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%!PS-Adobe-2.0 %%Creator: dvipsk 5.519a Copyright 1986, 1993 Radical Eye Software %%Title: spr.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: /usr/public/pkg/tex/bin/dvips spr %DVIPSSource: TeX output 1995.01.26:1326 %%BeginProcSet: tex.pro /TeXDict 250 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 4 get round 4 exch put dup dup 5 get round 5 exch put 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 /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin 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y(impressiv)n(e)21 b(to)h(con)n(vince)g(p)r(eople)g(of)h (the)f(infeasibilit)n(y)h(of)f(breaking)f(125-digit)g(RSA)i(k)n(eys.) 573 811 y(Since)j(1977)d(man)n(y)h(new)i(factoring)e(algorithms)f(ha)n (v)n(e)h(b)r(een)i(in)n(v)n(en)n(ted)f(and)f(the)i(a)n(v)-5 b(ail-)449 911 y(abilit)n(y)41 b(and)g(sp)r(eed)g(of)g(computing)g (resources)e(has)h(c)n(hanged)g(dramatically)-7 b(.)40 b(Riv)n(est's)449 1010 y(imaginary)e(`1)i(nanosecond)f(p)r(er)h(mo)r (dular)g(m)n(ultiplication')g(mac)n(hine,)g(ho)n(w)n(ev)n(er,)e(still) 449 1110 y(stretc)n(hes)32 b(the)h(imagination.)f(With)i(pip)r(elining) f(and)g(sp)r(ecial)f(purp)r(ose)h(hardw)n(are)d(with)449 1210 y(gate)c(dela)n(ys)g(of)g(ab)r(out)h(10)f(picoseconds,)g(it)h (migh)n(t)g(just)g(b)r(e)g(feasible,)g(ev)n(en)f(for)g(129-digit)449 1309 y(n)n(um)n(b)r(ers.)34 b(If)i(w)n(e)f(w)n(ere)f(to)h(build)g(suc)n (h)g(a)g(mac)n(hine,)f(then)i(w)n(e)f(estimate)g(that)g(it)h(has)e(a) 449 1409 y(fair)24 b(c)n(hance)h(to)g(\014nd)g(the)h(64-digit)d(factor) i(of)g Fo(r)i Fw(after)e(5)13 b Fl(\002)g Fw(10)2371 1379 y Fu(15)2465 1409 y Fw(mo)r(dular)25 b(m)n(ultiplications)449 1509 y(using)37 b(the)h Fj(elliptic)g(curv)n(e)e(metho)r(d)i Fw(\(ECM\).)g(This)g(w)n(ould)f(require)f(t)n(w)n(o)h(mon)n(ths:)g(240) 449 1608 y(quadrillion)28 b(times)h(faster)g(than)g(Riv)n(est's)g (estimate)g(|)g(progress)e(that)i(is)g(due)h(solely)e(to)449 1708 y(impro)n(v)n(ed)e(factoring)h(metho)r(ds.)573 1838 y(T)-7 b(o)r(da)n(y)g(,)26 b(\014nding)g(64-digit)g(prime)g(factors)f (using)h(ECM)g(is)g(b)r(eliev)n(ed)h(to)f(b)r(e)h(infeasible.)449 1938 y(On)35 b(a)f(Sparc)g(10)h(w)n(orkstation)e(the)i(5)23 b Fl(\002)g Fw(10)1875 1908 y Fu(15)1979 1938 y Fw(mo)r(dular)35 b(m)n(ultiplications)g(on)f(129-digit)449 2038 y(n)n(um)n(b)r(ers)c (that)h(w)n(ould)f(probably)f(b)r(e)i(needed)f(for)g(ECM)g(w)n(ould)h (tak)n(e)e(w)n(ell)i(in)f(excess)g(of)449 2137 y(15000)h(y)n(ears.)g (ECM)i(uses)g(mostly)g(computations)f(on)h(lo)r(cal)g(data)f(whic)n(h)i (can)e(b)r(e)i(k)n(ept)449 2237 y(in)26 b(the)g(cac)n(he,)f(and)g (hardly)g(an)n(y)g(access)f(to)i(main)f(memory)-7 b(.)25 b(Consequen)n(tly)-7 b(,)25 b(the)h(metho)r(d)449 2336 y(runs)20 b Fo(x)h Fw(times)g(faster)g(on)f(a)g(mac)n(hine)h(whose)f (`mips)h(rating')f(is)g Fo(x)i Fw(times)f(b)r(etter.)g(Therefore,)449 2436 y(a)33 b(statemen)n(t)g(of)g(the)g(kind)h(`\014nding)f(a)g (64-digit)f(prime)h(factor)f(of)h(a)g(129-digit)e(n)n(um)n(b)r(er)449 2536 y(using)37 b(ECM)g(w)n(ould)g(probably)f(tak)n(e)h(more)g(than)g (half)h(a)f(million)g(mips)h(y)n(ears')e(migh)n(t)449 2635 y(mak)n(e)28 b(some)g(sense.)g(Here,)g(w)n(e)g(rate)g(a)h(Sparc)e (10)h(w)n(orkstation)f(at)h(35)g(mips;)h(notice)f(that)449 2735 y(this)22 b(w)n(ould)g(imply)g(a)g(mips)g(rating)f(of)i(ab)r(out)f (3)f(million)i(for)e(Riv)n(est's)h(imaginary)e(mac)n(hine)449 2835 y(if)28 b(it)g(used)g(ordinary)d(32-bit)i(arithmetic.)573 2965 y(F)-7 b(or)30 b(the)h(quadratic)f(siev)n(e)g(factoring)f (algorithm)h(\(QS\),)h(the)g(metho)r(d)g(that)g(w)n(e)f(used)449 3065 y(to)f(factor)f Fo(r)r Fw(,)j(run-times)e(expressed)f(in)h(mips)h (y)n(ears)d(are)i(m)n(uc)n(h)g(harder)f(to)h(justify)-7 b(.)30 b(Siev-)449 3164 y(ing)24 b(algorithms)f(sp)r(end)i(most)g(of)f (their)h(time)g(accessing)e(more)h(or)g(less)g(random)g(lo)r(cations) 449 3264 y(in)39 b(memory)-7 b(.)38 b(On)h(most)f(w)n(orkstations,)f (pro)r(cessor)g(sp)r(eed)h(is)h(p)r(o)r(orly)f(correlated)f(with)449 3364 y(memory)29 b(sp)r(eed)g(and,)h(therefore,)e(the)i(mips)g(rating)f (of)g(the)h(pro)r(cessor)d(is)j(not)f(a)h(reliable)449 3463 y(measure)21 b(of)i(the)g(exp)r(ected)g(p)r(erformance)e(of)i(a)f (sieving)f(algorithm.)h(T)-7 b(o)22 b(giv)n(e)g(some)g(exam-)449 3563 y(ples:)f(on)h(a)f(Sparc)g(10)g(w)n(orkstation)f(QS)h(w)n(ould)g (ha)n(v)n(e)g(tak)n(en)g(ab)r(out)g(120)g(y)n(ears)f(to)h(factor)g Fo(r)r Fw(,)449 3662 y(whic)n(h)26 b(leads)g(to)g(an)g(estimate)g(of)g (ab)r(out)h(4000)d(mips)j(y)n(ears.)d(On)i(a)g(Dec)h(5000/240,)22 b(rated)449 3762 y(at)32 b(25)g(mips,)g(w)n(e)g(w)n(ould)g(ha)n(v)n(e)g (sp)r(en)n(t)g(237)f(y)n(ears,)g(whic)n(h)h(translates)g(in)n(to)g (almost)g(6000)449 3862 y(mips)i(y)n(ears.)f(On)g(a)h(Sun)h(3/50,)d(ho) n(w)n(ev)n(er,)g(rated)i(at)g(1.5)f(mips,)h(w)n(e)g(w)n(ould)g(ha)n(v)n (e)f(sp)r(en)n(t)449 3961 y(ab)r(out)e(1100)f(y)n(ears)g(but)j(only)e (1700)f(mips)i(y)n(ears.)e(Av)n(eraging)f(o)n(v)n(er)h(the)i(mac)n (hines)f(that)449 4061 y(w)n(ere)26 b(used,)i(and)g(k)n(eeping)e(in)i (mind)h(that)e(w)n(e)h(are)e(indulging)i(in)g(energetic)e(handw)n(a)n (ving,)449 4161 y(w)n(e)34 b(giv)n(e)g(a)g(v)n(ery)f(appro)n(ximate)g (estimate)i(of)f(4000)f(to)i(6000)d(mips)j(y)n(ears)e(for)h(the)h(time) 449 4260 y(sp)r(en)n(t)23 b(on)g(the)h(factorization)e(of)h Fo(r)r Fw(.)h(If)f(the)h(ratio)e(b)r(et)n(w)n(een)h(memory)g(and)g(pro) r(cessor)e(sp)r(eed)449 4360 y(of)35 b(Riv)n(est's)f(mac)n(hine)g(is)h (not)g(m)n(uc)n(h)f(w)n(orse)g(than)h(on)f(curren)n(t)g(w)n (orkstations,)f(it)i(could)449 4459 y(factor)c Fo(r)k Fw(in)d(less)g(than)g(a)g(da)n(y)-7 b(.)31 b(Ev)n(en)g(though)h (expressing)e(run-times)i(in)g(mips)h(y)n(ears)d(is)449 4559 y(far)h(from)g(ideal)h(for)f(memory-b)r(ound)f(pro)r(cesses,)h(it) h(is)f(common)g(practice)g(in)h(factoring)p eop %%Page: 5 5 5 4 bop 449 83 a Fw(literature.)37 b(It)h(pro)n(vides)e(a)h(con)n(v)n (enien)n(t)f(w)n(a)n(y)h(to)g(compare)g(and)g(predict)h(run-times)f(of) 449 183 y(factoring)26 b(problems,)h(and)h(to)f(c)n(hec)n(k)g(the)h (accuracy)e(of)h(theoretical)g(predictions.)573 286 y(Heuristically)g (it)h(can)g(b)r(e)g(argued)e(that)i(the)g(exp)r(ected)g(run-time)f(of)h (QS)f(b)r(eha)n(v)n(es)g(as)1196 477 y Fo(L)p Fw(\()p Fo(n)p Fw(\))c(=)g(exp\(\(1)18 b(+)g Fo(o)p Fw(\(1\)\))1990 402 y Fg(p)p 2074 402 477 4 v 2074 477 a Fw(log)c Fo(n)g Fw(log)g(log)g Fo(n)p Fw(\))449 668 y(for)34 b Fo(n)g Fl(!)g(1)p Fw(,)g(where)g Fo(n)g Fw(is)g(the)h(n)n(um)n(b)r(er)f(to)g (b)r(e)h(factored.)e(Often,)i(one)f(simply)g(ignores)449 768 y(the)c Fo(o)p Fw(\(1\),)g(and)g(in)n(terprets)e(the)i(resulting)g Fo(L)p Fw(\()p Fo(n)p Fw(\))f(as)g(the)h(n)n(um)n(b)r(er)g(of)f(op)r (erations)g(needed)449 867 y(to)f(factor)f Fo(n)p Fw(.)i(Since)f(the)h Fo(o)p Fw(\(1\))g(is)f(neither)g(0)g(nor)g(constan)n(t,)f(this)i (practice)e(hardly)h(mak)n(es)449 967 y(sense.)573 1070 y(This)22 b(naiv)n(e)g(approac)n(h)e(w)n(as)h(\014rst)h(used)g(in)g ([21:)g(Section)g(IX.A],)h(where)e Fo(L)p Fw(\()p Fo(n)p Fw(\),)h(without)449 1170 y(the)32 b Fo(o)p Fw(\(1\),)h(w)n(as)e(giv)n (en)g(as)g(the)i(run-time)f(of)g(Sc)n(hro)r(epp)r(el's)f(linear)g(siev) n(e)h(factoring)e(algo-)449 1269 y(rithm)21 b(\(cf.)g([22]\).)g(This)g (is)f(a)h(big)f(impro)n(v)n(emen)n(t)g(compared)f(with)j(Riv)n(est's)e (sligh)n(tly)g(earlier)449 1369 y(estimate)32 b(in)g([6].)g(The)g (run-time)g(from)g([21])f(did)h(not)g(include)h(the)f(matrix)g (elimination)449 1469 y(step,)25 b(b)r(ecause)f(this)i(step)f(w)n(as)e (considered)h(to)h(b)r(e)g(trivial)g(compared)e(to)i(the)g(sieving)g (step)449 1568 y(\(cf.)j([22]\).)f(Including)h(it)g(w)n(ould)f(ha)n(v)n (e)g(increased)f(the)i(run-time)g(to)1293 1771 y(exp\(\()p Fo(u=)p Fw(2)17 b(+)h Fo(o)p Fw(\(1\)\))1894 1696 y Fg(p)p 1978 1696 V 1978 1771 a Fw(log)c Fo(n)g Fw(log)g(log)g Fo(n)p Fw(\))449 1962 y(\(cf.)30 b([18:)f(Theorem)g(6.1]\),)g(where)g (a)g(linear)g(dep)r(endency)g(in)h(an)g(\()p Fo(s)19 b Fw(+)h(1\))f Fl(\002)h Fo(s)29 b Fw(bit-matrix)449 2061 y(can)35 b(b)r(e)g(found)h(in)f(time)h Fo(s)1310 2031 y Fn(u)1353 2061 y Fw(.)g(A)n(t)f(the)h(time)f(of)h(the)f (publication)g(of)g([21],)g(the)h(b)r(est)f(one)449 2161 y(could)h(do)g(w)n(as)g Fo(u)h Fl(\031)g Fw(2)p Fo(:)p Fw(795.)e(Although)h(for)g(small)g Fo(n)g Fw(the)h(matrix)f(step)h(is)f (in)h(practice)449 2261 y(negligible)g(compared)h(to)g(the)g(sieving)g (step,)g(this)h(is)f(not)g(the)h(case)e(for)h(the)h(larger)d Fo(n)449 2360 y Fw(in)f([21:)g(T)-7 b(able)35 b(1],)g(th)n(us)h(making) e(these)i(v)-5 b(alues)35 b(strictly)g(sp)r(eaking)f(\(and)i (disregarding)449 2460 y(the)27 b Fo(o)p Fw(\(1\)-issue\))g(incorrect,) f(for)h(1978.)e(It)i(w)n(as)g(later)f(sho)n(wn)g(that)i(the)f (sparseness)e(of)i(the)449 2560 y(matrix)g(can)g(b)r(e)h(exploited)g (and)f(that)h Fo(u)22 b Fw(=)h(2)18 b(+)g Fo(o)p Fw(\(1\))28 b(can)f(indeed)h(b)r(e)g(used.)573 2663 y(This)37 b(do)r(es)g(not)g (imply)g(that)g(there)g(are)f(an)n(y)g(signi\014can)n(t)g(digits)h(in)g ([21:)f(T)-7 b(able)37 b(1].)449 2762 y(The)28 b(table)g(illustrates,)f (ho)n(w)n(ev)n(er,)f(the)i(appro)n(ximate)f(gro)n(wth)f(rate)h(of)h Fo(L)p Fw(\()p Fo(n)p Fw(\),)g(whic)n(h)g Fj(can)449 2862 y Fw(b)r(e)40 b(used)g(in)g(an)f(attempt)i(to)e(predict)h (run-times:)g(if)g(factoring)e Fo(n)2658 2874 y Fu(1)2735 2862 y Fw(tak)n(es)h(time)h Fo(t)g Fw(for)449 2962 y(some)34 b(QS-implemen)n(tation)h(and)f Fl(j)14 b Fw(log)q(\()p Fo(n)1798 2974 y Fu(2)1835 2962 y Fo(=n)1927 2974 y Fu(1)1964 2962 y Fw(\))p Fl(j)35 b Fw(is)g(reasonably)e(small,)h(then)i (factoring)449 3061 y Fo(n)499 3073 y Fu(2)564 3061 y Fw(using)28 b(the)g(same)g(implemen)n(tation)g(can)g(b)r(e)g(exp)r (ected)h(to)f(tak)n(e)f(time)i(appro)n(ximately)449 3161 y Fo(t)18 b Fl(\001)h Fo(L)p Fw(\()p Fo(n)678 3173 y Fu(2)715 3161 y Fw(\))p Fo(=L)p Fw(\()p Fo(n)928 3173 y Fu(1)964 3161 y Fw(\),)28 b(where)f(b)r(oth)h Fo(o)p Fw(\(1\)'s)g(are)f(simply)g(omitted.)573 3264 y(Using)h(this)h(metho)r (d)g(it)g(w)n(as)e(estimated)h(in)h([2])f(that)g(factoring)g(a)f (120-digit)g(n)n(um)n(b)r(er)449 3364 y(using)i(QS)g(w)n(ould)g(tak)n (e)g(ab)r(out)g(950)f(mips)i(y)n(ears,)e(based)h(on)g(the)g(observ)-5 b(ation)28 b(from)i([16])449 3463 y(that)i(factoring)f(a)g(116-digit)f (n)n(um)n(b)r(er)i(to)r(ok)f(ab)r(out)h(400)e(mips)i(y)n(ears.)e(The)i (actual)f(120-)449 3563 y(digit)f(factorization)e(to)r(ok)h(ab)r(out)h (825)e(mips)i(y)n(ears,)e(sligh)n(tly)i(less)f(than)h(the)g (prediction,)449 3662 y(due)35 b(to)g(sev)n(eral)f(impro)n(v)n(emen)n (ts)f(in)j(the)f(program)e(and)i(in)n(ten)n(tional)g(sub)r(optimalit)n (y)g(of)449 3762 y(the)h(parameter)f(c)n(hoices)g(for)g(the)i (116-digit)d(factorization.)h(Similar)g(extrap)r(olation)g(of)449 3862 y(the)d(120-digit)d(result)i(to)h Fo(r)i Fw(w)n(ould)d(lead)g(to)g (ab)r(out)g(4200)f(mips)h(y)n(ears,)f(whic)n(h)h(is)g(on)g(the)449 3961 y(lo)n(w)h(side)i(of)f(the)h(4000)d(to)j(6000)d(mips)j(y)n(ears)d (that)j(w)n(e)f(sp)r(en)n(t.)h(This)f(ma)n(y)g(b)r(e)g(due)h(to)f(a)449 4061 y(com)n(bination)22 b(of)i(t)n(w)n(o)e(factors.)g(In)i(the)g (\014rst)f(place,)g(the)g(e\013ect)h(of)g(the)f(unkno)n(wn)g(b)r(eha)n (vior)449 4161 y(of)38 b(the)g Fo(o)p Fw(\(1\))h(in)f(the)h(9)e(digit)i (jump)g(from)e(120)g(to)h(129)f(digits)h(is)g(signi\014can)n(tly)f (bigger)449 4260 y(than)31 b(in)g(the)g(jump)g(from)g(116)e(to)i(120)e (digits.)i(Apparen)n(tly)-7 b(,)30 b(neglecting)g(the)h Fo(o)p Fw(\(1\))g(leads)449 4360 y(to)j(an)h(o)n(v)n(erly)d(optimistic) j(estimate.)g(Secondly)-7 b(,)34 b(the)i(e\013ects)e(of)h(small)f (\(and)h(relativ)n(ely)449 4459 y(slo)n(w\))30 b(memories)h(should)g(b) r(e)g(ev)n(en)g(w)n(orse)f(for)h Fo(r)j Fw(than)d(for)g(the)h (120-digit)d(factorization)449 4559 y(from)e([2].)p eop %%Page: 6 6 6 5 bop 573 83 a Fw(The)30 b(reason)f(wh)n(y)g(these)h(delib)r (erations)f(are)g(considered)g(to)h(b)r(e)g(useful)h(is)e(that)i(they) 449 183 y(migh)n(t)h(help)g(us)g(to)g(assess)e(the)i(securit)n(y)f(of)h (widely-used)g(512-bit)f(RSA)h(mo)r(duli.)h(Since)449 282 y Fo(L)p Fw(\(2)580 252 y Fu(512)683 282 y Fw(\))p Fo(=L)p Fw(\(10)930 252 y Fu(129)1032 282 y Fw(\))23 b Fl(\031)g Fw(92)h(\(without)j(the)e(t)n(w)n(o)g Fo(o)p Fw(\(1\)'s\),)h(w)n(e)g(\014nd)g(that)f(factoring)g(a)g(512-bit)449 382 y(RSA)31 b(mo)r(dulus)g(using)g(our)f(curren)n(t)g(QS-implemen)n (tation)h(can)f(probably)g(not)h(b)r(e)g(done)449 482 y(in)26 b(less)f(than)h(500)14 b(000)22 b(mips)k(y)n(ears.)e(An)i(upp)r (er)g(b)r(ound)g(is)f(harder)g(to)g(giv)n(e,)g(but)h(a)g(million)449 581 y(mips)i(y)n(ears)d(is)j(probably)e(the)i(righ)n(t)f(order)g(of)g (magnitude.)573 682 y(W)-7 b(e)35 b(should)e(stress)g(that)h(this)g (estimate)g(migh)n(t)g(not)g(b)r(e)g(a)g(go)r(o)r(d)f(indication)h(of)f (the)449 782 y(di\016cult)n(y)22 b(to)f(break)g(a)g(512-bit)f(RSA)i(mo) r(dulus.)g(It)g(tells)g(us)f(ho)n(w)g(hard)g(it)h(w)n(ould)f(probably) 449 881 y(b)r(e)g(for)f(a)g(fairly)g(w)n(ell)g(understo)r(o)r(d,)g (reliable)g(and)g(b)n(y)g(no)n(w)g(rather)f(ancien)n(t)h(algorithm.)g (The)449 981 y(new)n(er)29 b Fj(n)n(um)n(b)r(er)i(\014eld)f(siev)n(e)k Fw(factoring)c(algorithm)f(\(NFS\))j(\(cf.)f([13]\))f(is)g (asymptotically)449 1081 y(sup)r(erior)39 b(to)h(QS.)g(Recen)n(t)g(exp) r(erimen)n(ts)g(ha)n(v)n(e)f(suggested)g(that)h(the)h(crosso)n(v)n(er)c (p)r(oin)n(t)449 1180 y(b)r(et)n(w)n(een)e(QS)h(and)f(NFS)h(lies)f(w)n (ell)h(b)r(elo)n(w)f(116)f(digits)h(\(cf.)i([5;)e(7]\),)g(but)h(it)g (remains)f(to)449 1280 y(b)r(e)30 b(seen)f(ho)n(w)f(w)n(ell)i(the)f (memory)g(requiremen)n(ts)f(of)h(NFS)i(for)d(512-bit)h(n)n(um)n(b)r (ers)f(can)h(b)r(e)449 1379 y(handled.)i(In)g(an)n(y)f(case,)g(w)n(e)h (susp)r(ect)g(that)g(with)h(the)f(factorization)f(of)h Fo(r)j Fw(w)n(e)c(ha)n(v)n(e)g(seen)449 1479 y(the)c(last)f(big)g (QS-factorization.)f(F)-7 b(urthermore,)24 b(giv)n(en)h(ho)n(w)g(hard)f (it)i(w)n(as,)f(17)f(y)n(ears)g(ago,)449 1579 y(to)35 b(mak)n(e)f(a)g(decen)n(t)h(prediction)f(ab)r(out)h(the)g(di\016cult)n (y)h(of)e(factoring)g Fo(r)r Fw(,)i(it)f(is)g(probably)449 1678 y(b)r(etter)28 b(to)f(a)n(v)n(oid)f(making)h(an)n(y)g(\014rm)h (prediction)f(ab)r(out)g(512-bit)g(n)n(um)n(b)r(ers.)449 1967 y Fp(3)115 b(F)-10 b(actoring)39 b Fk(r)449 2172 y Fw(W)-7 b(e)29 b(used)g(the)h(double)f(large)e(prime)i(m)n(ultiple)h (p)r(olynomial)e(v)-5 b(ariation)28 b(of)h(the)h(quadratic)449 2272 y(siev)n(e)20 b(factoring)g(metho)r(d)h(\(QS\))g(to)g(factor)f Fo(r)r Fw(.)i(The)f(goal)e(of)i(QS)g(is)f(to)h(\014nd)g(pairs)f(of)h (in)n(tegers)449 2371 y Fo(x)p Fw(,)j Fo(y)i Fw(suc)n(h)d(that)g Fo(x)1015 2341 y Fu(2)1076 2371 y Fl(\021)g Fo(y)1208 2341 y Fu(2)1268 2371 y Fw(mo)r(d)g Fo(r)r Fw(.)h(Because)e Fo(r)k Fw(divides)e Fo(x)2235 2341 y Fu(2)2282 2371 y Fl(\000)10 b Fo(y)2401 2341 y Fu(2)2460 2371 y Fw(=)23 b(\()p Fo(x)10 b Fl(\000)g Fo(y)s Fw(\)\()p Fo(x)g Fw(+)g Fo(y)s Fw(\),)23 b(w)n(e)g(\014nd)449 2471 y(that)e Fo(r)26 b Fw(=)d(gcd)o(\()p Fo(r)n(;)14 b(x)5 b Fl(\000)g Fo(y)s Fw(\))14 b(gcd)q(\()p Fo(r)n(;)g(x)5 b Fw(+)g Fo(y)s Fw(\).)22 b(If)g(the)f(pairs)f(are)g(generated)g(in)h(a)g(random)f (fashion,)449 2571 y(then)30 b(there)g(is)g(a)g(probabilit)n(y)f(of)g (1)p Fo(=)p Fw(2)g(that)h(this)h(factorization)d(is)i(non-trivial)f (\(b)r(ecause)449 2670 y Fo(r)34 b Fw(has)d(only)g(t)n(w)n(o)f (distinct)i(prime)g(factors\))e([3].)h(In)h(QS,)f(pairs)f Fo(x)p Fw(,)i Fo(y)i Fw(are)d(constructed)g(in)449 2770 y(t)n(w)n(o)25 b(steps,)g(the)h Fj(sieving)f(step)j Fw(and)d(the)h Fj(matrix)f(step)h Fw([12:)e(4.16;)h(16;)f(18;)h(24].)g(W)-7 b(e)26 b(giv)n(e)e(an)449 2870 y(outline)h(of)h(these)f(steps)g(as)g (applied)g(to)g Fo(r)r Fw(.)h(In)g(the)g(full)g(pap)r(er)f(w)n(e)g (will)g(giv)n(e)f(more)h(details.)449 3125 y Fm(3.1)94 b(The)32 b(sieving)h(step)449 3297 y Fw(First,)18 b(w)n(e)g(select)g(a) g Fj(m)n(ultiplier)p Fw(,)h(a)f(small)g(p)r(ositiv)n(e)g(in)n(teger)f Fo(m)i Fw(suc)n(h)f(that)g Fo(mr)k Fw(is)c(a)g(quadratic)449 3397 y(residue)28 b(mo)r(dulo)h(man)n(y)f(small)h(primes)f(\(cf.)i([9:) e(4.5.4]\).)h(Next,)g(w)n(e)f(compute)h(the)h Fj(factor)449 3496 y(base)j Fo(P)12 b Fw(,)35 b(consisting)e(of)h Fl(\000)p Fw(1)g(and)g(the)g(primes)g Fo(p)g Fl(\024)g Fo(B)2225 3508 y Fu(1)2296 3496 y Fw(for)g(whic)n(h)g Fo(mr)j Fw(is)d(a)g (quadratic)449 3596 y(residue)h(mo)r(dulo)h Fo(p)p Fw(,)g(for)f(some)g (b)r(ound)i Fo(B)1825 3608 y Fu(1)1862 3596 y Fw(.)f(In)g(the)g (sieving)f(step)h(w)n(e)g(collect)f(a)h(set)g(of)449 3695 y Fj(relations)p Fw(:)26 b(in)n(teger)h(tuples)h(\()p Fo(v)s(;)14 b(q)1478 3707 y Fu(1)1516 3695 y Fo(;)g(q)1590 3707 y Fu(2)1627 3695 y Fo(;)g Fw(\()p Fo(e)1735 3707 y Fn(p)1773 3695 y Fw(\))1805 3707 y Fn(p)p Ff(2)p Fn(P)1940 3695 y Fw(\))28 b(suc)n(h)f(that)449 3893 y(\(3)p Fo(:)p Fw(2\))780 b Fo(v)1443 3859 y Fu(2)1504 3893 y Fl(\021)22 b Fo(q)1628 3905 y Fu(1)1684 3893 y Fl(\001)c Fo(q)1762 3905 y Fu(2)1818 3893 y Fl(\001)1872 3814 y Fg(Y)1860 3992 y Fn(p)p Ff(2)p Fn(P)2004 3893 y Fo(p)2046 3859 y Fn(e)2077 3867 y Fe(p)2139 3893 y Fw(mo)r(d)23 b Fo(r)n(;)449 4161 y Fw(where)39 b(eac)n(h)h Fo(q)938 4173 y Fn(i)1006 4161 y Fw(is)g(either)g(1)g(or)f(a)h Fj(large)e(prime)j Fl(2)j Fw(\()p Fo(B)2286 4173 y Fu(1)2323 4161 y Fo(;)14 b(B)2423 4173 y Fu(2)2461 4161 y Fw(],)40 b(for)g(some)f Fj(large)g(prime)449 4260 y(b)r(ound)27 b Fo(B)767 4272 y Fu(2)804 4260 y Fw(.)g(If)g Fo(q)973 4272 y Fu(1)1033 4260 y Fw(=)c Fo(q)1158 4272 y Fu(2)1218 4260 y Fw(=)g(1)j(the)h (relation)f(is)g(called)g Fj(full)p Fw(,)i(if)f(only)f(one)g(of)h(the)g Fo(q)2984 4272 y Fn(i)3038 4260 y Fw(equals)f(1)449 4360 y(it)35 b(is)h(a)e Fj(partial)p Fw(,)h(and)g(a)g Fj(double)g(partial)i Fw(otherwise.)e(If)g(the)h(large)e(primes)g(matc)n(h,)h(the)449 4459 y(\(double\))d(partial)e(relations)g(can)h(b)r(e)h(com)n(bined)f (in)n(to)g Fj(cycles)g Fw([16].)f(The)i(sieving)e(step)i(is)449 4559 y(complete)27 b(as)g(so)r(on)g(as)g(#fulls+#cycles)14 b Fo(>)22 b Fw(#)p Fo(P)12 b Fw(.)p eop %%Page: 7 7 7 6 bop 573 83 a Fw(W)-7 b(e)28 b(used)f Fo(m)c Fw(=)g(5)k(as)g(m)n (ultiplier.)g(Asymptotically)-7 b(,)28 b(the)f(factor)g(base)g(size)g (in)g(QS)g(ap-)449 183 y(plied)f(to)g Fo(n)g Fw(b)r(eha)n(v)n(es)f(as) 1234 112 y Fg(p)p 1317 112 171 4 v 71 x Fo(L)p Fw(\()p Fo(n)p Fw(\),)h(for)f Fo(n)e Fl(!)g(1)p Fw(.)k(A)f(factor)f(base)g (size)h(of)g(245)14 b(810)23 b(elemen)n(ts)449 282 y(w)n(as)f(close)g (to)h(optimal)g(for)f(the)i(120-digit)d(n)n(um)n(b)r(er)i(RSA-120)f (factored)g(in)i([2],)e(with)i(m)n(ul-)449 382 y(tiplier)32 b(7.)f(Therefore)g(a)g(factor)g(base)g(of)h(245)14 b(810)2044 311 y Fg(p)p 2124 311 820 4 v 2124 382 a Fo(L)p Fw(\()p Fo(mr)r Fw(\))p Fo(=L)p Fw(\(7)k Fl(\001)g Fw(RSA-120)o(\))30 b Fl(\031)g Fw(554)14 b(933)449 482 y(\(omitting)27 b(b)r(oth)h Fo(o)p Fw(\(1\)'s\))f(elemen)n(ts)g(should)g(b)r(e)g(reasonable)e(but)j (on)e(the)i(small)f(side)f(\(b)r(e-)449 581 y(cause)c(the)h Fo(o)p Fw(\(1\))h(gro)n(ws)d(with)i Fo(n)p Fw(\).)h(After)f(exp)r (erimen)n(ts)g(with)g(other)g(v)-5 b(alues,)22 b(w)n(e)h(settled)g(for) 449 681 y(#)p Fo(P)36 b Fw(=)25 b(524)14 b(339)21 b(=)j(51)18 b(+)h(2)1301 651 y Fu(19)1399 681 y Fw(\(and)29 b Fo(B)1657 693 y Fu(1)1719 681 y Fw(=)24 b(16)14 b(333)g(609\).)24 b(Our)k(c)n(hoice)g(is)g(certainly)g(within)449 780 y(20\045)c(of)g (optimal,)h(and)f(probably)g(sligh)n(tly)g(sub)r(optimal.)h(W)-7 b(e)25 b(felt)g(that)g(this)g(w)n(as)f(less)g(un-)449 880 y(desirable)31 b(than)h(the)g(memory-related)e(problems)h(caused)g (b)n(y)h(an)f(ev)n(en)g(larger)f(c)n(hoice.)3294 850 y Fu(7)449 980 y Fw(W)-7 b(e)28 b(used)f Fo(B)844 992 y Fu(2)905 980 y Fw(=)22 b(2)1034 950 y Fu(30)1104 980 y Fw(.)573 1084 y(Based)32 b(on)h(our)f(exp)r(erience)h(with)g(man)n(y) g(other)f(factorizations,)g(w)n(e)g(exp)r(ected)h(that)449 1183 y(w)n(e)27 b(w)n(ould)f(need)h(ab)r(out)g(110)14 b(000)24 b(fulls)k(b)r(efore)f(w)n(e)g(had)f(enough)h(cycles)f(to)h (complete)g(the)449 1283 y(sieving)35 b(step.)h(Exp)r(erimen)n(ts)f (indicated)h(that)f(this)h(w)n(ould)g(tak)n(e)f(less)g(than)h(6000)e (mips)449 1383 y(y)n(ears.)23 b(More)g(than)i(8)f(million)h(relations)e (w)n(ould)i(ha)n(v)n(e)e(to)h(b)r(e)h(collected)g(\(since)f(only)h(1)f (out)449 1482 y(of)35 b(ev)n(ery)e Fl(\031)i Fw(75)f(is)g(full\),)i (using)f(a)f(program)f(whic)n(h)i(w)n(ould)f(need)h(appro)n(ximately)e (10)p Fo(:)p Fw(5)449 1582 y(MBytes)h(to)g(run.)h(W)-7 b(e)34 b(opted)h(for)f(the)g(idle-cycle)g(collection)g(metho)r(d)h (from)f([15].)g(This)449 1682 y(sounds)h(easier)f(than)h(it)h(w)n(as,)e (ev)n(en)h(though)g(w)n(e)g(could)g(re-use)f(the)i(soft)n(w)n(are,)d(b) r(ecause)449 1781 y(our)23 b(e\013ort)h(w)n(ould)g(need)g(an)g(order)f (of)h(magnitude)g(more)f(time,)i(memory)-7 b(,)24 b(and)g(disk)g(space) 449 1881 y(to)31 b(run)h(the)g(sieving)f(program)e(and)j(to)f(store)g (the)h(relations.)e(W)-7 b(e)32 b(giv)n(e)f(more)f(details)i(of)449 1980 y(the)27 b(resource)e(managemen)n(t)h(in)h(Section)f(5.)h(W)-7 b(e)27 b(estimate)g(that)g(w)n(e)f(had)h(appro)n(ximately)449 2080 y(600)k(con)n(tributors)g(using)h(more)g(than)h(1600)d(mac)n (hines)i(and)h(pro)r(ducing)e(ab)r(out)i(80\045)f(of)449 2180 y(the)h(relations.)e(The)i(other)f(20\045)f(w)n(as)h(con)n (tributed)g(b)n(y)h(sev)n(eral)d(MasP)n(ars)g(running)i(the)449 2279 y(program)25 b(from)j([4].)573 2383 y(On)41 b(Marc)n(h)f(21)g (1994)f(w)n(e)h(had)h(ab)r(out)g(8)p Fo(:)p Fw(25)e(million)i (relations,)f(with)i(more)e(than)449 2483 y(108)14 b(000)22 b(fulls)j(and)g(417)14 b(000)22 b(cycles.)i(Because)g(108)14 b(000)f(+)g(417)h(000)j Fo(>)23 b Fw(524)14 b(339)20 b(=)i(#)p Fo(P)12 b Fw(,)25 b(the)449 2583 y(`cease)34 b(and)i(desist')f(message)f(w)n(as)g(mailed)i(out)f(on)g(Marc)n(h)g (22.)f(The)i(\014nal)f(coun)n(ts,)g(on)449 2682 y(Marc)n(h)c(26,)g(w)n (ere:)g(112)14 b(011)29 b(fulls,)j(1)14 b(431)g(337)28 b(partials,)j(and)h(6)14 b(881)g(138)28 b(double)k(partials,)449 2782 y(for)24 b(a)g(total)g(of)h(8)14 b(424)g(486)21 b(relations)i(after)h(less)g(than)h(220)e(da)n(ys)g(of)i(sieving.)f (The)g(\(double\))449 2882 y(partials)k(that)h(could)g(o)r(ccur)f(in)h (cycles)g(w)n(ere)f(written)h(to)g(tap)r(e)g(b)n(y)g(the)g(\014rst)g (author)f(and)449 2981 y(sen)n(t)33 b(to)h(the)g(third)g(\(b)n(y)f (regular)f(o)n(v)n(ernigh)n(t)g(mail\).)i(They)f(resulted)h(in)g(457)14 b(455)30 b(cycles,)449 3081 y(constructed)24 b(using)h(the)h(metho)r(d) f(from)g([16].)f(With)i(the)g(fulls,)f(whic)n(h)g(w)n(ere)g Fd(ftp)p Fw(-ed)e(from)449 3181 y Fd(hot-spare.mit.ed)o(u)g Fw(to)29 b Fd(flash.bellcore.)o(com)22 b Fw(via)29 b Fd(tbird.cc.iastat)o(e.e)o(du)o Fw(,)23 b(this)449 3280 y(led)28 b(to)f(569)14 b(466)25 b(sparse)h(524)14 b(339-dimensional)24 b(v)n(ectors)i(\()p Fo(e)2326 3292 y Fn(p)2364 3280 y Fw(\))2396 3292 y Fn(p)p Ff(2)p Fn(P)2558 3280 y Fw(as)h(in)h(\(3.2\).) 449 3552 y Fm(3.3)94 b(The)32 b(matrix)h(step)449 3741 y Fw(Here)49 b(w)n(e)g(\014nd)h(linear)f(dep)r(endencies)h(mo)r(dulo)f (2)g(among)g(the)g Fo(>)60 b Fw(#)p Fo(P)h Fw(bit-v)n(ectors)449 3840 y(\()p Fo(e)520 3852 y Fn(p)581 3840 y Fw(mo)r(d)23 b(2\))837 3852 y Fn(p)p Ff(2)p Fn(P)972 3840 y Fw(.)41 b(Eac)n(h)f(dep)r(endency)i(corresp)r(onds)e(to)h(a)g(set)h Fo(W)58 b Fw(=)46 b Fl(f)p Fw(\()p Fo(w)r(;)14 b Fw(\()p Fo(e)3024 3852 y Fn(p)3063 3840 y Fw(\))3095 3852 y Fn(p)p Ff(2)p Fn(P)3229 3840 y Fw(\))47 b(:)449 3940 y Fo(w)510 3910 y Fu(2)571 3940 y Fl(\021)658 3878 y Fg(Q)737 3965 y Fn(p)p Ff(2)p Fn(P)885 3940 y Fo(p)927 3910 y Fn(e)958 3918 y Fe(p)1020 3940 y Fw(mo)r(d)24 b Fo(r)r Fl(g)h Fw(for)f(whic)n(h)1668 3878 y Fg(P)1755 3965 y Fn(W)1831 3940 y Fw(\()p Fo(e)1902 3952 y Fn(p)1940 3940 y Fw(\))1972 3952 y Fn(p)p Ff(2)p Fn(P)2130 3940 y Fw(=)f(\(2)p Fo(w)2351 3952 y Fn(p)2389 3940 y Fw(\))2421 3952 y Fn(p)p Ff(2)p Fn(P)2581 3940 y Fw(for)h(in)n(tegers)g Fo(w)3069 3952 y Fn(p)3107 3940 y Fw(.)h(Con-)449 4052 y(sequen)n(tly)-7 b(,)27 b Fo(x)d Fl(\021)990 3990 y Fg(Q)1069 4077 y Fn(W)1158 4052 y Fo(w)i Fw(mo)r(d)d Fo(r)30 b Fw(and)e Fo(y)e Fl(\021)1808 3990 y Fg(Q)1887 4077 y Fn(p)p Ff(2)p Fn(P)2035 4052 y Fo(p)2077 4022 y Fn(w)2125 4030 y Fe(p)2187 4052 y Fw(mo)r(d)d Fo(r)30 b Fw(satisfy)d Fo(x)2742 4022 y Fu(2)2803 4052 y Fl(\021)c Fo(y)2935 4022 y Fu(2)2995 4052 y Fw(mo)r(d)g Fo(r)r Fw(.)p 449 4184 473 4 v 449 4238 a Ft(7)511 4270 y Fs(Because)30 b(#)p Fh(P)38 b Fc(\021)27 b Fs(51)22 b(mo)r(d)f(2)1351 4238 y Ft(14)1445 4270 y Fs(w)n(e)30 b(could)f(use)g(the)g(QS-implemen)n(tation)e(from)h([4])j(on)e(a)g(16K) 511 4361 y(MasP)n(ar)i(massiv)n(ely)f(parallel)h(computer.)e(The)h(c)n (hoice)g(is)g(on)g(the)f(high)h(side)g(for)h(this)f(imple-)511 4452 y(men)n(tation,)22 b(whic)n(h)g(p)r(erformed)f(noticeably)i(w)n (orse)g(than)e(exp)r(ected)h(based)g(on)g(the)g(exp)r(erience)511 4544 y(from)k([2].)p eop %%Page: 8 8 8 7 bop 573 83 a Fw(T)-7 b(o)37 b(\014nd)h(dep)r(endencies)g(among)e (the)i(ro)n(ws)e(of)h(the)h(569)14 b(466)22 b Fl(\002)j Fw(524)14 b(339)34 b(bit-matrix)449 183 y(with,)39 b(on)f(a)n(v)n (erage,)d Fl(\031)41 b Fw(47)d(bits)g(p)r(er)h(ro)n(w,)e(w)n(e)h(in)n (tended)h(to)f(use)g(the)h(same)f(approac)n(h)449 282 y(as)e(in)i(previous)e(large)g(scale)h(factorizations.)e(The)j(largest) e(matrix)g(pro)r(cessed)h(so)f(far,)449 382 y(ho)n(w)n(ev)n(er,)25 b(had)i(less)f(than)i(half)f(the)g(n)n(um)n(b)r(er)g(of)g(ro)n(ws)f (and)g(columns)h(\(252)14 b(222)h Fl(\002)i Fw(245)d(810)449 482 y(with)28 b(the)g(same)f(densit)n(y)-7 b(,)28 b(cf.)g([2]\),)g (making)f(the)h(presen)n(t)f(e\013ort)g(an)h(order)e(of)i(magnitude)449 581 y(more)21 b(di\016cult.)h(Using)f(structured)h(Gaussian)e (elimination)i(\(cf.)g([10;)f(14;)f(19]\))h(the)h(matrix)449 681 y(w)n(as)37 b(reduced)h(to)g(a)g(dense)g(188)14 b(614)22 b Fl(\002)j Fw(188)14 b(160)35 b(bit-matrix.)j(This)g(to)r(ok)g(less)f (than)i(12)449 780 y(CPU-hours)d(on)g(a)h(Sparc)g(10)f(w)n(orkstation,) f(97\045)i(of)g(whic)n(h)g(w)n(as)f(sp)r(en)n(t)i(building)f(the)449 880 y(4)14 b(436)g(201)g(280)22 b(b)n(yte)28 b(dense)f(matrix,)g(in)h (268)e(separate)g(\014les)i(of)f(ab)r(out)h(16)f(MBytes)g(eac)n(h.)573 980 y(T)-7 b(o)40 b(\014nd)g(a)g(dep)r(endency)g(among)f(the)i(ro)n(ws) d(of)i(the)g(dense)g(matrix,)g(w)n(e)f(used)h(the)449 1080 y(incremen)n(tal)32 b(v)n(ersion)e(from)i([1])h(of)f(the)h(MasP)n (ar)d(dense)i(matrix)g(eliminator)f(from)h([11].)449 1180 y(The)26 b(dense)g(matrix)g(w)n(as)f(pro)r(cessed)g(in)h(5)g(blo)r (c)n(ks.)g(With)h(a)e(core)h(size)f(of)i(1GByte,)e(41)14 b(595)449 1279 y(ro)n(ws)23 b(could)h(b)r(e)h(pro)r(cessed)e(p)r(er)h (blo)r(c)n(k.)g(Eac)n(h)f(new)i(blo)r(c)n(k)e(w)n(as)h(\014rst)g (eliminated)h(with)f(the)449 1379 y(piv)n(ots)g(found)i(in)f(the)h (previous)e(blo)r(c)n(ks,)h(then)g(with)h(the)g(new)f(piv)n(ots)f(in)i (the)g(blo)r(c)n(k)e(itself,)449 1479 y(after)j(whic)n(h)g(the)h (result)f(w)n(as)g(written)h(to)f(disk.)g(Eac)n(h)g(of)g(the)h(\014rst) f(four)g(blo)r(c)n(ks)g(resulted)449 1578 y(in)f(a)g(\014le)g(of)g Fl(\031)d Fw(980)i(MBytes.)h(Notice)g(that)g(one)g(wrong)f(bit,)h (either)g(in)h(the)f(dense)g(matrix)449 1678 y(or)35 b(in)h(the)g(in)n(termediate)f(\014les,)h(can)f(render)g(the)h(en)n (tire)g(computation)f(w)n(orthless.)f(F)-7 b(or)449 1777 y(this)26 b(reason,)f(the)h(dense)g(matrix)g(in)n(ten)n(tionally)f(con) n(tained)h(268)e(spurious)h(dep)r(endencies)449 1877 y(whic)n(h)38 b(w)n(ere)g(supp)r(osed)g(to)g(b)r(e)h(found)g(at)f (regular)f(in)n(terv)-5 b(als)37 b(during)h(the)h(elimination)449 1977 y(pro)r(cess)26 b(to)i(signal)e(p)r(ossible)i(errors.)573 2077 y(The)c(en)n(tire)e(pro)r(cess)g(to)r(ok)h(45)f(hours)h(on)f(a)h (16K)f(MasP)n(ar)f(MP-1:)h(75)g(min)n(utes)i(to)f(read)449 2177 y(the)34 b(dense)f(matrix,)g(100)f(min)n(utes)h(to)g(write)h(the)f (\014rst)h(four)e(blo)r(c)n(ks,)h(and)g(42)g(hours)f(for)449 2276 y(the)25 b(eliminations.)f(This)h(is)g(b)r(etter)g(than)g(w)n(e)f (had)h(exp)r(ected,)2397 2246 y Fu(8)2459 2276 y Fw(b)r(ecause,)f (unlik)n(e)h(previous)449 2376 y(eliminations,)j(most)g(piv)n(ot)g(ro)n (ws)f(had)i(to)f(b)r(e)h(read)e(from)h(disk)h(b)r(efore)f(eliminating)g (with)449 2476 y(them)j(and)g(they)g(could)f(not)h(b)r(e)g(stored)f(in) h(registers)e(during)h(the)h(elimination.)g(All)g(268)449 2575 y(spurious)24 b(dep)r(endencies)i(w)n(ere)e(found)h(at)g(the)h (righ)n(t)f(momen)n(t)g(during)f(the)i(computation,)449 2675 y(and)35 b(205)g(`true')g(dep)r(endencies)h(w)n(ere)f(found)h(at)g (the)g(end.)g(It)g(to)r(ok)f(a)g(few)h(min)n(utes)g(to)449 2775 y(con)n(v)n(ert)27 b(the)j(205)e(dep)r(endencies)h(mo)r(dulo)g(2)g (in)g(the)h(dense)f(matrix)f(to)h(dep)r(endencies)h(in)449 2874 y(the)23 b(sparse)e(one.)h(Computing)g(the)h(pairs)f Fo(x)p Fw(,)h Fo(y)i Fw(with)e Fo(x)2157 2844 y Fu(2)2218 2874 y Fl(\021)f Fo(y)2349 2844 y Fu(2)2409 2874 y Fw(mo)r(d)h Fo(r)j Fw(to)r(ok)c(a)g(few)h(min)n(utes)449 2974 y(p)r(er)31 b(dep)r(endency)-7 b(.)32 b(The)f(\014rst)g(three)g(dep)r(endencies)h (led)f(to)g Fo(r)h Fw(=)d(1)20 b Fl(\001)h Fo(r)r Fw(.)32 b(A)n(t)g(18:15)e(UT)h(on)449 3073 y(April)26 b(2)g(1994)e(the)j (fourth)f(one)g(led)h(to)f(the)g(factorization)f Fo(r)h Fw(=)d Fo(p)15 b Fl(\001)h Fo(q)s Fw(,)26 b(with)h Fo(p)f Fw(the)h(64-digit)449 3173 y(prime)661 3348 y Fu(3490)c(5295)g(1084)g (7650)g(9491)g(4784)g(9619)g(9038)g(9813)g(3417)g(7646)g(3849)g(3387)g (8439)g(9082)g(0577)449 3524 y Fw(and)k Fo(q)k Fw(the)d(65-digit)e (prime)623 3699 y Fu(3)d(2769)g(1329)g(9326)g(6709)g(5499)g(6198)g (8190)g(8344)g(6141)g(3177)g(6429)g(6799)g(2942)g(5397)g(9828)g(8533)p Fn(:)449 3875 y Fw(Fiv)n(e)i(of)f(the)i(\014rst)f(ten)g(dep)r (endencies)g(w)n(ere)f(`unluc)n(ky'.)h(It)g(migh)n(t)g(b)r(e)h(in)n (teresting)e(to)h(note)449 3974 y(that)f Fo(p)12 b Fl(\000)g Fw(1)22 b(=)g(2)949 3944 y Fu(5)998 3974 y Fl(\001)12 b Fw(3)1075 3944 y Fu(2)1123 3974 y Fl(\001)19 b Fw(\026)-49 b Fo(p)24 b Fw(for)g(a)g(62-digit)f(prime)31 b(\026)-49 b Fo(p)23 b Fw(=)g(1)12 b(+)g(2)g Fl(\001)g Fw(7)2362 3944 y Fu(2)2408 3974 y Fl(\001)19 b Fw(^)-49 b Fo(p)24 b Fw(and)g(a)g(60-digit)f(prime)31 b(^)-49 b Fo(p)p Fw(;)449 4074 y(and)31 b(that)g Fo(q)24 b Fl(\000)c Fw(1)28 b(=)g(2)1148 4044 y Fu(2)1206 4074 y Fl(\001)21 b Fw(41)e Fl(\001)27 b Fw(\026)-48 b Fo(q)34 b Fw(for)d(a)f(63-digit)g(prime)37 b(\026)-48 b Fo(q)31 b Fw(=)e(1)20 b(+)g(2)2569 4044 y Fu(2)2627 4074 y Fl(\001)g Fw(53)g Fl(\001)27 b Fw(^)-48 b Fo(q)34 b Fw(and)d(another)449 4174 y(60-digit)h(prime)40 b(^)-48 b Fo(q)s Fw(.)33 b(W)-7 b(e)34 b(susp)r(ect)g(that)g Fo(p)22 b Fl(\000)g Fw(1)33 b(and)g Fo(q)26 b Fl(\000)c Fw(1)33 b(w)n(ere)f(consciously)g(c)n(hosen)h(to)449 4273 y(ha)n(v)n(e)g(large)f(prime)i(factors,)g(so)f(that)h Fo(r)j Fw(w)n(ould)d(withstand)g(a)g(P)n(ollard)e Fo(p)23 b Fl(\000)f Fw(1)34 b(factoring)p 449 4371 473 4 v 449 4424 a Ft(8)511 4456 y Fs(The)17 b(4)h(hours)f(for)h(the)f(245)d(810)k (column)e(matrix)g(from)h([2])i(extrap)r(olate)f(to)f(4)p Fc(\001)p Fs(\(524)d(339)p Fh(=)p Fs(245)i(810\))3308 4424 y Ft(3)511 4547 y Fc(\031)21 b Fs(39)27 b(hours.)p eop %%Page: 9 9 9 8 bop 449 83 a Fw(attac)n(k.)25 b(The)h(large)e(prime)h(factors)g(of) 33 b(\026)-49 b Fo(p)14 b Fl(\000)h Fw(1)25 b(and)32 b(\026)-48 b Fo(q)18 b Fl(\000)c Fw(1)25 b(migh)n(t)h(ha)n(v)n(e)f(b)r (een)h(based)f(on)g(the)449 183 y(widespread)f(b)r(elief)h(that)g(they) g(w)n(ould)g(b)r(e)g(necessary)e(to)i(prev)n(en)n(t)f(a)g(decryption)g (attempt)449 282 y(using)g(iterated)f(encryption)h([17].)f(The)h (resistance)f(of)h Fo(r)j Fw(against)c(a)h(P)n(ollard)e Fo(p)11 b Fw(+)g(1)24 b(attac)n(k,)449 382 y(whic)n(h)i(had)h(not)f(y)n (et)h(b)r(een)g(published)g(b)n(y)f(1977,)f(w)n(as)g(probably)h(a)g (coincidence:)g Fo(p)17 b Fw(+)f(1)22 b(=)449 482 y(2)13 b Fl(\001)g Fw(1)h(376)g(164)g(939)g(3)o(07)f(94)o(9)g(99)o(6)h(6)o(50) f(93)o(3)g Fl(\001)g Fd(p)-7 b Fw(40,)24 b(and)h Fo(q)16 b Fw(+)d(1)22 b(=)g(2)13 b Fl(\001)g Fw(3)2485 451 y Fu(4)2534 482 y Fl(\001)g Fw(11)g Fl(\001)g Fw(79)g Fl(\001)g Fw(197)g Fl(\001)g Fw(227)g Fl(\001)g Fd(p)-5 b Fw(55,)449 581 y(with)28 b Fd(p)p Fo(j)k Fw(denoting)27 b(a)h Fo(j)5 b Fw(-digit)27 b(prime.)449 863 y Fp(4)115 b(Analysis)39 b(of)g(the)e(data)f(collected)449 1061 y Fw(Let)c Fo(Q)d Fw(=)g Fl(f)p Fo(q)j Fw(:)44 b Fo(q)34 b Fw(prime,)e Fo(B)1366 1073 y Fu(1)1433 1061 y Fo(<)d(q)k Fl(\024)c Fo(B)1754 1073 y Fu(2)1791 1061 y Fo(;)1842 994 y Fg(\000)1890 1029 y Fn(mr)p 1890 1043 92 4 v 1919 1090 a(q)1991 994 y Fg(\001)2059 1061 y Fw(=)g(1)p Fl(g)i Fw(b)r(e)h(the)g(set)f(of)h(26) 14 b(679)g(473)27 b(large)449 1172 y(primes)22 b(whic)n(h)g(can)g(p)r (ossibly)f(o)r(ccur)h(in)g(the)h(\(double\))f(partial)g(relations.)f (Let)h Fo(p)p Fw(\()p Fo(q)s Fw(\))h(b)r(e)f(the)449 1271 y(probabilit)n(y)c(that)h Fo(q)26 b Fl(2)d Fo(Q)c Fw(o)r(ccurs)f(in)h(a)f(partial)g(relation.)g(In)h([16])f(it)h(is)g (sho)n(wn)f(that)h Fo(t)g Fw(partial)449 1371 y(relations)k(can)g(b)r (e)i(exp)r(ected)f(to)g(generate)e Fo(c)11 b Fl(\001)g Fo(t)1920 1341 y Fu(2)1982 1371 y Fw(cycles,)24 b(where)f Fo(c)g Fw(=)g(\()2655 1309 y Fg(P)2743 1396 y Fn(q)r Ff(2)p Fn(Q)2890 1371 y Fo(p)p Fw(\()p Fo(q)s Fw(\))3036 1341 y Fu(2)3074 1371 y Fw(\))p Fo(=)p Fw(2.)g(T)-7 b(o)449 1471 y(b)r(e)31 b(able)e(to)i(predict)f(the)h(n)n(um)n(b)r(er)f(of)g (cycles)g(among)f(partials)g(it)i(is)f(therefore)f(useful)i(to)449 1570 y(get)22 b(more)g(insigh)n(t)h(in)g(the)g(b)r(eha)n(vior)e(of)i Fo(p)p Fw(\()p Fo(q)s Fw(\).)g(It)g(has)g(b)r(een)g(suggested)f(in)h ([16])f(that)h Fo(p)p Fw(\()p Fo(q)s Fw(\))g(is)449 1670 y(prop)r(ortional)j(to)j(1)p Fo(=q)1154 1640 y Fn(\013)1200 1670 y Fw(,)g(for)e(some)h(p)r(ositiv)n(e)g Fo(\013)d(<)f Fw(1.)k(In)h(the)f(full)h(pap)r(er)f(w)n(e)g(will)h(analyse)449 1770 y(whic)n(h)39 b Fo(\013)g Fw(b)r(est)g(\014ts)g(our)f(data,)g(and) h(discuss)f(ho)n(w)g(w)n(ell)h(the)g(theoretically)f(estimated)449 1869 y(n)n(um)n(b)r(er)29 b(of)g(cycles)g(among)f(the)h(partials)f (agrees)g(with)h(our)g(\014ndings.)g(F)-7 b(urthermore,)28 b(w)n(e)449 1969 y(will)g(presen)n(t)f(the)h Fo(\013)1091 1981 y Fu(1)1157 1969 y Fw(and)f Fo(\013)1371 1981 y Fu(2)1436 1969 y Fw(corresp)r(onding)f(to)i Fo(p)2114 1981 y Fu(1)2151 1969 y Fw(\()p Fo(q)s Fw(\))h(and)e Fo(p)2487 1981 y Fu(2)2524 1969 y Fw(\()p Fo(q)s Fw(\),)i(where)e Fo(p)2962 1981 y Fu(1)2999 1969 y Fw(\()p Fo(q)s Fw(\))i(is)e(the)449 2068 y(probabilit)n(y)32 b(that)i Fo(q)i Fl(2)d Fo(Q)h Fw(o)r(ccurs)e(as)h(smallest)g(large)f(prime)i(in)g(a)f(double)g (partial)g(and)449 2168 y Fo(p)491 2180 y Fu(2)528 2168 y Fw(\()p Fo(q)s Fw(\))26 b(is)f(the)g(probabilit)n(y)g(that)g(it)h (app)r(ears)e(as)g(largest)g(large)g(prime)h(in)h(a)e(double)i(partial) 449 2268 y(relation.)573 2367 y(Di\013eren)n(t)43 b(factorizations)d (with)j(similar)e Fo(\013)p Fw('s)i(usually)e(ha)n(v)n(e)g(similar)h (cycle-yields,)449 2467 y(whic)n(h)36 b(mak)n(es)e(the)i Fo(\013)p Fw('s)g(go)r(o)r(d)g(indicators)e(of)i(the)g(cycle-yield.)f (Because)g(the)h(b)r(eha)n(vior)449 2567 y(of)d(the)g Fo(\013)p Fw('s)g(can)f(b)r(e)i(deriv)n(ed)e(long)g(b)r(efore)g(the)i (factorization)d(is)i(complete,)g(this)g(migh)n(t)449 2666 y(b)r(e)27 b(helpful)h(to)e(predict)h(the)g(cycle-yield)g(in)g (other)f(factorizations.)f(The)i(b)r(eha)n(vior)f(of)g(the)449 2766 y(cycle-yield)f(is)i(of)f(particular)f(in)n(terest)h(for)g(future) g(NFS)h(factorizations)e(where)h(relations)449 2865 y(can)32 b(ha)n(v)n(e)f(more)h(than)h(2)f(large)f(primes)h(\(cf.)h([7]\).)g (Understanding)f(their)g(cycle-yield)g(is)449 2965 y(crucial)27 b(to)g(b)r(e)h(able)f(to)h(predict)f(the)h(run-times)g(\(cf.)g([5]\).) 573 3065 y(The)j(estimates)e(that)i(w)n(ere)e(sen)n(t)h(to)g(the)h(con) n(tributors)d(\(cf.)j(Section)f(5\))g(w)n(ere)f(based)449 3164 y(on)e(extrap)r(olations)f(of)h(the)h(cycle-curv)n(e.)e(Initially) h(a)g(quadratic)f(curv)n(e)h(ga)n(v)n(e)e(a)i(go)r(o)r(d)g(\014t,)449 3264 y(and)j(w)n(as)f(therefore)h(used)g(to)h(extrap)r(olate.)e(W)-7 b(e)31 b(lost)f(our)f(con\014dence)h(in)h(quadratic)e(ex-)449 3364 y(trap)r(olation)34 b(in)i(late)f(Decem)n(b)r(er)g(1993,)f(when)h (the)h(extrap)r(olation)e(predicted)h(that)h(w)n(e)449 3463 y(w)n(ould)29 b(need)i(more)e(than)h(13)g(million)g(relations.)f (F)-7 b(ortunately)g(,)30 b(w)n(e)f(noticed)h(in)h(Jan)n(uary)449 3563 y(that)f(a)g(quadratic)f(no)h(longer)f(ga)n(v)n(e)f(a)i(close)g (\014t,)g(and)g(that)h(the)f(cycle-curv)n(e)f(w)n(as)g(sho)n(w-)449 3662 y(ing)c(stronger)f(than)h(quadratic)f(gro)n(wth.)g(Exp)r(erimen)n (ts)h(indicated,)h(quite)f(unexp)r(ectedly)-7 b(,)449 3762 y(that)27 b(a)f(quartic)g(function)h(ga)n(v)n(e)e(a)i(v)n(ery)e (go)r(o)r(d)h(appro)n(ximation.)f(Ho)n(w)n(ev)n(er)g(the)i(residuals,) 449 3862 y(although)h(small,)g(w)n(ere)g(quite)h(w)n(ell)f(serially)g (correlated,)f(whic)n(h)h(probably)g(implies)g(that)449 3961 y(the)34 b(cycle-curv)n(e)d(do)r(es)i Fj(not)g Fw(b)r(eha)n(v)n(e) g(as)g(a)f(quartic.)h(Since)h(the)f(error)f(terms)h(w)n(ere)f(rela-)449 4061 y(tiv)n(ely)24 b(small,)g(and)g(b)r(ecause)f(the)i(\014rst)f(few)h (extrap)r(olations)d(turned)i(out)h(to)f(b)r(e)g(un)n(usually)449 4161 y(accurate,)e(w)n(e)g(k)n(ept)h(using)g(the)g(quartic)g(an)n(yw)n (a)n(y)-7 b(.)21 b(W)-7 b(e)23 b(predicted)g(in)h(early)e(F)-7 b(ebruary)21 b(that)449 4260 y(8)p Fo(:)p Fw(15)j(million)h(relations)f (w)n(ould)g(su\016ce.)h(The)g(quartic)g(extrap)r(olation)e(pro)n(v)n (ed)h(to)h(b)r(e)g(v)n(ery)449 4360 y(sligh)n(tly)e(optimistic)h(|)g (it)h(turned)f(out)g(that)g(w)n(e)f(actually)h(needed)g(8)p Fo(:)p Fw(2)f(million)h(relations.)449 4459 y(In)34 b(the)g(full)h(pap) r(er)e(w)n(e)h(will)g(pro)n(vide)f(more)g(details)h(and)g(graphs)e(of)i (the)h(actual)e(yields)449 4559 y(and)27 b(of)h(the)g(quadratic)e(and)i (quartic)f(extrap)r(olations.)p eop %%Page: 10 10 10 9 bop 449 83 a Fp(5)115 b(Resource)37 b(managemen)m(t)449 329 y Fw(T)-7 b(o)35 b(factor)h Fo(r)j Fw(w)n(e)c(assem)n(bled)g(the)i (largest)d(collab)r(oration)h(y)n(et)g(seen)h(in)h(computational)449 428 y(n)n(um)n(b)r(er)20 b(theory)f(and,)h(p)r(ossibly)-7 b(,)20 b(p)r(erformed)f(the)i(largest)d(single)i(computation)f(ev)n(er) g(com-)449 528 y(pleted.)24 b(In)h(sev)n(eral)d(imp)r(ortan)n(t)i(resp) r(ects,)f(the)i(resources)c(w)n(e)j(had)g(a)n(v)-5 b(ailable)23 b(w)n(ere)g(barely)449 627 y(adequate)i(for)h(the)h(task.)f(Consequen)n (tly)-7 b(,)26 b(ingen)n(uit)n(y)f(and)i(diplomacy)e(w)n(ere)h (required)f(for)449 727 y(the)37 b(successful)f(completion)g(of)g(the)h (pro)5 b(ject.)36 b(F)-7 b(ull)37 b(details)f(of)g(the)h(resource)d (manage-)449 827 y(men)n(t)29 b(asp)r(ects)f(of)h(the)g(computation)g (will)g(b)r(e)g(giv)n(en)f(in)h(the)h(full)f(pap)r(er;)f(here)h(w)n(e)f (giv)n(e)g(a)449 926 y(summary)f(of)g(the)h(salien)n(t)f(p)r(oin)n(ts.) 573 1035 y(Based)49 b(on)f(exp)r(erience)h(with)h(earlier)d (factorizations,)h(esp)r(ecially)g(those)h(in)h([15])449 1135 y(and)23 b([2],)g(w)n(e)g(could)g(mak)n(e)g(reasonably)e(precise)i (estimates)g(of)g(the)h(disk)f(space,)g(amoun)n(t)g(of)449 1234 y(computation)h(and)g(memory)g(usage)f(whic)n(h)i(w)n(ould)f(b)r (e)h(required.)e(Before)h(w)n(e)g(started,)g(w)n(e)449 1334 y(estimated)c(that)h(w)n(e)f(w)n(ould)g(need)h(ab)r(out)g(8)f (million)g(relations,)g(eac)n(h)f(of)i(whic)n(h)f(w)n(ould)g(tak)n(e) 449 1434 y(ab)r(out)27 b(350)g(b)n(ytes)g(to)h(store.)f(W)-7 b(e)28 b(knew)f(that)h(a)g(few)g(mips-millenia)f(w)n(ould)h(b)r(e)g (required.)449 1533 y(W)-7 b(e)37 b(also)e(estimated)i(that)g(the)g (factor)f(base)g(w)n(ould)g(need)h(to)f(ha)n(v)n(e)g(ab)r(out)g(400)14 b(000)34 b(to)449 1633 y(600)14 b(000)22 b(elemen)n(ts,)j(implying)g (that)g(the)h(sieving)e(programs)f(w)n(ould)i(tak)n(e)f(o)n(v)n(er)f(8) i(MBytes)449 1733 y(of)g(activ)n(e)g(memory)-7 b(.)24 b(W)-7 b(e)26 b(did)g(not)f(kno)n(w,)f(b)r(efore)h(w)n(e)g(started,)g (ho)n(w)g(man)n(y)f(mac)n(hines)h(and)449 1832 y(of)g(what)g(p)r(o)n(w) n(er)e(w)n(ould)i(b)r(e)g(a)n(v)-5 b(ailable)24 b(to)g(us)h(but)h(w)n (e)e(did)i(ha)n(v)n(e)d(some)i(encouraging)e(o\013ers)449 1932 y(of)j(supp)r(ort.)g(It)g(w)n(as)f(clear,)g(ho)n(w)n(ev)n(er,)g (that)h(an)g(un)n(usually)f(large)g(n)n(um)n(b)r(er)h(of)g(p)r(eople)g (and)449 2031 y(mac)n(hines)h(w)n(ould)g(b)r(e)h(in)n(v)n(olv)n(ed)e (with)i(the)g(pro)5 b(ject.)573 2140 y(The)27 b(\014rst)f(action)f(w)n (as)h(to)g(settle)g(the)h(size)f(of)g(the)h(factor)e(base.)h(As)g (explained)g(in)h(3.1,)449 2240 y(a)g(\014gure)g(of)g(ab)r(out)h(half)f (a)g(million)h(primes)f(w)n(as)g(close)f(to)i(optimal.)f(A)h(somewhat)f (larger)449 2340 y(n)n(um)n(b)r(er)20 b(w)n(ould)g(ha)n(v)n(e)f (reduced)i(the)f(theoretical)g(amoun)n(t)g(of)g(computation)h (required,)e(but)449 2439 y(w)n(e)30 b(w)n(ere)g(already)f(using)h (10.5)g(MBytes)g(of)h(activ)n(e)e(virtual)i(memory)-7 b(.)30 b(Giv)n(en)g(that)h(most)449 2539 y(of)f(the)g(mac)n(hines)f(on) h(the)g(In)n(ternet)g(ha)n(v)n(e)f(less)g(than)h(16)f(MBytes)h(of)f(ph) n(ysical)g(memory)-7 b(,)449 2638 y(and)32 b(man)n(y)h(ha)n(v)n(e)e(8)h (or)g(less,)h(ev)n(en)f(10.5)f(MBytes)i(is)f(uncomfortably)g(large)g (and)g(w)n(ould)449 2738 y(seriously)26 b(restrict)h(the)h(n)n(um)n(b)r (er)f(of)h(p)r(eople)f(capable)g(of)h(helping)f(us.)573 2847 y(Before)33 b(going)e(public,)j(the)g(\014rst)e(and)h(fourth)h (authors)e(p)r(orted)h(the)g(soft)n(w)n(are)e(to)i(as)449 2947 y(man)n(y)28 b(di\013eren)n(t)g(kinds)g(of)h(w)n(orkstations)d (and)i(PCs)g(as)f(w)n(ere)h(a)n(v)-5 b(ailable)27 b(to)h(us.)h(The)f (\014rst)449 3046 y(author)21 b(p)r(ersuaded)g(MIT)h(to)f(lend)h(us)g (a)f(\014le)h(serv)n(er)e(for)h(the)h(duration.)f(This)h(mac)n(hine)f (w)n(as)449 3146 y(a)j(DEC)g(5000/240)d(with)k(32)e(MBytes)h(RAM)h(and) g(t)n(w)n(o)e(\(later)h(three\))h(975)e(MBytes)h(disks.)449 3245 y(MIT)i(systems)f(sta\013)g(also)g(agreed)f(to)i(p)r(erform)f (regular)f(bac)n(kups)g(of)i(the)g(information)f(on)449 3345 y(disk.)i(F)-7 b(or)27 b(these)h(reasons)e(MIT)h(w)n(as)g(c)n (hosen)g(as)g(collection)g(site.)573 3454 y(Also)j(b)r(efore)h(the)f (computations)g(started,)g(the)h(second)f(and)g(fourth)g(authors)g (wrote)449 3554 y(do)r(cumen)n(ts)22 b(explaining)f(what)h(w)n(e)g(w)n (ere)f(prop)r(osing)f(to)i(do,)g(ho)n(w)f(w)n(e)h(w)n(ere)f(going)g(to) h(do)f(it,)449 3653 y(and)h(ho)n(w)h(to)f(get)h(further)f(information)g (should)h(the)g(reader)e(wish)i(to)g(join)g(in.)g(The)f(second)449 3753 y(author)27 b(set)i(up)f(sev)n(eral)f(mailing)h(lists)g(and)g (email)h(aliases)e(for)g(comm)n(unications)h(to)g(and)449 3852 y(from)23 b(our)g(w)n(ork)n(ers)e(and)i(the)h(four)f(co)r (ordinators;)e(he)j(also)e(to)r(ok)h(on)g(the)h(role)f(of)g(fron)n (t-line)449 3952 y(con)n(tact)k(p)r(erson,)f(dealing)h(with)h (virtually)f(all)g(the)h(600)e(con)n(tributors)g(and)i(man)n(y)f (others)449 4052 y(who)g(expressed)g(an)g(in)n(terest)g(but)h(did)g (not)g(join)g(in)f(for)g(whatev)n(er)g(reason.)573 4161 y(The)34 b(p)r(ostings)e(to)i(p)r(oten)n(tial)f(con)n(tributors)f(w)n (en)n(t)h(out)g(on)g(August)g(19)g(1993)e(to)i(the)449 4260 y(`n)n(um)n(b)r(er)g(theory)g(net',)h(on)f(August)h(23)e(to)i(the) g(`cypherpunks')f(and)g(the)h(`PGP)f(dev)n(el-)449 4360 y(opmen)n(t)43 b(group',)f(and)h(on)g(August)g(24)g(to)g(the)g(follo)n (wing)f(newsgroups:)g(alt.hac)n(k)n(ers,)449 4459 y(alt.securit)n(y)-7 b(,)35 b(alt.securit)n(y)-7 b(.pgp,)34 b(alt.securit)n(y)-7 b(.rip)r(em,)35 b(comp.arc)n(h,)g(comp.securit)n(y)-7 b(.misc,)449 4559 y(sci.crypt,)37 b(and)g(sci.math.)g(The)g(initial)h (resp)r(onse)e(to)h(our)g(p)r(ostings)f(and)h(the)h(n)n(um)n(b)r(er)p eop %%Page: 11 11 11 10 bop 449 83 a Fw(of)31 b(incoming)g(relations)f(w)n(as)g (su\016cien)n(tly)h(encouraging)e(to)i(conclude)g(that)h(the)f(pro)5 b(ject)449 183 y(w)n(ould)27 b(b)r(e)h(feasible.)573 301 y(Once)c(con)n(tributors)e(started)h(o\013ering)g(their)h (services,)e(w)n(e)i(had)f(to)h(p)r(ort)g(our)f(soft)n(w)n(are)449 400 y(to)h(their)f(platforms,)h(if)g(w)n(e)g(had)f(not)h(already)e (done)i(so.)f(Although)h(one)g(Unix)g(b)r(o)n(x)f(is)h(v)n(ery)449 500 y(m)n(uc)n(h)30 b(lik)n(e)g(another,)f(there)i(are)e(su\016cien)n (t)h(di\013erences)g(that)h(w)n(e)f(often)h(had)f(to)g(ask)g(our)449 600 y(con)n(tributors)h(to)h(p)r(erform)g(the)h(p)r(ort)g(for)f(us)g (and)h(to)f(send)h(bac)n(k)e(the)i(mo)r(di\014cations)f(so)449 699 y(that)24 b(w)n(e)g(could)g(pass)g(them)h(on)f(to)g(others)f(with)i (the)f(same)g(platform.)g(Our)g(co)r(de)g(w)n(as)f(run)449 799 y(successfully)33 b(on)g(mac)n(hines)f(as)h(disparate)f(as)g(16MHz) h(80386sx)d(PCs)j(and)g(Cra)n(y)f(C90s.)449 899 y(An)22 b(attempted)h(p)r(ort)e(to)h(a)g(Thinking)f(Mac)n(hines)g(CM-5)h (failed,)g(but)g(one)g(US)g(corp)r(oration)449 998 y(managed)k(to)i (get)f(the)h(sieving)f(co)r(de)g(running)h(on)f(a)g(couple)h(of)f(fax)g (mac)n(hines!)573 1116 y(The)21 b(relativ)n(ely)f(sev)n(ere)g(memory)g (requiremen)n(ts)g(and)h(the)g(commonness)f(of)h(8)g(MBytes)449 1216 y(mac)n(hines)32 b(ga)n(v)n(e)f(us)i(a)g(strong)e(incen)n(tiv)n(e) i(to)g(pro)r(duce)f(memory-frugal)f(v)-5 b(arian)n(ts)32 b(of)g(the)449 1316 y(siev)n(er.)24 b(By)i(reducing)e(the)i(siev)n(e)f (batc)n(h)h(size)f(and)g(b)n(y)h(reading)e(the)i(ro)r(ots)f(of)g Fo(mr)k Fw(mo)r(d)c(the)449 1415 y Fo(p)p Fw('s)k(from)g(a)f(\014le,)i (w)n(e)f(w)n(ere)f(able)h(to)g(reduce)g(the)g(space)g(required)f(from)h (the)g(default)h(10.5)449 1515 y(MBytes)23 b(to)g(6.5)f(MBytes,)h(alb)r (eit)h(at)f(a)g(cost)f(of)i(running)e(o)n(v)n(er)g(t)n(wice)h(as)f(slo) n(wly)-7 b(.)23 b(Ho)n(w)n(ev)n(er,)449 1614 y(half)30 b(a)f(computer)g(is)h(b)r(etter)g(than)g(none)g(and)f(b)n(y)h(allo)n (wing)e(more)h(p)r(eople)h(to)f(tak)n(e)g(part,)449 1714 y(w)n(e)f(shortened)f(the)i(o)n(v)n(erall)d(time)j(tak)n(en)f(for)g (the)h(sieving)e(phase.)h(Ev)n(en)g(so,)g(6.5)f(MBytes)449 1814 y(do)r(es)e(not)g(lea)n(v)n(e)e(m)n(uc)n(h)i(ro)r(om)g(for)f(an)h (op)r(erating)f(system)h(and)g(most)g(of)g(our)f(con)n(tributors)449 1913 y(with)k(8)f(MBytes)g(mac)n(hines)g(w)n(ere)g(only)g(able)g(to)h (run)f(our)g(co)r(de)h(o)n(v)n(ernigh)n(t.)573 2031 y(Once)21 b(w)n(e)f(w)n(ere)f(up)i(to)g(sp)r(eed)g(w)n(e)f(receiv)n(ed)f(on)i(a)n (v)n(erage)c(ab)r(out)k(40)f(thousand)g(relations)449 2131 y(p)r(er)32 b(da)n(y)-7 b(,)32 b(with)h(p)r(eaks)f(of)g(o)n(v)n (er)e(50)i(thousand)g(during)g(the)g(holida)n(y)g(seasons.)f(Daily)-7 b(,)32 b(all)449 2231 y(newly)e(receiv)n(ed)f(relations)g(that)h (satis\014ed)g(\(3.2\))g(w)n(ere)f(sorted)h(and)g(merged)f(with)i (older)449 2330 y(data;)36 b(fault)n(y)g(or)f(duplicate)i(relations)e (w)n(ere)g(thro)n(wn)g(out.)i(This)f(to)r(ok)g(at)g(most)g(a)f(few)449 2430 y(hours)22 b(p)r(er)h(da)n(y)-7 b(,)22 b(ev)n(en)g(when)h(the)g (accum)n(ulated)g(data)f(had)h(reac)n(hed)e(a)i(couple)f(of)h(GBytes.) 449 2530 y(Ab)r(out)29 b(one)f(relation)g(in)h(ten)f(thousand)h(receiv) n(ed)e(w)n(as)h(unacceptable)g(in)g(some)g(w)n(a)n(y)-7 b(.)28 b(Of)449 2629 y(these,)22 b(man)n(y)g(had)h(su\013ered)f(in)h (transit)f(through)g(mail)g(gatew)n(a)n(ys)e(but)j(w)n(e)f(also)g (receiv)n(ed)f(a)449 2729 y(n)n(um)n(b)r(er)29 b(of)g(relations)e(in)n (tended)j(for)e(another)g(factoring-b)n(y-email)f(computation)i(b)r (eing)449 2828 y(run)e(at)h(the)g(same)f(time)h(|)g(these)f(relations)g (w)n(ere)f(sen)n(t)i(on)f(to)h(the)g(correct)e(destination,)449 2928 y(together)36 b(with)h(a)f(p)r(olite)h(note)g(to)g(suggest)e(that) i(their)g(con)n(tributors)e(b)r(e)i(informed)g(of)449 3028 y(their)27 b(error.)573 3146 y(Ab)r(out)e(half-w)n(a)n(y)c (through)i(the)h(computation,)f(it)h(b)r(ecame)g(clear)e(that)i(w)n(e)f (w)n(ould)g(run)449 3245 y(out)g(of)f(disk)h(space)f(if)h(w)n(e)f(con)n (tin)n(ued)h(to)f(store)g(the)h(relations)f(in)h(plain)g(text.)g (Accordingly)-7 b(,)449 3345 y(the)31 b(fourth)h(author)e(wrote)h(a)f (simple)i(but)g(e\016cien)n(t)f(compressor,)e(and)i(the)h(bulk)f(of)g (the)449 3445 y(data)25 b(w)n(as)g(k)n(ept)h(in)h(a)e(more)h(compact)f (format.)h(Eac)n(h)f(da)n(y's)g(data)g(w)n(as)g(left)i(as)f(ASCI)r(I)g (for)449 3544 y(con)n(v)n(enience)k(\(few)n(er)i(pro)r(cessing)e (programs)g(had)i(to)f(b)r(e)i(re-written\))e(and)h(a)f(com)n(bined)449 3644 y(uncompress-sort-merge-compress)15 b(of)21 b(the)g(total)f(data)g (set)g(p)r(erformed)h(ev)n(ery)e(few)i(w)n(eeks)449 3744 y(as)33 b(the)h(disk)g(b)r(egan)g(to)g(get)f(uncomfortably)g(full.)i (Unfortunately)-7 b(,)34 b(w)n(e)g(did)g(not)g(notice)449 3843 y(the)e(disk)f(o)n(v)n(er\015o)n(wing)d(in)k(early)e(Jan)n(uary)f (1994.)h(Ab)r(out)i(three)f(w)n(eeks')f(data)h(had)g(b)r(een)449 3943 y(corrupted;)h(fortunately)-7 b(,)33 b(the)h(outputs)f(from)g(the) 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b(Details)g(will)h(b)r(e)g(giv)n(en)f(in)h(the)g (full)g(pap)r(er.)573 283 y(W)-7 b(e)42 b(ha)n(v)n(e)e(already)f (describ)r(ed)i(in)g(Section)g(4)g(ho)n(w)f(the)h(quadratic)f(extrap)r (olation)449 383 y(b)r(egan)34 b(to)h(indicate)g(that)g(w)n(e)g(w)n (ould)f(need)h(man)n(y)f(more)h(relations)e(than)i(w)n(e)g(had)f(pre-) 449 482 y(dicted)c(at)f(the)h(start)f(of)g(the)h(pro)5 b(ject.)29 b(W)-7 b(e)30 b(b)r(ecame)f(seriously)f(concerned)h(at)g (this)h(p)r(oin)n(t)449 582 y(that)d(w)n(e)g(w)n(ould)f(ha)n(v)n(e)g (di\016cult)n(y)h(in)h(main)n(taining)e(the)h(in)n(terest)g(of)g(our)f (con)n(tributors)f(for)449 681 y(sev)n(eral)k(mon)n(ths)i(longer)e (than)i(an)n(ticipated.)g(F)-7 b(urther,)30 b(although)h(w)n(e)f(had)h (enough)f(disk)449 781 y(space)e(\(just!\))j(to)e(pro)r(cess)f(9)h (million)g(relations,)f(an)h(extra)f(50\045)h(w)n(ould)f(ha)n(v)n(e)g (caused)h(us)449 881 y(signi\014can)n(t)h(problems.)g(Luc)n(kily)-7 b(,)31 b(our)f(disco)n(v)n(ery)f(of)i(the)h(m)n(uc)n(h)e(more)h 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b(ailable)20 b(is)h(around)f(1)p Fo(:)p Fw(5)6 b Fl(\002)g Fw(10)1640 3632 y Fu(7)1697 3662 y Fw(mips)21 b(|)g(ab)r(out)h(\014v)n(e)f(times)g(the)h(p)r(o)n(w) n(er)e(of)h(Riv)n(est's)449 3761 y(h)n(yp)r(othetical)31 b(mac)n(hine.)g(In)h(principle,)g(the)g(In)n(ternet)f(could)h(ha)n(v)n (e)e(factored)h Fo(r)j Fw(in)e(three)449 3861 y(hours!)27 b(The)g(implemen)n(tation)h(details)f(are)g(left)h(as)f(an)g(exercise)g (for)g(the)h(reader.)573 3961 y(Let)i(us)f(attempt)g(to)g(giv)n(e)g(a)f (more)g(plausible)h(estimate.)g(W)-7 b(e)30 b(b)r(eliev)n(e)f(that)g(w) n(e)g(could)449 4061 y(acquire)g(100)g(thousand)i(mac)n(hines)e (without)i(sup)r(erh)n(uman)f(or)g(unethical)h(e\013orts.)f(That)449 4161 y(is,)22 b(w)n(e)g(w)n(ould)g Fj(not)h Fw(set)f(free)h(an)f(In)n (ternet)g(w)n(orm)g(or)f(virus)h(to)h(\014nd)g(resources)d(for)i(us.)g (Man)n(y)449 4260 y(organizations)31 b(ha)n(v)n(e)i(sev)n(eral)f (thousand)i(mac)n(hines)f(eac)n(h)g(on)h(the)g(net.)h(Making)e(use)h (of)449 4360 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b(0.558)i(Bruce)f(K.)g(Martin)g(Jr.,)f(0.556)i(John)g(\016tc)n(h,)g (0.538)g(Jac)n(k)h(Rep)r(enning,)f(0.521)g(Je\013rey)449 3523 y(Blohm,)25 b(0.515)i(Jim)e(Castleb)r(erry)-6 b(,)27 b(0.514)g(John)h(Kohl,)e(0.497)h(Billy)e(Barron,)h(0.472)i(Andrew)e (Odlyzk)n(o,)449 3603 y(0.471)c(Boris)f(Hemk)n(emeier,)f(0.468)i(Bruce) g(Rossiter,)f(0.458)h(Nur)g(Serink)n(en,)g(0.445)g(P)n(eter)g(Conrad,)g (0.443)449 3682 y(Ken)g(Rose,)f(0.442)h(Christer)f(Berg,)g(0.429)h(Bob) f(Devine,)h(0.428)g(Dana)g(Jacobsen,)h(0.423)f(olson@lxt.com,)449 3762 y(0.414)28 b(Lee)h(Na)n(v)n(e,)f(0.409)h(Unclaimed,)d(0.370)j(P)n (aul)f(Ho)r(dgkinson,)g(0.367)h(Unclaimed,)d(0.365)j(Sylv)n(estre)449 3842 y(Matusc)n(hk)l(a,)22 b(0.354)f(Bill)f(Rea,)h(0.353)g(Con\014rmed) f(Anon)n(ymous,)h(0.349)g(Da)n(vid)g(Cro)n(wley)-6 b(,)20 b(0.347)i(Hugh)f(D.)449 3921 y(Gam)n(ble,)27 b(0.333)h(Bruce)h(Do)r (dson,)f(0.333)g(Herb)g(Sa)n(v)l(age,)h(0.326)g(Andrew)f(Mossb)r(erg,)f (0.321)h(Eric)g(P)n(ost-)449 4001 y(pisc)n(hil,)21 b(0.310)i (Unclaimed,)e(0.304)i(Daniel)f(L.)g(Carroll,)f(0.296)i(Unclaimed,)e (0.295)i(Ian)g(T.)f(M.)g(Flanigan,)449 4081 y(0.284)31 b(Con\014rmed)e(Anon)n(ymous,)h(0.281)h(Mitc)n(hell)f(P)n(erilstein,)f (0.270)i(Con\014rmed)e(Anon)n(ymous,)h(0.268)449 4161 y(Eric)f(Prestemon,)h(0.257)h(Mic)n(hael)f(J.)g(Gri\016n,)g(0.257)g (Dan)h(Nac)n(h)n(bar,)g(Applied)f(Computing)g(Systems)449 4240 y(Institute)e(of)f(Massac)n(h)n(usetts,)g(Cen)n(ter)h(for)e(In)n (telligen)n(t)i(Information)e(Retriev)l(al,)g(0.253)i(Roland)f(Kalte-) 449 4320 y(\015eiter,)c(0.250)g(Ric)n(hard)g(A.)g(Lethin,)g(0.248)h (Erik)e(Sc)n(ho)r(enfelder,)i(0.245)f(Mic)n(hael)g(K.)g(Cep)r(ek,)g (0.244)h(Alan)449 4400 y(Ba)n(wden,)33 b(0.243)g(Chris)f(Clab)r(orne,)g (A)-6 b(T&T)33 b(Global)f(Information)g(Systems,)g(0.240)i(Da)n(v)n(e)f (Sill,)e(0.238)449 4479 y(Je\013)i(Ha)n(yw)n(ard,)f(0.237)h(Mark)f(Eic) n(hin,)g(0.236)h(Bill)e(Broadley)-6 b(,)32 b(0.236)h(Con\014rmed)f (Anon)n(ymous,)g(0.227)449 4559 y(Lee)d(Ric)n(hardson,)g(0.223)h(T)-6 b(om\023)-35 b(as)28 b(Oliv)n(eira)g(e)h(Silv)l(a,)g(0.222)h(Theo)r (dore)g(Ts'o,)e(0.214)h(R.)g(Loader,)g(0.213)p eop %%Page: 14 14 14 13 bop 449 83 a Fa(Mark)30 b(V.)h(Shaney)-6 b(,)32 b(0.212)f(Roger)h(M.)e(Lev)l(asseur,)h(0.212)h(Keith)f(Bostic,)g(0.211) g(P)n(eter)h(Wido)n(w,)f(0.210)449 163 y(Da)n(vid)25 b(Hollen)n(b)r(erg,)f(0.198)h(Ric)n(hard)g(P)-6 b(.)24 b(Bren)n(t,)g(0.198)i(James)e(R.)g(F)-6 b(o)n(wler,)24 b(0.197)h(Leo)g(Broukhis,)f(0.196)449 242 y(Bill)f(Sommerfeld,)g(0.194) i(Da)n(vid)h(Lee,)f(0.190)g(Unclaimed,)f(0.189)i(Chris)e(Sieb)r (enmann,)h(0.189)h(Brian)e(P)-6 b(.)449 322 y(Fitzgerald,)26 b(0.186)h(Da)n(vid)f(Buc)n(kley)-6 b(,)27 b(0.180)g(Je\013)g(W)-6 b(o)r(olsey)g(,)27 b(0.174)g(Nic)n(k)f(Hollo)n(w)n(a)n(y)-6 b(,)26 b(0.173)h(Brad)f(Jones,)449 402 y(0.169)36 b(Andreas)f (Ste\013en,)h(0.169)g(Serge)g(P)n(ac)n(hk)n(o)n(vsky)-6 b(,)37 b(0.167)e(P)n(eter)h(Iliev)n(e,)f(0.165)h(Stefan)g(Gaertner,)449 482 y(0.165)25 b(Henri)e(Cohen,)i(0.164)g(T)-6 b(atu)25 b(Mannisto,)f(0.160)h(Will)e(Gilb)r(ert,)h(0.158)h(Andrew)f(S.)g (Krzywdzinski,)449 561 y(0.155)k(Da)n(vid)g(J.)f(W)-6 b(righ)n(t,)28 b(0.155)g(Henry)f(Ellingw)n(orth,)g(0.154)h(ro)r (ot@cse.nau.edu,)h(0.151)f(Marc)f(Brett,)449 641 y(0.150)j(Alan)g (Barrett,)g(0.147)h(Scott)g(Bennett,)h(0.146)e(anon)n(ymous)g(grad)h (studen)n(t)g(at)g(New)f(Y)-6 b(ork)30 b(Uni-)449 721 y(v)n(ersit)n(y)-6 b(,)29 b(0.145)h(Mik)n(e)f(Olson,)g(0.143)h(Alfred)e (W)-6 b(assermann,)29 b(0.142)h(Janne)g(Himank)l(a,)f(0.141)h(Da)n(vid) f(L.)449 800 y(Blac)n(k,)h(0.136)i(Matthew)g(Martin,)d(0.135)j(Jim)d (McKim,)f(0.134)j(Craig)g(Stein)n(b)r(erger,)g(0.133)g(Ross)f(Fin-)449 880 y(la)n(yson,)d(0.132)h(Con\014rmed)f(Anon)n(ymous,)f(0.129)i(Narin) f(Suphasindh)n(u,)h(0.129)g(W)-6 b(o)r(o)r(dy)29 b(W)-6 b(ea)n(v)n(er,)28 b(0.129)449 960 y(Llo)n(yd)c(Miller,)e(0.127)j (Con\014rmed)e(Anon)n(ymous,)h(0.126)h(Thomas)e(Denn)n(y)-6 b(,)25 b(0.125)f(Ro)n(y)h(Go)r(o)r(dman,)f(Digi-)449 1039 y(tal)d(Equipmen)n(t)f(Corp)r(oration,)h(0.124)g(Mic)n(hael)f(G.)h (Reed,)f(0.123)i(Lance)f(Cottrell,)g(0.123)g(Da)n(vid)f(Coma)n(y)-6 b(,)449 1119 y(0.123)30 b(Ronald)h(M.)e(Sorli,)g(0.123)h(Arc)n(hie)g (L.)g(Cobbs,)g(0.121)g(Jim)f(Hutc)n(hins,)h(0.120)h(Kelly)e(Hall,)g (0.118)449 1199 y(Unclaimed,)19 b(0.118)i(Allan)f(Gottlieb,)i(0.116)f (T)-6 b(omas)20 b(T)-6 b(engling,)21 b(0.116)g(Otto)g(J.)g(Mak)n(ela,)f (0.114)h(Matthew)449 1279 y(Lyle,)26 b(0.113)h(Bret)g(Ketc)n(h)n(um,)f (0.113)h(Unclaimed,)f(0.111)h(Milt)e(Mallory)-6 b(,)26 b(Lincoln)g(Ong,)h(Ron)g(Rob)r(erts,)449 1358 y(0.111)d(Je\013rey)h(I.) f(Sc)n(hiller,)f(0.110)i(P)n(aul)f(Rouse,)g(0.110)h(Mikk)n(o)f(Siren,)f (0.105)i(John)g(Osw)n(alt,)e(0.102)i(Mik)n(e)449 1438 y(Chang,)17 b(0.101)g(Gary)g(Ob)r(erbrunner,)g(0.097)g(Ric)n(hard)g (Dot)n(y)-6 b(,)17 b(0.095)g(Con\014rmed)f(Anon)n(ymous,)g(0.095)i (Don-)449 1518 y(ald)28 b(Nic)n(hols,)g(0.095)h(Bill)e(Danielson,)h 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(Gri\016n,)f(0.018)i(James)e(Lein)n(w)n(eb)r(er,)i(0.018)f(Christian)g (W)-6 b(eisgerb)r(er,)29 b(0.017)449 3749 y(Markus)g(Lamminmaki,)d (0.016)k(Bob)g(Proulx,)f(0.016)i(Mark)e(Edw)n(ard)h(Stahl,)g(0.016)g (Uw)n(e)g(Hollerbac)n(h,)449 3829 y(0.016)e(Matthias)h(Rosen)n(b)r (erger,)f(0.016)g(Stew)n(art)h(A.)e(Levin,)h(0.015)g(Mik)n(e)f(P)n (earce,)i(0.015)f(Cornelis)f(v)l(an)449 3909 y(der)20 b(Laan,)h(0.015)h(Unclaimed,)d(0.014)i(Rob)r(ert)g(A.)f(Ha)n(yden,)i (0.013)f(Jerome)f(Ab)r(ela,)g(0.013)h(Kenneth)h(S.A.)449 3988 y(Oksanen,)k(0.013)g(Da)n(vid)g(Metcalfe,)g(0.013)g(Unclaimed,)e (0.013)j(Lance)f(M.)f(Cottrell,)g(0.013)i(Allen)e(Con-)449 4068 y(dit,)20 b(0.012)h(Claude)f(Barb)r(e,)g(0.012)h(Unclaimed,)e (0.012)i(Da)n(vid)g(L.)f(Johnson,)h(0.012)f(Da)n(v)n(e)h(Mitc)n(hell,)f (0.011)449 4148 y(Bob)g(Kupiec,)g(0.011)g(Juergen)h(Hannk)n(en-Illjes,) e(0.011)i(John)f(Co)n(wnie,)g(0.011)g(P)n(eter)g(Clark,)f(0.011)i (Stefan)449 4227 y(Larsson,)d(0.011)i(Ken)g(Hardy)-6 b(,)19 b(0.010)g(Nelson)h(Boly)n(ard,)e(0.010)i(Unclaimed,)e(0.010)i (Timo)e(Suhonen,)i(0.010)449 4307 y(Andrew)i(Gra)n(y)-6 b(,)22 b(0.010)g(Unclaimed,)f(0.010)i(Malcolm)d(Beattie,)j(0.009)g (James)e(Beal,)h(0.009)g(Simon)f(F)-6 b(oley)g(,)449 4387 y(0.009)28 b(Seth)h(Fink)n(elstein,)f(0.009)g(John)h(Osw)n(alt,)e (0.009)i(Unclaimed,)d(0.009)j(Ric)n(hard)f(G.)f(v)n(on)i(Bluc)n(her,) 449 4467 y(0.009)19 b(Don)g(Hejna,)g(0.008)g(Craig)g(Gro)r(esc)n(hel,)g (0.008)g(P)n(er)g(Bo)t(jsen,)f(0.008)i(Carl)e(Witt)n(y)-6 b(,)19 b(0.008)h(D.J.)e(James,)449 4546 y(0.007)24 b(P)n(atric)n(k)g(W) -6 b(endt,)25 b(0.007)f(Unclaimed,)f(0.007)h(Jason)g(Y)-6 b(ano)n(witz,)25 b(and)f(0.007)g(Mark)f(Delan)n(y)-6 b(.)p eop %%Page: 15 15 15 14 bop 573 83 a Fw(Ac)n(kno)n(wledgemen)n(ts)27 b(are)h(due)g(to)h (Bruce)e(Do)r(dson)h(for)g(factoring)g Fo(p)18 b Fw(+)h(1)28 b(using)g(P)n(eter)449 183 y(Mon)n(tgomery's)23 b(ECM)j(program,)d(to)j (Allen)g(McIn)n(tosh)f(for)g(his)h(assistance)e(with)i(our)f(ini-)449 282 y(tial)19 b(curv)n(e-\014tting)e(exp)r(erimen)n(ts,)i(to)g(Ron)f (Riv)n(est)h(for)f(his)h(p)r(ermission)f(to)h(quote)f(from)g([20],)449 382 y(and)27 b(to)h(Ric)n(h)f(Sc)n(hro)r(epp)r(el)h(for)f(his)g (helpful)i(commen)n(ts)e(on)g(the)h(run-times)f(in)h([21].)573 482 y(W)-7 b(e)32 b(are)d(particularly)h(grateful)g(to)h(Je\013)f(Sc)n (hiller)h(at)f(MIT,)h(who)g(made)f(a)n(v)-5 b(ailable)30 b(to)449 582 y(us)24 b(computing)g(resources,)d(GBytes)j(of)g(disk)g (space)f(and)h(frequen)n(t)g(bac)n(kups)f(of)h(our)f(hard-)449 681 y(earned)i(data.)g(The)h(fourth)g(author)f(is)h(esp)r(ecially)f (thankful)h(for)f(b)r(eing)h(p)r(ermitted)h(to)f(log)449 781 y(in)32 b(to)g Fd(hot-spare.mit.ed)o(u)26 b Fw(in)32 b(order)f(to)h(monitor)f(progress)f(and)i(to)g(test)g(and)g(install)449 880 y(new)27 b(soft)n(w)n(are)f(to)i(handle)f(the)h(ev)n(er-increasing) d(torren)n(t)h(of)i(incoming)f(data.)449 1164 y Fp(References)487 1357 y Fs(1.)39 b(D.)13 b(J.)27 b(Bernstein,)f(A.)13 b(K.)25 b(Lenstra,)i Fi(A)e(general)i(n)n(um)n(b)r(er)c(\014eld)j(siev) n(e)g(implemen)n(tation)p Fs(,)e(103{)585 1449 y(126)j(in:)f([13].)487 1540 y(2.)39 b(T.)25 b(Denn)n(y)-6 b(,)23 b(B.)i(Do)r(dson,)g(A.)13 b(K.)24 b(Lenstra,)h(M.)13 b(S.)25 b(Manasse,)h Fi(On)e(the)g (factorization)i(of)f(RSA-)585 1632 y(120)p Fs(,)20 b(Adv)l(ances)d(in) h(Cryptology)-6 b(,)18 b(Crypto)g('93,)h(Lecture)f(Notes)g(in)g (Comput.)g(Sci.)g Fq(773)g Fs(\(1994\))585 1723 y(166{174.)487 1815 y(3.)39 b(J.)14 b(D.)25 b(Dixon,)f Fi(Asymptotically)g(fast)i (factorization)i(of)d(in)n(tegers)p Fs(,)h(Math.)g(Comp.)e Fq(36)h Fs(\(1981\))585 1906 y(255{260.)487 1998 y(4.)39 b(B.)25 b(Dixon,)f(A.)13 b(K.)24 b(Lenstra,)h Fi(F)-6 b(actoring)25 b(in)n(tegers)f(using)h(SIMD)e(siev)n(es)p Fs(,)i(Adv)l(ances)e(in)h(Cryp-)585 2089 y(tology)-6 b(,)27 b(Euro)r(crypt)e('93,)i(Lecture)f(Notes)g(in)f(Comput.)g(Sci.)h Fq(765)h Fs(\(1994\))f(28{39.)487 2181 y(5.)39 b(B.)31 b(Do)r(dson,)g(A.)12 b(K.)30 b(Lenstra,)h Fi(NFS)e(with)i(four)f(large) i(primes:)d(an)i(explosiv)n(e)f(exp)r(erimen)n(t)p Fs(,)585 2272 y(in)c(preparation.)487 2364 y(6.)39 b(M.)28 b(Gardner,)f Fi(Mathematical)h(games,)f(A)f(new)h(kind)f(of)h(cipher)g(that)f(w)n (ould)i(tak)n(e)e(millions)585 2455 y(of)h(y)n(ears)f(to)g(break)p Fs(,)f(Scien)n(ti\014c)h(American,)f(August)g(1977,)i(120{124.)487 2547 y(7.)39 b(R.)29 b(Golliv)n(er,)i(A.)12 b(K.)29 b(Lenstra,)h(K.)13 b(S.)29 b(McCurley)-6 b(,)29 b Fi(Lattice)h(sieving)g(and)e(trial)i (division)p Fs(,)g(Al-)585 2638 y(gorithmic)g(n)n(um)n(b)r(er)d(theory) h(symp)r(osium,)g(Lecture)h(Notes)h(in)f(Comput.)f(Sci.)i Fq(877)f Fs(\(1994\))585 2729 y(18{27.)487 2821 y(8.)39 b(R.)13 b(K.)31 b(Guy)-6 b(,)30 b Fi(Ho)n(w)g(to)h(factor)h(a)f(n)n(um) n(b)r(er)p Fs(,)e(Pro)r(c.)j(Fifth)f(Manitoba)h(Conf.)f(Numer.)f (Math.,)585 2912 y(Congressus)e(Numeran)n(tium)22 b Fq(16)k Fs(\(1976\))h(49{89.)487 3004 y(9.)39 b(D.)13 b(E.)32 b(Kn)n(uth,)e Fi(The)i(art)g(of)g(computer)f(programming)p Fs(,)g(v)n(olume)f(2,)i Fi(Semin)n(umerical)f(algo-)585 3095 y(rithms)p Fs(,)25 b(second)h(edition,)h(Addison-W)-6 b(esley)g(,)24 b(Reading,)i(Massac)n(h)n(usetts,)i(1981.)449 3187 y(10.)39 b(B.)14 b(A.)38 b(LaMacc)n(hia,)h(A.)13 b(M.)39 b(Odlyzk)n(o,)e Fi(Computation)g(of)i(discrete)f(logarithms)h (in)f(prime)585 3278 y(\014elds)p Fs(,)26 b(Designs,)h(Co)r(des)g(and)e (Cryptograph)n(y)g Fq(1)h Fs(\(1991\))h(47{62.)449 3370 y(11.)39 b(A.)13 b(K.)29 b(Lenstra,)g Fi(Massiv)n(ely)h(parallel)g (computing)e(and)g(factoring)p Fs(,)j(Pro)r(ceedings)f(Latin'92,)585 3461 y(Lecture)c(Notes)g(in)f(Comput.)g(Sci.)h Fq(583)h Fs(\(1992\))g(344{355.)449 3553 y(12.)39 b(A.)13 b(K.)32 b(Lenstra,)h(H.)13 b(W.)33 b(Lenstra,)g(Jr.,)g Fi(Algorithms)f(in)g(n)n (um)n(b)r(er)f(theory)p Fs(,)h(Chapter)g(12)h(in:)585 3644 y(J.)25 b(v)l(an)e(Leeu)n(w)n(en)g(\(ed.\),)h Fi(Handb)r(o)r(ok)f (of)h(theoretical)h(computer)e(science)p Fs(,)i(V)-6 b(olume)22 b(A,)h Fi(Algo-)585 3736 y(rithms)i(and)g(complexit)n(y)p Fs(,)g(Elsevier,)i(Amsterdam,)d(1990.)449 3827 y(13.)39 b(A.)13 b(K.)21 b(Lenstra,)i(H.)12 b(W.)22 b(Lenstra,)g(Jr.)g(\(eds\),) g Fi(The)f(dev)n(elopmen)n(t)f(of)i(the)f(n)n(um)n(b)r(er)f(\014eld)h (siev)n(e)p Fs(,)585 3919 y(Lecture)26 b(Notes)g(in)f(Math.)i Fq(1554)p Fs(,)f(Springer-V)-6 b(erlag,)26 b(Berlin,)h(1993.)449 4010 y(14.)39 b(A.)13 b(K.)26 b(Lenstra,)h(H.)13 b(W.)27 b(Lenstra,)g(Jr.,)g(M.)14 b(S.)26 b(Manasse,)i(J.)14 b(M.)27 b(P)n(ollard,)h Fi(The)e(factorization)585 4102 y(of)h(the)e(nin)n(th)g(F)-6 b(ermat)25 b(n)n(um)n(b)r(er)p Fs(,)f(Math.)i(Comp.)f Fq(61)h Fs(\(1993\))h(319{349.)449 4193 y(15.)39 b(A.)13 b(K.)28 b(Lenstra,)g(M.)13 b(S.)28 b(Manasse,)h Fi(F)-6 b(actoring)29 b(b)n(y)d(electronic)j(mail)p Fs(,)f(Adv)l(ances)e(in)i(Cryptol-)585 4285 y(ogy)-6 b(,)26 b(Euro)r(crypt)g('89,)h(Lecture)e(Notes)h(in)g(Comput.)f(Sci.)h Fq(434)g Fs(\(1990\))h(355{371.)449 4376 y(16.)39 b(A.)13 b(K.)23 b(Lenstra,)i(M.)13 b(S.)23 b(Manasse,)j Fi(F)-6 b(actoring)24 b(with)g(t)n(w)n(o)g(large)g(primes)p Fs(,)g(Adv)l(ances) e(in)h(Cryp-)585 4468 y(tology)-6 b(,)36 b(Euro)r(crypt)e('90,)i (Lecture)e(Notes)h(in)f(Comput.)g(Sci.)h Fq(473)g Fs(\(1990\))g(72{82;) i(Math.)585 4559 y(Comp.,)26 b(to)g(app)r(ear.)p eop %%Page: 16 16 16 15 bop 449 83 a Fs(17.)39 b(U.)13 b(M.)36 b(Maurer,)g Fi(F)-6 b(ast)35 b(generation)h(of)g(prime)e(n)n(um)n(b)r(ers)g(and)h (secure)g(public-k)n(ey)e(crypto-)585 174 y(graphic)27 b(parameters)p Fs(,)e(Journal)i(of)g(Cryptology)-6 b(,)26 b(to)g(app)r(ear.)449 266 y(18.)39 b(C.)27 b(P)n(omerance,)f Fi(Analysis)g(and)g(comparison)g(of)h(some)e(in)n(teger)i(factoring)g (algorithms)p Fs(,)g(pp.)585 357 y(89{139)d(in:)e(H.)13 b(W.)22 b(Lenstra,)g(Jr.,)h(R.)f(Tijdeman)g(\(eds\),)f Fi(Computational)h(metho)r(ds)f(in)h(n)n(um-)585 448 y(b)r(er)29 b(theory)p Fs(,)g(Math.)h(Cen)n(tre)f(T)-6 b(racts)30 b Fq(154)p Fs(/)p Fq(155)p Fs(,)g(Mathematisc)n(h)f(Cen)n (trum,)f(Amsterdam,)585 540 y(1983.)449 631 y(19.)39 b(C.)32 b(P)n(omerance,)f(J.)13 b(W.)31 b(Smith,)e Fi(Reduction)h(of)h (h)n(uge,)g(sparse)g(matrices)g(o)n(v)n(er)f(\014nite)h(\014elds)585 722 y(via)26 b(created)g(catastrophes)p Fs(,)h(Exp)r(erimen)n(t.)e (Math.)h Fq(1)g Fs(\(1992\))h(89{94.)449 814 y(20.)39 b(R.)13 b(L.)26 b(Riv)n(est,)f(letter)h(to)g(Martin)h(Gardner,)f(1977.) 449 905 y(21.)39 b(R.)13 b(L.)32 b(Riv)n(est,)g(A.)g(Shamir,)g(L.)g (Adleman,)f Fi(A)h(metho)r(d)f(for)i(obtaining)f(digital)i(signatures) 585 996 y(and)26 b(public-k)n(ey)e(cryptosystems)p Fs(,)h(Comm.)f(A)n (CM)i Fq(21)g Fs(\(1978\))h(120{126.)449 1088 y(22.)39 b(R.)13 b(C.)26 b(Sc)n(hro)r(epp)r(el,)g(p)r(ersonal)h(comm)n (unication,)e(Ma)n(y)h(1994.)449 1179 y(23.)39 b(A.)26 b(Shamir,)f(p)r(ersonal)i(comm)n(unication,)d(April)i(1994.)449 1270 y(24.)39 b(R.)13 b(D.)22 b(Silv)n(erman,)f Fi(The)h(m)n(ultiple)f (p)r(olynomial)h(quadratic)g(siev)n(e)p Fs(,)g(Math.)h(Comp.)e Fq(48)h Fs(\(1987\))585 1362 y(329{339.)449 1702 y(This)h(article)g(w)n (as)g(pro)r(cessed)g(b)n(y)f(the)g(authors)g(using)h(the)e(T)2212 1718 y(E)2256 1702 y(X)h(macro)g(pac)n(k)l(age)g(from)g(Springer-)449 1793 y(V)-6 b(erlag.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF