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Fl(is)g(200)h(bits)g(long.)164 1683 y(The)d(device)g(generates)g (primes)h Fk(p)f Fl(of)h(the)f(form)686 1793 y Fk(p)e Fl(=)g Fk(r)q(A)d Fl(+)g Fk(q)k Fl(=)f Fk(r)q Fl(\()p Fk(q)r(;)8 b(A)p Fl(\))p Fk(A)h Fl(+)i Fk(q)r(;)459 b Fl(\(1\))164 1903 y(where)16 b Fk(q)i Fl(is)g(at)f(most)g(ab)q(out)h(100)g(bits)f(long,)h(and)f Fk(r)h Fl(is)g(56)f(bits)h(long)g(and)f(a)g(function)164 1963 y(of)h Fk(A)f Fl(and)h Fk(q)r Fl(.)24 b(T)l(o)18 b(factor)g(the)f(RSA)g(mo)q (dulus)i Fk(n)d Fl(=)g Fk(pp)1194 1945 y Fj(0)1207 1963 y Fl(,)h(the)g(man)o (ufacturer)g(reduces)164 2023 y(the)e(mo)q(dulus)i(mo)q(dulo)g Fk(A)d Fl(to)i(reco)o(v)o(er)e(the)h(pro)q(duct)h Fk(q)r(q)1206 2005 y Fj(0)1216 2023 y Fl(,)f(follo)o(wing)j(the)d(relationship)584 2133 y Fk(n)f Fl(=)g Fk(pp)727 2113 y Fj(0)753 2133 y Fl(=)g Fk(r)q(r)851 2113 y Fj(0)863 2133 y Fk(A)900 2113 y Fi(2)931 2133 y Fl(+)d(\()p Fk(r)q(q)1046 2113 y Fj(0)1068 2133 y Fl(+)g Fk(r)1140 2113 y Fj(0)1152 2133 y Fk(q)r Fl(\))p Fk(A)f Fl(+)h Fk(q)r(q)1339 2113 y Fj(0)1349 2133 y Fk(:)358 b Fl(\(2\))164 2243 y(The)14 b(200-bit)i(pro)q(duct)f Fk(q)r(q)654 2225 y Fj(0)679 2243 y Fl(is)g(easily)g(factored,)f(and)h(the)f(man)o(ufacturer)f (reco)o(v)o(ers)g(the)164 2303 y(primes)j Fk(p)h Fl(and)g Fk(p)481 2285 y Fj(0)509 2303 y Fl(according)h(to)e(Equation)i(1.)p 164 2347 648 2 v 220 2378 a Fh(\003)239 2393 y Fg(RSA)e(Lab)q(oratories,)i (100)e(Marine)h(P)o(arkw)o(a)o(y)m(,)f(Redw)o(o)q(o)q(d)h(Cit)o(y)m(,)f(CA)h (94065.)25 b(Email)15 b(address:)164 2443 y Ff(burt@rsa.com)p Fg(.)961 2599 y Fl(1)p eop %%Page: 2 2 bop 237 307 a Fl(While)18 b(the)e(trap)q(do)q(or)j(is)e(indeed)g(practical,)g (it)g(can)g(b)q(e)g(brok)o(en:)k(F)l(actoring)d(suc)o(h)164 367 y(\\trapp)q(ed")k(mo)q(duli)g(is)f(easy)l(.)34 b(Let)21 b Fk(n)877 374 y Fi(0)897 367 y Fk(;)8 b(:)g(:)g(:)f(;)h(n)1035 374 y Fe(k)1077 367 y Fl(b)q(e)21 b(a)f(set)h(of)g(suc)o(h)f(mo)q(duli,)j (and)e(let)164 428 y Fk(r)186 435 y Fi(0)206 428 y Fk(;)8 b(r)251 409 y Fj(0)250 440 y Fi(0)269 428 y Fk(;)g(:)g(:)g(:)g(;)g(r)401 435 y Fe(k)422 428 y Fk(;)g(r)467 409 y Fj(0)466 440 y Fe(k)502 428 y Fl(b)q(e)14 b(the)h(corresp)q(onding)i(parameters)d(from)h(Equation)h (2.)21 b(It)14 b(is)i(easy)164 488 y(to)h(sho)o(w)f(the)g(follo)o(wing)j (inequalities)g(for)d(the)g(giv)o(en)h(parameter)f(lengths:)597 546 y Fd(\015)597 571 y(\015)597 596 y(\015)597 621 y(\015)620 608 y Fk(r)642 615 y Fi(0)661 608 y Fk(r)684 587 y Fj(0)683 620 y Fi(0)711 574 y Fk(n)740 581 y Fe(i)p 708 596 49 2 v 708 642 a Fk(n)737 649 y Fi(0)773 608 y Fc(\000)11 b Fk(r)845 615 y Fe(i)859 608 y Fk(r)882 587 y Fj(0)881 620 y Fe(i)895 546 y Fd(\015)895 571 y(\015)895 596 y(\015)895 621 y(\015)932 608 y Fc(\024)i Fl(2)1008 587 y Fj(\000)p Fi(41)1074 608 y Fk(;)56 b Fl(1)14 b Fc(\024)g Fk(i)f Fc(\024)h Fk(k)r(:)362 b Fl(\(3\))164 733 y(Suc)o(h)12 b(inequalities)i(are)e(called)h(\\sim)o (ultaneous)h(Diophan)o(tine)f(appro)o(ximations,")i(and)164 793 y(they)k(are)h(classi\014ed)h(as)g(\\un)o(usually)h(go)q(o)q(d")g(if)e (the)f(error)h(term)f(is)i(less)f(than)h Fk(n)1699 767 y Fj(\000)p Fi(1)p Fe(=k)1699 804 y Fi(0)164 853 y Fl([3].)g(F)l(or)16 b(the)g(giv)o(en)g(parameter)g(lengths,)h(this)g(is)g(so)g(when)g Fk(k)h Fl(is)f(13)g(or)g(more.)237 913 y(Giv)o(en)j(a)h(set)f(of)g(mo)q(duli) i(kno)o(wn)e(to)h(ha)o(v)o(e)e(suc)o(h)h(appro)o(ximations,)j(\014nding)e (the)164 973 y(appro)o(ximations)f(is)e(straigh)o(tforw)o(ard.)27 b(F)l(ollo)o(wing)20 b(tec)o(hniques)d(for)h(breaking)g(knap-)164 1034 y(sac)o(k)h(cryptosystems)f(\(see)h([2],)g([3],)f([4]\),)h(one)g (\014nds)h(a)f(set)g(of)g(short)h(v)o(ectors)e(in)i(the)164 1094 y(lattice)d(generated)f(b)o(y)g(the)g(basis)610 1163 y Fd(0)610 1236 y(B)610 1261 y(B)610 1286 y(B)610 1311 y(B)610 1336 y(B)610 1360 y(B)610 1385 y(B)610 1412 y(@)687 1194 y Fk(\025n)744 1201 y Fi(0)871 1194 y Fl(0)104 b(0)89 b Fc(\001)8 b(\001)g(\001)71 b Fl(0)713 1275 y(0)107 b Fk(\025n)901 1282 y Fi(0)999 1275 y Fl(0)89 b Fc(\001)8 b(\001)g(\001)1247 1242 y Fl(.)1247 1258 y(.)1247 1275 y(.)719 1323 y(.)719 1339 y(.)719 1356 y(.)138 b(0)986 1327 y(.)1005 1339 y(.)1024 1352 y(.)1129 1356 y(0)1247 1323 y(.)1247 1339 y(.)1247 1356 y(.)713 1416 y(0)117 b Fc(\001)8 b(\001)g(\001)87 b Fl(0)79 b Fk(\025n)1159 1423 y Fi(0)1241 1416 y Fl(0)668 1476 y Fc(\000)p Fk(\025n)764 1483 y Fi(1)825 1476 y Fc(\000)p Fk(\025n)921 1483 y Fi(2)983 1476 y Fc(\001)8 b(\001)g(\001)41 b(\000)p Fk(\025n)1178 1483 y Fe(k)1241 1476 y Fl(1)1287 1163 y Fd(1)1287 1236 y(C)1287 1261 y(C)1287 1286 y(C)1287 1311 y(C)1287 1336 y(C)1287 1360 y(C)1287 1385 y(C)1287 1412 y(A)1331 1336 y Fk(;)376 b Fl(\(4\))164 1593 y(where)16 b Fk(\025)h Fl(is)g(an)f(in)o(teger)h(near)f Fk(n)765 1568 y Fj(\000)p Fi(1)p Fe(=k)765 1604 y Fi(0)849 1593 y Fl(.)21 b(In)16 b(most)h(cases,)f(the)g(short)h(v)o(ector)405 1654 y Fd(\020)450 1702 y Fk(\025)p Fl(\()p Fk(r)519 1709 y Fi(1)540 1702 y Fk(r)563 1684 y Fj(0)562 1714 y Fi(1)581 1702 y Fk(n)610 1709 y Fi(0)641 1702 y Fc(\000)11 b Fk(r)713 1709 y Fi(0)733 1702 y Fk(r)756 1684 y Fj(0)755 1714 y Fi(0)774 1702 y Fk(n)803 1709 y Fi(1)823 1702 y Fl(\))42 b Fc(\001)8 b(\001)g(\001)41 b Fk(\025)p Fl(\()p Fk(r)1052 1709 y Fe(k)1074 1702 y Fk(r)1097 1684 y Fj(0)1096 1714 y Fe(k)1117 1702 y Fk(n)1146 1709 y Fi(0)1177 1702 y Fc(\000)11 b Fk(r)1249 1709 y Fi(0)1269 1702 y Fk(r)1292 1684 y Fj(0)1291 1714 y Fi(0)1310 1702 y Fk(n)1339 1709 y Fe(k)1361 1702 y Fl(\))41 b Fk(r)1443 1709 y Fi(0)1463 1702 y Fk(r)1486 1684 y Fj(0)1485 1714 y Fi(0)1526 1654 y Fd(\021)1721 1702 y Fl(\(5\))164 1812 y(is)24 b(a)g(mem)o(b)q(er)e(of)i(the)f(set.)42 b(The)23 b(secret)g(v)m(alue)h Fk(A)f Fl(follo)o(ws)i(from)e Fk(r)1489 1819 y Fi(0)1509 1812 y Fk(r)1532 1794 y Fj(0)1531 1824 y Fi(0)1551 1812 y Fl(,)i(since,)f(b)o(y)164 1872 y(Equation)18 b(2,)e(the)g(in)o(teger)g(nearest)g(to)h Fk(n)931 1879 y Fi(0)951 1872 y Fk(=)p Fl(\()p Fk(r)1016 1879 y Fi(0)1036 1872 y Fk(r)1059 1854 y Fj(0)1058 1884 y Fi(0)1078 1872 y Fl(\))f(is)h Fk(A)1199 1854 y Fi(2)1218 1872 y Fl(.)237 1932 y(One)24 b(w)o(a)o(y)g(to)g(o)o(v)o (ercome)e(this)j(attac)o(k)f(is)h(to)f(assign)i(a)f(di\013eren)o(t)f(secret)f (v)m(alue)164 1993 y(to)e(eac)o(h)g(device,)g(a)g(precaution)h(Anderson)f (has)g(suggested)h(for)f(another)g(purp)q(ose.)164 2053 y(Then)c(a)g(user)g (can)g(only)g(factor)h(his)f(or)h(her)e(o)o(wn)h(mo)q(duli.)25 b(The)17 b(user)f(do)q(es)i(not)f(need)164 2113 y(14)h(mo)q(duli)h(to)f (\014nd)g Fk(A)p Fl(,)e(ho)o(w)o(ev)o(er.)24 b(Tw)o(o)17 b(prime)h(factors)g Fk(p)g Fl(and)g Fk(p)1404 2095 y Fj(0)1433 2113 y Fl(su\016ce,)f(since)g(the) 164 2173 y(fraction)24 b Fk(r)374 2155 y Fj(0)386 2173 y Fk(=r)g Fl(is)f(suc)o(h)g(a)g(go)q(o)q(d)i(appro)o(ximation)f(to)f(the)g(fraction)g Fk(p)1499 2155 y Fj(0)1511 2173 y Fk(=p)h Fl(that)f(it)g(is)164 2233 y(guaran)o(teed)d(to)f(b)q(e)h(a)g(con)o(v)o(ergen)o(t)e(in)h(the)h(con) o(tin)o(ued)f(fraction)h(expansion)h(of)e Fk(p)1708 2215 y Fj(0)1720 2233 y Fk(=p)p Fl(.)164 2293 y(The)g(user)g(can)h(therefore)e (detect)h(a)g(trap)q(do)q(or)j(ev)o(en)c(if)h(the)g(device)g(generates)g(eac) o(h)164 2354 y(mo)q(dulus)f(with)f(a)f(di\013eren)o(t)h(secret)e(v)m(alue.) 237 2414 y(The)d(man)o(ufacturer's)g(only)g(recourse,)h(at)f(least)h(as)g (far)f(as)h(the)f(prop)q(osed)h(trap)q(do)q(or)164 2474 y(is)24 b(concerned,)g(is)g(for)f(the)g(device)f(to)i(generate)f(eac)o(h)g(mo)q (dulus)h(with)g(a)g(di\013eren)o(t)961 2599 y(2)p eop %%Page: 3 3 bop 164 307 a Fl(secret)19 b(v)m(alue)i(and)f(to)g(k)o(eep)f(the)g(prime)h (factors)g(secret.)31 b(In)20 b(suc)o(h)f(a)i(situation,)h(the)164 367 y(man)o(ufacturer)f(ma)o(y)h(as)g(w)o(ell)h(preload)g(the)e(device)g (with)i(the)f(primes)g(and)g(escro)o(w)164 428 y(copies|a)14 b(practical)h(\\trap)q(do)q(or")h(to)e(whic)o(h)g(all)h(cryptosystems,)e(not) h(just)g(RSA,)f(are)164 488 y(vulnerable.)164 654 y Fb(References)164 764 y Fl([1])24 b(Ross)f(Anderson.)39 b(A)22 b(practical)i(RSA)e(trap)q(do)q (or.)41 b Fa(Ele)n(ctr)n(onics)24 b(L)n(etters)p Fl(,)f(June)240 824 y(1993.)164 926 y([2])h(E.F.)18 b(Bric)o(k)o(ell)h(and)g(A.M.)f(Odlyzk)o (o.)29 b(Cryptanalysis:)g(A)19 b(surv)o(ey)f(of)h(recen)o(t)f(re-)240 986 y(sults.)k Fa(Pr)n(o)n(c)n(e)n(e)n(dings)16 b(of)i(the)g(IEEE)p Fl(,)d(76:578{593,)k(1988.)164 1087 y([3])24 b(J.C.)18 b(Lagarias.)30 b(Knapsac)o(k)18 b(public)h(k)o(ey)e(cryptosystems)h(and)h(diophan)o(tine)h (ap-)240 1148 y(pro)o(ximation.)k(In)16 b(D.)h(Chaum,)f(editor,)h Fa(A)n(dvanc)n(es)i(in)f(Cryptolo)n(gy:)k(Pr)n(o)n(c)n(e)n(e)n(dings)240 1208 y(of)17 b(CR)l(YPTO)h('83)p Fl(,)d(pages)i(3{23,)h(Plen)o(um)e(Press,)g (New)g(Y)l(ork,)f(1984.)164 1310 y([4])24 b(A.K.)18 b(Lenstra,)i(H.W.)e (Lenstra)j(Jr.,)e(and)h(L.)f(Lo)o(v)m(asz.)31 b(F)l(actoring)21 b(p)q(olynomials)240 1370 y(with)c(rational)h(co)q(e\016cien)o(ts.)j Fa(Math.)c(A)o(nnalen)p Fl(,)h(261:513{534,)h(1982.)164 1471 y([5])24 b(R.L.)14 b(Riv)o(est,)g(A.)g(Shamir,)i(and)f(L.)g(Adleman.)k(A)14 b(metho)q(d)h(for)g(obtaining)i(digital)240 1532 y(signatures)i(and)e (public-k)o(ey)g(cryptosystems.)24 b Fa(Communic)n(ations)18 b(of)g(the)h(A)o(CM)p Fl(,)240 1592 y(21\(2\):120{126,)g(F)l(ebruary)d(1978.) 961 2599 y(3)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF