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f(has)h(the)g(sets)g(of)g(strings)g Fk(f)p Fl(s)1261 262 y Fj(1)1281 255 y Fl(;)8 b(:)g(:)g(:)f(;)h(s)1413 262 y Fi(n)1437 255 y Fk(g)22 b Fo(and)g(the)g(set)f(of)i(bits)51 315 y(whic)o(h)18 b(generated)h(eac)o(h)f(string,)h Fk(fB)743 322 y Fj(1)762 315 y Fl(;)8 b(:)g(:)g(:)f(;)h Fk(B)904 322 y Fi(n)927 315 y Fk(g)p Fo(.)28 b(The)19 b(implem)o(en)o(tati)o(on)e(describ)q(ed)h(in)g([W) l(a)o(y91])51 376 y(will)f(p)q(erm)o(ute)f(the)h(mapping)g(b)q(et)o(w)o(een)f (the)i(bits)f(and)h(the)g(pro)q(ductions)g(in)f(order)h(to)g(in)o(tro)q(duce) 51 436 y(further)k(complexit)o(y)l(.)37 b(Assume)20 b(that)j(this)f(is)g(not) h(turned)f(on)h(and)g(that)g(the)f(same)f(bits)i(will)51 496 y(alw)o(a)o(ys)17 b(sp)q(ecify)e(the)h(same)f(sequence)g(of)i(pro)q(ductions) g(and)g(consequen)o(tally)e(the)h(same)f(strings.)125 556 y(A)22 b(straigh)o(t-forw)o(ard)h(attac)o(k)f(on)h(the)f(system)f(can)h(tak)o(e)g(b) q(e)h(done)f(b)o(y)g(\014nding)h(all)f(sets)g(of)51 616 y(terminals)10 b(that)j(are)f(rep)q(eated)f(in)h(some)f(subset)h(of)g(the)g(strings)g(from)f Fl(L)p Fo(\()p Fl(G)p Fo(\).)20 b(Since)11 b(eac)o(h)h(terminal)51 677 y(o)q(ccurs)18 b(in)f(only)g(one)g(pro)q(duction,)h(it)f(is)g(relativ)o (ely)d(easy)k(to)f(\014nd)h(the)f(common)e(bits.)24 b(It)17 b(is)g(only)51 737 y(necessary)d(to)f(ha)o(v)o(e)g(enough)h Fl(s)619 744 y Fi(i)647 737 y Fo(so)g(that)f(a)h(signi\014can)o(t)f(ma)s (jorit)o(y)f(of)h(the)g(terminal)f(strings)h(app)q(ear)51 797 y(at)g(least)g(t)o(wice.)19 b(More)12 b(duplication)g(mak)o(es)f(it)h(easier) g(to)h(solv)o(e)f(the)g(problem.)18 b(It)13 b(ma)o(y)e(b)q(e)h(p)q(ossible)51 857 y(to)19 b(pro)o(vide)e(a)i(go)q(o)q(d)h(estimate)c(of)j(the)f(\\w)o (eakness")h(of)f(this)g(system)f(using)i(statistics,)f(but)g(it)g(is)51 917 y(probably)e(not)g(w)o(orth)g(the)f(e\013ort.)21 b(Although)16 b(the)f(size)f(of)i Fl(L)p Fo(\()p Fl(G)p Fo(\))g(ma)o(y)e(gro)o(w)i(exp)q (onen)o(tially)e(with)51 978 y(resp)q(ect)20 b(to)g(the)g(size)f(of)h Fl(G)p Fo(,)h(the)e(n)o(um)o(b)q(er)f(of)i(terminals)e(is)i(b)q(ounded)h(b)o (y)e(the)h(size)f(of)h Fl(G)p Fo(.)32 b(This)51 1038 y(means)16 b(that)h(the)f(terminals)f(will)g(app)q(ear)j(often)e(and)h(a)g(linear)f(n)o (um)o(b)q(er)f(of)h(strings)h(is)g(necessary)51 1098 y(to)f(reco)o(v)o(er)d (sev)o(eral)h(instances)h(of)h(a)f(terminal)e(and)i(disco)o(v)o(er)f(the)h (bits)g(whic)o(h)f(generated)h(it.)20 b(The)51 1158 y(n)o(um)o(b)q(er)12 b(of)i(strings)g(necessary)f(is)h(in)o(v)o(ersely)d(prop)q(ortional)j(to)g (the)f(probabilit)o(y)g(a)h(giv)o(en)f(terminal)51 1218 y(will)j(app)q(ear.) 51 1363 y Fm(A)23 b(CFL)h(that)e(ma)n(y)h(b)r(e)g(Hard)h(to)f(Disco)n(v)n(er) 51 1455 y Fo(If)d(it)f(is)h(easy)g(to)g(disco)o(v)o(er)f(the)h(mapping)f(b)q (et)o(w)o(een)g(terminals)f(and)i(bits)g(when)g(eac)o(h)g(terminal)51 1515 y(app)q(ears)15 b(only)f(once,)g(do)q(es)g(it)f(get)h(harder)g(when)g (terminals)e(app)q(ear)j(more)d(than)i(once)g(in)f(a)h(CF)o(G?)51 1576 y(What)k(if)f(an)g(attac)o(k)o(er)f(has)i(access)f(to)g(enough)h (strings)f(from)f(the)h(language)h(to)g(determine)c(that)51 1636 y(eac)o(h)23 b(terminal)e Fl(t)388 1643 y Fi(i)427 1636 y Fk(2)26 b Fo(\006)d(could)g(ha)o(v)o(e)f(b)q(een)h(pro)q(duced)g(b)o(y)g (one)g(of)g(sev)o(eral)f(patterns)i(of)f(bits:)51 1696 y Fl(\014)82 1678 y Fj(1)79 1708 y Fi(i)101 1696 y Fl(;)8 b(:)g(:)g(:)g(;)g(\014)242 1678 y Fi(n)239 1708 y(i)265 1696 y Fo(?)30 b(The)20 b(attac)o(k)o(er)e(m)o (ust)g(con)o(v)o(ert)g(eac)o(h)h(string)h(of)f(terminals)f(in)o(to)h(bits)g (b)o(y)g(c)o(ho)q(osing)51 1756 y(one)g(of)f(the)g Fl(\014)317 1732 y Fi(j)314 1768 y(i)353 1756 y Fo(for)g(eac)o(h)f Fl(t)558 1763 y Fi(i)590 1756 y Fo(in)h(the)g(string.)27 b(If)17 b(there)h(are)g Fl(n)g Fo(c)o(hoices)f(for)i(eac)o(h)e(terminal)f(and)j Fl(m)51 1816 y Fo(terminals)f(in)g(an)i(a)o(v)o(erage)e(string,)i(then)f(there)f(are) h Fl(m)1100 1798 y Fi(n)1142 1816 y Fo(di\013eren)o(t)g(p)q(ossible)g(in)o (terpretations)f(of)51 1876 y(the)e(string{)h(a)g(daun)o(ting)g(n)o(um)o(b)q (er)d(when)i Fl(m)g Fo(and)h Fl(n)g Fo(gro)o(w)f(large.)125 1937 y(But)22 b(is)h(this)g(a)h(go)q(o)q(d)h(measure)c(of)j(the)f(burden)g (on)g(the)g(shoulders)g(of)h(the)f(attac)o(k)o(er?)41 b(It)51 1997 y(only)18 b(dep)q(ends)f(up)q(on)i(whether)e(there)f(are)i(an)o(y)f (correlations)g(b)q(et)o(w)o(een)f(the)h(bits)h(pro)q(ducing)g(the)51 2057 y(terminals)10 b(in)g(a)i(string.)19 b(If,)12 b(for)f(instance,)g Fl(t)848 2064 y Fj(1)879 2057 y Fo(w)o(as)g(generated)g(b)o(y)g Fl(\014)1278 2039 y Fj(1)1275 2069 y(1)1308 2057 y Fo(if)f(and)i(only)f(if)f Fl(t)1596 2064 y Fj(2)1627 2057 y Fo(app)q(ears)i(next)51 2117 y(to)i(it)g(and)g(w)o(as)g(generated)g(b)o(y)f Fl(\014)654 2099 y Fj(1)651 2130 y(2)673 2117 y Fo(,)h(then)f(there)g(the)h(assumption)f (ab)q(out)i(exp)q(onen)o(tial)e(complexit)o(y)51 2177 y(do)q(es)18 b(not)g(hold.)24 b(On)17 b(the)g(other)h(hand,)f(if)g Fl(t)876 2184 y Fj(1)912 2177 y Fo(is)g(generated)h(b)o(y)e Fl(\014)1285 2159 y 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b(enco)q(de)f(v)o(ery)f (complex)f(correlations)i(b)q(et)o(w)o(een)g(tok)m(ans)h(as)g(the)f(examples) 30 b(and)k(sho)o(w.)125 2659 y(It)17 b(is)h(p)q(ossible)g(to)h(calculate)e (the)h(m)o(utual)e(dep)q(endencies)h(for)h(reasonable)h(sized)e(grammars.)961 2828 y(5)p eop %%Page: 6 6 bop 51 195 a Fo(Let)12 b Fl(\032)p Fo(\()p Fl(a)204 202 y Fi(i)217 195 y Fo(\))f(b)q(e)g(the)g(probabilit)o(y)f(that)h(a)h(particular)e (terminal)f Fl(a)1202 202 y Fi(i)1227 195 y Fo(will)g(app)q(ear)j(in)f(a)g (string)g(generated)51 255 y(b)o(y)16 b(a)f(grammar)f(and)j(let)d Fl(\032)p Fo(\()p Fl(a)600 262 y Fi(i)614 255 y Fo(\))633 237 y Fj(\()p Fi(j)r Fj(\))694 255 y Fo(b)q(e)i(the)f(probabilit)o(y)g(that)h (the)f Fl(j)s Fo(-th)h(duplicate)f(of)h(a)g(particular)51 322 y(terminal)h Fl(a)275 329 y Fi(i)308 322 y Fo(will)h(app)q(ear)i(in)e(a)i (string.)29 b(Also)19 b(let)f Fl(\032)p Fo(\()p Fl(a)1091 297 y Fj(\()p Fi(j)r Fj(\))1091 334 y Fi(i)1136 322 y Fk(j)p Fl(a)1176 297 y Fj(\()p Fi(l)p Fj(\))1176 335 y 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1525 y Fl(;)8 b(t)1327 1532 y Fj(2)1346 1525 y Fl(;)g(:)g(:)g(:)f(;)h(t)1473 1532 y Fi(m)1506 1525 y Fo(,)17 b(then)g(the)g(n)o(um)o(b)q(er)51 1585 y(of)g(p)q(ossible)f(messages)g(is:) 904 1668 y Fi(m)893 1680 y Fg(Y)891 1771 y Fi(i)p Fj(=0)957 1721 y Fo(2)981 1701 y Fi(E)r Fj(\()p Fi(t)1036 1706 y Fd(i)1048 1701 y Fj(\))1833 1721 y Fo(\(2\))125 1841 y(Is)h(this)h(a)g(reasonable)h (estimate)d(for)i(the)g(n)o(um)o(b)q(er)e(of)i(p)q(ossible)g(com)o(binations) f(the)g(attac)o(k)o(er)51 1901 y(m)o(ust)d(consider?)20 b(Y)l(es,)14 b(if)g(w)o(e)g(also)i(b)q(eliev)o(e)c(that)j(the)f(size)g(of)h Fl(L)p Fo(\()p Fl(G)p Fo(\))g(is)f(so)i(large)e(that)h(the)f(attac)o(k)o(er) 51 1961 y(is)20 b(not)f(capable)h(of)f(k)o(eeping)g(trac)o(k)f(of)i(the)f (substrings)h(of)g(terminals)d(from)i(the)g(system.)29 b(That)51 2022 y(is)22 b(they)g(cannot)h(main)o(tain)d(an)o(y)i(partial)g(kno)o(wledge) g(ab)q(out)h(a)g(the)e(probabilit)o(y)h(that)g(certain)51 2082 y(instances)15 b(of)h(the)f(terminals)e(ma)o(y)g(co-exist.)20 b(The)15 b(strength)h(of)f(the)g(system)f(against)i(this)f(sort)g(of)51 2142 y(partial)i(kno)o(wledge)e(can)i(also)g(b)q(e)f(estimated)e(with)j (information)e(theory)l(.)21 b(Let)16 b Fl(E)s Fo(\()p Fl(a)1635 2149 y Fi(i)1649 2142 y Fl(;)8 b(a)1697 2149 y Fi(j)1715 2142 y Fo(\))16 b(b)q(e:)623 2231 y Fg(X)633 2323 y Ff(8)p Fi(k)691 2231 y Fg(X)705 2323 y Ff(8)p Fi(l)759 2272 y Fl(\032)p Fo(\()p Fl(a)829 2247 y Fj(\()p Fi(k)q Fj(\))829 2284 y Fi(i)878 2272 y Fk(j)p Fl(a)918 2247 y Fj(\()p Fi(l)p Fj(\))918 2284 y Fi(j)957 2272 y Fo(\))8 b(log)1048 2284 y Fj(2)1067 2272 y Fo(\()1188 2238 y(1)p 1091 2260 218 2 v 1091 2316 a Fl(\032)p Fo(\()p Fl(a)1161 2290 y Fj(\()p Fi(k)q Fj(\))1161 2327 y Fi(i)1210 2316 y Fk(j)p Fl(a)1250 2290 y Fj(\()p Fi(l)p Fj(\))1250 2327 y Fi(j)1290 2316 y Fo(\))1314 2272 y(\))500 b(\(3\))125 2399 y(It)18 b(is)h(p)q(ossible)g(to)h(rew)o(ork)e(equation)h(2)h(to)f(use)g(this) 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Fk(j)h Fo(Thelma)f(and)i(Louise)g Fk(j)f Fo(Bob)g(and)h(Ra)o(y)174 1898 y Fn(what)50 b Fl(*)f Fo(w)o(en)o(t)16 b(h)o(un)o(ting)g(b)q(ear)g Fk(j)h Fo(w)o(en)o(t)e (sur\014ng)i Fk(j)f Fo(decided)f(to)i(do)g(nothing)g Fk(j)f Fo(slept)g(late)152 1958 y Fn(wher)n(e)50 b Fl(*)f Fo(in)16 b(Laguna)i Fk(j)e Fo(in)g(Bro)q(oklyn)g Fk(j)h Fo(in)e(Vienna)h Fk(j)g Fo(in)g(Tim)o(buktu)125 2045 y(Here)f(it)h(is)g(after)g(the)g (expansions:)961 2828 y(8)p eop %%Page: 9 9 bop 254 196 a Fl(H)294 203 y Fj(4)314 196 y Fo(:)22 b(S)50 b Fl(*)f Fo(Bill)15 b(and)h(T)l(ed)h Fn(what)924 203 y Fj(1)977 196 y Fn(wher)n(e)1097 203 y Fj(2)1151 196 y Fk(j)f Fo(La)o(v)o(erne)f(and)i (Shirley)e Fn(what)1721 203 y Fj(2)1775 196 y Fn(wher)n(e)1895 203 y Fj(1)1948 196 y Fk(j)525 256 y Fo(Thelma)g(and)i(Louise)g Fn(what)1050 263 y Fj(1)1086 256 y Fn(wher)n(e)1206 263 y Fj(1)1242 256 y Fk(j)f Fo(Bob)g(and)h(Ra)o(y)f Fn(c)n(omp)n(osite)259 316 y(what)357 323 y Fj(1)427 316 y Fl(*)49 b Fo(w)o(en)o(t)16 b Fn(somewher)n(e)34 b 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Fn(somewher)n(e)g Fo(in)g Fn(plac)n(e)1046 865 y Fj(1)1083 858 y Fk(j)g Fo(w)o(en)o(t)f Fn(somewher)n(e)i Fo(in)f Fn(plac)n(e)1634 865 y Fj(2)1670 858 y Fk(j)525 918 y Fn(nada)h Fo(in)e(Bro)q(oklyn)i Fk(j)f Fo(decided)f(to)h(do)h(nothing)g(in) f(Laguna)i Fk(j)525 978 y Fo(slept)e(late)g(in)g(Bro)q(oklyn)g Fk(j)525 1038 y Fo(slept)g(late)g(in)g Fn(plac)n(e)898 1045 y Fj(2)934 1038 y Fk(j)g Fo(decided)g(to)g(do)h(nothing)g(in)f Fn(plac)n(e)1608 1045 y Fj(2)125 1125 y Fo(The)21 b(practical)g(e\013ect)g (of)h(the)f(duplication)g(should)h(b)q(e)g(ob)o(vious)f(The)h(w)o(a)o(y)f (that)h(the)f(v)o(erb)51 1185 y(and)c(ob)s(ject)f(phrase)h(of)f(the)g(sen)o (tences)g(in)g Fl(L)p Fo(\()p Fl(H)951 1192 y Fj(4)971 1185 y Fo(\))g(dep)q(ends)h(up)q(on)g(the)f(noun.)22 b(An)o(y)15 b(information)51 1246 y(ab)q(out)i(the)d(bits)h(whic)o(h)f(pro)q(duced)h(the) g(sen)o(tence)f(\\Bill)f(and)i(T)l(ed)g(w)o(en)o(t)f(sur\014ng)i(in)e (Laguna")k(will)51 1306 y(not)e(help)f(deco)q(de)g(the)g(sen)o(tence)f(\\Bob) h(and)h(Ra)o(y)f(w)o(en)o(t)f(sur\014ng)j(in)d(Laguna.")23 b(Straigh)o(t-forw)o(ard)51 1366 y(co)q(de-b)q(o)q(ok)c(attac)o(ks)d(lik)o(e) e(the)i(one)h(describ)q(ed)e(in)h(a)h(previous)f(section)g(are)g(not)h(p)q (ossible.)125 1426 y(Moreo)o(v)o(er,)i(the)h(size)f(of)i(the)f(language)h Fl(L)p Fo(\()p Fl(G)p Fo(\))g(gro)o(ws)g(exp)q(onen)o(tially)e(with)h(the)g (n)o(um)o(b)q(er)e(of)51 1486 y(m)o(ulti-pro)q(duction)f(v)m(ariables)g(b)q (et)o(w)o(een)g(the)h(start)g(sym)o(b)q(ol)f(the)g(terminal.)24 b(This)18 b(means)f(that)h(a)51 1547 y(w)o(ell-designed)g(grammar)e(can)j (gro)o(w)g(linearly)l(,)e(but)h(the)g(size)g(of)g(its)g(language)i(will)d (gro)o(w)i(exp)q(o-)51 1607 y(nen)o(tially)l(.)h(A)c(large)g(grammar)f(could) h(pro)q(duce)g(v)o(ery)f(in)o(tractible)g(problems)g(for)h(the)g(attac)o(k)o (er.)125 1667 y(In)i(practice,)h(grammars)e(lik)o(e)g(the)i(ones)h(in)e (\014gure)i([1])e(can)h(b)q(e)g(expanded,)h(duplicated)e(and)51 1727 y(con)o(tracted)12 b(sev)o(eral)f(times)f(to)j(pro)q(duce)f(a)h(new)f (grammar)f(where)g(man)o(y)g(of)h(the)g(terminals)e(app)q(ear)51 1787 y(in)19 b(t)o(w)o(o)g(or)h(more)e(of)h(the)g(pro)q(ductions.)31 b(Expansion)20 b(can)g(lead)f(to)g(exp)q(onen)o(tial)g(gro)o(wth)h(in)f(the) 51 1847 y(grammar,)d(but)h(this)g(can)g(b)q(e)g(con)o(tained)g(if)f(the)h (expansions)h(only)e(e\013ect)h(a)g(few)g(pro)q(ductions)h(at)51 1908 y(a)f(time.)51 2048 y Fm(Probabilit)n(y)22 b(and)i(Mimic)n(kry)51 2140 y Fo(The)17 b(last)f(section)g(sho)o(w)o(ed)g(ho)o(w)h(to)f(use)g(three) g(basic)g(op)q(erations)h(to)g(construct)f(grammars)f(that)51 2200 y(are)j(more)f(secure.)25 b(Naturally)l(,)17 b(w)o(eaknesses)g(can)h (emerge)e(in)h(some)g(situations)h(when)g(probabil-)51 2261 y(ities)h(are)h(not)g(preserv)o(ed)f(w)o(ell.)30 b(Ordinarily)l(,)19 b(the)g(de\014nition)g(of)h(mimic)c(grammars)i(includes)h(a)51 2321 y(probabilit)o(y)h(that)i(go)o(v)o(erns)f(ho)o(w)g(often)g(a)g (particular)g(pro)q(duction)g(will)f(b)q(e)h(c)o(hosen.)35 b(The)21 b(last)51 2381 y(section)f(did)g(not)h(men)o(tion)d(the)i(e\013ects) f(of)i(these)e(for)i(the)f(sak)o(e)f(of)i(clarit)o(y)l(.)31 b(Occasionally)l(,)20 b(the)51 2441 y(e\013ects)g(of)f(these)g(probabilities) g(can)h(conspire)f(to)h(n)o(ullify)d(man)o(y)h(of)i(the)f(e\013ects)g(of)h (expansions)51 2501 y(and)d(duplications.)125 2562 y(Consider)f(this)g(v)o (ersion)g(of)g Fl(H)684 2569 y Fj(5)704 2562 y Fo(:)549 2704 y Fl(H)589 2711 y Fj(5)623 2704 y Fo(:)d Fl(S)45 b(*)c(aZ)874 2711 y Fj(1)894 2704 y Fk(f)p Fl(:)p Fo(999)p Fk(gj)p Fl(bZ)1098 2711 y Fj(1)1118 2704 y Fk(f)p Fl(:)p Fo(001)p Fk(g)961 2828 y Fo(9)p eop %%Page: 10 10 bop 630 195 a Fl(Z)663 202 y Fj(1)725 195 y Fl(*)41 b(a)p Fk(f)p Fl(:)p Fo(25)p Fk(gj)p Fl(b)p Fk(f)p Fl(:)p Fo(25)p Fk(gj)p Fl(c)p Fk(f)p Fl(:)p Fo(25)p Fk(gj)p Fl(d)p Fk(f)p Fl(:)p Fo(25)p Fk(g)125 305 y Fo(The)21 b(v)m(alues)h(in)f(brac)o(k)o(ets)g(\()p Fk(f)p Fl(;)8 b Fk(jg)p Fo(\))21 b(signify)g(the)g(probabilit)o(y)g(that)h (one)g(pro)q(duction)g(will)e(b)q(e)51 365 y(tak)o(en.)h(If)16 b(the)g(pro)q(duction)h Fl(S)g(*)c(bZ)747 372 y Fj(1)783 365 y Fo(is)j(expanded,)g(the)g(language)h(b)q(ecomes:)537 535 y Fl(H)577 542 y Fj(6)597 535 y Fo(1)d(:)g Fl(S)44 b(*)d(aZ)886 542 y Fj(1)906 535 y Fk(f)p Fl(:)p Fo(999)p Fk(gj)p Fl(Z)1089 542 y Fj(2)1109 535 y Fk(f)p Fl(:)p Fo(001)p Fk(g)642 608 y Fl(Z)675 615 y Fj(1)737 608 y Fl(*)g(a)p Fk(f)p Fl(:)p Fo(25)p Fk(gj)p Fl(b)p Fk(f)p Fl(:)p Fo(25)p Fk(gj)p Fl(c)p Fk(f)p Fl(:)p Fo(25)p Fk(gj)p Fl(d)p Fk(f)p Fl(:)p Fo(25)p Fk(g)642 681 y Fl(Z)675 688 y Fj(2)737 681 y Fl(*)g(ba)p Fk(f)p Fl(:)p Fo(25)p Fk(gj)p Fl(bb)p Fk(f)p Fl(:)p Fo(25)p Fk(gj)p Fl(Z)t Fo(3)p Fk(f)p Fl(:)p Fo(5)p Fk(g)642 753 y Fl(Z)675 760 y Fj(3)737 753 y Fl(*)g(bc)p Fk(f)p Fl(:)p Fo(25)p Fk(gj)p Fl(bd)p Fk(f)p Fl(:)p Fo(25)p Fk(g)125 863 y Fo(The)17 b(expansion)h(will)f(only)g(e\013ect) g(.1\045)g(of)h(the)g(sen)o(tences)e(created)h(b)o(y)g(this)h(grammar.)23 b(It)18 b(is)51 924 y(probably)c(an)f(un)o(wise)f(use)h(of)g(the)g(resources) f(to)h(mo)q(dify)f(the)g(grammar)g(b)o(y)g(expansion.)20 b(It)13 b(follo)o(ws)51 984 y(that)k(the)f(b)q(est)h(c)o(hoices)e(for)h(expansion)h (are)f(the)g(pro)q(ductions)h(with)g(the)f(highest)g(probabilities.)125 1044 y(If)10 b(there)g(are)h(go)q(o)q(d)i(c)o(hoices,)d(then)h(it)f(also)i (follo)o(ws)e(that)i(there)e(are)h(some)f(expansion/duplication)51 1104 y(c)o(hoices)k(that)i(are)f(b)q(etter)g(than)g(others.)21 b(The)15 b(en)o(tire)f(purp)q(ose)i(of)f(expanding)g(and)h(duplicating)e(a)51 1164 y(grammar)g(is)h(to)h(presen)o(t)e(an)i(attac)o(k)o(er)e(with)i(am)o (biguous)e(v)m(alues)i(of)f(eac)o(h)g(tok)o(en.)20 b(They)15 b(can't)g(b)q(e)51 1225 y(sure)h(if)f(it)g(has)h(one)g(v)m(alue)f(or)h (another.)21 b(The)16 b(result)f(is)g(that)h(they)f(m)o(ust)f(confron)o(t)i (exp)q(onen)o(tially)51 1285 y(man)o(y)f(strings.)51 1429 y Fm(A)23 b(Practical)e(Key)i(System)f(to)h(Making)h(a)f(Grammar)g(More)g (Secure)51 1522 y Fo(The)18 b(last)g(t)o(w)o(o)f(sections)g(ha)o(v)o(e)g (discussed)g(ho)o(w)h(to)g(re-arrange)g(a)g(grammar)e(so)i(that)g(eac)o(h)f (tok)o(en)51 1582 y(can)e(b)q(e)f(pro)q(duced)h(b)o(y)f(sev)o(eral)f (di\013eren)o(t)h(patterns)g(of)h(bits.)20 b(This)15 b(forces)f(the)g(attac)o (k)o(er)f(to)i(c)o(ho)q(ose)51 1642 y(b)q(et)o(w)o(een)h(b)q(oth)h(in)o (terpretations)f(when)h(confron)o(ted)f(with)g(a)h(tok)o(en.)22 b(When)16 b(there)g(are)h(no)g(cross-)51 1702 y(correlations)j(b)q(et)o(w)o (een)g(the)g(tok)o(ens,)g(then)g(this)g(ma)o(y)f(presen)o(t)g(a)i(fairly)e (formidable)f(barrier)i(to)51 1762 y(attac)o(k.)125 1823 y(Here)14 b(is)h(an)h(algorithm)e(for)i(con)o(v)o(erting)e(a)h(giv)o(en)g(grammar)e Fl(G)j Fo(in)o(to)f(a)h(strong)g(grammar)e(that)51 1883 y(will)i(resist)f (this)i(form)e(of)h(brute-force)g(attac)o(k:)125 1943 y(Rep)q(eat)g(this)g (next)g(step)g(as)h(often)f(as)h(desired:)111 2045 y(1.)24 b(Cho)q(ose)18 b(a)f(random)f(pro)q(duction)g(and)h(expand)g(it.)111 2146 y(2.)24 b(The)12 b(expansion)g(ma)o(y)e(pro)q(duce)h Fl(n)h Fo(di\013eren)o(t)f(copies)g(of)g(the)h(original)f(pro)q(duction:)19 b Fl(A)1742 2153 y Fi(i)1770 2146 y Fl(*)14 b(\013)1864 2153 y Fi(j)1882 2146 y Fo(.)173 2206 y(F)l(or)h(eac)o(h)f(v)m(ariable)h Fl(A)586 2213 y Fi(k)621 2206 y Fo(in)g Fl(\013)708 2213 y Fi(j)726 2206 y Fo(,)g(c)o(ho)q(ose)g(a)g(di\013eren)o(t)f(v)o(ersion)g(of)h (the)f(v)m(ariable)h Fl(A)1657 2213 y Fi(k)1692 2206 y Fo(to)g(app)q(ear)173 2267 y(in)h(eac)o(h)g(of)h(the)f(duplicated)f(v)o(ersions)h(of)g Fl(\013)989 2274 y Fi(j)1008 2267 y Fo(.)21 b(Create)16 b(these)g Fl(A)1362 2274 y Fi(k)1399 2267 y Fo(via)g(duplication.)111 2368 y(3.)24 b(Cho)q(ose)18 b(a)f(random)f(v)m(ariable.)111 2470 y(4.)24 b(Con)o(tract)h(some)e(of)h(the)f(pro)q(ductions)i(if)f(this)f (can)h(b)q(e)g(done.)45 b(Cho)q(ose)25 b(the)f(subset)g(of)173 2530 y(pro)q(ductions)18 b(at)e(random.)125 2632 y(The)d(selection)g(of)h(a)h 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b(implem)o(e)o(n)o(tation)d(is)i(in)o(tended)f (to)h(enco)q(de)h(information)e(in)h(English-lik)o(e)e(text.)51 733 y(This)17 b(is)g(useful,)f(but)g(it)h(isn't)f(alw)o(a)o(ys)g(e\016cien)o (t.)21 b(Man)o(y)16 b(of)h(the)f(exp)q(erimen)o(tal)e(grammars)h(w)o(ould)51 793 y(pro)q(duce)h(an)f(\\encrypted")g(\014le)g(that)g(w)o(as)h(20)f(to)h(30) g(times)d(larger)i(than)g(the)g(original.)21 b(This)15 b(isn't)51 853 y(practical.)125 913 y(It)10 b(is)h(p)q(ossible,)h(ho)o(w)o(ev)o(er,)e (to)h(rely)f(up)q(on)i(the)f(theoretical)f(results)g(describ)q(ed)h(here)f (to)h(construct)51 974 y(e\016cien)o(t)19 b(grammars)g(with)h(a)h(set)g(of)f (tok)o(ens)h(that)f(is)h(equiv)m(alen)o(t)e(to)h(the)h(set)f(of)h(c)o (haracters)f(or)51 1034 y(w)o(ords.)25 b(In)16 b(the)h(ASCI)q(I)g(w)o(orld,)g (this)f(means)h(using)g(all)g(256)h(or)f(128)h(c)o(haracters)f(as)h(tok)o (ens.)23 b(The)51 1094 y(output)14 b(of)f(these)g(grammars)f(will)g(not)h(b)q (e)g(readible,)f(but)i(they)e(will)g(inherit)g(all)g(of)i(the)e(theoretical) 51 1154 y(securit)o(y)j(describ)q(ed)h(in)g(the)g(last)g(t)o(w)o(o)g (sections.)125 1214 y(F)l(or)g(simplicit)o(y)l(,)d(assume)j(that)h(the)f (data)i(comes)d(in)h(blo)q(c)o(ks)h(of)g(one)f(b)o(yte)g(and)i(there)e(are)g (256)51 1275 y(tok)o(ens)j(in)f(the)g(grammar)e Fk(f)p Fl(a)612 1282 y Fj(0)632 1275 y Fl(;)8 b(:)g(:)g(:)f(;)h(a)767 1282 y Fj(255)821 1275 y Fk(g)p Fo(.)28 b(The)18 b(grammar)f(consists)h(of)h (these)f(tok)o(ens,)g(a)h(set)f(of)51 1335 y(v)m(ariables,)e Fk(f)p Fl(v)316 1342 y Fj(0)335 1335 y Fl(;)8 b(:)g(:)g(:)g(;)g(v)469 1342 y Fi(n)492 1335 y Fk(g)16 b Fo(and)h(a)f(set)g(of)h(pro)q(ductions)g(of) g(the)f(form:)744 1445 y Fl(v)768 1452 y Fi(i)795 1445 y Fl(*)e(a)884 1452 y Fj(0)903 1445 y Fl(V)931 1452 y Fj(0)951 1445 y Fk(j)8 b Fl(:)g(:)g(:)g(a)1065 1452 y Fj(255)1120 1445 y Fl(V)1148 1452 y Fj(255)51 1555 y Fo(,)18 b(where)f(the)g Fl(V)338 1562 y Fi(i)370 1555 y Fo(are)h(p)q(ossibly)g(empt)o(y)d(strings)j(of)g(v)m (ariables.)25 b(One)17 b(of)h(the)f Fl(v)1503 1562 y Fi(i)1535 1555 y Fo(is)g(designated)h(the)51 1615 y(start)f(sym)o(b)q(ol.)125 1675 y(Ho)o(w)d(is)g(the)g(comp)q(osition)f(of)i Fl(V)715 1682 y Fi(i)729 1675 y Fo('s)f(determined?)19 b(It)14 b(can)g(either)f(b)q(e)h (generated)h(at)f(random)g(or)51 1735 y(through)j(the)e(expansion)h(and)g (con)o(traction)f(op)q(erations.)22 b(The)15 b(a)o(v)o(erage)g(n)o(um)o(b)q (er)f(of)h(v)m(ariables)h(in)51 1796 y(the)e Fl(V)161 1803 y Fi(i)176 1796 y Fo('s)g(determines)e(the)h(a)o(v)o(erage)h(length)g(of)g (the)g(strings)h(in)f(the)g(language.)21 b(If)14 b(the)g(a)o(v)o(erage)f (size)51 1856 y(is)18 b(1)g(or)h(larger,)e(then)h(there)f(will)g(b)q(e)h (non-trivial)g(n)o(um)o(b)q(er)e(of)i(in\014nite)f(strings)h(in)g(the)f (language)51 1916 y(and)f(it)f(will)g(not)h(b)q(e)f(practically)f(useful.)21 b(If)15 b Fl(\016)h Fo(is)g(the)f(a)o(v)o(erage)g(length)g(of)h(the)f Fl(V)1540 1923 y Fi(i)1554 1916 y Fo('s,)g(then)g Fl(\016)1750 1898 y Fi(i)1779 1916 y Fo(is)h(the)51 1976 y(probabilit)o(y)g(that)g(a)h (giv)o(en)e(string)i(will)e(b)q(e)h Fl(i)g Fo(tok)o(ens)g(long.)51 2121 y Fm(T)-6 b(uring)24 b(Mac)n(hines)e(and)i(Mimic)n(kry)51 2213 y Fo(The)15 b(last)h(sections)e(discussed)h(using)h(CF)o(G's)e(for)i (mim)o(ic)n(kry)l(.)h(In)e([W)l(a)o(y90],)g(the)f(mimic)d(functions)51 2273 y(that)19 b(pro)q(duced)f(all)g(recursiv)o(ely)d(en)o(umerable)g (languages)20 b(w)o(ere)d(presen)o(ted)g(as)h(VW-grammars.)51 2333 y(This)g(construct)g(\014tted)g(nicely)e(with)i(the)f(other)h(items)e (in)i(the)f(pap)q(er)i(and)f(it)g(also)g(allo)o(ws)g(us)g(to)51 2393 y(apply)h(all)g(of)g(the)g(results)g(in)f(this)h(pap)q(er)h(to)f(the)g (larger)g(classes)g(of)g(functions)g(computable)f(b)o(y)51 2454 y(T)l(uring)f(Mac)o(hine.)i(This)d(section,)f(though,)i(will)e(address)h (the)g(class)g(of)g(r.e.)k(mimi)o(c)13 b(functions)j(as)51 2514 y(turing)g(mac)o(hines)d(running)j(bac)o(kw)o(ards.)21 b(Man)o(y)15 b(of)g(the)g(classical)g(facts)g(ab)q(out)i(turing)e(mac)o (hines)51 2574 y(are)g(directly)e(applicable)h(to)h(this)g(problem,)e(but)i (there)f(a)i(enough)f(details)g(that)g(mak)o(e)e(the)h(use)h(of)51 2634 y(these)h(general)g(theorems)f(problematic.)949 2828 y(11)p eop %%Page: 12 12 bop 125 195 a Fo(Let)20 b(a)h(turing)g(mac)o(hine)d(consists)j(of)g(an)f (in\014nite)g(tap)q(e)h(con)o(taining)f(sym)o(b)q(ols)g(from)f(a)i(tap)q(e)51 255 y(language,)16 b(\006,)f(and)h(a)f(\014nite)f(con)o(trol)h(with)g(a)g (set)g(of)g(states,)h Fl(S)s Fo(.)k(The)15 b(in\014nite)f(tap)q(e)i(has)f(a)h (p)q(oin)o(ter)51 315 y(to)d(the)e(sym)o(b)q(ol)g(at)h(one)g(lo)q(cation)g (and)g(this)g(head)g(can)g(write)f(a)h(new)g(sym)o(b)q(ol)e(at)i(a)g(lo)q (cation,)h(and/or)51 376 y(mo)o(v)o(e)i(left)h(or)i(righ)o(t.)23 b(A)o(t)16 b(the)h(b)q(eginning,)g(the)g(tap)q(e)g(con)o(tains)h(an)f (initial)f(string)h(of)g(sym)o(b)q(ols.)23 b(A)51 436 y(function,)c Fl(\016)r Fo(\()p Fl(s;)8 b(t)p Fo(\),)17 b(go)o(v)o(erns)h(the)h(action)f (of)h(the)f(turing)h(mac)o(hine)d(based)j(on)g(the)f(curren)o(t)g(state,)51 496 y Fl(s)d Fk(2)f Fl(S)s Fo(,)i(and)h(sym)o(b)q(ol)e(at)i(the)f(tap)q(e)h (head,)g Fl(t)d Fk(2)g Fo(\006.)22 b(It)16 b(p)q(erforms)g(three)g(actions.) 22 b(First)16 b(it)g(c)o(hanges)51 556 y(the)d(\014nite)f(con)o(trol)h(to)g (a)h(new)f(state,)g(second)g(it)g(ma)o(y)e(write)h(a)i(new)f(sym)o(b)q(ol)e (at)i(the)g(lo)q(cation)g(of)h(the)51 616 y(tap)q(e)j(head)f(and)g(third)g (it)f(ma)o(y)g(mo)o(v)o(e)e(the)j(tap)q(e)g(head)g(left)f(or)i(righ)o(t.)j(A) c(subset)g(of)g(the)g(state)g(set,)51 677 y Fl(S)s Fo(,)f(are)f(mark)o(ed)f (as)i(either)e(start)i(states)g(or)g(end)f(states.)21 b(If)14 b(a)h(mac)o(hine)d(b)q(egins)j(with)f(a)h(particular)51 737 y(string)j(on)g(its)f(tap)q(e)h(and)f(\014nds)h(itself)e(in)h(an)h(end)f (state)h(with)f(an)h(empt)o(y)d(tap)q(e,)i(then)g(the)g(turing)51 797 y(mac)o(hine)f(is)h(said)g(to)h Fn(ac)n(c)n(ept)f Fo(that)h(string.)24 b Fl(L)p Fo(\()p Fl(M)5 b Fo(\))18 b(is)f(the)g(sym)o(b)q(olic)e(represen)o (tation)i(for)h(the)f(the)51 857 y(set)g(of)f(strings)h(accepted)e(b)o(y)h (the)g(turing)g(mac)o(hine,)e Fl(M)5 b Fo(.)125 917 y(A)15 b Fn(mimic)i Fo(TM,)e Fl(M)487 899 y Ff(0)516 917 y Fo(can)h(b)q(e)g(built)g (from)f(a)h(turing)h(mac)o(hine,)c Fl(M)22 b Fo(b)o(y)16 b(running)g(it)g (bac)o(kw)o(ards.)51 978 y(Let)h Fl(\016)r Fo(\()p Fl(s)205 985 y Fi(i)219 978 y Fl(;)8 b(t)p Fo(\))14 b Fl(*)g Fo(\()p Fl(s)397 985 y Fi(j)415 978 y Fl(;)8 b(u;)g Fk(f)p Fl(l)q(ef)d(t;)j(r)q(ig)r (ht)p Fk(g)p Fo(\))15 b(mean)g(if)h(the)g(TM)h(is)f(at)h(state,)g Fl(s)1412 985 y Fi(i)1426 978 y Fo(,)f(and)h(the)g(tap)q(e)g(head)g(is)51 1038 y(reading)d(sym)o(b)q(ol)e Fl(t)p Fo(,)h(then)g(mo)o(v)o(e)f(to)h (state,)h Fl(s)873 1045 y Fi(j)891 1038 y Fo(,)g(write)f(sym)o(b)q(ol)f Fl(u)h Fo(on)h(the)f(tap)q(e)h(and)g(mo)o(v)o(e)d(the)i(head)51 1098 y(to)i(either)e(the)h(left)f(or)h(the)g(righ)o(t.)20 b(Call)14 b(the)g(rev)o(erse)e(of)j Fl(\016)r Fo(,)e Fl(\016)1157 1080 y Ff(0)1168 1098 y Fo(.)21 b(If)13 b Fl(\016)r Fo(\()p Fl(s)1315 1105 y Fi(i)1329 1098 y Fl(;)8 b(t)p Fo(\))13 b Fl(*)h Fo(\()p Fl(s)1506 1105 y Fi(j)1524 1098 y Fl(;)8 b(u;)g Fk(f)p Fl(l)q(ef)d(t;)j(r)q (ig)r(ht)p Fk(g)p Fo(\),)51 1158 y(then)18 b Fl(\016)188 1140 y Ff(0)199 1158 y Fo(\()p Fl(s)241 1165 y Fi(j)259 1158 y Fl(;)8 b(u)p Fo(\))15 b Fl(*)g Fo(\()p Fl(s)449 1165 y Fi(i)464 1158 y Fl(;)8 b(t;)g Fk(f)p Fl(r)q(ig)r(ht;)g(l)q(ef)d(t)p Fk(g)p Fo(\),)15 b(where)i(the)g(mo)o(v)o(em)o(en)n(t)e(left)h(or)i(righ)o(t)f(is)g (rev)o(ersed.)23 b(All)51 1218 y(the)16 b(start)h(states)f(of)g Fl(M)22 b Fo(b)q(ecome)14 b(the)i(end)g(states)g(of)g Fl(M)1109 1200 y Ff(0)1137 1218 y Fo(and)h(all)e(the)h(end)g(states)g(of)h Fl(M)k Fo(b)q(ecome)51 1279 y(the)16 b(start)h(states)g(of)g Fl(M)499 1260 y Ff(0)511 1279 y Fo(.)125 1339 y(The)k(only)g(real)g (di\013erence)g(b)q(et)o(w)o(een)f(a)i(mimi)o(c)c(TM,)j Fl(M)1223 1321 y Ff(0)1257 1339 y Fo(and)h(its)f(predicessor,)h Fl(M)k Fo(is)c(the)51 1399 y(starting)16 b(and)g(ending)f(condition)g(of)g(the)g (tap)q(e.)21 b Fl(M)f Fo(starts)c(with)f(a)h(string)f(on)g(the)g(tap)q(e)h (and)f(ends)51 1459 y(when)k(the)g(tap)q(e)g(is)g(empt)o(y)l(.)26 b Fl(M)655 1441 y Ff(0)686 1459 y Fo(starts)20 b(with)e(an)i(empt)o(y)c(tap)q (e)k(and)f(ends)g(up)g(with)g(a)g(string)g(in)51 1519 y Fl(L)p Fo(\()p Fl(M)5 b Fo(\))17 b(on)g(the)f(tap)q(e.)125 1579 y(Naturally)l(,)d (there)g(are)h(some)f(functions)h(that)g(are)h(not)f(in)o(v)o(ertable)e(b)q (ecause)i(there)f(are)i(sev)o(eral)51 1640 y(inputs)21 b(to)g Fl(\016)h Fo(that)f(pro)q(duce)g(the)f(same)f(result.)34 b(This)21 b(is)f(where)g(data)h(can)g(b)q(e)g(enco)q(ded.)34 b(One)51 1700 y(actual)19 b(algorithm)f(for)h(con)o(v)o(erting)e(a)i(set)g(of)g(c)o (hoices)e(to)i(bits)g(is)f(sho)o(wn)i(in)e(the)g(app)q(endix,)h(but)51 1760 y(there)e(ma)o(y)f(b)q(e)i(others.)25 b(The)17 b(data)h(is)f(reco)o(v)o (ered)f(when)h(the)h(TM)f(executes)f(this)h(one)h(branc)o(h)f(of)51 1820 y(the)f(delta)g(function.)125 1880 y(One)e(string)g(in)g Fl(L)p Fo(\()p Fl(M)5 b Fo(\))15 b(can)g(b)q(e)f(pro)q(duced)h(with)f Fl(M)1079 1862 y Ff(0)1106 1880 y Fo(b)o(y)f(starting)i(in)f(an)h(end)g (state)f(of)h Fl(M)20 b Fo(with)51 1941 y(an)d(empt)o(y)d(tap)q(e)i(and)h (running)f Fl(M)705 1923 y Ff(0)733 1941 y Fo(with)g Fl(\016)868 1923 y Ff(0)895 1941 y Fo(un)o(til)f(a)i(start)f(state)h(of)f Fl(M)22 b Fo(\(an)16 b(end)g(state)g(of)h Fl(M)1816 1923 y Ff(0)1828 1941 y Fo(\))f(is)51 2001 y(reac)o(hed.)22 b(When)16 b(a)h(c)o(hoice)e(m)o(ust)g(b)q(e)i(made)e(b)o(y)h Fl(\016)987 1983 y Ff(0)999 2001 y Fo(,)g(then)g(data)h(can)g(b)q(e)g(enco)q(ded.)22 b(An)16 b(en)o(tire)f(\014le)51 2061 y(can)i(b)q(e)f(pro)q(duced)h(b)o(y)f (generating)g(strings)h(in)f Fl(L)p Fo(\()p Fl(M)5 b Fo(\))17 b(un)o(til)e(all)g(the)h(data)i(is)e(used.)125 2121 y(This)11 b(data)g(can)g(b)q(e)g(reco)o(v)o(ered)e(b)o(y)i(running)g Fl(M)16 b Fo(on)11 b(the)g(strings.)20 b(If)10 b Fl(\016)1376 2103 y Ff(0)1387 2121 y Fo(\()p Fl(s)1429 2128 y Fi(i)1443 2121 y Fl(;)e(u)p Fo(\))14 b Fl(*)f Fo(\()p Fl(s)1630 2128 y Fi(j)1648 2121 y Fl(;)8 b(t)1688 2128 y Fi(j)1706 2121 y Fl(;)g(L)p Fo(\))p Fk(j)g Fl(:)g(:)g(:)g Fk(j)p Fo(\()p Fl(s)1924 2128 y Fi(k)1945 2121 y Fl(;)g(t)1985 2128 y Fi(k)2006 2121 y Fl(;)g(R)p Fo(\))51 2181 y(and)20 b(the)g(algorithm)e(in)h(the)g(app)q (endix)h(is)f(used)h(to)f(mak)o(e)f(the)h(c)o(hoices,)g(then)g(the)g(bits)h (can)f(b)q(e)51 2242 y(reco)o(v)o(ered)d(when)h(the)g(result)g(of)h Fl(\016)h Fo(is)e(state)g Fl(s)909 2249 y Fi(i)923 2242 y Fo(.)25 b(These)17 b(bits)g(will)f(b)q(e)i(pro)q(duced)g(in)f(the)g(opp)q(osite)51 2302 y(order.)51 2446 y Fm(Theoretical)k(Bounds)j(and)g(Practical)d (Questions)51 2539 y Fo(Classical)12 b(computer)e(science)g(b)q(orro)o(w)o (ed)i(man)o(y)e(results)h(ab)q(out)i(turing)f(mac)o(hines)d(from)i(the)g (realm)51 2599 y(of)20 b(logic)f(and)h(philosoph)o(y)l(.)31 b([HU69)o(])19 b(pro)o(vides)g(a)h(go)q(o)q(d)h(discussion)f(of)g(this)f (area.)31 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b(the)e(sender)h(place)f(fak)o(e)51 556 y(strings)f(in)e(the)g(message)g(that)h(are)g(not)g(in)f Fl(L)p Fo(\()p Fl(M)5 b Fo(\)?)22 b(Not)14 b(ones)h(that)g(will)f(cause)g Fl(M)20 b Fo(to)15 b(fail)f(to)h(halt)51 616 y(b)q(ecause)i(the)f(sender)g (itself)f(w)o(ould)h(fail)g(to)h(deco)q(de)f(them.)125 677 y(But,)g(there)g(are)g(other)h(features)g(of)g(undecidabilit)o(y)d(that)j(ma) o(y)e(pro)o(v)o(e)h(v)o(exing.)22 b(Rice's)15 b(theo-)51 737 y(rem)c(states)i(that)g(ev)o(ery)e(non-trivial)h(prop)q(ert)o(y)g(of)g (recursiv)o(ely)e(en)o(umerable)g(sets)j(is)f(also)h(undecid-)51 797 y(able.)22 b(This)17 b(means)e(that)i(it)f(is)h(undecidable)e(whether)h (t)o(w)o(o)h(TM's)f(accept)g(the)g(same)g(languages.)51 857 y(Theoretically)l(,)g(it)g(means)g(that)i(breaking)f(suc)o(h)g(a)g(system)f (with)h(a)g(mac)o(hine)e(is)i(an)g(undecidable)51 917 y(prop)q(osition.)125 978 y(In)12 b(realit)o(y)l(,)g(this)h(problem)e(can)j(only)e(arise)h(when)g (there)g(are)g(in\014nite)f(strings)h(in)g(the)g(language.)51 1038 y(These)18 b(cases)f(are)h(certainly)e(p)q(ossible,)h(but)h(they)f(can)g (b)q(e)h(a)o(v)o(oided)e(b)o(y)h(mo)q(delling)f(them)g(with)h(a)51 1098 y(\014nite)i(appro)o(ximation.)27 b(The)19 b(samples)f(from)f(real)i (systems)e(will)h(all)g(ha)o(v)o(e)g(\014nite)g(strings.)29 b(It)19 b(is)51 1158 y(unclear)h(if)g(an)o(y)f(of)i(these)f(results)f(from)g (computabilit)o(y)f(ha)o(v)o(e)h(an)o(y)h(application)g(here.)32 b(Nev)o(er-)51 1218 y(the-less,)22 b(the)f(results)g(from)f(the)h(section)g (of)g(the)g(pap)q(er)h(on)g(Con)o(text-F)l(ree)e(Mimic)f(F)l(unctions)51 1279 y(also)e(applies)e(here)h(b)q(ecause)g(the)f(recursiv)o(ely)f(en)o (umerable)f(languages)k(can)f(b)q(e)h(mo)q(deled)d(as)i(V)l(an)51 1339 y(Wijngaarden)h(grammars.)j(And)c(these)g(are)g(certainly)f(more)g (complex)f(for)j(an)g(attac)o(k)o(er.)51 1483 y Fm(Conclusion)22 b(and)i(Notes)e(ab)r(out)i(No)n(v)n(elt)n(y)51 1576 y Fo(Most)c(of)f(the)g(b) q(est)h(w)o(ork)f(on)h(cryptograph)o(y)f(has)h(tried)e(to)i(guaran)o(tee)f (securit)o(y)f(b)o(y)h(relying)f(on)51 1636 y(the)f(assumptions)f(that)h (certain)f(problems)g(lik)o(e)e(factoring)j(large)g(in)o(tegers)f([RRA78)o(]) g(or)h(solving)51 1696 y(knapsac)o(ks)f([)p Fb(?)p Fo(])k(is)15 b(a)h(di\016cult)d(problem.)19 b(If)c(these)f(problems)g(are)h(hard,)g(then)g (the)g(cipher)f(system)51 1756 y(ma)o(y)21 b(b)q(e)h(secure.)39 b(This)22 b(w)o(ork)g(tries)g(to)g(enco)q(de)h(information)e(in)h(three)f (di\013eren)o(t)g(theoretical)51 1816 y(mo)q(dels)14 b(of)h(general)f (computation)g(and)h(these)f(mo)q(dels)g(are)g(m)o(uc)o(h)e(more)i(general)g (than)h(the)f(other)51 1876 y(sp)q(eci\014c)i(cases.)21 b(This)16 b(is)f(b)q(oth)i(a)f(blessing)f(and)i(a)f(curse.)21 b(In)15 b(one)h(sense)g(the)f(classes)h(of)g(problems)51 1937 y(are)c(larger)f(and)g (therefore)g(m)o(ust)e(b)q(e)j(at)f(least)g(more)f(complicated)f(to)i(solv)o (e.)19 b(T)l(uring)11 b(mac)o(hines)e(can)51 1997 y(mo)q(del)j(factoring,)h (but)f(factoring)h(cannot)g(mo)q(del)e(turing)i(mac)o(hines.)18 b(On)12 b(the)g(other)h(hand)g(it)f(ma)o(y)51 2057 y(not)18 b(b)q(e)f(p)q(ossible)g(to)g(\014nd)g(hard-to-break)h(examples)d(of)i(the)g (set.)23 b(The)17 b(\014rst)g(theoretical)f(section)51 2117 y(sho)o(w)o(ed)23 b(that)g(if)f(a)h(general)g(metho)q(d)f(for)h(breaking)f (Mimic)e(F)l(unctions)j(could)f(b)q(e)h(disco)o(v)o(ered)51 2177 y(through)f(a)e(probabilistic)f(learning)h(theory)l(,)g(then)g(it)g(w)o (ould)g(b)q(e)g(p)q(ossible)g(to)g(break)g(RSA)g(and)51 2238 y(factor)d(Blum)d(in)o(tegers.)125 2298 y(But)k(these)h(theoretical)e (reductions)i(are)g(only)f(half)h(of)g(the)g(problem.)27 b(The)19 b(Knapsac)o(k)g(sys-)51 2358 y(tem)f(of)h(Merkle)e(and)j(Hellman)c(taugh)o(t) k(the)f(lesson)g(that)g(it)g(ma)o(y)e(not)j(b)q(e)f(p)q(ossible)g(to)g(iden)o (tify)51 2418 y(signi\014can)o(t)g(classes)h(of)f(di\016cult)f(to)i(solv)o(e) e(NP-complete)f(problems.)29 b(The)19 b(same)g(ma)o(y)e(b)q(e)j(true)51 2478 y(for)f(grammars.)25 b(The)19 b(construction)f(here)g(attempts)f(to)h (pro)q(duce)h(a)f(set)g(that)h(ma)o(y)e(b)q(e)h(di\016cult)51 2539 y(to)j(crac)o(k)f(and)h(it)e(o\013ers)j(a)e(straigh)o(t-forw)o(ard)i (metho)q(d)d(for)i(estimating)e(the)h(brute-force)g(e\013ort)51 2599 y(that)f(w)o(ould)f(b)q(e)g(needed)f(to)h(moun)o(t)f(an)h(attac)o(k)g (on)h(a)f(particular)f(grammar.)25 b(It)17 b(is)h(not)h(certain,)51 2659 y(ho)o(w)o(ev)o(er,)e(whether)g(this)g(classi\014cation)h(of)f (di\016cult)g(to)h(crac)o(k)e(grammars)g(is)i(a)g(solid)f(one.)25 b(This)949 2828 y(13)p eop %%Page: 14 14 bop 51 195 a Fo(job)15 b(is)f(left)f(to)h(the)g(readers)g(who)g(will,)f(I)h (hop)q(e,)g(b)q(e)g(able)g(to)g(dev)o(ote)f(some)g(time)f(to)i(p)q(osing)h (attac)o(ks)51 255 y(on)i(the)f(system.)125 315 y(The)g(last)g(section)g(of)h (the)f(pap)q(er)g(discussed)h(turing-complete)d(mimi)o(c)f(functions)j(whic)o (h)g(are,)51 376 y(in)e(a)h(sense,)f(as)g(big)h(as)f(all)g(cryptograph)o(y)l (.)20 b(All)13 b(encryption)h(systems)e(can)j(b)q(e)f(mo)q(deled)e(as)j(T)l (uring)51 436 y(Mac)o(hines)j(and)h(breaking)f(a)g(system)f(is)h(equiv)m (alen)o(t)e(to)j(infering)e(the)h(mac)o(hine.)25 b(Although)19 b(this)51 496 y(is)f(tec)o(hnically)d(an)j(undecidable)e(problem,)g(it)h(is)g (unclear)g(whether)g(these)h(results)f(can)g(add)i(an)o(y)51 556 y(real)13 b(kno)o(wledge)g(to)h(the)f(securit)o(y)f(of)i(the)f(system.)18 b(It)13 b(sho)o(ws)i(that)e(that)h(some)e(theoretical)h(results)51 616 y(are)k(not)f(immediatel)o(y)d(v)m(aluable.)21 b(Nev)o(er-the-less,)14 b(there)i(is)g(no)h(reason)g(not)f(to)h(expand)f(the)g(use)51 677 y(of)j(the)f(system)f(b)q(ey)o(ond)i(con)o(text-free)e(grammars)g(to)h(T) l(uring)h(Mac)o(hines.)27 b(All)17 b(of)i(the)f(previous)51 737 y(results)c(hold)g(and)g(it)f(should)i(b)q(e)f(ob)o(vious)g(that)g(the)f (grammars)g(are)g(more)g(robust,)h(more)f(realistic)51 797 y(and)k(at)g(least)f(as)h(di\016cult)e(to)h(break.)51 941 y Fm(App)r(endix:)31 b(One)22 b(W)-6 b(a)n(y)24 b(of)f(Enco)r(ding)g(Data)g(in) g(Non-determanism)51 1034 y Fo(This)d(is)g(a)g(general)g(tec)o(hnique)e(that) i(con)o(v)o(erts)f(data)i(in)o(to)e(a)h(stream)f(of)h(c)o(hoices)f(that)h(is) g(based)51 1094 y(Hu\013man)h(co)q(ding.)36 b([Huf51)o(,)21 b(Sto88)q(].)35 b(This)21 b(can)g(b)q(e)g(used)g(to)h(turn)f(non-determanism) d(in)o(to)j(a)51 1154 y(tec)o(hnique)15 b(for)h(enco)q(ding)h(information.) 125 1214 y(Start)h(with)h(a)f(set)h(of)f(c)o(hoices,)g Fk(f)p Fl(c)770 1221 y Fj(1)790 1214 y Fl(;)8 b(:)g(:)g(:)f(;)h(c)920 1221 y Fi(m)953 1214 y Fk(g)p Fo(.)28 b(Let,)18 b Fl(p)p Fo(\()p Fl(c)1187 1221 y Fi(i)1202 1214 y Fo(\))g(b)q(e)h(the)f(probabilit)o(y)f (that)i(c)o(hoice,)51 1275 y Fl(c)72 1282 y Fi(i)87 1275 y Fo(,)c(should)g(b)q(e)h(c)o(hosen.)21 b(This)15 b(algorithm)f(will)h(pro)q (duce)g(a)h(tree,)e(that)i(can)g(b)q(e)f(con)o(v)o(erted)f(in)o(to)h(an)51 1335 y(enco)q(ding)i(device:)111 1449 y(1.)24 b(Construct)19 b(a)f(no)q(de,)h Fl(n)608 1456 y Fi(i)622 1449 y Fo(,)f(for)g(eac)o(h)f(c)o (hoice,)g Fl(c)1023 1456 y Fi(i)1037 1449 y Fo(.)26 b(Eac)o(h)18 b(no)q(de)g(will)f(ha)o(v)o(e)g(a)h(n)o(um)o(b)q(er,)e Fl(P)7 b Fo(\()p Fl(n)1848 1456 y Fi(i)1863 1449 y Fo(\),)173 1509 y(asso)q(ciated)19 b(with)e(it.)24 b(F)l(or)17 b(no)q(des,)h Fk(f)p Fl(n)884 1516 y Fj(1)904 1509 y Fl(;)8 b(:)g(:)g(:)f(;)h(n)1042 1516 y Fi(m)1075 1509 y Fk(g)p Fo(,)17 b(set)h Fl(P)7 b Fo(\()p Fl(n)1295 1516 y Fi(i)1309 1509 y Fo(\))17 b(equal)g(to)h Fl(p)p Fo(\()p Fl(c)1600 1516 y Fi(i)1614 1509 y Fo(\).)24 b(Eac)o(h)18 b(no)q(de)173 1569 y(can)g(also)g(ha)o(v)o(e)f(t)o(w)o(o)g(descenden)o(ts,)g (whic)o(h)g(in)g(the)g(case)h(of)g(no)q(des,)g Fk(f)p Fl(n)1505 1576 y Fj(1)1525 1569 y Fl(;)8 b(:)g(:)g(:)f(;)h(n)1663 1576 y Fi(m)1697 1569 y Fk(g)p Fo(,)17 b(are)h(set)173 1629 y(to)f(b)q(e)f(empt)o (y)l(.)111 1731 y(2.)24 b(Construct)17 b(a)g(set,)f Fl(S)s Fo(.)21 b(Place)15 b(all)h(the)g(no)q(des)h(in)f Fl(S)s Fo(.)111 1833 y(3.)24 b(While)16 b Fl(S)j Fo(has)e(more)e(than)i(one)f(no)q(de)h(in)f (it,)f(do:)194 1947 y(\(a\))25 b(Find)16 b(the)g(t)o(w)o(o)g(no)q(des,)g Fl(n)752 1954 y Fi(i)783 1947 y Fo(and)h Fl(n)907 1954 y Fi(j)941 1947 y Fo(in)f Fl(S)j Fo(with)d(the)g(smallest)f(v)m(alues)h(of)h Fl(P)7 b Fo(.)191 2028 y(\(b\))25 b(Remo)o(v)o(e)13 b(these)j(no)q(des)h (from)e Fl(S)s Fo(.)197 2109 y(\(c\))24 b(Construct)18 b(a)g(new)g(no)q(de,)g Fl(n)816 2116 y Fi(k)838 2109 y Fo(,)f(with)h Fl(n)1011 2116 y Fi(i)1043 2109 y Fo(and)g Fl(n)1168 2116 y Fi(j)1204 2109 y Fo(as)g(its)g(descenden)o(ts.)25 b(Let)18 b Fl(P)7 b Fo(\()p Fl(n)1801 2116 y Fi(k)1822 2109 y Fo(\))16 b(=)281 2169 y Fl(P)7 b Fo(\()p Fl(n)367 2176 y Fi(i)381 2169 y Fo(\))k(+)g Fl(P)c Fo(\()p Fl(n)546 2176 y Fi(j)565 2169 y Fo(\).)21 b(Place)16 b Fl(n)779 2176 y Fi(k)816 2169 y Fo(in)g Fl(S)s Fo(.)111 2283 y(4.)24 b(The)17 b(one)f(remaining)f(no)q(de)i(is)f(the)g(ro)q(ot)h(of)f(the) g(tree.)21 b(A)o(t)15 b(the)h(lea)o(v)o(es)f(are)i(the)f(c)o(hoices.)125 2397 y(Bits)f(can)i(b)q(e)f(con)o(v)o(erted)f(to)h(c)o(hoices)g(with)g(this)g (algorithm:)111 2512 y(1.)24 b(Start)16 b(at)g(the)g(ro)q(ot.)22 b(Let)16 b Fl(N)k Fo(b)q(e)c(the)f(curren)o(t)g(no)q(de)h(of)g(the)g(searc)o (h,)f(whic)o(h)f(at)i(this)g(p)q(oin)o(t)g(is)173 2572 y(the)g(ro)q(ot.)111 2673 y(2.)24 b(While)16 b Fl(N)21 b Fo(is)16 b(not)h(a)g(leaf)e(no)q(de)i (do:)949 2828 y(14)p eop %%Page: 15 15 bop 194 195 a Fo(\(a\))25 b(Sample)14 b(one)j(bit)f(from)f(the)h(data)h (stream.)191 276 y(\(b\))25 b(Use)18 b(this)g(one)h(bit)f(to)h(c)o(ho)q(ose)g (one)f(of)h(the)f(t)o(w)o(o)h(descenden)o(ts)e(for)i(the)f(new)h(v)m(alue)f (of)281 336 y Fl(N)5 b Fo(.)25 b(In)17 b(the)g(simplest)f(form)g(this)i(c)o (hoice)e(can)i(b)q(e)f(hard-co)q(ded.)26 b(I.E.)17 b(zero)g(means)g(go)281 396 y(left,)c(one)i(means)f(go)h(righ)o(t.)21 b(But)14 b(a)h(complex)d(mac)o (hine)h(could)h(also)i(b)q(e)e(used)h(to)g(mak)o(e)281 457 y(the)h(c)o(hoice)f(based)h(up)q(on)i(the)e(no)q(de,)g Fl(n)1013 464 y Fi(i)1044 457 y Fo(and)g(some)g(pseudo-random)g(state.)125 571 y(When)e(the)f(data)i(is)f(ev)o(enly)e(distributed)h(b)q(et)o(w)o(een)g (0's)h(and)h(1's)f(and)g(the)g(function)g(that)g(maps)51 631 y(the)g(bits)g(to)g(the)g(c)o(hoices)e(is)i(un)o(biased,)g(then)f(f)h Fl(D)q Fo(\()p Fl(n)1013 638 y Fi(i)1028 631 y Fo(\))g(is)g(the)f(distance)h (b)q(et)o(w)o(een)f Fl(n)1592 638 y Fi(i)1620 631 y Fo(and)h(the)g(ro)q(ot,) 51 691 y(then)j(the)f(probabilit)o(y)f(of)h(a)h(c)o(hoice,)e Fl(c)772 698 y Fi(i)786 691 y Fo(,)g(b)q(eing)i(made)e(is)h(2)1150 673 y Fi(D)q Fj(\()p Fi(n)1215 678 y Fd(i)1229 673 y Fj(\))1244 691 y Fo(.)51 858 y Fa(References)51 967 y Fo([AHU83])23 b(A.V.)15 b(Aho,)g(J.E.)g(Hop)q(croft,)h(and)g(J.D.)g(Ullman.)i Fn(Data)f(Structur)n (es)g(and)h(A)o(lgorithms)p Fo(.)261 1027 y(Addison-W)l(esley)l(,)d(Reading,) h(Massac)o(h)o(usetts,)g(1983.)51 1129 y([HU69])60 b(J.E.)23 b(Hop)q(croft)h(and)h(J.D.)e(Ullman.)41 b Fn(F)l(ormal)24 b(L)n(anguages)h (and)f(their)h(R)n(elation)f(to)261 1189 y(A)o(utomata)p Fo(.)e(Addison-W)l (esley)l(,)15 b(Reading,)h(Massac)o(h)o(usetts,)g(1969.)51 1291 y([Huf51])55 b(D)19 b(Hu\013man.)27 b(A)18 b(metho)q(d)g(for)g(the)g (construction)h(of)g(minim)n(um)14 b(redundancy)k(co)q(des.)261 1351 y Fn(Pr)n(o)n(c)n(e)n(e)n(dings)f(of)g(the)h(Institute)h(of)e(R)n(adio)f (Engine)n(ers)p Fo(,)h(40:1098{1101)q(,)h(1951.)51 1453 y([JKM90])26 b(H.N.)21 b(Jendal,)j(Y.)e(J.)g(B.)g(Kuhn,)i(and)f(J.)g(L.)g(Massey)l(.)40 b(An)22 b(information-theoretic)261 1513 y(treatmen)o(t)14 b(of)i(homophonic)f(substitution.)21 b(In)15 b Fn(A)n(dvanc)n(es)j(in)f (Cryptolo)n(gy{Eur)n(o)n(crypt)261 1573 y(`89)p Fo(,)f(New)g(Y)l(ork,)f (1990.)i(Springer-V)l(erlag,)f(Lecture)g(Notes)g(in)g(Computer)f(Science.)51 1675 y([Kea89])50 b(Mic)o(hael)16 b(Kearns.)24 b Fn(The)19 b(Computational)f(Complexity)h(of)f(Machine)h(L)n(e)n(arning)p Fo(.)24 b(PhD)261 1735 y(thesis,)h(Harv)m(ard)f(Univ)o(ersit)o(y)d(Cen)o(ter) i(for)h(Researc)o(h)f(in)g(Computing)g(T)l(ec)o(hnology)l(,)261 1795 y(Ma)o(y)16 b(1989.)51 1897 y([KV89])59 b(Mic)o(hael)21 b(Kearns)i(and)g(Leslie)f(V)l(alien)o(t.)39 b(Cryptographic)23 b(limitations)e(on)i(learning)261 1957 y(b)q(o)q(olean)g(form)o(ulae)d(and)i (\014nite)f(automata.)37 b(In)22 b Fn(Pr)n(o)n(c)n(e)n(e)n(dings)f(of)h(the)h (Twenty-First)261 2017 y(A)o(nnual)k(A)o(CM)e(Symp)n(osium)g(on)h(The)n(ory)e (of)h(Computing)p Fo(,)j(pages)e(433{444,)j(Seat-)261 2077 y(tle,W)l(ashington,)16 b(Ma)o(y)g(1989.)51 2179 y([Riv91])58 b(Ron)17 b(Riv)o(est.)j(Cryptograph)o(y)d(and)g(mac)o(hine)d(learning.)21 b(In)16 b Fn(Asia)h(Crypt)g(91)p Fo(,)e(1991.)51 2281 y([RRA78])25 b(Adi)12 b(Shamir)g(Ron)h(Riv)o(est)e(and)j(Len)f(Adleman.)g(A)g(metho)q(d)e (for)i(obtaining)h(digital)e(sig-)261 2341 y(natures)k(and)g(public-k)o(ey)d (cryptosystems.)18 b Fn(Communic)n(ations)e(of)g(the)h(A)o(CM)p Fo(,)d(21\(11\),)261 2401 y(1978.)51 2503 y([Sim84])51 b(Gus)23 b(J.)f(Simmons.)37 b(The)22 b(prisoner's)h(problem)d(and)j(the)f(subliminal)e (c)o(hannel.)39 b(In)261 2563 y(Da)o(vid)22 b(Chaum,)h(editor,)f Fn(A)n(dvanc)n(es)i(in)f(Cryptolo)n(gy:)32 b(Pr)n(o)n(c)n(e)n(e)n(dings)22 b(of)g(Crypto)g(`83)p Fo(,)261 2623 y(pages)17 b(51{67,)h(New)e(Y)l(ork,)f (1984.)j(Plen)o(um.)949 2828 y(15)p eop %%Page: 16 16 bop 51 195 a Fo([Sto88])64 b(James)15 b(Storer.)20 b Fn(Data)d(Compr)n (ession)p Fo(.)j(Computer)14 b(Science)h(Press,)g(Ro)q(c)o(kville,)e(Mary-) 261 255 y(land,)j(1988.)51 357 y([V)l(al84])63 b(Leslie)18 b(G.)g(V)l(alien)o(t.)26 b(A)18 b(theory)h(of)f(the)g(learnable.)28 b Fn(Communic)n(ations)19 b(of)g(the)h(A)o(CM)p Fo(,)261 417 y(27:1134{114)q(2,)f(1984.)51 519 y([W)l(a)o(y90])39 b(P)o(eter)21 b(W)l(a)o(yner.)34 b(Mimic)19 b(functions.)35 b(T)l(ec)o(hnical)19 b(Rep)q(ort)j(???,)g(Cornell)e(Univ)o(ersit)o(y)261 579 y(Departmen)o(t)15 b(of)i(Computer)e(Science,)f(Decem)o(b)q(er)g(1990.)51 681 y([W)l(a)o(y91])39 b(P)o(eter)14 b(W)l(a)o(yner.)j(If)d(sb266)h(w)o(an)o(ts)g (plain)o(text,)e(giv)o(e)g(them)f(plain)o(text...)k Fn(RISKS)g(Digest)p Fo(,)261 741 y(11\(71\),)i(Ma)o(y)d(1991.)949 2828 y(16)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF