%!PS-Adobe-2.0 %%Creator: dvips 5.521 Copyright 1986, 1993 Radical Eye Software %%Title: BlockLanczos.dvi %%CreationDate: Wed Feb 22 20:17:27 1995 %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips BlockLanczos.dvi %DVIPSSource: TeX output 1995.02.21:0553 %%BeginProcSet: texc.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 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b(of)i(Gaussian)365 4066 y(elimination)d([9)o(,)g(p.)f (425],)f(requiring)g(ab)r(out)i Fm(mn)f Fx(bits.)h(When)g Fp(B)f Fx(is)g(sparse,)f(one)h(can)h(\014rst)365 4166 y(apply)32 b(structured)g(Gaussian)e(elimination)i([10)o(,)h Fo(x)o Fx(5],)f(replacing)f Fp(B)h Fx(b)n(y)f(a)h(dense)g(matrix)365 4265 y(with)i(ab)r(out)g(one)f(third)h(as)f(man)n(y)g(ro)n(ws)f(and)i (columns.)f(A)h(gigab)n(yte)e(do)r(es)i(not)f(hold)h(a)365 4365 y(dense)28 b(10)676 4335 y Fl(5)730 4365 y Fo(\002)19 b Fx(10)898 4335 y Fl(5)961 4365 y Fx(matrix,)28 b(whereas)e(w)n(e)h(w) n(an)n(t)g(to)h(solv)n(e)e(systems)h(with)h Fm(n)d Fx(around)f(10)3134 4335 y Fl(6)3170 4365 y Fx(.)490 4465 y(LaMacc)n(hia)k(and)i(Odlyzk)n (o)f([10)o(])h(implemen)n(t)h(v)-5 b(arian)n(ts)29 b(to)h(the)h (Lanczos)d(and)i(conju-)365 4565 y(gate)e(gradien)n(t)f(metho)r(ds,)i (as)f(previously)f(suggested)h(b)n(y)g(Odlyzk)n(o)f(et)e(al.)k([7)o(,)g (12)o(].)g(These)365 4664 y(metho)r(ds)37 b(rep)r(eatedly)f(apply)g(a)f (symmetric)h Fm(n)24 b Fo(\002)g Fm(n)36 b Fx(matrix)g(to)g(a)g(v)n (ector.)f(They)h(store)365 4764 y(only)25 b(a)h(few)f(temp)r(orary)g(v) n(ectors)f(and)h(the)h(original)e(matrix,)h(th)n(us)h(relieving)e(the)i (storage)365 4863 y(problem)36 b(if)h(the)f(matrix)g(is)g(sparse.)e (The)j(metho)r(ds)f(w)n(ere)f(dev)n(elop)r(ed)h(for)f(use)h(on)g(real)p eop %%Page: 2 2 2 1 bop 681 387 a Fx(matrices,)22 b(but)i(w)n(ork)d(o)n(v)n(er)g(other) i(\014elds)g(unless)f(one)h(encoun)n(ters)f(a)g(v)n(ector)g(orthogonal) f(to)681 487 y(itself.)k(T)-7 b(o)25 b(a)n(v)n(oid)e(self{orthogonal)f (v)n(ectors,)i(they)h(w)n(ork)e(in)i(an)g(extension)f(\014eld)h(of)g (GF)q(\(2\))681 587 y(rather)h(than)i(GF\(2\))g(itself.)805 690 y(Wiedemann)20 b([15)o(])f(prop)r(oses)f(another)g(iterativ)n(e)h (algorithm.)f(His)h(algorithm)f(applies)h(a)681 790 y(\(not)25 b(necessarily)d(symmetric\))j Fm(n)12 b Fo(\002)g Fm(n)24 b Fx(matrix)g Fp(B)h Fx(to)f(a)g(v)n(ector)f(appro)n(ximately)h(2)p Fm(n)f Fx(times,)681 890 y(and)34 b(constructs)g(the)i(minimal)f(p)r (olynomial)f(of)h Fp(B)p Fx(.)g(Using)f(this)h(minimal)g(p)r (olynomial,)681 989 y(one)24 b(can)g(\014nd)g(v)n(ectors)f(in)i(the)g (n)n(ull)f(space)g(of)g Fp(B)p Fx(,)h(if)g Fp(B)f Fx(is)g(singular.)f (The)i(metho)r(d)f(lik)n(ewise)681 1089 y(requires)i(storage)g(only)h (for)g(the)h(matrix)c Fp(B)k Fx(and)f(for)g(a)g(few)h(temp)r(orary)f(v) n(ectors.)805 1193 y(The)32 b(Lanczos,)e(conjugate)g(gradien)n(t,)h (and)g(Wiedemann)g(algorithms)f(all)h(apply)g(the)681 1292 y(giv)n(en)37 b(matrix)g(\(or)h(its)g(transp)r(ose\))f(to)h Fo(O)r Fx(\()p Fm(n)p Fx(\))h(v)n(ectors.)d(On)i(a)f(binary)h(computer) f(with)681 1392 y Fm(N)c Fx(bits)c(p)r(er)g(w)n(ord,)e(one)i(can)f (apply)g(a)g(matrix)g(to)h Fm(N)37 b Fx(indep)r(enden)n(t)29 b(v)n(ectors)e(o)n(v)n(er)g(GF)q(\(2\))681 1491 y(at)39 b(once,)g(using)g(the)g(mac)n(hine's)g(bit)n(wise)g(op)r(erators.)e(W) -7 b(e)40 b(w)n(ould)f(lik)n(e)f(to)i(reduce)e(the)681 1591 y(iteration)26 b(coun)n(t)g(from)h Fo(O)r Fx(\()p Fm(n)p Fx(\))g(to)g Fo(O)r Fx(\()p Fm(n=)-5 b(N)9 b Fx(\),)27 b(b)n(y)f(accomplishing)g Fm(N)34 b Fx(times)27 b(as)f(m)n(uc)n(h)g(w)n (ork)681 1691 y(p)r(er)40 b(iteration.)f(Ev)n(en)g(if)i(w)n(e)e(do)h Fm(N)33 b Fx(times)41 b(as)e(man)n(y)g(op)r(erations)g(p)r(er)h (iteration)f(after)681 1790 y(applying)27 b(the)h(matrix,)f(the)h (total)f(cost)g(of)h(applying)f(the)h(matrix)f(will)h(drop)f Fm(N)9 b Fx({fold.)805 1894 y(Our)22 b(v)-5 b(ariation)21 b(of)h(Lanczos)f(ac)n(hiev)n(es)g(this)h(ob)5 b(jectiv)n(e)22 b(b)n(y)g(decomp)r(osing)f(GF\(2\))3354 1864 y Fn(n)3422 1894 y Fx(in)n(to)681 1994 y(sev)n(eral)36 b(subspaces)i(of)g (dimension)h(almost)24 b Fm(N)47 b Fx(whic)n(h)38 b(are)g(pairwise)f (orthogonal)f(with)681 2111 y(resp)r(ect)23 b(to)g(the)h(symmetric)f Fm(n)10 b Fo(\002)g Fm(n)22 b Fx(matrix)h Fp(A)g Fx(=)g Fp(B)2321 2075 y Fx(T)2385 2111 y Fp(B)p Fx(.)g(The)h(resulting)f (algorithm)f(tak)n(es)681 2229 y(ab)r(out)i Fm(n=)p Fx(\()p Fm(N)d Fo(\000)13 b Fx(0)p Fm(:)p Fx(76\))23 b(iterations.)h(Eac)n(h)g (iteration)g(applies)g(the)h(matrices)f Fp(B)h Fx(and)f Fp(B)3400 2193 y Fx(T)3489 2229 y(to)681 2329 y(an)j Fm(n)17 b Fo(\002)g Fm(N)36 b Fx(matrix)27 b(and)f(do)r(es)h(a)g(few)g (supplemen)n(tary)g(op)r(erations)e(\(i.e.,)j(inner)f(pro)r(ducts)681 2428 y(of)f(t)n(w)n(o)g Fm(n)16 b Fo(\002)g Fm(N)36 b Fx(matrices,)26 b(m)n(ultiplication)g(of)h(an)f Fm(n)17 b Fo(\002)f Fm(N)35 b Fx(matrix)26 b(b)n(y)g(an)h Fm(N)e Fo(\002)16 b Fm(N)35 b Fx(matrix,)681 2528 y(m)n(ultiplication)28 b(and)f(in)n(v)n(ersion)f(of)h Fm(N)h Fo(\002)18 b Fm(N)36 b Fx(matrices\).)805 2632 y(Don)31 b(Copp)r(ersmith)g(published)h(a)e (blo)r(c)n(k)h(Wiedemann)g(algorithm)f([6])h(whic)n(h)g(needs)681 2731 y(ab)r(out)25 b(3)p Fm(n=)-5 b(N)34 b Fx(applications)24 b(of)h Fp(B)p Fx(.)h(The)f(presen)n(t)g(w)n(ork)f(w)n(as)h(inspired)g (b)n(y)g(a)g(commen)n(t)g([6,)681 2831 y(p.)g(334])j(that)h(Copp)r (ersmith)g(had)g(previously)f(found)h(a)g(blo)r(c)n(k)f(Lanczos)g (algorithm,)g(but)681 2930 y(b)r(efore)34 b(this)g(author)f(had)h(seen) g([5].)g(When)h Fm(N)43 b Fo(\025)34 b Fx(16,)f(the)h(presen)n(t)g (algorithm)f(and)h([5])681 3030 y(eac)n(h)c(need)i(ab)r(out)f(2)p Fm(=)p Fx(3)f(as)g(man)n(y)h(sparse)f(matrix)g(op)r(erations)g(as)h([6) o(],)h(ev)n(en)f(if)g Fp(B)h Fx(is)f(not)681 3130 y(symmetric.)24 b(This)h(algorithm)e(constructs)h(the)h(orthogonal)d(v)n(ectors)h (di\013eren)n(tly)h(than)h([5])681 3229 y(and)i(needs)h(ab)r(out)f (40\045)g(as)g(man)n(y)g(supplemen)n(tary)g(op)r(erations)f(as)h([5].) 805 3333 y(Except)39 b(for)f(Gaussian)g(elimination,)g(these)h (algorithms)e(are)h(probabilistic.)g(They)681 3433 y(mak)n(e)31 b(random)h(c)n(hoices,)f(and)i(ma)n(y)e(fail)i(for)f(some)g(of)g(these) h(c)n(hoices.)e(I)i(ha)n(v)n(e)e(tried)h(the)681 3532 y(prop)r(osed)26 b(metho)r(d)i(on)g(ab)r(out)f(50)d(matrices,)j(and)g (ha)n(v)n(e)g(not)g(exp)r(erienced)h(failure.)805 3636 y(The)j(metho)r(ds)h(herein)e(w)n(ork)g(o)n(v)n(er)f(other)i(\014nite)g 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Fp(v)f Fo(2)g(V)7 b(g)41 b Fm(;)1432 4867 y Fo(V)1490 4837 y Fx(T)1554 4867 y Fo(W)30 b Fx(=)22 b Fo(f)p Fp(v)1845 4837 y Fx(T)1909 4867 y Fp(w)h Fx(:)g Fp(v)h Fo(2)f(V)35 b Fx(and)27 b Fp(w)c Fo(2)h(W)7 b(g)40 b Fm(:)p eop %%Page: 3 3 3 2 bop 365 391 a Fx(Tw)n(o)30 b(subspaces)f Fo(V)37 b Fx(and)30 b Fo(W)37 b Fx(of)30 b Fm(K)1482 361 y Fn(n)1557 391 y Fx(are)f(said)g(to)h(b)r(e)h Fp(A)p Fx({orthogonal)d(if)i Fp(v)2752 361 y Fx(T)2816 391 y Fp(Aw)d Fx(=)g(0)j(for)365 503 y(all)e Fp(v)23 b Fo(2)g(V)35 b Fx(and)27 b Fp(w)c Fo(2)h(W)7 b Fx(;)27 b(this)h(is)g(equiv)-5 b(alen)n(t)27 b(to)g Fo(V)1988 472 y Fx(T)2052 503 y Fp(A)p Fo(W)j Fx(=)23 b Fo(f)p Fx(0)p Fo(g)p Fx(.)490 603 y(If)28 b Fp(V)h Fx(is)f(an)g Fm(n)924 615 y Fl(1)980 603 y Fo(\002)18 b Fm(n)1113 615 y Fl(2)1179 603 y Fx(matrix,)27 b(then)i Fo(h)p Fp(V)p Fo(i)g Fx(denotes)f(the)g(subspace)g(of)g Fm(K)2791 573 y Fn(n)2832 581 y Fg(1)2896 603 y Fx(generated)365 703 y(b)n(y)g(the)g(column)f(v)n(ectors)f(of)f Fp(V)p Fx(.)490 804 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Fx(nonzero)681 1259 y(en)n(tries)30 b(p)r(er)i(ro)n(w)e(or)g(column,)h(then)h(the)g(cost)f(p)r(er)g (iteration)g(is)g Fo(O)r Fx(\()p Fm(dn)p Fx(\))h(to)f(m)n(ultiply)h(b)n (y)681 1358 y Fp(A)c Fx(and)f Fo(O)r Fx(\()p Fm(n)p Fx(\))i(for)e(the)h (other)f(v)n(ector)f(arithmetic.)h(The)h(total)f(time)h(is)g Fo(O)r Fx(\()p Fm(dn)3148 1328 y Fl(2)3186 1358 y Fx(\))19 b(+)f Fo(O)r Fx(\()p Fm(n)3470 1328 y Fl(2)3507 1358 y Fx(\).)805 1458 y(The)j(storage)d(requiremen)n(t)i(is)g(nominal:)h Fo(O)r Fx(\(1\))g(temp)r(orary)e(v)n(ectors)g(of)h(length)25 b Fm(n)p Fx(,)c(plus)681 1558 y(the)28 b(matrix)f Fp(A)h Fx(itself.)681 1798 y Fq(4)112 b(Lanczos)38 b(o)m(v)m(er)f(Other)h (Algebraic)e(Domains)681 1969 y Fx(The)28 b(Lanczos)f(iterations)g (\(1\),)h(\(3\),)g(and)g(\(5\))g(use)g(only)f(rational)g(arithmetic)h (op)r(erations)681 2069 y(\(no)22 b(square)f(ro)r(ots)h(or)f (transcenden)n(tal)g(functions\).)i(If)g(there)f(is)g(no)g(round{o\013) g(error,)e(then)681 2168 y(the)28 b(\014nal)g(v)n(ector)c Fp(x)k Fx(is)g(an)g(exact)f(solution,)h(not)g(an)g(appro)n(ximation.)e (As)i(Odlyzk)n(o)f(et)e(al.)681 2268 y([7)o(])31 b([10)o(,)g Fo(x)p Fx(3])f([12)o(])h(observ)n(e,)e(this)i(mak)n(es)e(Lanczos)h (usable)g(in)h(other)f(algebraic)e(domains,)681 2367 y(although)f(the)h(metho)r(d)g(w)n(as)e(disco)n(v)n(ered)g(b)n(y)h(n)n (umerical)g(analysts.)805 2467 y(Let)h Fp(A)g Fx(b)r(e)g(a)g(symmetric) f Fm(n)18 b Fo(\002)h Fm(n)27 b Fx(matrix)g(o)n(v)n(er)f(a)i(\014eld)d Fm(K)6 b Fx(.)27 b(Assume)h(that)g Fp(w)3249 2479 y Fn(i)3300 2467 y Fo(6)p Fx(=)23 b Fp(0)k Fx(for)681 2567 y(0)35 b Fo(\024)h Fm(i)f(<)h(m)f Fx(and)g(that)h Fp(w)1558 2579 y Fn(m)1657 2567 y Fx(=)f Fp(0)p Fx(.)g(When)h Fm(K)41 b Fx(=)36 b(I)-14 b(R)36 b(and)f Fp(A)g Fx(is)h(p)r(ositiv)n(e)e (de\014nite,)i(the)681 2666 y(Lanczos)26 b(v)n(ectors)g Fo(f)p Fp(w)1389 2678 y Fn(i)1416 2666 y Fo(g)1458 2631 y Fn(m)p Fi(\000)p Fl(1)1458 2689 y Fn(i)p Fl(=0)1633 2666 y Fx(satisfy)1118 2827 y Fp(w)1187 2797 y Fx(T)1187 2849 y Fn(i)1251 2827 y Fp(A)q(w)1392 2839 y Fn(i)1443 2827 y Fo(6)p Fx(=)d(0)236 b(\(0)23 b Fo(\024)f Fm(i)h(<)g(m)p Fx(\))41 b Fm(;)1118 2944 y Fp(w)1187 2914 y Fx(T)1187 2965 y Fn(j)1251 2944 y Fp(A)q(w)1392 2956 y Fn(i)1443 2944 y Fx(=)23 b(0)236 b(\()p Fm(i)23 b Fo(6)p Fx(=)f Fm(j)5 b Fx(\))42 b Fm(;)1259 3046 y Fp(A)p Fo(W)30 b(\022)23 b(W)7 b Fm(;)166 b Fx(where)82 b Fo(W)30 b Fx(=)23 b Fo(h)p Fp(w)2405 3058 y Fl(0)2442 3046 y Fm(;)28 b Fp(w)2562 3058 y Fl(1)2599 3046 y Fm(;)g Fo(\001)14 b(\001)g(\001)g Fm(;)27 b Fp(w)2880 3058 y Fn(m)p Fi(\000)p Fl(1)3028 3046 y Fo(i)42 b Fm(:)3457 2929 y Fx(\(6\))805 3189 y(If)25 b Fp(b)e Fo(2)h(W)31 b Fx(and)24 b Fp(x)h Fx(is)f(de\014ned)h(b)n(y)f (\(3\),)h(then)g(w)n(e)f(claim)g(\(6\))h(implies)f Fp(A)q(x)f Fx(=)g Fp(b)p Fx(,)h(without)681 3289 y(further)35 b(assumptions)g(ab)r (out)g(the)h(\014eld)25 b Fm(K)41 b Fx(\(the)36 b(pro)r(of)f(in)g Fo(x)p Fx(3)g(assumes)g Fm(K)41 b Fx(=)36 b(I)-14 b(R\).)36 b(As)681 3405 y(b)r(efore,)27 b Fo(W)1042 3375 y 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Fx({orthogonal)e(to)j(themselv)n(es,)f (and)h(need)g(to)g(v)-5 b(ary)35 b(our)i(metho)r(ds.)g(The)681 4565 y(\014eld)26 b(GF)q(\(2\))g(has)g(its)h(adv)-5 b(an)n(tages,)24 b(since)j(one)f(can)g(apply)g(the)h(matrix)e Fp(A)i Fx(to)f Fm(N)36 b Fx(di\013eren)n(t)681 4664 y(v)n(ectors)30 b(in)j(GF\(2\))1294 4634 y Fn(n)1372 4664 y Fx(at)f(once,)g(using)g (bit)n(wise)g(op)r(erators.)e(W)-7 b(e)33 b(generalize)d(\(6\))j(to)f (allo)n(w)681 4764 y(a)j(sequence)h(of)g(subspaces)f(in)h(place)f(of)h (the)g(v)n(ectors)f Fo(f)p Fp(w)2582 4776 y Fn(i)2610 4764 y Fo(g)p Fx(,)g(and)h(adapt)g(the)g(Lanczos)681 4863 y(iteration)27 b(\(1\))g(to)h(the)g(new)g(framew)n(ork.)p eop %%Page: 5 5 5 4 bop 365 387 a Fq(5)112 b(Sequences)39 b(of)f(Orthogonal)f (Subspaces)365 591 y Fx(Let)23 b Fp(A)f Fx(b)r(e)h(a)f(symmetric)g (matrix)f(o)n(v)n(er)g(a)h(\014eld)j Fm(K)6 b Fx(.)22 b(Blo)r(c)n(k)f(Lanczos)g(algorithms)g([8)o(,)i(Chap-)365 690 y(ter)i(7])h(mo)r(dify)h(\(6\))f(to)h(pro)r(duce)f(a)g(sequence)f (of)i(subspaces)e Fo(fW)2415 702 y Fn(i)2442 690 y Fo(g)2484 655 y Fn(m)p Fi(\000)p Fl(1)2484 713 y Fn(i)p Fl(=0)2658 690 y Fx(of)i Fm(K)2829 660 y Fn(n)2900 690 y Fx(whic)n(h)f(are)365 808 y(pairwise)31 b Fp(A)q Fx({orthogonal.)e(The)j(condition)g Fp(w)1863 778 y Fx(T)1863 830 y Fn(i)1927 808 y Fp(A)q(w)2069 820 y Fn(i)2127 808 y Fo(6)p Fx(=)e(0)i(in)h(\(6\))f(is)g(replaced)f(b) n(y)h(a)g(re-)365 908 y(quiremen)n(t)c(that)f(no)h(nonzero)e(v)n(ector) g(in)i Fo(W)1790 920 y Fn(i)1846 908 y Fx(b)r(e)g Fp(A)p Fx({orthogonal)d(to)i(all)g(of)h Fo(W)2881 920 y Fn(i)2909 908 y Fx(.)365 1095 y Fp(De\014nition)15 b(1.)41 b Fx(A)31 b(subspace)g Fo(W)k(\022)29 b Fm(K)1643 1065 y Fn(n)1718 1095 y Fx(is)i(said)g(to)g(b)r(e)g Fp(A)p Ff({invertible)i Fx(if)e(it)h(has)e(a)h(basis)365 1212 y Fp(W)d Fx(of)f(column)h(v)n (ectors)e(suc)n(h)h(that)h Fp(W)1625 1176 y Fx(T)1689 1212 y Fp(AW)f Fx(is)h(in)n(v)n(ertible.)490 1398 y(The)38 b(prop)r(ert)n(y)f(of)h(b)r(eing)h Fp(A)p Fx({in)n(v)n(ertible)e(is)h 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y Fo(W)819 2237 y Fx(T)812 2289 y Fn(j)883 2267 y Fp(A)p Fo(W)1037 2279 y Fn(i)1088 2267 y Fx(=)22 b Fo(f)p Fx(0)p Fo(g)152 b Fx(\()p Fm(i)23 b Fo(6)p Fx(=)g Fm(j)5 b Fx(\))42 b Fm(;)904 2370 y Fp(A)p Fo(W)30 b(\022)22 b(W)7 b Fm(;)166 b Fx(where)83 b Fo(W)30 b Fx(=)22 b Fo(W)2030 2382 y Fl(0)2086 2370 y Fx(+)c Fo(W)2251 2382 y Fl(1)2307 2370 y Fx(+)g Fo(\001)c(\001)g (\001)k Fx(+)g Fo(W)2670 2382 y Fn(m)p Fi(\000)p Fl(1)2860 2370 y Fm(:)3141 2261 y Fx(\(7\))490 2547 y(Assume)30 b(\(7\).)h(Giv)n(en)f Fp(b)e Fo(2)g(W)7 b Fx(,)30 b(w)n(e)g(can)g (construct)g(an)g Fp(x)e Fo(2)g(W)38 b Fx(suc)n(h)30 b(that)g Fp(A)q(x)e Fx(=)f Fp(b)p Fx(.)365 2655 y(Let)i Fp(x)c Fx(=)679 2593 y Fk(P)766 2613 y Fn(m)p Fi(\000)p Fl(1)766 2680 y Fn(j)s Fl(=0)928 2655 y Fp(w)997 2667 y Fn(j)1032 2655 y Fx(,)k(where)f Fp(w)1394 2667 y Fn(j)1453 2655 y Fo(2)d(W)1615 2667 y Fn(j)1678 2655 y Fx(is)k(c)n(hosen)e(so)h (that)h Fp(Aw)2454 2667 y Fn(j)2508 2655 y Fo(\000)18 b Fp(b)29 b Fx(is)f(orthogonal)e(to)365 2755 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3371 y Fx(\()p Fm(i)23 b Fo(\025)f Fx(2\))p Fm(:)3100 3471 y Fx(\(14\))365 3571 y(Although)f(\(14\))f(has) f(more)h(terms)g(than)g(\(5\),)h(the)f(time)h(p)r(er)f(iteration)g(and) g(the)h(temp)r(orary)365 3671 y(storage)26 b(requiremen)n(ts)h(will)h (remain)f(acceptable)g(using)g(\(14\).)56 b(Equation)27 b(\(14\))g(remains)365 3770 y(v)-5 b(alid)28 b(for)f Fm(i)c Fx(=)f(0)27 b(and)h Fm(i)23 b Fx(=)f(1)27 b(if)i(w)n(e)e (de\014ne)h Fp(V)1782 3782 y Fn(j)1840 3770 y Fx(=)22 b Fp(0)28 b Fx(and)f Fp(W)2263 3782 y Fn(j)2321 3770 y Fx(=)c Fp(0)k Fx(for)g Fm(j)h(<)23 b Fx(0.)490 3876 y(T)-7 b(o)34 b(ac)n(hiev)n(e)g(\(14\),)g(w)n(e)h(require)e(that)i (\(13\))g(v)-5 b(anish)34 b(whenev)n(er)g Fm(j)40 b Fo(\024)35 b Fm(i)22 b Fo(\000)h Fx(3.)35 b(That)f(is,)365 3976 y(w)n(e)g(require)f Fp(V)852 3988 y Fn(j)s Fl(+1)1006 3976 y Fx(to)h(b)r(e)g Fp(A)q Fx({orthogonal)d(to)j Fp(W)1976 3988 y Fn(j)s Fl(+3)2129 3976 y Fx(through)f Fp(W)2548 3988 y Fn(m)2611 3976 y Fx(.)i(W)-7 b(e)34 b(ac)n(hiev)n(e)f(this)365 4075 y(b)n(y)27 b(requiring)e(that)i(all)f(v)n(ectors)f(in)i Fp(V)1575 4087 y Fn(j)s Fl(+1)1720 4075 y Fx(b)r(e)g(used)g(either)g (in)f Fp(W)2450 4087 y Fn(j)s Fl(+1)2596 4075 y Fx(or)g(in)h Fp(W)2891 4087 y Fn(j)s Fl(+2)3010 4075 y Fx(.)g(More)365 4175 y(precisely)-7 b(,)27 b(w)n(e)g(require)927 4381 y Fo(h)p Fp(V)1032 4393 y Fn(j)s Fl(+1)1151 4381 y Fo(i)c(\022)g(W)1376 4393 y Fl(0)1431 4381 y Fx(+)18 b Fo(W)1596 4393 y Fl(1)1652 4381 y Fx(+)g Fo(\001)c(\001)g(\001)k Fx(+)g Fo(W)2015 4393 y Fn(j)s Fl(+2)2300 4381 y Fx(\()p Fm(j)29 b Fo(\025)22 b(\000)p Fx(1\))41 b Fm(:)415 b Fx(\(15\))490 4594 y(Assuming)27 b(\(15\),)h(w)n(e)f(try)g(to)h(simplify)g(the)g(matrix)f(equation)g (\(14\).)g(Denote)720 4842 y Fp(W)819 4806 y Fl(in)n(v)819 4863 y Fn(i)934 4842 y Fx(=)c Fp(S)1075 4854 y Fn(i)1117 4751 y Fk(\020)1166 4842 y Fp(W)1265 4806 y Fx(T)1265 4863 y Fn(i)1329 4842 y Fp(AW)1500 4854 y Fn(i)1528 4751 y Fk(\021)1577 4768 y Fi(\000)p Fl(1)1680 4842 y Fp(S)1733 4806 y Fx(T)1733 4863 y Fn(i)1820 4842 y Fx(=)g Fp(S)1961 4854 y Fn(i)2002 4751 y Fk(\020)2052 4842 y Fp(S)2105 4806 y Fx(T)2105 4863 y Fn(i)2169 4842 y Fp(V)2241 4806 y Fx(T)2241 4863 y Fn(i)2306 4842 y Fp(AV)2450 4854 y Fn(i)2478 4842 y Fp(S)2531 4854 y Fn(i)2558 4751 y Fk(\021)2608 4768 y Fi(\000)p Fl(1)2711 4842 y Fp(S)2764 4806 y Fx(T)2764 4863 y Fn(i)2869 4842 y Fm(:)3100 4834 y Fx(\(16\))p eop %%Page: 8 8 8 7 bop 681 387 a Fx(Eac)n(h)17 b Fp(W)977 351 y Fl(in)n(v)977 408 y Fn(i)1089 387 y Fx(is)i(a)f(symmetric)h Fm(N)10 b Fo(\002)q Fm(N)27 b Fx(matrix.)56 b(Eliminate)19 b(all)g(references)e (to)i Fp(W)3221 399 y Fn(i)3271 387 y Fx(=)k Fp(V)3431 399 y Fn(i)3459 387 y Fp(S)3512 399 y Fn(i)3540 387 y Fx(:)693 575 y Fp(V)766 587 y Fn(i)p Fl(+1)887 575 y Fx(=)f Fp(AV)1119 587 y Fn(i)1146 575 y Fp(S)1199 587 y Fn(i)1227 575 y Fp(S)1280 538 y Fx(T)1280 595 y Fn(i)1362 575 y Fx(+)c Fp(V)1518 587 y Fn(i)1564 575 y Fo(\000)g Fp(V)1719 587 y Fn(i)1747 575 y Fp(S)1800 587 y Fn(i)1827 575 y Fp(C)1896 587 y Fn(i)p Fl(+1)p Fn(;i)2070 575 y Fo(\000)g Fp(V)2213 587 y Fn(i)p Fi(\000)p Fl(1)2326 575 y Fp(S)2379 587 y Fn(i)p Fi(\000)p Fl(1)2492 575 y Fp(C)2561 587 y Fn(i)p Fl(+1)p Fn(;i)p Fi(\000)p Fl(1)2819 575 y Fo(\000)g Fp(V)2963 587 y Fn(i)p Fi(\000)p Fl(2)3076 575 y Fp(S)3129 587 y Fn(i)p Fi(\000)p Fl(2)3241 575 y Fp(C)3310 587 y Fn(i)p Fl(+1)p Fn(;i)p Fi(\000)p Fl(2)887 698 y Fx(=)k Fp(AV)1119 710 y Fn(i)1146 698 y Fp(S)1199 710 y Fn(i)1227 698 y Fp(S)1280 662 y Fx(T)1280 718 y Fn(i)1362 698 y Fx(+)c Fp(V)1518 710 y Fn(i)1564 698 y Fo(\000)g Fp(V)1719 710 y Fn(i)1747 698 y Fp(W)1845 662 y Fl(in)n(v)1845 718 y Fn(i)1938 698 y Fp(V)2010 662 y Fx(T)2010 718 y Fn(i)2074 698 y Fp(A)p Fx(\()p Fp(A)q(V)2323 710 y Fn(i)2351 698 y Fp(S)2404 710 y Fn(i)2431 698 y Fp(S)2484 662 y Fx(T)2484 718 y Fn(i)2567 698 y Fx(+)g Fp(V)2722 710 y Fn(i)2750 698 y Fx(\))1048 821 y Fo(\000)g Fp(V)1203 833 y Fn(i)p Fi(\000)p Fl(1)1316 821 y Fp(W)1414 785 y Fl(in)n(v)1414 842 y Fn(i)p Fi(\000)p 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Fl(2)975 1166 y Fp(V)1047 1178 y Fn(i)1075 1166 y Fx(,)35 b(and)48 b Fp(V)1388 1130 y Fx(T)1388 1186 y Fn(i)p Fi(\000)p Fl(2)1500 1166 y Fp(A)1572 1130 y Fl(2)1610 1166 y Fp(V)1682 1178 y Fn(i)1710 1166 y Fx(.)35 b(W)-7 b(e)35 b(can)f(express)g(the)h(latter)g(t)n(w)n(o)f (inner)h(pro)r(ducts)f(in)681 1266 y(terms)27 b(of)h(the)g(\014rst)f(t) n(w)n(o,)g(using)g(\(10\),)g(\(11\),)h(\(12\),)f(and)h(\(14\):)725 1459 y Fp(S)778 1422 y Fx(T)778 1479 y Fn(i)p Fi(\000)p Fl(1)891 1459 y Fp(V)963 1422 y Fx(T)963 1479 y Fn(i)p Fi(\000)p Fl(1)1075 1459 y Fp(A)1148 1422 y Fl(2)1185 1459 y Fp(V)1257 1471 y Fn(i)1308 1459 y Fx(=)22 b(\()p Fp(A)q(W)1599 1471 y Fn(i)p Fi(\000)p Fl(1)1711 1459 y Fx(\))1743 1429 y(T)1808 1459 y Fp(AV)1952 1471 y Fn(i)2003 1459 y Fx(=)g(\()q Fp(V)2195 1471 y Fn(i)2222 1459 y Fp(S)2275 1471 y Fn(i)p Fi(\000)p Fl(1)2407 1459 y Fx(+)c Fo(O)r Fx(\()p Fo(W)2672 1471 y Fl(0)2728 1459 y Fx(+)g Fo(\001)c(\001)g(\001)k Fx(+)g Fo(W)3091 1471 y Fn(i)p Fi(\000)p Fl(1)3204 1459 y Fx(\)\))3268 1417 y(T)3346 1459 y Fp(A)q(V)3491 1471 y Fn(i)1308 1582 y Fx(=)k Fp(S)1448 1545 y Fx(T)1448 1602 y Fn(i)p Fi(\000)p Fl(1)1561 1582 y Fp(V)1633 1545 y Fx(T)1633 1602 y Fn(i)1697 1582 y Fp(A)q(V)1842 1594 y Fn(i)1911 1582 y Fm(;)694 1781 y Fp(S)747 1745 y Fx(T)747 1802 y Fn(i)p Fi(\000)p Fl(2)860 1781 y Fp(V)932 1745 y Fx(T)920 1802 y Fn(i)p Fi(\000)p Fl(2)1033 1781 y Fp(A)1105 1745 y Fl(2)1142 1781 y Fp(V)1215 1793 y Fn(i)1265 1781 y Fx(=)h(\()p Fp(AW)1556 1793 y Fn(i)p Fi(\000)p Fl(2)1669 1781 y Fx(\))1701 1751 y(T)1765 1781 y Fp(AV)1909 1793 y Fn(i)1960 1781 y Fx(=)g(\()p Fp(V)2141 1793 y Fn(i)p Fi(\000)p Fl(1)2254 1781 y Fp(S)2307 1793 y Fn(i)p Fi(\000)p Fl(2)2438 1781 y Fx(+)18 b Fo(O)r Fx(\()p Fo(W)2703 1793 y Fl(0)2759 1781 y Fx(+)g Fo(\001)c(\001)g(\001) k Fx(+)g Fo(W)3122 1793 y Fn(i)p Fi(\000)p Fl(2)3235 1781 y Fx(\)\))3300 1739 y(T)3378 1781 y Fp(AV)3522 1793 y Fn(i)1265 1904 y Fx(=)23 b Fp(S)1406 1868 y Fx(T)1406 1925 y Fn(i)p Fi(\000)p Fl(2)1519 1904 y Fp(V)1591 1868 y Fx(T)1591 1925 y Fn(i)p Fi(\000)p Fl(1)1703 1904 y Fp(A)q(V)1848 1916 y Fn(i)1265 2055 y Fx(=)g Fp(S)1406 2018 y Fx(T)1406 2075 y Fn(i)p Fi(\000)p Fl(2)1519 2055 y Fp(V)1591 2018 y Fx(T)1591 2075 y Fn(i)p Fi(\000)p Fl(1)1703 2055 y Fp(A)1776 1937 y Fk(\022)1837 2055 y Fp(AW)2008 2067 y Fn(i)p Fi(\000)p Fl(1)2120 2055 y Fp(S)2173 2018 y Fx(T)2173 2075 y Fn(i)p Fi(\000)p Fl(1)2304 2055 y Fx(+)18 b Fp(V)2460 2067 y Fn(i)p Fi(\000)p Fl(1)1427 2254 y Fo(\000)g Fp(W)1608 2266 y Fn(i)p Fi(\000)p Fl(1)1721 2254 y Fp(C)1790 2266 y Fn(i;i)p Fi(\000)p Fl(1)1964 2254 y Fx(+)g Fo(O)r Fx(\()p Fo(W)2229 2266 y Fn(i)p Fi(\000)p Fl(2)2361 2254 y Fx(+)g Fo(W)2526 2266 y Fn(i)p Fi(\000)p Fl(3)2639 2254 y Fx(\))2671 2137 y Fk(\023)1265 2453 y Fx(=)23 b Fp(S)1406 2417 y Fx(T)1406 2474 y Fn(i)p Fi(\000)p Fl(2)1519 2453 y Fp(V)1591 2417 y Fx(T)1591 2474 y Fn(i)p Fi(\000)p Fl(1)1703 2453 y Fp(A)1776 2336 y Fk(\022)1823 2453 y Fp(I)1859 2465 y Fn(n)1923 2453 y Fo(\000)18 b Fp(V)2078 2465 y 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Fp(S)2280 2740 y Fx(T)2280 2797 y Fn(i)p Fi(\000)p Fl(1)2411 2777 y Fx(+)g Fp(V)2566 2740 y Fx(T)2566 2797 y Fn(i)p Fi(\000)p Fl(1)2679 2777 y Fp(AV)2823 2789 y Fn(i)p Fi(\000)p Fl(1)2936 2685 y Fk(\021)3041 2777 y Fm(:)681 2943 y Fx(Hence)28 b(\(17\))f(simpli\014es)g(to)1137 3131 y Fp(V)1209 3143 y Fn(i)p Fl(+1)1344 3131 y Fx(=)c Fp(AV)1576 3143 y Fn(i)1604 3131 y Fp(S)1657 3143 y Fn(i)1684 3131 y Fp(S)1737 3095 y Fx(T)1737 3152 y Fn(i)1820 3131 y Fx(+)18 b Fp(V)1975 3143 y Fn(i)2003 3131 y Fp(D)2076 3143 y Fn(i)p Fl(+1)2206 3131 y Fx(+)g Fp(V)2361 3143 y Fn(i)p Fi(\000)p Fl(1)2474 3131 y Fp(E)2537 3143 y Fn(i)p Fl(+1)2667 3131 y Fx(+)g Fp(V)2822 3143 y Fn(i)p Fi(\000)p Fl(2)2935 3131 y Fp(F)2995 3143 y Fn(i)p Fl(+1)3415 3131 y Fx(\(18\))681 3296 y(for)27 b Fm(i)c Fo(\025)f Fx(0,)27 b(where)891 3482 y Fp(D)964 3494 y Fn(i)p Fl(+1)1099 3482 y Fx(=)22 b Fp(I)1223 3494 y Fn(N)1304 3482 y Fo(\000)c Fp(W)1486 3445 y Fl(in)n(v)1486 3502 y Fn(i)1592 3390 y Fk(\020)1642 3482 y Fp(V)1714 3445 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Fm(n)1179 4577 y Fl(1)1222 4565 y Fo(\002)6 b Fm(n)1343 4577 y Fl(2)1401 4565 y Fx(matrix)21 b(o)n(v)n(er)e(the)j(\014eld)g Fm(K)28 b Fx(=)23 b(GF\(2\),)f(where)f Fm(n)2888 4577 y Fl(1)2948 4565 y Fm(<)i(n)3086 4577 y Fl(2)3123 4565 y Fx(.)e(Then)h(there)681 4664 y(exist)29 b(at)h(least)f Fm(n)1229 4676 y Fl(2)1286 4664 y Fo(\000)19 b Fm(n)1420 4676 y Fl(1)1487 4664 y Fx(linearly)29 b(indep)r(enden)n(t)h(v)n(ectors)e Fp(x)f Fo(2)f Fm(K)2778 4634 y Fn(n)2819 4642 y Fg(2)2885 4664 y Fx(suc)n(h)j(that)h Fp(Bx)d Fx(=)f Fp(0)p Fx(.)681 4764 y(Some)35 b(in)n(teger)f(factorization)f(algorithms)h([9,)h(pp.)25 b(380\013.][11)n(])35 b(require)f(\014nding)i(sev)n(eral)681 4863 y(\(p)r(erhaps)27 b(ten\))h(suc)n(h)g Fp(x)p Fx(.)g(In)g (practice,)e(the)i(matrix)f Fp(B)h Fx(is)g(large)e(but)i(v)n(ery)e (sparse.)p eop %%Page: 9 9 9 8 bop 490 387 a Fx(The)22 b(Lanczos)e(algorithm)h(requires)g(that)h (the)g(matrix)f(b)r(e)i(symmetric.)e(W)-7 b(e)22 b(let)h Fm(n)f Fx(=)h Fm(n)3210 399 y Fl(2)365 505 y Fx(and)31 b Fp(A)f Fx(=)e Fp(B)793 469 y Fx(T)857 505 y Fp(B)p Fx(.)j(This)h Fp(A)f Fx(is)g(symmetric,)g(and)g(an)n(y)f(solution)h(of) g Fp(Bx)f Fx(=)e Fp(0)j Fx(will)h(satisfy)365 605 y Fp(Ax)24 b Fx(=)f Fp(0)e Fx(\(although)h(the)g(con)n(v)n(erse)e(need)j(not)f(b)r (e)g(true)g(if)h(the)f(rank)f(of)h Fp(A)g Fx(is)g(less)g(than)j Fm(n)3155 617 y Fl(1)3192 605 y Fx(\).)490 710 y(Let)33 b Fm(N)42 b Fx(denote)32 b(the)i(computer)e(w)n(ord)g(size,)g(t)n (ypically)h(32)f(or)g(64.)g(Select)h(a)f(random)365 809 y Fm(n)10 b Fo(\002)g Fm(N)32 b Fx(matrix)24 b Fp(Y)g Fx(o)n(v)n(er)d(GF\(2\),)j(compute)f Fp(A)q(Y)p Fx(,)g(and)g(attempt)h (to)g(\014nd)f(an)g Fm(n)10 b Fo(\002)g Fm(N)32 b Fx(matrix)365 909 y Fp(X)24 b Fx(suc)n(h)f(that)g Fp(AX)h Fx(=)e Fp(AY)q Fx(.)h(If)h(w)n(e)e(succeed,)h(then)h(the)g(column)f(v)n(ectors)f(of)h Fp(X)10 b Fo(\000)g Fp(Y)22 b Fx(will)i(b)r(e)365 1008 y(random)19 b(v)n(ectors)g(in)i(the)f(n)n(ull)g(space)g(of)g Fp(A)p Fx(.)g(If)h(the)f(rank)g(of)g Fp(A)g Fx(is)g(at)g(least)g(rank)o (\()p Fp(B)p Fx(\))t Fo(\000)t Fm(N)13 b Fx(+)t(1,)365 1108 y(then)28 b(one)g(can)f(com)n(bine)g(columns)g(of)h Fp(X)18 b Fo(\000)h Fp(Y)27 b Fx(to)h(\014nd)g(v)n(ectors)e(in)i(the)g (n)n(ull)g(space)e(of)f Fp(B)p Fx(.)490 1213 y(After)k(selecting)f Fp(Y)q Fx(,)h(w)n(e)f(initialize)h Fp(V)1704 1225 y Fl(0)1767 1213 y Fx(=)24 b Fp(AY)29 b Fx(and)g(pro)r(ceed)f(through)g(the)i (Lanczos)365 1330 y(iterations)d(\(18\))g(un)n(til)h(some)f Fp(V)1386 1294 y Fx(T)1386 1351 y Fn(i)1451 1330 y Fp(AV)1595 1342 y Fn(i)1646 1330 y Fx(=)22 b Fp(0)p Fx(,)28 b(sa)n(y)e(for)h Fm(i)c Fx(=)g Fm(m)p Fx(.)k(Compute)713 1612 y Fp(X)c Fx(=)896 1508 y Fn(m)p Fi(\000)p Fl(1)908 1533 y Fk(X)914 1709 y Fn(i)p Fl(=0)1054 1612 y Fp(W)1152 1624 y Fn(i)1194 1519 y Fk(\020)1243 1612 y Fp(W)1342 1575 y Fx(T)1342 1632 y Fn(i)1406 1612 y Fp(AW)1577 1624 y Fn(i)1605 1519 y Fk(\021)1654 1537 y Fi(\000)p Fl(1)1757 1612 y Fp(W)1856 1575 y Fx(T)1856 1632 y Fn(i)1920 1612 y Fp(V)1992 1624 y Fl(0)2053 1612 y Fx(=)2140 1508 y Fn(m)p Fi(\000)p Fl(1)2152 1533 y Fk(X)2158 1709 y Fn(i)p Fl(=0)2298 1612 y Fp(V)2370 1624 y Fn(i)2398 1612 y Fp(W)2496 1575 y Fl(in)n(v)2496 1632 y Fn(i)2589 1612 y Fp(V)2661 1575 y Fx(T)2661 1632 y Fn(i)2725 1612 y Fp(V)2798 1624 y Fl(0)2876 1612 y Fm(:)3100 1611 y Fx(\(20\))490 1896 y(Denote)i Fo(W)k Fx(=)23 b Fo(W)1053 1908 y Fl(0)1103 1896 y Fx(+)12 b Fo(\001)i(\001)g(\001)e Fx(+)g Fo(W)1448 1908 y Fn(m)p Fi(\000)p Fl(1)1620 1896 y Fx(and)25 b Fo(W)1861 1908 y Fn(m)1947 1896 y Fx(=)d Fo(h)p Fp(V)2139 1908 y Fn(m)2202 1896 y Fo(i)p Fx(.)j(Then)g Fo(W)2578 1908 y Fn(m)2665 1896 y Fx(is)g Fp(A)p Fx({orthogonal)365 1996 y(to)34 b(itself)g(and)f(to)h Fo(W)7 b Fx(.)33 b(By)h (construction,)e Fp(A)q(X)22 b Fo(\000)g Fp(V)2067 2008 y Fl(0)2138 1996 y Fo(2)33 b(W)c Fx(+)23 b Fo(W)2507 2008 y Fn(m)2570 1996 y Fx(.)33 b(If)h Fp(V)2788 2008 y Fn(m)2884 1996 y Fx(=)e Fp(0)p Fx(,)i(then)365 2095 y Fp(AX)24 b Fx(=)e Fp(V)693 2107 y Fl(0)753 2095 y Fx(=)g 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Fo(W)1258 3346 y Fn(m)1344 3334 y Fo(\022)i Fp(A)p Fo(V)30 b(\022)23 b(V)29 b Fx(=)23 b Fo(W)11 b Fx(+)t Fo(W)2085 3346 y Fn(m)2147 3334 y Fx(.)21 b(Supp)r(ose)f Fp(w)2578 3346 y Fl(0)2620 3334 y Fx(+)t Fp(w)2757 3346 y Fl(1)2799 3334 y Fx(+)t Fo(\001)14 b(\001)g(\001)s Fx(+)t Fp(w)3106 3346 y Fn(m)3192 3334 y Fo(2)365 3433 y Fp(A)p Fo(W)519 3445 y Fn(m)583 3433 y Fx(,)31 b(where)g(eac)n(h)f Fp(w)1140 3445 y Fn(j)1204 3433 y Fo(2)f(W)1370 3445 y Fn(j)1405 3433 y Fx(.)j(W)-7 b(e)31 b(claim)g(that)h Fp(w)2084 3445 y Fn(j)2148 3433 y Fx(=)d Fp(0)i Fx(for)f(0)f Fo(\024)g Fm(j)34 b(<)28 b(m)p Fx(.)k(W)-7 b(e)31 b(c)n(hec)n(k)365 3533 y(that)735 3660 y Fp(w)804 3630 y Fx(T)804 3681 y Fn(j)868 3660 y Fp(A)p Fo(W)1022 3672 y Fn(j)1080 3660 y Fx(=)23 b(\()p Fp(w)1269 3672 y Fl(0)1325 3660 y Fx(+)18 b Fp(w)1477 3672 y Fl(1)1533 3660 y Fx(+)g Fo(\001)c(\001)g(\001)k Fx(+)g Fp(w)1883 3672 y Fn(m)1946 3660 y Fx(\))1978 3630 y(T)2042 3660 y Fp(A)q Fo(W)2197 3672 y Fn(j)2255 3660 y Fo(\022)k Fx(\()p Fp(A)q Fo(W)2529 3672 y Fn(m)2592 3660 y Fx(\))2624 3630 y(T)2688 3660 y Fp(A)p Fo(W)2842 3672 y Fn(j)1080 3779 y Fx(=)h Fo(W)1257 3749 y Fx(T)1250 3800 y Fn(m)1321 3779 y Fp(A)p Fx(\()p Fp(A)q Fo(W)1580 3791 y Fn(j)1615 3779 y Fx(\))g Fo(\022)g(W)1847 3749 y Fx(T)1840 3800 y Fn(m)1911 3779 y Fp(A)p Fx(\()p Fo(W)i Fx(+)18 b Fo(W)2287 3791 y Fn(m)2350 3779 y Fx(\))24 b(=)e Fo(f)p Fx(0)p Fo(g)40 b Fm(:)365 3944 y Fx(Since)28 b Fp(w)651 3956 y Fn(j)709 3944 y Fo(2)23 b(W)869 3956 y Fn(j)932 3944 y Fx(and)28 b Fo(W)1176 3956 y Fn(j)1238 3944 y Fx(is)g Fp(A)p Fx({in)n(v)n(ertible)e(for)h Fm(j)h(<)23 b(m)p Fx(,)28 b(this)g(sho)n(ws)e(that)i Fp(w)2852 3956 y Fn(j)2910 3944 y Fx(=)23 b Fp(0)o Fx(.)490 4048 y(If)i(this)f (heuristic)g(argumen)n(t)g(ab)r(out)g Fo(V)31 b Fx(is)24 b(correct,)f(then)i(the)g(images)e Fp(A)q Fx(\()p Fp(X)12 b Fo(\000)g Fp(Y)p Fx(\))24 b(and)365 4148 y Fp(AV)510 4160 y Fn(m)601 4148 y Fx(are)j(b)r(oth)h(in)g Fo(W)1115 4160 y Fn(m)1178 4148 y 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b(whose)g(columns)g(span)h (the)g(n)n(ull)f(space)g(of)d Fp(BZ)p Fx(.)365 4764 y(Then)f(output)g (a)f(basis)g(for)h Fp(Z)q(U)p Fx(.)g(In)f(practice,)g(one)g(do)r(es)h (not)f(compute)h Fp(U)g Fx(explicitly)-7 b(,)24 b(but)365 4863 y(applies)j(the)h(same)f(column)h(op)r(erations)e(to)i Fp(Z)g Fx(and)d Fp(B)o(Z)q Fx(.)p eop %%Page: 10 10 10 9 bop 681 387 a Fq(8)112 b(Selecting)36 b(S)1387 402 y Fd(i)1453 387 y Fq(and)i(W)1785 402 y Fd(i)681 594 y Fx(Recurrence)26 b(\(9\))i(do)r(es)f(not)h(sp)r(ecify)g(ho)n(w)f(to)g (select)h Fp(S)2409 606 y Fn(i)2464 594 y Fx(and)g Fp(W)2724 606 y Fn(i)2775 594 y Fx(=)23 b Fp(V)2935 606 y Fn(i)2962 594 y Fp(S)3015 606 y Fn(i)3043 594 y Fx(,)28 b(except)f(that)733 788 y Fp({)41 b(W)921 752 y Fx(T)921 809 y Fn(i)985 788 y Fp(AW)1156 800 y Fn(i)1211 788 y Fx(m)n(ust)28 b(b)r(e)g(in)n(v)n (ertible;)733 889 y Fp({)41 b Fx(rank)o(\()p Fp(W)1117 901 y Fn(i)1145 889 y Fx(\))27 b(should)h(b)r(e)g(as)f(large)f(as)h(p)r (ossible;)733 990 y Fp({)41 b Fx(An)n(y)27 b(column)h(of)g Fp(V)1457 1002 y Fn(i)p Fi(\000)p Fl(1)1597 990 y Fx(whic)n(h)g(w)n(as) e(not)i(used)f(in)h Fp(W)2527 1002 y Fn(i)p Fi(\000)p Fl(1)2667 990 y Fx(m)n(ust)g(b)r(e)g(used)g(no)n(w)e(\(15\).)805 1160 y(Let)40 b Fp(Q)f Fx(b)r(e)h(a)f(symmetric)g Fm(N)c Fo(\002)25 b Fm(N)49 b Fx(matrix)38 b(o)n(v)n(er)g(a)h(\014eld)g Fm(K)6 b Fx(.)39 b(Let)h Fm(r)45 b Fx(=)d(rank)o(\()p Fp(Q)p Fx(\).)681 1260 y(W)-7 b(e)34 b(claim)f(that)h(if)g(w)n(e)g (select)f(an)n(y)g Fm(r)j Fx(linearly)d(indep)r(enden)n(t)i(columns)e (of)25 b Fp(Q)o Fx(,)34 b(then)g(the)681 1360 y(symmetric)d Fm(r)24 b Fo(\002)c Fm(r)35 b Fx(submatrix)c(of)24 b Fp(Q)32 b Fx(with)g(the)g(same)e(ro)n(w)h(indices)g(is)g(in)n(v)n (ertible.)g(After)681 1504 y(ren)n(um)n(b)r(ering)23 b(the)j(ro)n(ws)d(and)h(columns,)h(w)n(e)f(ma)n(y)g(write)h Fp(Q)e Fx(=)2680 1387 y Fk(\024)2736 1454 y Fp(Q)2808 1474 y Fl(11)2901 1454 y Fp(Q)2973 1474 y Fl(12)2736 1553 y Fp(Q)2808 1574 y Fl(21)2901 1553 y Fp(Q)2973 1574 y Fl(22)3055 1387 y 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y Fn(k)2795 3570 y Fm(;)27 b(c)2881 3582 y Fn(j)2941 3570 y Fx(+)d Fm(N)9 b Fx(])36 b(is)h(nonzero,)681 3670 y(where)d Fm(k)k Fo(\025)c Fm(j)5 b Fx(.)35 b(A)n(t)g(all)f(times,)h(if)g Fp(M)1880 3682 y Fl(1)1952 3670 y Fx(denotes)f(the)h(left)g(half)g(of)g Fp(M)f Fx(and)g Fp(M)3218 3682 y Fl(2)3290 3670 y Fx(denotes)681 3770 y(its)h(righ)n(t)f(half,)h(then)h Fp(M)1493 3782 y Fl(1)1566 3770 y Fx(=)f Fp(M)1756 3782 y Fl(2)1794 3770 y Fp(T)g Fx(since)g(only)g(ro)n(w)e(op)r(erations)h(are)g(p)r (erformed.)g(W)-7 b(e)681 3869 y(attempt)36 b(to)f(get)g Fp(I)1298 3881 y Fn(N)1397 3869 y Fx(on)g(the)g(left)h(of)g Fp(M)o Fx(,)g(but)g(o)r(ccasionally)d(zero)h(an)i(en)n(tire)e(ro)n(w.)h (Let)p 681 3902 56 4 v 681 3969 a Fm(S)49 b Fx(=)44 b Fo(f)p Fm(c)968 3981 y Fl(1)1005 3969 y Fm(;)28 b Fo(\001)14 b(\001)g(\001)f Fm(;)28 b(c)1253 3981 y Fn(j)s Fi(\000)p Fl(1)1373 3969 y Fo(g)f(n)f Fm(S)46 b Fx(denote)40 b(the)h(ro)n(ws)e (of)h Fp(M)h Fx(whic)n(h)f(ha)n(v)n(e)g(b)r(een)h(zero)r(ed.)e(W)-7 b(e)681 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Fm(c)935 2364 y Fn(k)976 2352 y Fm(;)28 b(c)1063 2364 y Fn(j)1116 2352 y Fx(+)18 b Fm(N)9 b Fx(])23 b Fo(6)p Fx(=)f(0\))111 b(Exc)n(hange)26 b(ro)n(ws)g Fm(c)2196 2364 y Fn(j)2258 2352 y Fx(and)i Fm(c)2456 2364 y Fn(k)2524 2352 y Fx(of)g Fp(M)o Fx(.)531 2452 y Fp(end)g(do)531 2551 y(assert)56 b Fx(\()p Fp(M)p Fx([)p Fm(c)1011 2563 y Fn(j)1046 2551 y Fm(;)27 b(c)1132 2563 y Fn(j)1186 2551 y Fx(+)18 b Fm(N)9 b Fx(])23 b Fo(6)p Fx(=)f(0\))531 2651 y(Add)29 b(m)n(ultiples)e(of)h(ro)n(w)e Fm(c)1361 2663 y Fn(j)1424 2651 y Fx(to)h(other)g(ro)n(ws,)g(to)g(zero) g(rest)g(of)g(column)h Fm(c)2818 2663 y Fn(j)2871 2651 y Fx(+)18 b Fm(N)9 b Fx(.)531 2751 y(Zero)27 b(ro)n(w)f Fm(c)916 2763 y Fn(j)979 2751 y Fx(of)h Fp(M)p Fx(.)194 b Fp(Cm)m(t)p Fx(.)24 b(Will)k(b)r(e)h(zero)d(for)h(rest)g(of)h (algorithm.)448 2850 y Fp(end)k(if)448 2950 y(Cm)m(t)p Fx(.)25 b(Column)g Fm(c)1029 2962 y Fn(j)1092 2950 y Fx(will)i(remain)g(unc)n(hanged)g(for)g(the)h(duration)f(of)h(the)g (algorithm.)365 3050 y Fp(end)g(do)365 3155 y Fx(Cop)n(y)f(righ)n(t)g (half)h(of)f Fp(M)h Fx(in)n(to)f Fp(W)1427 3119 y Fl(in)n(v)1427 3176 y Fn(i)1519 3155 y Fx(.)p 365 3188 V 1010 3379 a Fp(Fig.)15 b(1.)27 b Fx(Pseudo)r(co)r(de)f(for)h(selecting)g Fp(S)2259 3391 y Fn(i)2315 3379 y Fx(and)g Fp(W)2575 3391 y Fn(i)490 3975 y Fx(Sets)e(1)g(and)g(2)g(ha)n(v)n(e)f(a)h(total)g (of)h Fo(j)p 1524 3908 56 4 v Fm(S)t Fo(j)15 b Fx(+)e(\()p Fm(N)24 b Fo(\000)13 b Fm(j)19 b Fx(+)14 b(1\))23 b(=)g Fm(N)g Fo(\000)13 b(j)p Fm(S)5 b Fo(j)26 b Fx(linearly)e(indep)r(enden) n(t)365 4075 y(v)n(ectors.)34 b(The)i Fo(j)p Fm(S)5 b Fo(j)36 b Fx(indep)r(enden)n(t)g(v)n(ectors)e(in)i(Set)25 b(3)35 b(are)g(orthogonal)e(to)j(all)f(v)n(ectors)f(in)365 4174 y(Sets)25 b(1)i(and)e(2,)i(and)h(therefore)e(span)i(the)g (orthogonal)d(complemen)n(t.)490 4365 y(If)38 b(\()p Fp(M)706 4377 y Fl(2)743 4365 y Fp(T)q Fx(\)[)p Fm(c)901 4377 y Fn(k)942 4365 y Fm(;)27 b(c)1028 4377 y Fn(j)1063 4365 y Fx(])40 b(=)g(0)d(for)g(all)h Fm(k)k Fo(\025)e Fm(j)5 b Fx(,)38 b(then)g(column)25 b Fm(c)2386 4377 y Fn(j)2458 4365 y Fx(of)g Fp(T)38 b Fx(is)g(orthogonal)d(to)365 4465 y(ev)n(erything)24 b(in)g(Sets)h(1)f(and)h(2.)f(This)h(column)f(m) n(ust)h(b)r(e)g(a)f(linear)g(com)n(bination)f(of)i(v)n(ectors)365 4565 y(in)32 b(Set)26 b(3.)31 b(Let)h(the)h(dep)r(endency)f(v)n(ector)f (for)g Fp(T)h Fx(b)r(e)26 b Fp(v)k Fx(=)g Fp(v)2280 4577 y Fn(c)2310 4585 y Fj(j)2344 4565 y Fx(.)j(By)e(symmetry)-7 b(,)32 b(this)g Fp(v)g Fx(is)365 4664 y(orthogonal)26 b(to)i(all)f(of)h(Set)e(3)h(and)h(m)n(ust)g(b)r(e)g(a)g(linear)f(com)n (bination)g(of)h(v)n(ectors)f(in)h(Sets)d(1)365 4764 y(and)g(2.)31 b(Since)h(elemen)n(t)25 b Fm(c)1181 4776 y Fn(j)1248 4764 y Fx(of)32 b Fp(v)g Fx(is)f(nonzero,)g(some)g(v)n (ector)f(in)i(Set)25 b(1)32 b(or)f(Set)25 b(2)31 b(m)n(ust)h(b)r(e)365 4863 y(nonzero)27 b(there.)g(This)h(translates)e(in)n(to)h Fp(M)1738 4875 y Fl(2)1775 4863 y Fx([)p Fm(c)1834 4875 y Fn(k)1875 4863 y Fm(;)h(c)1962 4875 y Fn(j)1997 4863 y Fx(])23 b Fo(6)p Fx(=)f(0)28 b(for)f(some)g Fm(k)f Fo(\025)c Fm(j)5 b Fx(,)28 b(as)f(required.)p eop %%Page: 12 12 12 11 bop 681 387 a Fp(8.1)94 b(Rank)33 b(of)e(W)1359 399 y Fn(i)681 543 y Fx(Let)24 b Fm(f)867 555 y Fn(N)930 543 y Fx(\()p Fm(X)7 b Fx(\))24 b(b)r(e)h(the)f(generating)f(function)i (for)f(the)g(corank)f(of)h(a)g(random)f Fm(N)e Fo(\002)12 b Fm(N)32 b Fx(matrix)681 643 y(o)n(v)n(er)26 b(the)j(\014eld)f(GF)q (\()p Fm(p)p Fx(\).)g(That)g(is,)h(the)f(co)r(e\016cien)n(t)g(of)g Fm(X)2478 612 y Fn(m)2569 643 y Fx(in)g Fm(f)2707 655 y Fn(N)2770 643 y Fx(\()p Fm(X)7 b Fx(\))28 b(is)g(the)h(probabilit)n (y)681 742 y(that)f(the)g(matrix)f(has)g(rank)f(exactly)f Fm(N)i Fo(\000)18 b Fm(m)p Fx(.)27 b(If)h Fm(q)f Fx(=)22 b(1)p Fm(=p)p Fx(,)27 b(then)838 901 y Fm(f)879 913 y Fl(0)915 901 y Fx(\()p Fm(X)7 b Fx(\))23 b(=)g(1)41 b Fm(;)838 1001 y(f)879 1013 y Fl(1)915 1001 y Fx(\()p Fm(X)7 b Fx(\))23 b(=)g(1)18 b Fo(\000)g Fm(q)j Fx(+)d Fm(q)s(X)48 b(;)812 1101 y(f)853 1113 y Fn(N)915 1101 y Fx(\()p Fm(X)7 b Fx(\))23 b(=)g(\(1)18 b Fo(\000)g Fm(q)k Fx(+)c Fm(q)1523 1071 y Fn(N)1586 1101 y Fm(X)7 b Fx(\))p Fm(f)1735 1113 y Fn(N)f Fi(\000)p Fl(1)1882 1101 y Fx(\()p Fm(X)h Fx(\))18 b(+)g(\()p Fm(q)k Fo(\000)c Fm(q)2337 1071 y Fn(N)2400 1101 y Fx(\))p Fm(f)2473 1113 y Fn(N)6 b Fi(\000)p Fl(2)2621 1101 y Fx(\()p Fm(X)h Fx(\))166 b(\()p Fm(N)32 b Fo(\025)23 b Fx(2\))41 b Fm(:)3415 1002 y Fx(\(22\))681 1262 y(Equation)e(\(22\))h(is)h(deriv)n(ed)e(b)n (y)h(c)n(hec)n(king)g(whic)n(h)g(elemen)n(ts)g(in)h(the)g(\014rst)f(ro) n(w)f(of)h(the)681 1361 y(random)f(matrix)h(are)f(zero,)h(using)g(the)g (metho)r(ds)h(of)f([2].)g(It)h(implies)g(the)f(recurrence)681 1461 y Fm(f)722 1473 y Fn(N)784 1461 y Fx(\()p Fm(pX)7 b Fx(\))24 b(=)f Fm(f)1119 1473 y Fn(N)1182 1461 y Fx(\()p Fm(X)7 b Fx(\))18 b(+)h(\(1)f Fo(\000)h Fm(q)1640 1431 y Fn(N)1703 1461 y Fx(\))p Fm(X)7 b(f)1852 1473 y Fn(N)f Fi(\000)p Fl(1)1999 1461 y Fx(\()p Fm(X)h Fx(\))28 b(for)f Fm(N)33 b Fo(\025)23 b Fx(1.)28 b(As)g Fm(N)k Fo(!)24 b(1)p 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Fx(\)\(1)h Fo(\000)f Fm(q)1808 1959 y Fl(5)1845 1989 y Fx(\))c Fo(\001)g(\001)g(\001)56 b Fx(to)27 b(force)g Fm(f)9 b Fx(\(1\))23 b(=)f(1.)805 2089 y(The)28 b(exp)r(ected)g(rank)f(Er)1603 2101 y Fn(N)1689 2089 y Fx(=)22 b Fm(N)27 b Fo(\000)18 b Fm(f)2003 2059 y Fi(0)1994 2112 y Fn(N)2057 2089 y Fx(\(1\))28 b(satis\014es)789 2250 y(Er)878 2262 y Fl(0)938 2250 y Fx(=)23 b(0)41 b Fm(;)789 2350 y Fx(Er)878 2362 y Fl(1)938 2350 y Fx(=)23 b(1)18 b Fo(\000)g Fm(q)44 b(;)763 2449 y Fx(Er)852 2461 y Fn(N)938 2449 y Fx(=)23 b(\(1)18 b Fo(\000)g Fm(q)s Fx(\)\(1)h(+)f(Er)1538 2461 y Fn(N)6 b Fi(\000)p Fl(1)1686 2449 y Fx(\))18 b(+)g Fm(q)1859 2419 y Fn(N)1922 2449 y Fx(Er)2011 2461 y Fn(N)6 b Fi(\000)p Fl(1)2178 2449 y Fx(+)18 b(\()p Fm(q)j Fo(\000)e Fm(q)2475 2419 y Fn(N)2538 2449 y Fx(\)\(2)f(+)g(Er)2834 2461 y Fn(N)6 b Fi(\000)p Fl(2)2982 2449 y Fx(\))938 2549 y(=)23 b(\(1)18 b(+)g Fm(q)j Fo(\000)d Fx(2)p Fm(q)1424 2519 y Fn(N)1487 2549 y Fx(\))h(+)f(\(1)g Fo(\000)g Fm(q)j Fx(+)e Fm(q)1978 2519 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Fx(for)21 b(large)j Fm(N)30 b Fx(and)25 b Fm(n)p Fx(,)d(sub)5 b(ject)22 b(to)g Fm(N)32 b Fo(\034)23 b Fm(n)g Fo(\034)g Fx(2)2159 3129 y Fn(N)2218 3104 y Fg(2)2249 3129 y Fn(=)p Fl(2)2321 3159 y Fx(.)f(The)g(exp)r (erimen)n(tal)f(data)g(in)i(T)-7 b(able)24 b(1)681 3259 y(of)j Fo(x)p Fx(10)g(supp)r(ort)g(this)h(conjecture.)681 3517 y Fq(9)112 b(Cost)37 b(of)h(Blo)s(c)m(k)e(Lanczos)681 3706 y Fx(Eac)n(h)19 b(iteration)i(computes)f Fp(A)q(V)1718 3718 y Fn(i)1745 3706 y Fx(,)h Fp(V)1862 3670 y Fx(T)1862 3727 y Fn(i)1926 3706 y Fp(AV)2070 3718 y Fn(i)2098 3706 y Fx(,)g(and)f Fp(V)2369 3670 y Fx(T)2369 3727 y Fn(i)2433 3706 y Fp(A)2505 3670 y Fl(2)2542 3706 y Fp(V)2614 3718 y Fn(i)2665 3706 y Fx(=)j(\()p Fp(AV)2929 3718 y Fn(i)2957 3706 y Fx(\))2989 3676 y(T)3053 3706 y Fp(A)q(V)3198 3718 y Fn(i)3246 3706 y Fx(from)i Fp(V)3512 3718 y Fn(i)3540 3706 y Fx(.)681 3806 y(After)20 b(c)n(ho)r(osing)f Fp(S)1273 3818 y Fn(i)1301 3806 y Fx(,)h(it)h(computes)f Fp(W)1881 3769 y Fl(in)n(v)1994 3806 y 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b(if)681 4449 y Fm(i)22 b Fo(\025)h Fx(2,)k(then)h Fp(V)1174 4412 y Fx(T)1174 4469 y Fl(0)1238 4449 y Fp(AV)1382 4461 y Fn(i)1433 4449 y Fx(=)22 b Fp(0)28 b Fx(b)n(y)f(\(11\).)g(By)h (\(18\),)934 4695 y Fp(V)1007 4658 y Fx(T)1007 4715 y Fn(i)p Fl(+1)1118 4695 y Fp(V)1190 4707 y Fl(0)1251 4695 y Fx(=)1338 4578 y Fk(\022)1400 4695 y Fp(AV)1544 4707 y Fn(i)1572 4695 y Fp(S)1625 4707 y Fn(i)1652 4695 y Fp(S)1705 4658 y Fx(T)1705 4715 y Fn(i)1788 4695 y Fx(+)18 b Fp(V)1943 4707 y Fn(i)1971 4695 y Fp(D)2044 4707 y Fn(i)p Fl(+1)2174 4695 y Fx(+)g Fp(V)2329 4707 y Fn(i)p Fi(\000)p Fl(1)2442 4695 y Fp(E)2505 4707 y Fn(i)p Fl(+1)2635 4695 y Fx(+)g Fp(V)2790 4707 y Fn(i)p Fi(\000)p Fl(2)2903 4695 y Fp(F)2963 4707 y Fn(i)p Fl(+1)3074 4578 y Fk(\023)3135 4595 y Fx(T)3200 4695 y Fp(V)3272 4707 y Fl(0)1251 4867 y Fx(=)k Fp(D)1412 4830 y Fx(T)1412 4887 y Fn(i)p Fl(+1)1523 4867 y Fp(V)1596 4830 y Fx(T)1596 4887 y Fn(i)1660 4867 y Fp(V)1732 4879 y Fl(0)1788 4867 y Fx(+)c Fp(E)1933 4830 y Fx(T)1933 4887 y Fn(i)p Fl(+1)2045 4867 y Fp(V)2117 4830 y Fx(T)2117 4887 y Fn(i)p Fi(\000)p Fl(1)2230 4867 y Fp(V)2302 4879 y Fl(0)2358 4867 y Fx(+)g Fp(F)2501 4830 y Fx(T)2501 4887 y Fn(i)p Fl(+1)2612 4867 y Fp(V)2685 4830 y Fx(T)2685 4887 y Fn(i)p Fi(\000)p Fl(2)2797 4867 y Fp(V)2869 4879 y Fl(0)2948 4867 y Fm(:)p eop %%Page: 13 13 13 12 bop 365 398 a Fx(The)28 b(inner)f(pro)r(ducts)h Fp(V)1165 361 y Fx(T)1165 418 y Fn(i)1229 398 y Fp(V)1301 410 y Fl(0)1338 398 y Fx(,)g Fp(V)1461 361 y Fx(T)1461 418 y Fn(i)p Fi(\000)p Fl(1)1574 398 y Fp(V)1646 410 y Fl(0)1683 398 y Fx(,)g(and)g Fp(V)1968 361 y Fx(T)1968 418 y Fn(i)p Fi(\000)p Fl(2)2080 398 y Fp(V)2153 410 y Fl(0)2217 398 y Fx(are)f(kno)n(wn)g(b)n(y)g(induction.)490 501 y(With)h(this)g(impro)n(v)n(emen)n(t,)f(the)h(cost)f(of)g(\(16\),)h (\(18\),)f(\(19\),)g(and)h(\(20\))f(is)g(\(for)h Fm(i)22 b(>)h Fx(2\):)417 704 y Fp({)41 b Fx(One)28 b(application)f(of)g Fp(A)c Fx(=)g Fp(B)1456 668 y Fx(T)1520 704 y Fp(B)28 b Fx(to)f Fp(V)1789 716 y Fn(i)1845 704 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Fn(i)2700 1128 y Fx(,)g(as)f(in)h Fo(x)o Fx(8.)417 1231 y Fp({)41 b Fx(A)28 b(few)g Fm(N)f Fo(\002)19 b Fm(N)36 b Fx(matrix)27 b(op)r(erations)f(to)i(compute)f Fp(D)2213 1243 y Fn(i)p Fl(+1)2324 1231 y Fx(,)h Fp(E)2438 1243 y Fn(i)p Fl(+1)2550 1231 y Fx(,)f(and)h Fp(F)2822 1243 y Fn(i)p Fl(+1)2958 1231 y Fx(in)d(\(19\).)417 1334 y Fp({)41 b Fx(F)-7 b(our)22 b(m)n(ultiplications)g(of)g Fm(n)8 b Fo(\002)g Fm(N)30 b Fx(matrices)21 b(b)n(y)h Fm(N)16 b Fo(\002)8 b Fm(N)31 b Fx(matrices)21 b(and)h(four)g(additions)506 1434 y(of)j Fm(n)12 b Fo(\002)g Fm(N)33 b Fx(matrices)24 b(when)g(computing)h Fp(V)1860 1446 y Fn(i)p Fl(+1)1996 1434 y Fx(and)g(the)f(new)h(partial) f(sum)g(of)h Fp(X)12 b Fo(\000)g Fp(Y)p Fx(.)506 1555 y(\(The)22 b(p)r(ost{m)n(ultiplication)f(b)n(y)g Fp(S)1588 1567 y Fn(i)1615 1555 y Fp(S)1669 1519 y Fx(T)1669 1575 y Fn(i)1754 1555 y Fx(in)g(\(18\))g(is)g(easy)g(since)g Fp(S)2517 1567 y Fn(i)2544 1555 y Fp(S)2597 1519 y Fx(T)2597 1575 y Fn(i)2683 1555 y Fx(is)g(diagonal)f(with)506 1655 y(zeros)27 b(and)g(ones.)g(When)h Fm(K)h Fx(=)22 b(GF)q(\(2\),)28 b(it)g(can)f(b)r(e)h(done)f(via)g(bit)n(wise)h(ANDs.\))365 1832 y(The)c(conjectured)g(iteration)f(coun)n(t)h(is)g Fm(n=)p Fx(Er)1792 1844 y Fn(N)1878 1832 y Fx(=)f Fo(O)r Fx(\()p Fm(n=)-5 b(N)9 b Fx(\))25 b(if)f(the)h(initial)f Fp(V)2806 1844 y Fl(0)2867 1832 y Fx(is)g(random.)490 1935 y(If)h(the)g(matrix)f Fp(B)h Fx(a)n(v)n(erages)c Fm(d)k Fx(nonzero)e(en)n(tries)h(p)r(er)h(column,)f(then)i(eac)n(h)d(m) n(ultiplica-)365 2035 y(tion)f(b)n(y)g Fp(A)g Fx(tak)n(es)f(time)i Fo(O)r Fx(\()p Fm(dn)p Fx(\).)g(An)f Fo(O)r Fx(\()p Fm(N)9 b(n)p Fx(\))23 b(algorithm)e(for)g(inner)h(pro)r(ducts)g(o)n(v)n(er)h (GF\(2\))365 2134 y(circularly)i(shifts)i(the)f(\014rst)g(argumen)n(t)g (b)n(y)f Fm(k)j Fx(bits,)f(and)f(uses)g Fm(n)e Fx(ANDs)j(and)g Fm(n)15 b Fo(\000)g Fx(1)25 b(X)n(ORs)365 2234 y(to)f(construct)e Fm(N)32 b Fx(bits)24 b(of)f(the)h(inner)f(pro)r(duct;)h(this)f(step)h (is)f(rep)r(eated)g(for)g(0)f Fo(\024)h Fm(k)j Fo(\024)c Fm(N)d Fo(\000)10 b Fx(1.)365 2333 y(The)32 b(m)n(ultiplications)f(of)g (an)g Fm(n)20 b Fo(\002)h Fm(N)40 b Fx(matrix)30 b(b)n(y)h(an)g Fm(N)f Fo(\002)20 b Fm(N)40 b Fx(matrix)31 b(can)g(similarly)f(b)r(e) 365 2433 y(done)e(with)g Fo(O)r Fx(\()p Fm(N)9 b(n)p Fx(\))28 b(op)r(erations)e(on)i Fm(N)9 b Fx({bit)27 b(w)n(ords.)490 2536 y(The)d(net)h(time)h(is)e Fo(O)r Fx(\()p Fm(dn)p Fx(\))12 b(+)g Fo(O)r Fx(\()p Fm(N)d(n)p Fx(\))27 b(p)r(er)d(iteration) g(and)h Fo(O)r Fx(\()p Fm(dn)2491 2506 y Fl(2)2529 2536 y Fm(=)-5 b(N)9 b Fx(\))j(+)g Fo(O)r Fx(\()p Fm(n)2913 2506 y Fl(2)2951 2536 y Fx(\))25 b(for)f(the)365 2636 y(algorithm.)i(Gaussian)h(elimination)g(tak)n(es)f(time)i Fo(O)r Fx(\()p Fm(n)2107 2606 y Fl(3)2145 2636 y Fx(\).)f(Blo)r(c)n(k)g (Lanczos)f(is)h(asymptoti-)365 2736 y(cally)e(sup)r(erior)f(\(in)h (time)h(and)f(space\))f(to)h(Gaussian)f(elimination)h(if)h Fm(d)d Fo(\034)g Fm(n)i Fx(and)g(remains)365 2835 y(comp)r(etitiv)n(e)j (with)g(Gaussian)f(elimination)g(if)h Fm(d)c Fx(=)e Fo(O)r Fx(\()p Fm(n)p Fx(\).)490 2938 y(Copp)r(ersmith)j([6,)h(pp.)f(342{343]) d(sho)n(ws)j(ho)n(w)g(to)h(compute)f(the)i(inner)e(pro)r(ducts)g(and) 365 3038 y(the)j(pro)r(ducts)g(of)f Fm(n)18 b Fo(\002)g Fm(N)37 b Fx(b)n(y)27 b Fm(N)g Fo(\002)18 b Fm(N)37 b Fx(matrices)27 b(more)f(e\016cien)n(tly)-7 b(.)490 3141 y(When)28 b Fp(S)785 3153 y Fn(i)p Fi(\000)p Fl(1)921 3141 y Fx(=)22 b Fp(I)1044 3153 y Fn(N)1135 3141 y Fx(\(so)27 b(that)h Fp(W)1548 3153 y Fn(i)p Fi(\000)p Fl(1)1683 3141 y Fx(=)23 b Fp(V)1843 3153 y Fn(i)p Fi(\000)p Fl(1)1956 3141 y Fx(\),)28 b(the)g(form)n(ula)e(for)h Fp(F)2673 3153 y Fn(i)p Fl(+1)2813 3141 y Fx(simpli\014es)g(to)365 3241 y(zero)g(and)g(the)h(term)g Fp(V)1116 3253 y Fn(i)p Fi(\000)p Fl(2)1229 3241 y Fp(F)1289 3253 y Fn(i)p Fl(+1)1428 3241 y Fx(can)f(b)r(e)h(omitted)g(from)g(\(18\).)490 3344 y(Storage)d(requiremen)n(ts)g(are)g(lo)n(w.)h(Other)g(than)g(the)h (matrix)f Fp(A)h 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b(w)n(ork)f(w)n(as)g(supp)r(orted)g(b)n(y)h(Stieltjes) h(Institute)g(for)e(Mathematics,)h(Leiden)g(and)g(b)n(y)365 692 y(Cen)n(trum)38 b(v)n(o)r(or)f(Wiskunde)i(en)f(Informatica,)g (Amsterdam.)g(The)g(CPU)g(time)h(on)f(the)365 791 y(Cra)n(y)24 b(C90)f(w)n(as)g(pro)n(vided)g(b)n(y)h(the)h(Dutc)n(h)g(National)e (Computing)h(F)-7 b(acilities)25 b(F)-7 b(oundation)365 891 y(NCF,)36 b(with)h(\014nancial)e(supp)r(ort)g(from)g(the)h (Netherlands)g(Organization)d(for)i(Scien)n(ti\014c)365 990 y(Researc)n(h)c(NW)n(O.)h(Thanks)f(to)h(Scott)g(Con)n(tini)f(of)h (Bellcore)f(for)g(his)h(constructiv)n(e)f(com-)365 1090 y(men)n(ts)d(on)f(an)h(earlier)e(v)n(ersion)g(of)h(this)h(man)n (uscript.)365 1364 y Fq(References)404 1561 y Fw(1.)42 b(Henk)24 b(Bo)r(ender,)j(Priv)l(ate)f(comm)n(unication,)f(1994.)404 1653 y(2.)42 b(Ric)n(hard)24 b(P)-6 b(.)25 b(Bren)n(t)g(and)f(Brendan)h (D.)g(McKa)n(y)-6 b(,)24 b Fc(On)j(determinants)h(of)f(r)l(andom)g (symmetric)505 1744 y(matric)l(es)i(over)f Fb(Z)-31 b(Z)1046 1752 y Fs(m)1105 1744 y Fw(,)26 b(Ars)f(Com)n(binatoria)i Fv(26A)f Fw(\(1988\),)h(57{64.)404 1837 y(3.)42 b(J.P)-6 b(.)31 b(Buhler,)f(H.W.)g(Lenstra,)g(Jr.,)h(and)e(Carl)i(P)n(omerance,) f Fc(F)-6 b(actoring)33 b(inte)l(gers)g(with)e(the)505 1928 y(numb)l(er)36 b(\014eld)f(sieve)p Fw(,)g(The)f(Dev)n(elopmen)n(t) f(of)h(the)g(Num)n(b)r(er)e(Field)i(Siev)n(e)g(\(Berlin\))g(\(A.K.)505 2020 y(Lenstra)f(and)g(H.W.)g(Lenstra,)h(Jr.,)g(eds.\),)g(Lecture)f (Notes)g(in)g(Mathematics,)h(v)n(ol.)f(1554,)505 2111 y(Springer{V)-6 b(erlag,)27 b(Berlin,)g(1993,)g(pp.)e(50{94.)404 2203 y(4.)42 b(Scott)18 b(Con)n(tini)g(and)f(Arjen)25 b(K.)18 b(Lenstra,)g Fc(Implementation)j(of)f(blo)l(cke)l(d)h(L)l (anczos)h(and)f(Wie)l(de-)505 2295 y(mann)28 b(algorithms)p Fw(,)e(In)f(preparation,)i(1995.)404 2387 y(5.)42 b(Don)27 b(Copp)r(ersmith,)g Fc(Solving)i(line)l(ar)f(e)l(quations)j(over)f(GF)p Fw(\(2\))p Fc(:)f(Blo)l(ck)g(L)l(anczos)h(algorithm)p Fw(,)505 2478 y(Linear)c(Algebra)g(and)g(its)g(Applications)g Fv(192)g Fw(\(1993\),)h(33{60.)404 2571 y(6.)p 505 2571 231 4 v 285 w(,)33 b Fc(Solving)h(homo)l(gene)l(ous)i(line)l(ar)e(e)l (quations)i(over)f(GF)p Fw(\(2\))f Fc(via)g(blo)l(ck)h(Wie)l(demann)505 2662 y(algorithm)p Fw(,)26 b(Mathematics)h(of)f(Computation)f Fv(62)h Fw(\(1994\),)i(no.)d(205,)i(333{350.)404 2754 y(7.)42 b(Don)33 b(Copp)r(ersmith,)h(Andrew)24 b(M.)34 b(Odlyzk)n(o,)f(and)g(Ric)n(hard)g(Sc)n(hro)r(epp)r(el,)h Fc(Discr)l(ete)i(lo)l(ga-)505 2846 y(rithms)28 b(in)f(GF\()p Fu(p)p Fc(\))p Fw(,)g(Algorithmica)f Fv(1)g Fw(\(1986\),)h(1{15.)404 2938 y(8.)42 b(Jane)26 b(K.)20 b(Cullum)g(and)h(Ralph)j(A.)d(Willough)n (b)n(y)-6 b(,)21 b Fc(L)l(anczos)j(algorithms)g(for)f(lar)l(ge)h (symmetric)505 3029 y(eigenvalue)k(c)l(omputations.)h(V)-6 b(ol.)27 b(I)g(The)l(ory)p Fw(,)h(Birkh\177)-38 b(auser,)26 b(Boston,)h(1985.)404 3122 y(9.)42 b(Donald)25 b(E.)d(Kn)n(uth,)e Fc(Seminumeric)l(al)k(algorithms)p Fw(,)f(The)f(Art)f(of)h(Computer)f (Programming,)505 3213 y(v)n(ol.)k(2,)i(Addison{W)-6 b(esley)g(,)25 b(Reading,)h(MA,)g(2nd)f(ed.,)i(1981.)365 3306 y(10.)43 b(B.A.)24 b(LaMacc)n(hia)i(and)d(A.M.)i(Odlyzk)n(o,)e Fc(Solving)j(lar)l(ge)h(sp)l(arse)g(systems)h(over)f(\014nite)f (\014elds)p Fw(,)505 3397 y(Adv)l(ances)31 b(in)h(Cryptology)-6 b(,)32 b(CR)-6 b(YPTO)31 b('90)i(\(A.J.)g(Menezes)g(and)e(S.A.)h(V)-6 b(anstone,)32 b(eds.\),)505 3488 y(Lecture)26 b(Notes)f(in)h(Computer)f (Science,)h(v)n(ol.)g(537,)h(Springer{V)-6 b(erlag,)27 b(pp.)e(109{133.)365 3581 y(11.)43 b(A.K.)26 b(Lenstra,)i(H.W.)f (Lenstra,)g(Jr.,)h(M.S.)f(Manasse,)h(and)f(J.M.)h(P)n(ollard,)g Fc(The)h(factoriza-)505 3672 y(tion)23 b(of)g(the)h(ninth)g(F)-6 b(ermat)24 b(numb)l(er)p Fw(,)f(Mathematics)e(of)h(Computation)e Fv(61)i Fw(\(1993\),)g(no.)k(203,)505 3763 y(319{349.)365 3856 y(12.)43 b(A.M.)32 b(Odlyzk)n(o,)e Fc(Discr)l(ete)k(lo)l(garithms) g(in)e(\014nite)h(\014elds)g(and)g(their)h(crypto)l(gr)l(aphic)h (signif-)505 3947 y(ic)l(anc)l(e)p Fw(,)k(Adv)l(ances)e(in)g (Cryptology:)i(Pro)r(ceedings)g(of)g(EUR)n(OCR)-6 b(YPT)36 b(84)i(\(New)g(Y)-6 b(ork\))505 4038 y(\(T.)26 b(Beth,)i(N.)d(Cot,)k (and)f(I.)d(Ingemarsson,)k(eds.\),)g(Lecture)f(Notes)h(in)f(Computer)g (Science,)505 4130 y(v)n(ol.)e(209,)h(Springer{V)-6 b(erlag,)27 b(pp.)d(224{314.)365 4222 y(13.)43 b(Carl)j(P)n(omerance,)f Fc(The)g(quadr)l(atic)i(sieve)f(factoring)g(algorithm)p Fw(,)f(Adv)l(ances)f(in)h(Cryp-)505 4313 y(tology)-6 b(,)49 b(Pro)r(ceedings)h(of)e(EUR)n(OCR)-6 b(YPT)47 b(84)i(\(New)f(Y)-6 b(ork\))47 b(\(T.)26 b(Beth,)49 b(N.)24 b(Cot,)49 b(and)505 4405 y(I.)25 b(Ingemarsson,)41 b(eds.\),)g(Lecture) g(Notes)f(in)h(Computer)f(Science,)h(v)n(ol.)g(209,)h(Springer{)505 4496 y(V)-6 b(erlag,)26 b(pp.)f(169{182.)365 4588 y(14.)43 b(Rob)r(ert)25 b(D.)38 b(Silv)n(erman,)g Fc(The)h(multiple)g(p)l (olynomial)g(quadr)l(atic)i(sieve)p Fw(,)e(Mathematics)g(of)505 4680 y(Computation)25 b Fv(48)h Fw(\(1987\),)i(no.)d(177,)i(329{339.) 365 4772 y(15.)43 b(Douglas)26 b(H.)35 b(Wiedemann,)f Fc(Solving)i(sp)l(arse)i(line)l(ar)e(e)l(quations)h(over)g(\014nite)g (\014elds)p Fw(,)f(IEEE)505 4863 y(T)-6 b(rans.)26 b(Inform.)g(Theory)f Fv(32)h Fw(\(1986\),)i(no.)d(1,)h(54{62.)p eop %%Page: 16 16 16 15 bop 681 387 a Fw(This)26 b(article)h(w)n(as)g(pro)r(cessed)f (using)g(the)f(L)1936 377 y Fa(a)1971 387 y Fw(T)2013 410 y(E)2056 387 y(X)h(macro)f(pac)n(k)l(age)h(with)g(LLNCS)f(st)n(yle) p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF