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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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