(\))31 b(is)f(b)s(ounded)e(b)m (y)1545 336 y Ft(\034)35 b Fu(:=)1741 180 y Fl(")1794 336 y Fu(log)1972 222 y Fs(\024)1927 249 y Fl(X)1935 445 y Fs(i)p Fr(=0)2073 208 y Fl(\022)2140 274 y Ft(n)2152 398 y(i)2195 208 y Fl(\023)2262 180 y(#)2335 336 y Fn(\000)20 b Ft(m)0 603 y Fu(for)30 b(all)f Ft(M)10 b Fu(.)41 b(Then)30 b(w)m(e)g(sho)m(w:)0 746 y Fh(Theorem)k(5.1)46 b Fj(The)29 b(derive)-5 b(d)30 b(signatur)-5 b(e)30 b(scheme)g Fu(\()p Fi(sk)m(ey-gen)s Fj(,)f Fi(sig-gen)q Fj(,)h Fi(sig-ver)q Fu(\))f Fj(is)g Fu(\()p Ft(t)p Fj(,)h Ft(\017)13 b Fu(+)3255 710 y Fr(1)p 3226 725 94 4 v 3226 777 a(2)3261 758 y Fm(m)3330 746 y Fj(,)30 b Ft(q)s Fu(\))p Fj(-se)-5 b(cur)g(e)29 b(against)0 859 y(sele)-5 b(ctive)32 b(for)-5 b(gery)34 b(under)f(an)g(adaptive)h(chosen)g(message)f(attack.)0 1001 y Fu(The)i(pro)s(of)g(follo)m(ws)f(that)j(of)e(Theorem)h(3.1.)57 b(Because)37 b(of)f(the)f(restriction)g(on)g(the)h(size)f(of)h(the)g (coalition)f Ft(I)7 b Fu(,)37 b(the)0 1114 y(follo)m(wing)29 b(alternativ)m(e)i(to)g(Lemma)f(3.2)i(is)d(required.)0 1257 y Fh(Lemma)k(5.2)46 b Fj(F)-7 b(or)33 b(any)h(\014xe)-5 b(d)33 b(message)g Ft(M)10 b Fj(,)1551 1481 y Fu(Pr[\()p Ft(M)5 b(;)15 b(I)7 b Fu(\))34 b Fj(is)e(b)-5 b(ad)11 b Fu(])25 b Fn(\024)2385 1419 y Fu(1)p 2351 1460 113 4 v 2351 1543 a(2)2396 1517 y Fs(m)2473 1481 y Ft(:)0 1697 y Fj(wher)-5 b(e)34 b(the)f(pr)-5 b(ob)g(ability)35 b(is)d(over)h(the)g(choic)-5 b(e)33 b(of)g(a)g(r)-5 b(andom)35 b(c)-5 b(o)g(alition)35 b Ft(I)d Fn(\022)25 b(f)p Fu(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(n)p Fn(g)33 b Fj(satisfying)g Fn(j)p Ft(I)7 b Fn(j)26 b Ft(<)f(\024)p Fj(.)0 1840 y Fh(Pro)s(of)103 b Fu(F)-8 b(or)31 b(eac)m(h)h(tag)g Ft(T)39 b Fn(2)26 b(f)p Fu(0)p Ft(;)15 b Fu(1)p Fn(g)1280 1807 y Fs(\034)1325 1840 y Fu(,)31 b(there)h(is)d(exactly)j(one)f(set)h Ft(I)2370 1854 y Fs(T)2456 1840 y Fu(con)m(taining)e(precisely)f(those) j(receiv)m(ers)f Ft(i)g Fu(for)0 1953 y(whic)m(h)e Fi(mac-ver)q Fu(\()p Ft(M)5 b(;)15 b(T)8 b(;)15 b Fh(rk)928 1967 y Fs(i)956 1953 y Fu(\))26 b(=)f Fi(`y)m(es')q Fu(.)41 b(By)30 b(de\014nition,)f(the)h(pair)f(\()p Ft(M)5 b(;)15 b(I)2467 1967 y Fs(T)2523 1953 y Fu(\))31 b(is)e(bad.)40 b(The)30 b(collection)1571 2143 y Fn(f)p Fu(\()p Ft(M)5 b(;)15 b(I)1824 2157 y Fs(T)1880 2143 y Fu(\))26 b Fn(j)f Ft(T)38 b Fn(2)25 b(f)p Fu(0)p Ft(;)15 b Fu(1)p Fn(g)2388 2106 y Fs(\034)2434 2143 y Fn(g)0 2333 y Fu(completely)25 b(describ)s(es)f(all)g(bad)h(pairs)f(con)m(taining)h Ft(M)36 b Fu(in)24 b(the)h(\014rst)g(co)s(ordinate.)39 b(Since)24 b(there)i(are)g(only)e(2)3669 2300 y Fs(\034)3738 2333 y Fu(p)s(ossible)0 2446 y(v)-5 b(alues)30 b(for)g Ft(T)13 b Fu(,)30 b(this)f(set)i(is)f(of)g(size)h(at)g(most)1649 2701 y(2)1694 2664 y Fs(\034)1763 2701 y Fu(=)25 b(2)1904 2664 y Fg(\000)p Fs(m)2086 2587 y(\024)2041 2615 y Fl(X)2049 2810 y Fs(i)p Fr(=0)2187 2573 y Fl(\022)2254 2640 y Ft(n)2266 2763 y(i)2309 2573 y Fl(\023)2376 2701 y Ft(:)0 2968 y Fu(Since)k Ft(I)38 b Fu(is)29 b(c)m(hosen)i(indep)s(enden)m(tly)c(of) k Ft(M)10 b Fu(,)31 b(it)f(follo)m(ws)f(that)1205 3216 y(Pr)973 3310 y Ft(I)k Fn(\022)25 b(f)p Fu(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(n)p Fn(g)1118 3423 y(j)p Ft(I)7 b Fn(j)26 b(\024)f Ft(\024)1591 3216 y Fu([\()p Ft(M)5 b(;)15 b(I)7 b Fu(\))32 b(is)d(bad])c Fn(\024)2292 3149 y Fu(2)2337 3116 y Fg(\000)p Fs(m)2474 3081 y Fl(P)2570 3107 y Fs(\024)2570 3176 y(i)p Fr(=0)2704 3075 y Fl(\000)2745 3108 y Fs(n)2755 3180 y(i)2788 3075 y Fl(\001)p 2292 3195 538 4 v 2383 3215 a(P)2479 3241 y Fs(\024)2479 3310 y(i)p Fr(=0)2613 3210 y Fl(\000)2654 3243 y Fs(n)2664 3315 y(i)2697 3210 y Fl(\001)2865 3216 y Fu(=)3004 3155 y(1)p 2971 3195 113 4 v 2971 3278 a(2)3016 3252 y Fs(m)3093 3216 y Ft(;)0 3615 y Fu(establishing)j(the)j(lemma.)p 4011 3615 40 40 v 141 3805 a(With)f(this)f(lemma)h(in)f(place,)i(Theorem)f(5.1)h (follo)m(ws)f(just)g(as)g(Theorem)g(3.1.)141 3948 y(An)25 b(analogous)h(theorem)g(ma)m(y)g(b)s(e)f(sho)m(wn)g(for)g(securit)m(y)g (against)h(existen)m(tial)f(forgery)-8 b(.)40 b(Let)26 b(\()p Fi(k)m(ey-gen)s Fu(,)g Fi(mac-gen)r Fu(,)0 4061 y Fi(mac-ver)q Fu(\))36 b(b)s(e)f(a)h(MMA)m(C)h(that)f(is)f(\()p Ft(t)p Fu(,)j Ft(\017)p Fu(,)f Ft(q)s Fu(,)g Ft(\024)p Fu(\)-secure)g(against)f(existen)m(tial)f(forgery)h(under)f(an)g (adaptiv)m(e)h(c)m(hosen)0 4174 y(message)f(attac)m(k.)55 b(De\014ne)35 b(the)f(deriv)m(ed)g(signature)f(sc)m(heme)j(\()p Fi(sk)m(ey-gen)r Fu(,)g Fi(sig-gen)q Fu(,)f Fi(sig-ver)q Fu(\))g(as)f(in)f(Section)h(3,)i(with)0 4287 y(the)i(mo)s(di\014cation) f(that)h Fi(sk)m(ey-gen)r Fu(\()p Ft(k)s(;)15 b(n)p Fu(\))39 b(pic)m(ks)f(a)g(random)f(subset)h Ft(I)45 b Fn(\022)37 b(f)p Fu(1)p Ft(;)15 b(:)g(:)g(:)j(;)d(n)p Fn(g)38 b Fu(sub)5 b(ject)38 b(to)h(the)f(constrain)m(t)0 4400 y Fn(j)p Ft(I)7 b Fn(j)26 b(\024)f Ft(\024)p Fu(.)141 4543 y(Supp)s(ose)i(the)h(MMA)m(C)h(\()p Fi(k)m(ey-gen)s Fu(,)g Fi(mac-gen)q Fu(,)g Fi(mac-ver)q Fu(\))f(is)f(\()p Ft(t)p Fu(,)i Ft(\017)p Fu(,)g Ft(q)s Fu(\)-secure)g(against)f(existen) m(tial)g(forgery)g(under)0 4655 y(an)33 b(adaptiv)m(e)h(c)m(hosen)g (message)g(attac)m(k,)j(and)c(supp)s(ose)f(the)i(length)e(of)i(the)g (output)f(of)g Fi(mac-gen)r Fu(\()p Ft(M)10 b Fu(\))34 b(is)e(b)s(ounded)0 4768 y(b)m(y)51 b Ft(\034)35 b Fu(=)25 b(\(log)486 4700 y Fl(P)582 4726 y Fs(\024)582 4795 y(i)p Fr(=0)715 4695 y Fl(\000)757 4728 y Fs(n)766 4800 y(i)800 4695 y Fl(\001)841 4768 y Fu(\))10 b Fn(\000)g Ft(m)51 b Fu(for)26 b(all)e Ft(M)10 b Fu(.)39 b(F)-8 b(urthermore)25 b(assume)h(that)g Ft(H)32 b Fu(is)24 b(a)i(collision-resistan)m(t)e (hash)g(function)0 4881 y(with)29 b(securit)m(y)h(parameter)h Ft(\017)1018 4895 y Fs(H)1085 4881 y Fu(.)41 b(Then)29 b(one)i(can)g(sho)m(w)0 5024 y Fh(Theorem)j(5.3)46 b Fj(The)31 b(derive)-5 b(d)31 b(signatur)-5 b(e)32 b(scheme)f Fu(\()p Fi(sk)m(ey-gen)r Fj(,)g Fi(sig-gen)q Fj(,)g Fi(sig-ver)q Fu(\))g Fj(is)f Fu(\()p Ft(t)p Fj(,)h Ft(\017)16 b Fu(+)3316 4988 y Fr(1)p 3244 5003 179 4 v 3244 5060 a(2)3279 5041 y Fm(m)p Fq(\000)p Fm(h)3448 5024 y Fu(+)g Ft(\017)3572 5038 y Fs(H)3639 5024 y Fj(,)31 b Ft(q)s Fu(\))p Fj(-se)-5 b(cur)g(e)0 5137 y(against)34 b(sele)-5 b(ctive)32 b(for)-5 b(gery)34 b(under)f(an)g(adaptive)h(chosen)f(message)g(attack.)0 5280 y Fu(The)d(pro)s(of)f(is)h(similar)e(to)j(the)f(pro)s(of)g(of)g (Theorem)h(4.1)g(with)e(the)i(appropriate)e(mo)s(di\014cation)g(to)i (Lemma)f(4.2.)1980 5529 y(12)p eop %%Page: 13 13 13 12 bop 0 83 a Fv(6)135 b(Conclusions)0 316 y Fu(W)-8 b(e)45 b(ga)m(v)m(e)h(precise)d(de\014nitions)e(for)j(Multicast)f(MA)m (Cs)i(\(MMA)m(Cs\))g(secure)f(against)g(selectiv)m(e)g(and)f(existen)m (tial)0 429 y(forgeries.)38 b(Our)23 b(main)f(results)h(sho)m(w)h(that) g(a)g(short)g(collusion-resistan)m(t)e(m)m(ulticast)h(MA)m(C)h(can)g(b) s(e)g(easily)e(con)m(v)m(erted)0 542 y(in)m(to)32 b(a)h(signature)e(sc) m(heme.)47 b(This)30 b(sho)m(ws)i(a)g(gap)h(b)s(et)m(w)m(een)g(the)f (cryptographic)f(resources)i(needed)e(for)h(t)m(w)m(o)i(part)m(y)0 655 y(MA)m(Cs)e(\(where)g(signatures)g(are)g(not)g(needed\))g(and)g (the)g(resources)g(needed)g(for)g(Multicast)f(MA)m(Cs.)46 b(Our)31 b(b)s(ounds)0 768 y(justify)d(the)h(recen)m(t)h(e\013ort)g(in) m(to)f(designing)e(signature)i(sc)m(hemes)h(for)e(a)i(m)m(ulticast)f (en)m(vironmen)m(t)f([5)q(,)h(12)q(,)h(10)q(,)f(7,)h(12].)0 881 y(Suc)m(h)j(sc)m(hemes)h(require)e(minimal)f(bu\013ering)g(on)j (the)g(sender's)e(side)h(and)g(resist)f(pac)m(k)m(et)k(loss.)49 b(W)-8 b(e)35 b(also)f(note)g(the)0 993 y(constructions)39 b(of)g([10)q(,)h(11)q(])f(that)h(pro)m(vide)e(a)i(short)f(MMA)m(C)h (without)f(non-repudiation)d(b)m(y)k(using)d(some)j(w)m(eak)0 1106 y(timing)29 b(assumptions.)141 1249 y(F)-8 b(or)41 b(small)d(v)-5 b(alues)40 b(of)g Ft(\024)p Fu(,)j(our)c(lo)m(w)m(er)h (b)s(ound)e(for)i Ft(\024)p Fu(-secure)h(MMA)m(Cs)f(asymptotically)g (matc)m(hes)h(the)f(upp)s(er)0 1362 y(b)s(ound)26 b(construction)i(of)h (Canetti)f(et)h(al.)f([1)q(].)40 b(Hence,)30 b(the)e(Canetti)g(et)h (al.)40 b(construction)28 b(has)g(optimal)f(length)h(\(up)0 1475 y(to)j(a)g(small)e(constan)m(t)i(factor\))h(for)e(a)h(MMA)m(C)g (that)g(is)f(based)g(purely)e(on)j(pseudorandom)d(functions.)0 1791 y Fv(References)215 2024 y Fu([1])46 b(R.)33 b(Canetti,)h(J.)f (Gara)m(y)-8 b(,)36 b(G.)e(Itkis,)f(D.)h(Micciancio,)f(M.)h(Naor,)h (and)d(B.)i(Pink)-5 b(as,)48 b(\\Multicast)33 b(Securit)m(y:)356 2137 y(A)f(taxonom)m(y)h(and)d(some)i(e\016cien)m(t)g(constructions",) 45 b(in)30 b(IEEE)h(INF)m(OCOM)g('99,)j(v)m(ol.)d(2,)i(pp.)e(708{716,) 356 2250 y(1999.)215 2438 y([2])46 b(A.)28 b(De)h(San)m(tis)e(and)h(M.) g(Y)-8 b(ung,)37 b(\\On)28 b(the)g(design)f(of)h(pro)m(v)-5 b(ably-secure)27 b(cryptographic)h(hash)f(functions",)356 2551 y(in)i(Pro)s(c.)i(of)f(Euro)s(crypt)f('90,)j(LNCS)d(473,)j(pp.)e (412{431,)k(1990.)215 2738 y([3])46 b(Y.)35 b(Desmedt,)i(Y.)e(F)-8 b(rank)m(el,)36 b(and)e(M.)i(Y)-8 b(ung,)54 b(\\Multi-receiv)m (er/Multi-sender)34 b(net)m(w)m(ork)h(securit)m(y:)50 b(e\016-)356 2851 y(cien)m(t)31 b(authen)m(ticated)g(m)m (ulticast/feedbac)m(k",)42 b(in)29 b(IEEE)h(INF)m(OCOM)g('92,)i(pp.)e (2045{2054,)k(1992.)215 3039 y([4])46 b(F.)d(F)-8 b(ujii,)45 b(W.)e(Kac)m(hen,)j(and)c(K.)h(Kurosa)m(w)m(a,)81 b(\\Com)m(binatorial) 41 b(b)s(ounds)g(and)h(design)g(of)g(broadcast)356 3152 y(authen)m(tication",)f(in)29 b(IEICE)h(T)-8 b(rans.,)30 b(v)m(ol.)h(E79-A,)h(no.)e(4,)h(pp.)f(502{506,)j(1996.)215 3340 y([5])46 b(R.)31 b(Gennaro)f(and)g(P)-8 b(.)31 b(Rohatgi,)41 b(\\Ho)m(w)31 b(to)g(sign)f(digital)f(streams",)41 b(in)29 b(Pro)s(c.)h(of)h(Crypto)f('97,)h(1997.)215 3527 y([6])46 b(S.)30 b(Goldw)m(asser,)h(S.)g(Micali,)e(and)i(R.)f(Riv)m(est,)42 b(\\A)31 b(digital)e(signature)h(sc)m(heme)i(secure)f(against)g (adaptiv)m(e)356 3640 y(c)m(hosen-message)h(attac)m(ks",)44 b(SIAM)30 b(Journal)f(of)h(Computing,)f(v)m(ol.)i(17,)g(pp.)f(281{308,) k(1988.)215 3828 y([7])46 b(P)-8 b(.)29 b(Golle,)h(N.)f(Mo)s(dadugo,)39 b(\\Streamed)29 b(authen)m(tication)g(in)f(the)h(presence)g(of)g (random)f(pac)m(k)m(et)j(loss",)38 b(in)356 3941 y(Pro)s(c.)24 b(of)g(8th)g(Ann)m(ual)f(In)m(ternet)i(So)s(ciet)m(y)f(Symp)s(osium)d (on)j(Net)m(w)m(ork)h(and)f(Distributed)e(System)i(Securit)m(y)356 4053 y(\(NDSS)30 b('01\),)i(San)e(Diego,)i(2001.)215 4241 y([8])46 b(K.)31 b(Kurosa)m(w)m(a,)i(S.)e(Obana,)43 b(\\Characterization)32 b(of)f(\()p Ft(k)s(;)15 b(n)p Fu(\))32 b(m)m(ulti-receiv)m(er)f(authen)m(tication",)45 b(in)30 b(Infor-)356 4354 y(mation)g(Securit)m(y)g(and)g(Priv)-5 b(acy)d(,)30 b(A)m(CISP)g('97,)i(LNCS)d(1270,)k(pp.)c(205{215,)34 b(1997.)215 4542 y([9])46 b(M.)41 b(Naor)f(and)g(M.)g(Y)-8 b(ung,)72 b(\\Univ)m(ersal)40 b(one-w)m(a)m(y)i(hash)d(functions)g(and) g(their)g(cryptographic)h(appli-)356 4655 y(cations",)73 b(in)39 b(Pro)s(c.)h(of)h(21st)g(Ann)m(ual)e(A)m(CM)i(Symp)s(osium)c (on)j(Theory)g(of)h(Computing,)g(pp.)e(33{43,)356 4767 y(1989.)169 4955 y([10])47 b(A.)23 b(P)m(errig,)h(R.)f(Canetti,)h(D.)f (T)m(ygar,)i(D.)f(Song,)29 b(\\E\016cien)m(t)23 b(Authen)m(tication)g (and)f(Signature)f(of)i(Multicast)356 5068 y(Streams)33 b(o)m(v)m(er)i(Lossy)e(Channels",)49 b(in)32 b(Pro)s(c.)i(of)g(2000)h (IEEE)d(Symp)s(osium)f(on)i(Securit)m(y)g(and)g(Priv)-5 b(acy)d(,)356 5181 y(Oakland,)29 b(2000.)1980 5529 y(13)p eop %%Page: 14 14 14 13 bop 169 68 a Fu([11])47 b(A.)37 b(P)m(errig,)h(R.)f(Canetti,)h (D.)g(Song,)g(D.)f(T)m(ygar,)62 b(\\E\016cien)m(t)37 b(and)f(Secure)g(Source)h(Authen)m(tication)f(for)356 181 y(Multicast",)41 b(in)29 b(Pro)s(c.)h(of)h(8th)f(Ann)m(ual)f(In)m (ternet)i(So)s(ciet)m(y)g(Symp)s(osium)c(on)j(Net)m(w)m(ork)i(and)e (Distributed)356 294 y(System)g(Securit)m(y)g(\(NDSS)g('01\),)i(San)e (Diego,)i(2001.)169 482 y([12])47 b(P)-8 b(.)22 b(Rohatgi,)28 b(\\A)21 b(compact)i(and)d(fast)i(h)m(ybrid)d(signature)h(sc)m(heme)i (for)f(m)m(ulticast)g(pac)m(k)m(et)i(authen)m(tication",)356 595 y(in)29 b(Pro)s(c.)i(of)f(6th)h(A)m(CM)g(conference)g(on)f (Computer)g(and)f(Comm)m(unication)g(Securit)m(y)-8 b(,)31 b(1999.)169 782 y([13])47 b(J.)27 b(Romp)s(el,)34 b(\\One-w)m(a)m(y)28 b(functions)d(are)i(necessary)g(and)f(su\016cien)m(t)g(for)g(secure)h (signatures",)35 b(in)25 b(Pro)s(c.)i(of)356 895 y(22nd)j(Ann)m(ual)f (A)m(CM)i(Symp)s(osium)d(on)i(Theory)g(of)h(Computing,)e(pp.)g (387{394,)34 b(1990.)169 1083 y([14])47 b(R.)22 b(Safa)m(vi-Naini,)h (H.)g(W)-8 b(ang,)30 b(\\Multireceiv)m(er)22 b(authen)m(tication)g(co)s (des:)37 b(mo)s(dels,)23 b(b)s(ounds,)e(constructions)356 1196 y(and)30 b(extensions",)40 b(Information)30 b(and)f(Computation,)h (v)m(ol.)h(151,)h(no.)e(1/2,)i(pp.)e(148{172,)k(1999.)169 1383 y([15])47 b(R.)34 b(Safa)m(vi-Naini,)f(H.)h(W)-8 b(ang,)53 b(\\New)34 b(results)f(on)g(m)m(ultireceiv)m(er)g(authen)m (tication)h(co)s(des",)52 b(in)32 b(Pro)s(c.)i(of)356 1496 y(Euro)s(crypt)29 b('98,)j(LNCS)d(1403,)k(pp.)c(527{541,)34 b(1998.)169 1684 y([16])47 b(G.)42 b(Simmons,)74 b(\\A)41 b(cartesian)h(pro)s(duct)e(construction)h(for)g(unconditionally)c (secure)42 b(authen)m(tication)356 1797 y(co)s(des)31 b(that)f(p)s(ermit)f(arbitration",)40 b Fj(J.)32 b(Cryptolo)-5 b(gy)p Fu(,)33 b(v)m(ol.)e(2,)g(no.)g(2,)g(pp.)e(77{104,)k(1990.)169 1984 y([17])47 b(C.)30 b(K.)g(W)-8 b(ong,)32 b(S.)d(S.)h(Lam,)41 b(\\Digital)30 b(signatures)f(for)h(\015o)m(ws)g(and)f(m)m(ulticasts",) 40 b(IEEE)30 b(ICNP)g('98.)h(Also,)356 2097 y(Univ)m(ersit)m(y)f(of)g (T)-8 b(exas)31 b(at)g(Austin,)f(Computer)f(Science)h(T)-8 b(ec)m(hnical)30 b(rep)s(ort)g(TR)g(98-15.)1980 5529 y(14)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF