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b(og)f(som)g(i)g(praksis)g(ikk)o(e)h (har)f(virk)o(et.)p 3782 3169 V 2172 3172 1613 4 v eop %%Page: 1 3 1 2 bop 204 841 a FE(F)-5 b(or)l(ord)204 1273 y FC(Denne)30 b(rapport)f(er)i(resultatet)g(af)g(v)n(ores)g(f\370rste)f(projekt,)g (P0,)h(p\345)g(Aalbor)o(g)e(Uni)n(v)o(ersitet.)h(S\345)h(l\346seren)h (m\345)e(ha)n(v)o(e)h(den)204 1373 y(undsk)o(yldt.)328 1472 y(V)-5 b(i)29 b(har)e(i)h(projektet)e(lagt)i(stor)g(v\346gt)f (p\345)h(at)g(samle)g(os)g(en)f(s\345)i(stor)e(f)o(aglig)g(viden)g (inden)g(for)f(projektperioden)e(som)204 1572 y(o)o(v)o(erho)o(v)o (edet)13 b(muligt,)j(og)g(gennem)g(rapporten)f(fors\370ge)g(at)j (dokumentere)c(den)i(viden)g(vi)h(har)g(f\345et)g(og)f(fors\370ge)g(at) h(besv)n(are)204 1671 y(de)j(sp\370r)o(gsm\345l)e(som)j(vi)f(er)g (st\370dt)g(p\345.)328 1771 y(Rapporten)f(m\345)h(frit)g(distrib)n (ueres)g(i)h(en)f(hvilk)o(en)f(som)h(helst)h(form.)328 1871 y Fw(En)f(s\346rlig)i(tak)e(til)h(vor)m(es)f(vejleder)h(Hans)f 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(.)g(.)g(.)h(.)f(.)127 b(5)519 1937 y(2.1.2)98 b(Kryptogra\002)46 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.) g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)g(.)h(.)f(.)127 b(5)328 2037 y(2.2)86 b(Pri)n(v)n(ate)20 b(n\370gler)49 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)127 b(5)519 2136 y(2.2.1)98 b(Problemerne)18 b(med)i(symmetrisk)o(e)f (kryptosystemer)51 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)127 b(6)328 2236 y(2.3)86 b(Of)n(fentlige)19 b(n\370gler)66 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)127 b(6)519 2336 y(2.3.1)98 b(Asymmetrisk)o(e)19 b(kryptosystemer)39 b(.)j(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f 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b(20)204 5524 y(A)64 b(Br)o(e)o(v)20 b(fra)g(\223Micr)o(osoft\224)2620 b(22)1970 6016 y FC(2)p eop %%Page: 3 5 3 4 bop 3399 -143 a FA(INDHOLD)p 204 -110 3573 4 v 204 177 a Fx(B)69 b(Lidt)21 b(talteori)2933 b(23)328 277 y FC(B.1)73 b(Di)n(vision)20 b(og)g(rester)46 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)86 b(23)328 376 y(B.2)73 b(Modulo)63 b(.)41 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.) h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)86 b(23)328 476 y(B.3)73 b(Primtal)21 b(.)41 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)86 b(24)328 576 y(B.4)73 b(Den)20 b(kinesisk)o(e)g(rests\346tning)i (.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.) f(.)86 b(24)328 675 y(B.5)73 b(Reducering)19 b(med)g Fv(\036)p Ft(\()p Fv(n)p Ft(\))34 b FC(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.) h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)86 b(24)1970 6016 y(3)p eop %%Page: 4 6 4 5 bop 204 800 a Fs(Kapitel)43 b(1)204 1215 y FE(Indledning)204 1658 y FC(Kryptering)13 b(er)j(de)g(seneste)g(\345r)g(ble)n(v)o(et)f (mere)g(og)g(mere)h(hv)o(erdag)d(for)i(computerbrugere)d(v)o(erden)i(o) o(v)o(er)-5 b(.)14 b(I)i(takt)g(med)g(elektro-)204 1758 y(nisk)o(e)j(o)o(v)o(erf\370relser)d(af)k(f.eks.)e(bank)o(oplysninger)d (er)20 b(beho)o(v)o(et)d(for)h(sikk)o(er)h(kryptering)e(ble)n(v)o(et)h (st\370rre)h(og)g(st\370rre)f(\226)i(forestil)204 1858 y(dig)25 b(at)h(en)f(crack)o(er)f(kan)h(o)o(v)o(erv\345ge)d(og)j(endda) f(\346ndre)h(dine)g(bank)o(o)o(v)o(erf\370relser)-5 b(.)21 b(Dette)26 b(beho)o(v)d(stiller)j(stadig)f(strengere)204 1957 y(kra)n(v)19 b(til)i(de)g(systemudviklere)c(som)k(arbejder)d(med)i (o)o(v)o(erf\370relser)d(af)k(fortroligt)d(data.)328 2057 y(Men)g(f)o(aktisk)g(er)h(kryptering)c(langt)j(mere)g(end)g(blot)g (at)g(k)o(\370re)g(en)g(eller)g(\003ere)h(funktioner)c(i)k(sit)h (foretrukk)o(ede)15 b(program-)204 2156 y(meringssprog.)h(Bag)j (krypteringen)d(ligger)i(en)h(k)o(ompleks)f(og)g(sp\346ndende)g(teori.) g(T)-6 b(eorien)18 b(g\345r)g(helt)h(tilbage)g(til)h(oldtiden)204 2256 y(hv)n(or)c(man)h(s\345)h(de)f(f\370rste)g(simple)g (krypteringsformer)-5 b(.)13 b(Jo)k(st\370rre)g(re)o(gnekraft)e(man)i (senere)f(har)h(f\345et)g(til)h(r\345dighed,)d(jo)j(mere)204 2356 y(k)o(omplekse)g(algoritmer)h(ligger)h(der)f(bag)h(de)g(metoder)f (der)h(bruges.)204 2636 y Fr(1.1)119 b(Pr)n(oblemf)m(ormulering)204 2821 y FC(Rapporten)18 b(vil)j(bli)n(v)o(e)e(baseret)h(p\345)g (f\370lgende)f(ho)o(v)o(edsp\370r)o(gsm\345l:)328 2921 y Fw(Hvor)m(dan)g(fung)o(er)m(er)h(kryptering)g(o)o(g)f(hvilk)o(en)h (betydning)e(har)i(det)h(haft)f(o)o(g)g(vil)g(det)h(f\345)f(for)h (samfundet?)328 3021 y FC(Herunder)e(vil)h(vi)g(k)o(oncentrere)e(os)j (om)f(f\370lgende)e(sp\370r)o(gsm\345l:)308 3203 y(1.)40 b(Hv)n(ad)20 b(er)g(kryptering,)d(hv)n(ad)j(er)g(den)f(historisk)o(e)h (baggrund)e(og)h(hvilk)o(en)g(rolle)h(har)g(det)g(spillet)h(gennem)e (tiden?)411 3336 y(V)-5 b(i)22 b(v)o(ed)d(at)i(kryptering)d(spillede)j (en)f(v\346sentlig)h(rolle)f(under)f(2.)i(v)o(erdenskrig)c(og)k(vi)f (vil)h(fors\370ge)e(at)i(belyse)f(denne)411 3436 y(rolle.)26 b(Desuden)f(vil)h(vi)g(kigge)f(p\345)g(hv)n(ad)g(det)h(betyder)e(at)j (en)f(virksomhed)d(som)j(Tvind)f(\(n\346sten\))g(kunne)f(skjule)411 3535 y(be)n(vismateriale)19 b(for)h(politiet.)308 3701 y(2.)40 b(Hvilk)o(e)20 b(klassisk)o(e)h(krypteringsmetoder)16 b(\002ndes)k(der)g(og)g(hv)n(ordan)e(virk)o(er)h(disse?)411 3834 y(Af)31 b(klassisk)o(e)h(krypteringsmetoder)26 b(mener)k(vi)h(ikk) o(e)g(dem)f(man)g(brugte)g(for)g(lang)g(tid)h(siden.)g(V)-5 b(i)31 b(lader)g(v)n(ores)411 3934 y(tidslinje)d(g\345)g(frem)f(til)i (i)g(dag,)e(men)g(vi)h(kalder)f(de)h(krypteringsmetoder)23 b(som)28 b(udelukk)o(ende)d(an)m(v)o(ender)g(pri)n(v)n(ate)411 4034 y(n\370gler)19 b(for)h(klassisk)o(e)g(fordi)g(ideen)f(i)i(dem)f (har)g(v\346ret)g(k)o(endt)f(i)i(\345rtusinder)-5 b(.)308 4200 y(3.)40 b(Hvilk)o(e)20 b(moderne)f(krypteringsmetoder)c(\002ndes) 21 b(der)e(og)h(hv)n(ordan)e(virk)o(er)h(disse?)411 4332 y(Her)j(har)f(vi)h(is\346r)g(t\346nkt)g(os)g(at)g(k)o(omme)e(ind)h (p\345)h(RSA)g(som)g(er)f(af)h(de)f(mest)h(ben)o(yttede)e(moderne)f (krypteringsme-)411 4432 y(toder)h(\226)g(det)g(bli)n(v)o(er)f(for)h (omf)o(attende)e(at)j(kigge)e(p\345)h(andre)g(systemer)f(\(som)h (f.eks.)g(El)g(Gamal\).)308 4598 y(4.)40 b(Hvilk)o(e)20 b(teknisk)o(e)g(og)g(samfundsm\346ssige)f(o)o(v)o(erv)o(ejelser)e(er)k (der)e(i)i(forbindelse)e(med)g(kryptering)f(i)j(fremtiden?)411 4731 y(Her)e(vil)g(vi)g(fors\370ge)e(at)i(beskri)n(v)o(e)e(de)i (interessek)o(on\003ikter)e(der)h(kan)g(v\346re)h(i)g(forbindelse)e (med)h(krypteringslo)o(vgi)n(v-)411 4831 y(ning.)h(V)-5 b(i)21 b(t\346nk)o(er)f(b\345de)g(politisk)g(og)f(p\345)h(den)g(v)n (oksende)f(lobbyisme.)1970 6016 y(4)p eop %%Page: 5 7 5 6 bop 204 800 a Fs(Kapitel)43 b(2)204 1215 y FE(Intr)l(oduktion)204 1696 y Fr(2.1)119 b(De\002nitioner)204 1882 y FC(I)20 b(det)g(f\370lgende)f(vil)h(vi)g(beskri)n(v)o(e)f(nogle)g(generelle)g (de\002nitioner)m(,)f(som)j(vil)f(bli)n(v)o(e)f(brugt)g(me)o(get)h (igennem)f(rapporten.)204 2120 y Fq(2.1.1)99 b(Kryptologi)204 2275 y FC(Ordet)30 b Fw(kryptolo)o(gi)f FC(stammer)i(fra)f(latin,)g(og) g(er)h(sammensat)f(af)h(2)f(latinsk)o(e)h(ord;)f(krypto)e(\(at)j (skjule\))f(og)g(logi)h(\(l\346ren)204 2375 y(om\).)22 b(Det)i(d\346kk)o(er)e(o)o(v)o(er)g(tre)h(forsk)o(ellige)f(be)o(greber) f([Landrock)g(og)h(Nissen,)i(1997;)e(Bauer,)h(1997],)e Fw(ste)m(gano)o(gr)o(a\002)f FC(som)204 2474 y(drejer)i(sig)h(om)g (helt)g(at)g(skjule)g(at)g(der)g(o)o(v)o(erho)o(v)o(edet)c(bli)n(v)o (er)i(sendt)i(en)g(meddelelse,)f Fw(krypto)o(gr)o(a\002)f FC(som)i(drejer)f(sig)h(om)g(at)204 2574 y(forv)n(andle)f(en)i (meddelelse)g(til)h(noget)f(der)g(er)g(uforst\345eligt)f(for)h(alle)h (andre)e(end)h(de)h(tilsigtede)f(modtagere,)e(og)i(endelig)204 2674 y Fw(kryptoanalyse)18 b FC(som)i(handler)f(om)h(at)h(bryde)e (krypteringen)e(uden)i(at)h(v\346re)h(en)f(tilsigtet)h(modtager)-5 b(.)328 2773 y(Selv)o(e)17 b(ideen)f(i)i(at)g(kryptere)d(er)i(at)g(en)g (part)g(\370nsk)o(er)f(at)h(holde)f(en)h(eller)g(anden)f(form)g(for)g (information)f(hemmelig)h(under)204 2873 y(transporten)i(af)i (informationen.)328 2973 y(Hvis)e(en)f(meddelelse)f(ble)n(v)h (opsnappet)e(i)j(gamle)e(dage,)h(kunne)e(man)i(ofte)g(\002nde)g(ud)g (af)g(om)g(informationen)d(v)n(ar)j(f)o(aldet)204 3072 y(i)k(de)f(fork)o(erte)e(h\346nder)i(\226)g(b)n(udbringeren)d(dukk)o (ede)h(simpelthen)h(ikk)o(e)h(op)g(i)g(hel)h(tilstand.)328 3172 y(I)h(dag)g(er)g(det)g(lidt)g(anderledes.)e(T)m(ransportv)o(ejene) f(er)j(ogs\345)f(usikre)h(i)g(og)g(med)f(at)i(de)e(er)h(\345bne)g (\(det)f(er)h(f.eks.)f(sv\346rt)i(at)204 3271 y(k)o(ontrollere)d(hv)n (or)h(et)i(radiosignal)d(ha)n(vner)h(henne\),)g(og)g(ydermere)f(er)i (det)h(me)o(get)e(sv\346rere)h(at)g(\002nde)g(ud)g(af)g(om)g(meddel-) 204 3371 y(elsen)h(er)g(ble)n(v)o(et)f(opsnappet.)f(Desuden)h(er)g(der) h(i)g(dag)g(adgang)e(til)i(store)g(m\346ngder)e(re)o(gnekraft)g(i)i (form)f(af)h(f.eks.)f(pc'ere)204 3471 y(til)f(at)f(fors\370ge)f(at)i (bryde)e(krypteringen.)328 3570 y(Derfor)g(er)i(der)e(i)i(dag)f(brug)f (for)g(me)o(get)h(ef)n(fekti)n(v)o(e)e(krypteringsformer)-5 b(.)204 3808 y Fq(2.1.2)99 b(Kryptogra\002)204 3964 y FC(Et)22 b Fw(kryptosystem)h FC([Landrock)c(og)j(Nissen,)h(1997,)e(s.)i (12])e(best\345r)h(af)h(en)f(metode)f(til)i(at)g(foretage)e Fw(kryptering)h FC(der)g(g\370r)f(en)204 4063 y(gi)n(v)o(en)d (klartekst)g Fv(m)i FC(ul\346selig,)f(samt)g(en)g(metode)f(til)i Fw(dekryptering)e FC(der)h(o)o(v)o(ers\346tter)f(kryptoteksten)e Fv(c)k FC(til)g(den)e(oprindelig)204 4163 y(besk)o(ed)g Fv(m)i FC(igen.)e(Krypteringen)f Fv(E)5 b Ft(\()p Fv(m)p Ft(\))20 b FC(og)e(dekrypteringen)e Fv(D)r Ft(\()p Fv(c)p Ft(\))k FC(kan)f(e)n(vt.)f(v\346re)h(sammenf)o(aldende)e(eller)i (hinandens)204 4263 y(in)m(v)o(erse)g(funktioner)-5 b(.)328 4362 y(Oftest)23 b(skal)f(der)f(v)o(ed)g(kryptering)e(og)j (dekryptering)c(bruges)j(noget)g(ekstra)g(information,)e(en)j Fw(n\370)o(gle)f Fv(K)6 b FC(.)22 b(S\345)g Fv(E)3598 4374 y Fu(k)3639 4362 y Ft(\()p Fv(m)p Ft(\))204 4462 y FC(bete)o(gner)c Fv(m)j FC(krypteret)d(med)i(n\370glen)f Fv(K)26 b FC(og)20 b Fv(D)1604 4474 y Fu(k)1644 4462 y Ft(\()p Fv(c)p Ft(\))i FC(bete)o(gner)c Fv(c)j FC(dekrypteret)c (samme)j(n\370gle.)204 4742 y Fr(2.2)119 b(Pri)o(v)o(ate)29 b(n\370gler)204 4928 y FC(T)m(il)21 b(et)g(kryptosystem)d(med)i(pri)n (v)n(ate)f(n\370gler)g([Hank)o(erson)g(et)h(al.,)h(2000,)e(s.)i(230],)d (den)i(simpleste)h(og)f(klassisk)o(e)h(form)e(for)204 5027 y(kryptering,)d(skal)k(der)e(bruges)g(\351n)i(n\370gle.)e (N\370glen)g(man)g(bruger)g(til)i(at)f(kryptere)f(med,)g(er)h(den)g (samme)g(som)g(n\370glen)f(til)i(at)204 5127 y(dekryptere)e(med.)h (Systemet)h(kaldes)g(af)g(denne)f(grund)g(ogs\345)h(for)f(et)i (symmetrisk)e(kryptosystem.)328 5226 y(Ideen)28 b(i)g(symmetrisk)g (kryptering)d(er)k(me)o(get)e(ligetil)i(fordi)e(dekryptering)e(fore)o (g\345r)h(v)o(ed)h(at)i(g\370re)e(det)i(modsatte)e(af)204 5326 y(krypteringen,)16 b(og)j(eksempler)f(g\345r)h(helt)h(fra)f(de)h (simple)f(krypteringsmetoder)c(som)20 b(bogsta)n(v)o(erstatning)d(til)j (DES)g(som)f(er)204 5426 y(en)h(moderne)e(krypteringsstandard)e(der)k (stadig)g(bruges.)328 5525 y(De)h(pri)n(v)n(ate)e(n\370gler)g(skal)h (naturligvis)f(holdes)h(hemmelige,)e(sikk)o(erheden)g(i)j(systemet)g (hviler)e(p\345)h(dette.)1970 6016 y(5)p eop %%Page: 6 8 6 7 bop 2769 -143 a FA(2.3.)40 b(OFFENTLIGE)19 b(N\370GLER)p 204 -110 3573 4 v 204 177 a Fq(2.2.1)99 b(Pr)n(oblemer)o(ne)26 b(med)f(symmetrisk)o(e)g(kryptosystemer)204 333 y FC(Der)20 b(er)g(imidlertid)g(nogle)f(problemer)f([Bauer,)h(1997,)g(s.)i(171]:) 308 515 y(1.)40 b(N\370glerne)31 b(skal)i(o)o(v)o(erbringes)d(ad)j(en)f (k)o(ommunikationskanal)c(der)33 b(er)f(mere)g(sikk)o(er)h(end)f(den)g (der)g(bruges)f(til)411 615 y(almindelig)25 b(k)o(ommunikation)d(\226)j (hvilk)o(et)g(kan)g(v\346re)g(umuligt,)f(f.eks.)h(hvis)h(ens)f(k)o (ommunikationsmiddel)c(er)26 b(et)411 715 y(traditionelt)19 b(computernetv\346rk.)308 881 y(2.)40 b(Hvis)29 b(mange)e(personer)g (\370nsk)o(er)g(adgang)g(til)i(k)o(ommunikation)c(med)j(hinanden,)e (kan)h(antallet)i(af)f(n\370gler)f(bli)n(v)o(e)411 980 y(me)o(get)22 b(stort.)g(En)h(n\370gle)e(skal)i(jo)g(holdes)e(pri)n(v)n (at)h(mellem)g(to)h(parter)m(,)e(s\345)i(med)f(et)h(netv\346rk)e(med)h Fv(n)h FC(parter)f(der)g(alle)411 1080 y(\370nsk)o(er)k(at)h(kunne)e(k) o(ommunik)o(ere)f(med)i(hinanden,)e(bli)n(v)o(er)i(antallet)g(af)h (n\370gler)2805 1013 y Fp(\000)2843 1043 y Fu(n)2847 1109 y Fo(2)2884 1013 y Fp(\001)2922 1080 y FC(.)g(F)o(or)f(blot)h (10.000)d(brugere)411 1180 y(\(som)c(f.eks.)g(p\345)g(et)h(uni)n(v)o (ersitet\))d(drejer)i(det)g(om)g(i)g(alt)h(ca.)f(50)g(mill.)h (n\370gler)-5 b(.)308 1346 y(3.)40 b(Og)19 b(endelig)e(kan)h(man)g(ikk) o(e)g(bruge)f(systemet)i(til)g(at)g(signere)f(meddelelser)f(med)h (\(alts\345)g(som)h(en)f(erstatning)g(for)f(en)411 1445 y(underskrift\).)h(S\345)j(l\346nge)f(der)f(er)i(to)f(parter)f(som)i (besidder)e(n\370glen,)f(kan)i(be)o(gge)e(ha)n(v)o(e)i(krypteret)f (den.)204 1725 y Fr(2.3)119 b(Offentlige)30 b(n\370gler)204 1911 y FC(Disse)21 b(sv)n(agheder)d(v)o(ed)i(symmetrisk)f(kryptering)f (har)h(i)i(moderne)d(tid)j(skabt)f(beho)o(v)o(et)e(for)h(andre)h (l\370sninger)-5 b(.)204 2149 y Fq(2.3.1)99 b(Asymmetrisk)o(e)25 b(kryptosystemer)204 2304 y FC(Med)f(et)g Fw(asymmetrisk)i FC(kryptosystem)c(er)i(der)g(\351n)g(n\370gle)f(til)i(at)g(kryptere)d (med,)h(den)h Fw(of)o(fentlig)o(e)f(n\370)o(gle)p FC(,)g(og)g(\351n)h (n\370gle)f(til)204 2404 y(at)k(dekryptere)c(med)j(\226)g (sidstn\346vnte)g(skal)h(stadig)f(holdes)g(hemmelig.)e(Den)i(n\370gle)g (der)g(ben)o(yttes)f(til)i(at)g(kryptere,)d(kan)204 2503 y(derimod)c(of)n(fentligg\370res)f(og)j(bruges)f(af)h(alle)h(til)g(at)f (sende)g(en)g(meddelelse)f(der)h(ikk)o(e)g(kan)f(l\346ses)j(af)e(andre) f(modtageren)204 2603 y(som)f(jo)g(sidder)g(med)g(den)f(eneste)i (n\370gle)e(der)h(kan)f(\204l\345se\223)i(besk)o(eden)e(op)g(igen.)328 2703 y(Hvis)i Fv(K)577 2715 y Fu(p)636 2703 y FC(bete)o(gner)d(den)i (pri)n(v)n(ate)f(n\370gle)g(og)h Fv(K)1709 2715 y Fu(o)1766 2703 y FC(den)g(of)n(fentlige,)e(virk)o(er)h(systemet)h(alts\345)h(v)o (ed)1638 2885 y Fv(D)1707 2897 y Fu(K)1763 2905 y Fn(p)1802 2885 y Ft(\()p Fv(E)1895 2897 y Fu(K)1951 2905 y Fn(o)1989 2885 y Ft(\()p Fv(m)p Ft(\)\))j(=)e Fv(m)328 3068 y FC(Eftersom)k (kryptosystemet)e(er)i(of)n(fentligt)f(k)o(endt,)h(skal)g(det)h (v\346re)f(en)h(s\345)g(bere)o(gnsm\346ssig)e(tung)g(opga)n(v)o(e)g(at) i(bryde)204 3168 y(systemet)22 b(vha.)e(en)i(udt\370mmende)d (s\370gning,)h(at)i(det)g(ikk)o(e)f(kan)g(lade)h(sig)g(g\370re)f(inden) f(for)h(en)h(o)o(v)o(erskuelig)d(fremtid.)i(Man)204 3267 y(kan)k(alts\345)h(opstille)f(nogle)f(grundl\346ggende)e(kra)n(v)j(til) h(et)g(kryptosystem)d(baseret)i(p\345)g(of)n(fentlige)f(n\370gler)g ([Landrock)f(og)204 3367 y(Nissen,)d(1997,)f(s.)i(58]:)308 3550 y(1.)40 b(K)m(onstruktionen)18 b(af)i(den)g(of)n(fentlige)e(og)i (den)g(pri)n(v)n(ate)f(n\370gle)g(skal)h(v\346re)h(simpel.)308 3716 y(2.)40 b(Det)21 b(skal)f(v\346re)h(simpelt)f(at)h(foretage)d (krypteringen)f(vha.)j(den)f(of)n(fentlige)g(n\370gle)g(og)h (klarteksten.)308 3882 y(3.)40 b(Retablering)19 b(af)i(klarteksten)e (vha.)g(den)h(hemmelige)f(n\370gle)g(af)h(kryptoteksten)e(skal)i (v\346re)g(simpel.)308 4048 y(4.)40 b(At)22 b(bestemme)e(den)h (hemmelige)e(n\370gle)h(ud)h(fra)f(den)h(of)n(fentlige)e(n\370gle)h (skal)i(v\346re)e(et)i(bere)o(gningsm\346ssigt)d(sv\346rt)411 4147 y(problem.)308 4313 y(5.)40 b(Det)22 b(skal)f(v\346re)f(sv\346rt)h (at)h(retablere)d(klarteksten)h(ud)g(fra)h(et)g(k)o(endskab)e(til)i (kun)f(den)g(of)n(fentlige)f(n\370gle)h(og)g(krypto-)411 4413 y(teksten.)328 4596 y(Et)26 b(kryptosystem)e(der)h(besidder)f (disse)i(e)o(genskaber)m(,)d(l\370ser)i(de)h(to)f(f\370rste)g (problemer)f(med)h(symmetrisk)o(e)f(krypto-)204 4695 y(systemer:)k(n\345r)g(de)g(to)h(n\370gler)e(er)h(genereret,)f(kan)h (den)f(ene)h(distrib)n(ueres)g(o)o(v)o(er)f(\345bne)h(kanaler)f(uden)g (risik)o(o)i(for)e(k)o(om-)204 4795 y(promittering)21 b(af)i(systemet.)g(Og)h(der)f(er)g(ogs\345)g(kun)f(brug)g(for)h(\351t)h (n\370glepar)d(pr)-5 b(.)23 b(person,)f(alts\345)i Ft(2)p Fv(n)g FC(n\370gler)e(i)h(alt)h(i)g(forhold)204 4895 y(til)298 4827 y Fp(\000)336 4858 y Fu(n)340 4923 y Fo(2)377 4827 y Fp(\001)445 4895 y Ft(=)29 b Fv(n)21 b Fm(\001)g Ft(\()p Fv(n)h Fm(\000)e Ft(1\))p Fv(=)p Ft(2)k FC(v)o(ed)f(et)h (symmetrisk)f(system.)h(De)g(of)n(fentlige)f(n\370gler)f(kan)h(opbe)n (v)n(ares)f(i)j(et)f(f\346lles)h(katalog)3745 4864 y Fl(1)204 4994 y FC(som)c(man)f(s\345)i(kan)e(sl\345)i(en)f(gi)n(v)o (ens)f(person)f(n\370gle)h(op)h(i)g(n\345r)g(man)f(har)h(beho)o(v)e (for)h(at)i(sende)e(en)h(hemmelig)e(meddelelse)h(til)204 5094 y(v)o(edk)o(ommende,)c(lidt)21 b(p\345)f(samme)g(m\345de)f(som)h (en)h(telefonbog.)p 204 5165 1430 4 v 294 5220 a Fk(1)323 5244 y FD(Med)15 b(et)h(f\346lles)g(opslagskatalog)i(er)d(der)g(dog)g (naturligvis)i(brug)e(for)g(en)g(eller)h(anden)g(form)f(for)g(system)f (eller)j(myndighed)f(til)g(at)f(sikre)g(at)h(de)f(enk)o(elte)204 5323 y(personers)j(of)n(fentlige)i(n\370gler)e(f)o(aktisk)h (tilh\370rer)g(de)e(gi)n(vne)i(personer)l(.)f(Hvis)f(det)h(lykk)o(es)g (en)f(ondsindet)i(person)f(at)f(l\346gge)i(en)e(f)o(alsk)h(of)n (fentlig)i(n\370gle)e(ind)204 5402 y(i)f(en)g(andens)h(na)o(vn,)f(vil)h (det)g(jo)f(v\346re)h(muligt)g(for)f(v)o(edk)o(ommende)i(at)e(l\346se)h (de)f(krypterede)j(meddelelser)g(indtil)e(f)o(alskneriet)j(er)c (opdaget.)1970 6016 y FC(6)p eop %%Page: 7 9 7 8 bop 2769 -143 a FA(2.3.)40 b(OFFENTLIGE)19 b(N\370GLER)p 204 -110 3573 4 v 204 177 a Fq(2.3.2)99 b(Signer)n(et)26 b(kryptering)204 333 y FC(Hvis)20 b(kryptosystemet)e(ydermere)h(har)g (e)o(genskaben)1638 515 y Fv(E)1699 527 y Fu(K)1755 535 y Fn(o)1793 515 y Ft(\()p Fv(D)1894 527 y Fu(K)1950 535 y Fn(p)1989 515 y Ft(\()p Fv(m)p Ft(\)\))24 b(=)e Fv(m)1276 b FC(\(2.1\))204 698 y(kan)27 b(det)h(ogs\345)g(ben)o(yttes)f(til)h (signering)f([Bauer,)g(1997,)f(s.)j(173])d(hvilk)o(et)i(l\370ser)g(den) f(sidste)i(hage)e(v)o(ed)g(et)h(symmetrisk)204 798 y(system.)20 b(Kryptering)e(med)i(signering)e(virk)o(er)i(som)g(f\370lger:)328 897 y(Hvis)h Fv(A)g FC(\370nsk)o(er)e(at)i(sende)f(en)g(meddelelse)f (til)i Fv(B)t FC(,)g(krypterer)d Fv(A)j FC(f\370rst)f(klarteksten)f (med)g(sin)i(pri)n(v)n(ate)e(n\370gle:)1708 1080 y Fv(c)1744 1092 y Fu(A)1821 1080 y Ft(=)k Fv(D)1978 1092 y Fu(K)2034 1100 y Fn(p;A)2135 1080 y Ft(\()p Fv(m)p Ft(\))204 1263 y FC(Det)e(kan)g(virk)o(e)g(lidt)g(m\346rk)o(eligt)g(at)g(man)g(bruger) e(dekrypteringsfunktionen)c(p\345)21 b(klartekst,)g(men)g(det)g (f\370lger)f(af)h(kriteriet)204 1362 y(for)e(signeringen)e(gi)n(v)o(et) i(v)o(ed)g(\(2.1\))n(.)h(Derefter)f(tilf\370jer)g Fv(A)i FC(sit)g(na)n(vn,)d(\204)p Fv(A)p FC(\223,)i(og)g(krypterer)d(\204)p Fv(A)p FC(\223)j(og)g Fv(c)3076 1374 y Fu(A)3150 1362 y FC(med)f Fv(B)t FC(')-5 b(s)21 b(of)n(fentlige)204 1462 y(n\370gle)e(som)h Fv(A)h FC(har)f(sl\345et)h(op)f(til)h (lejligheden:)1590 1644 y Fv(c)1626 1656 y Fu(B)s(A)1756 1644 y Ft(=)i Fv(E)1905 1656 y Fu(K)1961 1664 y Fn(o;B)2063 1644 y Ft(\()p FC(\204)p Fv(A)p FC(\223)p Fv(;)14 b(c)2304 1656 y Fu(A)2358 1644 y Ft(\))204 1827 y FC(Nu)20 b(kan)g Fv(c)503 1839 y Fu(B)s(A)630 1827 y FC(sendes)g(o)o(v)o(er)f(en)h (\345ben)g(kanal)f(til)i Fv(B)t FC(.)328 1927 y(Herefter)f(dekrypterer) d Fv(B)25 b FC(kryptoteksten)18 b(med)h(sin)i(pri)n(v)n(ate)e (n\370gle:)1554 2109 y Fv(D)1623 2121 y Fu(K)1679 2129 y Fn(p;B)1781 2109 y Ft(\()p Fv(c)1849 2121 y Fu(B)s(A)1956 2109 y Ft(\))24 b(=)e(\()p FC(\204)q Fv(A)p FC(\223)p Fv(;)14 b(c)2341 2121 y Fu(A)2395 2109 y Ft(\))204 2292 y Fv(B)25 b FC(bem\346rk)o(er)19 b(ud)g(fra)h(\204)p Fv(A)p FC(\223)h(at)g(afsenderen)d(er)i Fv(A)p FC(,)h(og)f(kan)g(bruge) e Fv(A)p FC(')-5 b(s)22 b(of)n(fentlige)c(n\370gle)h(til)i(at)g (genoprette)d(klarteksten:)1713 2475 y Fv(E)1774 2487 y Fu(K)1830 2495 y Fn(o;A)1929 2475 y Ft(\()p Fv(c)1997 2487 y Fu(A)2052 2475 y Ft(\))23 b(=)g Fv(m)204 2657 y FC(Hvis)k(det)f(sidste)i(skridt)e(gi)n(v)o(er)f(en)i(meningsfuld)d (klartekst)i Fv(m)p FC(,)h(kan)f Fv(B)32 b FC(v\346re)26 b(sikk)o(er)g(p\345)h(at)g Fv(A)h FC(f)o(aktisk)e Fw(er)j FC(afsenderen)204 2757 y(fordi)19 b(kun)g Fv(A)p FC(')-5 b(s)22 b(pri)n(v)n(ate)d(n\370gle)g(kan)h(ha)n(v)o(e)f(v\346ret)h (brugt)f(til)i(at)g(kryptere)e(klarteksten)g(med.)1970 6016 y(7)p eop %%Page: 8 10 8 9 bop 204 800 a Fs(Kapitel)43 b(3)204 1215 y FE(Historisk)52 b(set)204 1646 y FC(De)17 b(f\370rste)g(k)o(endte)f(eksempler)g(p\345)h (kryptering)e(stammer)i(helt)g(tilbage)g(fra)g(500)f(f\370r)h(v)n(or)f (tidsre)o(gning)f(hv)n(or)i(det)g(ble)n(v)f(brugt)204 1746 y(af)k(den)g(spartansk)o(e)f(re)o(gering)f(til)j(at)g(sende)f (hemmelige)e(besk)o(eder)h(til)i(deres)f(generaler)f([Beutelspacher,)f (1994,)h(s.)i(3].)328 1846 y(Et)j(andet)e(eksempel)h(fra)f(oldtiden)g (er)h(C\346sars)i(rotering)c([Hank)o(erson)h(et)h(al.,)g(2000,)f(s.)i (231],)d(hv)n(or)h(en)h(v)n(ariation)f(af)204 1945 y(denne)d(stadig)h (bruges)f(i)i(dag.)204 2226 y Fr(3.1)119 b(C\346sars)29 b(r)n(otering)204 2411 y FC(C\346sars)20 b(rotering)e(v)n(ar)g(det)i (f\370rste)e(k)o(endte)g(eksempel)h(p\345)g(bogsta)n(v)o (erstatningskryptering.)13 b(Det)20 b(virk)o(er)e(v)o(ed)g(at)i(man)e (gi)n(v)o(er)204 2511 y(alle)28 b(bogsta)n(v)o(erne)d(i)k(alf)o(abetet) e(en)h(v\346rdi,)f(og)g(derefter)g(fore)o(g\345r)e(krypteringen)g(v)o (ed)i(at)h(l\346gge)g Fv(K)33 b FC(til)c(det)f(oprindelige)204 2610 y(bogsta)n(v)e Fv(m)i FC(fra)g(kildeteksten.)e(Hvis)i(tallet)g (s\345)g(bli)n(v)o(er)f(o)o(v)o(er)f(26)h(\(i)h(det)f(engelsk)o(e)g (alf)o(abet)g(er)h(der)f(kun)f(26)h(bogsta)n(v)o(er\),)204 2710 y(tr\346kk)o(es)20 b(26)g(fra)g(s\345)h(man)e(f\345r)h(et)h Fv(c)g FC(mellem)e(1)i(og)e(26)1722 2680 y Fl(1)1755 2710 y FC(.)328 2810 y(Eksempel)h(med)f Fv(K)29 b Ft(=)22 b(3)p FC(:)793 2989 y(Klartekst:)173 b(t)132 b(e)127 b(s)155 b(t)160 b(t)132 b(e)127 b(s)155 b(t)160 b(t)132 b(e)127 b(s)155 b(t)817 3088 y(Som)20 b(tal:)168 b(20)98 b(5)i(19)e(20)h(20)g(5)g(19)f(20)h(20)g(5)g(19)f(20)786 3188 y(Krypteret:)135 b(23)98 b(8)i(22)e(23)h(23)g(8)g(22)f(23)h(23)g (8)g(22)f(23)749 3288 y(Kryptotekst:)109 b(w)i(h)120 b(v)132 b(w)122 b(w)112 b(h)119 b(v)132 b(w)123 b(w)111 b(h)120 b(v)131 b(w)328 3469 y(C\346car)29 b(ben)o(yttede)e(efter)g (sigende)h(selv)g Fv(K)44 b Ft(=)37 b(3)p FC(,)28 b(men)g Fv(K)43 b Ft(=)38 b(13)28 b FC(er)g(mere)g(popul\346r)e(i)j(v)n(ore)f (dage,)f(m\345sk)o(e)h(fordi)204 3568 y(krypteringen)12 b(og)j(dekrypteringen)d(kan)i(fore)o(g\345)g(med)h(n\370jagtigt)f (samme)i(funktion.)d(Denne)h(specielle)i(kryptering)d(kaldes)204 3668 y(ogs\345)20 b(R)m(O)m(T13:)1333 3950 y Fv(R)q(O)r(T)1511 3962 y Fo(13)1582 3950 y Ft(\()p Fv(m)p Ft(\))j(=)g Fv(c)83 b FC(og)f Fv(R)q(O)r(T)2293 3962 y Fo(13)2363 3950 y Ft(\()p Fv(c)p Ft(\))24 b(=)e Fv(m)328 4133 y FC(Da)31 b(hv)o(ert)f(bogsta)n(v)g(i)h(kildeteksten)e(kun)h(kan)g(bli)n(v)o(e)g (o)o(v)o(ersat)f(til)i(\351t)h(bogsta)n(v)d(er)i(denne)e (krypteringsform)e(me)o(get)204 4233 y(s\345rbar)21 b(o)o(v)o(er)f(for) h(frekv)o(ensanalyse,)e(hv)n(or)i(man)g(t\346ller)h(frekv)o(ensen)e(af) h(de)h(enk)o(elte)f(bogsta)n(v)o(er)f(og)i(sammenholder)d(dem)204 4332 y(med)25 b(det)h(man)g(tror)g(teksten)f(er)-5 b(.)27 b(F)-7 b(.eks.)26 b(er)g(\223e\224)h(det)f(mest)g(brugte)f(bogsta)n(v)g (i)i(de)f(\003este)h(europ\346isk)o(e)e(sprog,)g(derfor)f(vil)204 4432 y(det)d(mest)g(brugte)f(bogsta)n(v)g(i)h(den)g(krypterede)d(tekst) j(sandsynligvis)f(sv)n(are)h(til)g(\223e\224.)g(Med)g(26)f(k)o (ombinationsmuligheder)204 4532 y(ville)g(en)g(udt\370mmende)e (s\370gning)g(dog)i(heller)f(ikk)o(e)h(tage)g(al)h(for)f(lang)f(tid.) 328 4631 y(Dette)30 b(g\370r)e(dog)g(ikk)o(e)g Fv(R)q(O)r(T)1181 4643 y Fo(13)1281 4631 y FC(ubrugelig)f(til)j(andre)d(form\345l.)h(F)-7 b(.eks.)29 b(bruges)f(det)h(i)h(n)o(yhedsgrupper)24 b(til)30 b(at)f(skjule)204 4731 y(b)n(udskaber)h(hvis)i(nogen)e(\002nder)h (materialet)h(forn\346rmende)d([Hank)o(erson)h(et)j(al.,)f(2000,)e(s.)j (231],)d(eller)i(til)h(at)f(skjule)204 4830 y(sv)n(aret)20 b(p\345)g(vittigheder)-5 b(.)204 5111 y Fr(3.2)119 b(Anden)31 b(v)o(erdenskrig)204 5296 y FC(Under)23 b(2.)g(v)o(erdenskrig)e(brugte) i(man)g(i)i(stor)e(stil)i(radioer)e(til)h(at)h(klare)e(langdistancek)o (ommunikationen)o(.)c(F)o(ordelen)j(v)n(ar)204 5396 y(naturligvis)d(at) j(man)e(ikk)o(e)g(beh\370)o(v)o(ede)e(at)j(ha)n(v)o(e)g(fysisk)f(k)o (ontakt,)g(men)g(samtidig)g(gi)n(v)o(er)g(radiob\370lgerne)d(u\370nsk)o (ede)i(perso-)204 5495 y(ner)e(mulighed)e(for)i(ubem\346rk)o(et)f(at)i (lytte)g(med.)e(S\345)i(nu)f(m\345tte)h(man)e(v)o(ed)h(alle)h (meddelelser)e(re)o(gne)g(med)h(at)h(fjenden)e(lyttede)204 5595 y(med.)p 204 5649 1430 4 v 294 5705 a Fk(1)323 5728 y FD(Egentlig)j(fore)o(g\345r)f(her)g(en)f(modulo-operation,)j(se)d (bilag)i(B)1970 6016 y FC(8)p eop %%Page: 9 11 9 10 bop 3352 -143 a FA(3.3.)41 b(TVIND)p 204 -110 3573 4 v 328 177 a FC(F)o(or)16 b(at)h(undg\345)e(at)i(fjenden)e(forstod)g (b)n(udskaberne,)f(det)i(kunne)f(f.eks.)h(v\346re)g(ret)h(kritisk)f (hvis)g(den)g(fremtidige)f(position)204 277 y(af)20 b(en)g(ub\345d)f (ble)n(v)h(r\370bet,)f(be)o(gyndte)f(man)i(at)g(udvikle)f(kryptering)f (i)j(stor)f(stil.)328 376 y(Enigma)i(er)g(et)h(k)o(endt)e(eksempel)h (p\345)g(en)g(af)h(krigstidens)e(frembringelser)f([Bauer,)h(1997,)g(s.) i(106\226109].)c(Det)k(v)n(ar)e(en)204 476 y(s\345kaldt)27 b(rotorkrypteringsmaskine)22 b(som)27 b(virk)o(ede)f(vha.)h(et)g(k)o (ompliceret)f(system)i(af)f(bogsta)n(v)o(erstatninger)-5 b(.)24 b(T)-7 b(ysk)o(erne)204 576 y(ben)o(yttede)24 b(anordningen)e(i)27 b(stor)f(stil)h(\226)f(man)f(re)o(gner)g(med)g(at) i(der)e(v)n(ar)h(m\345sk)o(e)f(omkring)f(10.000)g(maskiner)h(i)i (oml\370b)d(\226)204 675 y(men)29 b(uheldigvis)f(for)h(dem)g(ble)n(v)g (systemet)h(allerede)f(brudt)f(i)i(starten)g(af)f(krigen,)f(hjulpet)h (p\345)g(v)o(ej)h(af)f(nogle)g(kaprede)204 775 y(eksemplarer)-5 b(.)18 b(F)o(orsk)o(ellige)i(senere)g(modeller)e(form\345ede)h(ikk)o(e) h(at)g(rette)h(op)e(p\345)h(problemet)f(i)i(l\346ngere)e(tid)i(ad)f (gangen.)328 874 y(De)28 b(andre)d(stormagter)h(ben)o(yttede)f(sig)i (af)g(lignende)e(anordninger)m(,)e(og)k(visse)g(af)g(dem)g(ble)n(v)f (brudt)g(p\345)g(tilsv)n(arende)204 974 y(m\345de)19 b(af)i(aksemagterne.)328 1074 y(Af)27 b(modstandsbe)n(v\346gelserne)c (ble)n(v)j(der)g(ogs\345)h(ben)o(yttet)e(forsk)o(ellige)g(metoder)h (til)h(at)g(skjule)f(b)n(udskabet)g(for)g(u\370n-)204 1173 y(sk)o(ede)17 b(lyttere.)g(En)g(k)o(endt)g(og)g(noget)g (utraditionel)f(m\345de)h(bestod)f(i)j(helt)e(at)h(skjule)g(b)n (udskabet)e(v)o(ed)g(at)i(indrykk)o(e)e(usk)o(yldigt)204 1273 y(udseende)h(annoncer)m(,)g(f.eks.)h(kunne)f(\223Sk)o(yd)h(kl.)h (8\224)g(oms\346ttes)h(til:)g(\223S\370ren)e(Kristensen)h(yder)f(dig)g (og)h(klassen)g(l\346kk)o(er)g(og)204 1373 y(opk)o(ogt)c(tilberedt)i (tykste)o(g)f(efterbehag\224)f(\(forbogsta)n(v)o(erne)e(indeholder)i(b) n(udskabet)h(mens)h(\223og\224)f(sv)n(arer)h(til)h(mellemrum\).)328 1472 y(Under)i(2.)g(v)o(erdenskrig)d(kunne)i(krypteringens)e(ef)n (fekti)n(vitet)i(alts\345)i(betyde)e(forsk)o(ellen)g(mellem)h(li)n(v)g (og)g(d\370d.)204 1752 y Fr(3.3)119 b(Tvind)204 1938 y FC(Et)19 b(helt)f(aktuelt)g(eksempel)g(er)g(da)h(politiet)f(i)h (for\345ret)e(2001)g(fra)i(det)f(berygtede)e(Tvind-imperium)f (beslaglagde)i(en)i(r\346kk)o(e)204 2038 y(materialer)g(der)h(viste)g (sig)h(at)g(v\346re)f(krypterede.)328 2137 y(Flere)i(sikk)o (erhedseksperter)m(,)c(heriblandt)i(en)h(dansk)g(professor)e(med)i (speciale)h(i)g(kryptologi,)d(Peter)i(Landrock,)e(ud-)204 2237 y(talte)31 b(til)h(Computer)e(W)-7 b(orld)31 b([Thorhauge,)c (2001b])i(at)j(politiet)f(ikk)o(e)f(ville)i(kunne)d(bryde)h (krypteringen)d(som)k(skulle)204 2336 y(ben)o(ytte)20 b(sig)h(af)h(n\370gler)e(p\345)h(128)f(og)h(256)f(bit.)h(Det)h(lykk)o (edes)e(dog)g(allige)n(v)o(el)h(politiet)g(under)f(mystisk)o(e)h (omst\346ndigheder)204 2436 y(at)f(f\345)h(dekrypteret)d(dataene.)328 2536 y(Politiet)23 b(sagde)g(til)g(Computer)e(W)-7 b(orld)23 b([Thorhauge,)c(2001a])i(at)i(de)f(v)o(ed)g(hj\346lp)h(af)f (\223normale\224)f(hack)o(ermetoder)f(har)204 2635 y(brudt)31 b(den)g(kryptering)e(som)j(besk)o(yttede)e(fem)i(af)g(de)g(beslaglagte) f(computere.)e(Det)k(afviser)e(\003ere)h(kilder)f(dog:)g(det)204 2735 y(er)c(helt)h(usandsynligt)e(at)i(man)f(skulle)h(kunne)e(bryde)g (systemet.)i(Berlingsk)o(e)f(T)m(idende)f(har)h(if\370lge)g(Computer)f (W)-7 b(orld)204 2835 y([Thorhauge,)17 b(2001a])h(f\345et)i(oplyst)g (at)h(det)f(er)g(en)g(af)g(Tvinds)g(samle)n(v)o(ere)f(som)h(har)g(udle) n(v)o(eret)e(et)j(passw)o(ord)e(til)i(politiet.)204 3115 y Fr(3.4)119 b(V)l(irus)204 3300 y FC(T)m(raditionen)16 b(tro)i(skal)g(en)g(n)o(y)f(teknologi)f(naturligvis)h(udn)o(yttes)g (ondsindet)f([Bj\370rk,)h(2001].)f(Dette)j(v)n(ar)e(bestemt)h (tilf\346ldet)204 3400 y(da)i(uk)o(endte)f(virus-programm\370rer)c (udsendte)k(en)h(email,)g(se)h(bilag)e(A)i(p\345)f(side)h(22,)e(som)h (p\345stod)g(at)g(v\346re)h(fra)f(Microsoft.)328 3500 y(En)30 b(\002l)g(der)f(v)n(ar)g(v)o(edh\346ftet)f(bre)n(v)o(et,)g (indeholdt)f(en)j(virus)f(som)g(un\346gtelig)g(ha)n(vde)f(en)h(lidt)h (kreati)n(v)f(\370del\346ggende)204 3599 y(ef)n(fekt.)19 b(If\370lge)g(IT)-8 b(-A)i(visen.dk)17 b(krypterede)h(den)i(alle)g(k)o (\370rbare)f(\002ler)h(p\345)g(harddisk)o(en)e(med)i(en)g(vilk\345rlig) f(n\370gle.)g(Dette)h(g\370r)204 3699 y(dem)i(naturligt)f(nok)g (ubrugelige)f(for)i(brugeren.)e(Uden)h(at)i(\002nde)f(n\370glen)f(for)h (krypteringen)d(er)j(man)g(reelt)g(ude)g(af)g(stand)204 3798 y(til)f(at)f(redde)g(ramte)f(maskiner)-5 b(.)1970 6016 y(9)p eop %%Page: 10 12 10 11 bop 204 800 a Fs(Kapitel)43 b(4)204 1215 y FE(Klassisk)n(e)53 b(kryptosystemer)204 1696 y Fr(4.1)119 b(XOR)204 1882 y FC(XOR)23 b(er)g(en)f(boolsk)g(operator)e(ligesom)i(AND)h(eller)g(OR) g(\(som)f(man)g(normalt)f(skri)n(v)o(er)h(symbolsk)f(med)h Fm(^)h FC(og)g Fm(_)p FC(,)g(sym-)204 1981 y(bolet)g(for)g(XOR)h(er)g Fm(\010)p FC(\))f(\226)h(den)f(kan)g(de\002neres)g(bitvist)h(som)f Ft(1)e Fm(\010)g Ft(1)28 b(=)h(0)p FC(,)24 b Ft(1)c Fm(\010)h Ft(0)29 b(=)g(1)p FC(,)23 b Ft(0)e Fm(\010)f Ft(1)29 b(=)g(1)24 b FC(og)f Ft(0)d Fm(\010)h Ft(0)29 b(=)g(0)p FC(.)23 b(Ud)204 2081 y(fra)f(dette)h(kan)f(man)h(vise)g(at)g Ft(\()p Fv(k)1129 2093 y Fu(p)1188 2081 y Fm(\010)d Fv(m)p Ft(\))h Fm(\010)f Fv(k)1527 2093 y Fu(p)1593 2081 y Ft(=)28 b Fv(m)23 b FC(s\345)h(en)e(m\345de)h(at)g(kryptere)e(p\345,)h(er)h(at) h(XOR'e)f(en)f(gi)n(v)o(en)g(n\370gle)g(med)204 2181 y(klarteksten)d(hv)n(orefter)f(den)h(kan)h(dekrypteres)e(v)o(ed)h(at)i (XOR'e)f(den)f(igen)h(med)g(n\370glen.)e(P\345)j(lignende)d(m\345de)i (har)f(man)h(at)204 2280 y Ft(\()p Fv(k)279 2292 y Fu(p)328 2280 y Fm(\010)11 b Fv(m)p Ft(\))g Fm(\010)g Fv(m)22 b Ft(=)h Fv(k)822 2292 y Fu(p)860 2280 y FC(.)c(Dvs.)f(k)o(ender)f(man) g(kryptoteksten)f Fv(c)23 b Ft(=)g Fv(k)2139 2292 y Fu(p)2188 2280 y Fm(\010)11 b Fv(m)18 b FC(og)g(klarteksten,)e(kan)i(man)g(nemt)f (\002nde)h(n\370glen.)328 2380 y(Lad)23 b(os)g(pr\370)o(v)o(e)e (XOR-kryptering)f(i)k(praksis.)e(Lad)g(\223tim\224)h(v\346re)g Fv(m)g FC(og)g(\223mit\224)g(v\346re)f(v)n(ores)h(n\370gle)f Fv(K)3307 2392 y Fu(p)3345 2380 y FC(.)h(En)g(talv\346rdi)204 2480 y(for)d(bogsta)n(v)o(et)f(\223m\224)i(kunne)e(v\346re)i(109)e(der) i(sv)n(arer)f(til)h(det)g(bin\346re)f(tal)h(01101101,)c(for)j (\223i\224)h(105)f(sv)n(arende)f(til)j(01101001)204 2579 y(og)e(for)f(\223t\224)i(116)e(sv)n(arende)g(til)i(01110100.)1338 2758 y Fv(t)d Fm(\010)g Fv(m)p 1028 2792 628 4 v 1100 2861 a(t)p FC(:)121 b(01110100)1078 2961 y Fv(m)p FC(:)100 b(01101101)1078 3061 y Fv(c)1114 3073 y Fo(1)1151 3061 y FC(:)g(00011001)2009 2758 y Fv(i)18 b Fm(\010)g Fv(i)p 1676 2792 V 1749 2861 a(i)p FC(:)121 b(01101001)1749 2961 y Fv(i)p FC(:)g(01101001)1726 3061 y Fv(c)1762 3073 y Fo(2)1799 3061 y FC(:)100 b(00000000)2634 2758 y Fv(m)19 b Fm(\010)f Fv(t)p 2325 2792 V 2375 2861 a(m)p FC(:)100 b(01101101)2396 2961 y Fv(t)p FC(:)122 b(01110100)2375 3061 y Fv(c)2411 3073 y Fo(3)2448 3061 y FC(:)100 b(00011001)328 3242 y(Hvis)21 b(vi)f(g\345r)g(ud)g(fra)g(at)g(vi)h(kun)e(k)o(ender)g Fv(c)i FC(og)e Fv(m)p FC(,)i(kan)e(vi)i(\002nde)f Fv(K)2251 3254 y Fu(p)2288 3242 y FC(:)1334 3427 y Fv(c)1370 3439 y Fo(1)1426 3427 y Fm(\010)e Fv(t)p 990 3460 663 4 v 1079 3530 a(t)p FC(:)138 b(01110100)1057 3629 y Fv(c)1093 3641 y Fo(1)1130 3629 y FC(:)117 b(00011001)1040 3729 y Fv(K)1111 3741 y Fo(1)1147 3729 y FC(:)100 b(01101101)1404 3828 y(m)2018 3427 y Fv(c)2054 3439 y Fo(2)2109 3427 y Fm(\010)18 b Fv(i)p 1673 3460 V 1740 3530 a(c)1776 3542 y Fo(2)1813 3530 y FC(:)117 b(00000000)1762 3629 y Fv(i)p FC(:)139 b(01101001)1723 3729 y Fv(K)1794 3741 y Fo(2)1830 3729 y FC(:)100 b(01101001)2108 3828 y(i)2651 3427 y Fv(c)2687 3439 y Fo(3)2743 3427 y Fm(\010)18 b Fv(m)p 2356 3460 635 4 v 2409 3530 a(c)2445 3542 y Fo(3)2482 3530 y FC(:)104 b(00011001)2409 3629 y Fv(m)p FC(:)g(01101101)2406 3729 y Fv(k)2449 3741 y Fo(2)2486 3729 y FC(:)c(01110100)2763 3828 y(t)328 4010 y(Som)18 b(ses)h(i)g(de)e(o)o(v)o(enst\345ende)f(to)i (eksempler)e(er)i(der)g(med)f(XOR)i(tale)f(om)g(en)g(ret)g(simpel)g(re) o(gneform,)d(mens)i(resultatet)204 4109 y(bli)n(v)o(er)j(relati)n(vt)i (godt)f(skjult.)g(F)o(aktisk)h(s\345)g(godt)f(at)h(hvis)g(n\370glen)e (kun)h(bruges)g(en)g(gang,)f(og)h(hvis)h(n\370glen)e(er)i (tilf\346ldig,)g(s\345)204 4209 y(er)e(kryptoteksten)e(ubrydelig,)f (uanset)j(hv)n(or)f(me)o(get)h(bere)o(gningskraft)c(man)k(har)g(til)h (r\345dighed)d([Hank)o(erson)g(et)j(al.,)f(2000,)204 4308 y(s.)h(236].)328 4408 y(Dette)h(h\346nger)d(sammen)h(med)h(at)g (alt)g(efter)g(hvilk)o(en)f(n\370gle)f(man)i(har)m(,)f(kan)g(man)g(f)o (aktisk)h(la)n(v)o(e)g(et)g(hv)o(ert)f(stykk)o(e)g(tekst)204 4508 y(ud)15 b(fra)h(en)g(gi)n(v)o(en)f(kryptotekst,)f(blot)i(v)o(ed)f (at)i(\346ndre)e(n\370glen)g(tilstr\346kk)o(eligt.)g(Derfor)g(har)h (man)g(ingen)f(grund)f(til)j(at)f(f)o(a)n(v)n(orere)204 4607 y(den)24 b(ene)h(tekst)g(frem)f(for)g(den)h(anden.)e(Brugen)h(af)h (engangsn\370gler)c(kaldes)j(one-time)g(pads)g(\226)h(det)g(er)g (naturligvis)f(ikk)o(e)204 4707 y(praktisk)19 b(til)i(almindeligt)e (brug.)204 4987 y Fr(4.2)119 b(DES)204 5173 y FC(DES,)26 b(Data)g(Encryption)d(Standard,)i(ble)n(v)g(optaget)g(som)h(standard)f (i)h(1977,)f(og)g(er)h(siden)g(ble)n(v)o(et)f(til)i(den)e(mest)h(k)o (endte)204 5272 y(symmetrisk)o(e)d(krypteringsmetode)e([Hank)o(erson)h (et)j(al.,)f(2000,)f(s.)i(243].)e(Den)h(bygger)f(p\345)h(et)h(antal)f (permutationer)e(og)204 5372 y(en)e(metode)f(kaldet)h(en)g (Feistel-stige)h(og)e(XOR.)328 5472 y(Ligesom)h(med)g(andre)f (krypteringsmetoder)d(starter)k(man)g(med)g(en)g(besk)o(ed)g Fv(m)p FC(,)g(og)g(denne)f(deles)i(op)e(i)i(blokk)o(e)e(af)i(64)204 5571 y(bit)j(som)f(hv)o(er)f(gennemg\345r)f(en)j(kryptering.)c(Der)k (er)g(s\345)g(forsk)o(ellige)e(m\345der)h(at)h(samle)f(de)h(krypterede) d(blokk)o(e)h(p\345,)h(hv)o(er)204 5671 y(med)d(deres)f(fordele.)1949 6016 y(10)p eop %%Page: 11 13 11 12 bop 3454 -143 a FA(4.2.)40 b(DES)p 204 -110 3573 4 v 1199 5101 a @beginspecial 0 @llx 0 @lly 558 @urx 1636 @ury 1899 @rwi @setspecial %%BeginDocument: DES.eps [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis] /isolatin1encoding exch def /Times-Roman-latin1 /Times-Roman findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /Times-Italic-latin1 /Times-Italic findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /Times-Bold-latin1 /Times-Bold findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /Times-BoldItalic-latin1 /Times-BoldItalic findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /AvantGarde-Book-latin1 /AvantGarde-Book findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding isolatin1encoding def currentdict end definefont pop /AvantGarde-BookOblique-latin1 /AvantGarde-BookOblique findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding 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FC(s\345ledes)g(at)g Fv(d)g FC(opfylder)d Fv(ed)c Ft(\(mo)r(d)g Fv(\036)p Ft(\()p Fv(n)p Ft(\)\))24 b(=)e(1)p FC(.)204 343 y(RSA-n\370gles\346ttet)h(best\345r)f(s\345)i(af)f(den)f (of)n(fentlig)f(n\370gle)h(i)h(form)f(af)h(talparret)f Ft(\()p Fv(n;e)p Ft(\))i FC(og)e(den)g(pri)n(v)n(at)g(n\370gle)g(i)i (form)d Ft(\()p Fv(n;d)p Ft(\))p FC(.)204 443 y(Krypteringen)c(fore)o (g\345r)h(v)o(ed:)1712 625 y Fv(c)23 b Ft(=)g Fv(m)1932 591 y Fu(e)1981 625 y Ft(\(mo)r(d)14 b Fv(n)p Ft(\))204 808 y FC(Dekrypteringen)j(fore)o(g\345r)h(v)o(ed:)1711 991 y Fv(m)k Ft(=)h Fv(c)1930 956 y Fu(d)1983 991 y Ft(\(mo)r(d)14 b Fv(n)p Ft(\))204 1173 y FC(Dette)20 b(kan)g(vi)g(sammenf)o(atte)g (til)h(grundstenen)c(i)k(RSA:)1070 1356 y Fv(D)1139 1368 y Fu(k)1174 1376 y Fn(p)1213 1356 y Ft(\()p Fv(E)1306 1368 y Fu(k)1341 1376 y Fn(o)1379 1356 y Ft(\()p Fv(m)p Ft(\)\))j(=)e(\()p Fv(m)1764 1322 y Fu(e)1800 1356 y Ft(\))1832 1322 y Fu(d)1885 1356 y Ft(\(mo)r(d)14 b Fv(n)p Ft(\))23 b(=)g Fv(m)2356 1322 y Fu(ed)2440 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5233 y(=)1283 5116 y Fp(\022)1344 5233 y Fv(m)g Ft(\(mo)r(d)g Fv(q)s Ft(\))1722 5141 y Fp(\020)1772 5233 y Fv(m)1845 5198 y Fo(\()p Fu(q)r Fd(\000)p Fo(1\))2032 5233 y Ft(\(mo)r(d)g Fv(q)s Ft(\))2309 5141 y Fp(\021)2359 5158 y Fu(k)q Fo(\()p Fu(p)p Fd(\000)p Fo(1\))2599 5233 y Ft(\(mo)r(d)g Fv(q)s Ft(\))2876 5116 y Fp(\023)2965 5233 y Ft(\(mo)r(d)g Fv(q)s Ft(\))1195 5440 y(=)1283 5348 y Fp(\020)1333 5440 y Fv(m)g Ft(\(mo)r(d)g Fv(q)s Ft(\))k Fm(\001)h Ft(1)1799 5406 y Fu(k)q Fo(\()p Fu(p)p Fd(\000)p Fo(1\))2024 5440 y Ft(\(mo)r(d)14 b Fv(q)s Ft(\))2301 5348 y Fp(\021)2379 5440 y Ft(\(mo)r(d)g Fv(q)s Ft(\))1195 5597 y(=)23 b Fv(m)14 b Ft(\(mo)r(d)g Fv(q)s Ft(\))1949 6016 y FC(15)p eop %%Page: 16 18 16 17 bop 3059 -143 a FA(5.3.)41 b(ET)20 b(EKSEMPEL)p 204 -110 3573 4 v 204 177 a Fr(5.3)119 b(Et)29 b(eksempel)204 363 y FC(En)20 b(n\370gle)o(generering)c(i)21 b(RSA)g(kan)e(fore)o (g\345)g(p\345)h(f\370lgende)e(m\345de:)308 545 y(1.)40 b Fv(p)32 b Ft(=)g(37)25 b FC(og)f Fv(q)36 b Ft(=)31 b(89)25 b FC(hvilk)o(et)f(gi)n(v)o(er)g Fv(n)32 b Ft(=)g Fv(pq)j Ft(=)d(3293)p FC(.)23 b Fv(p)j FC(og)e Fv(q)29 b FC(v\346lges)c(her)g(sm\345)g(for)g(at)h(lette)f(o)o(v)o(erblikk)o (et)e(og)411 645 y(udre)o(gningerne)308 811 y(2.)40 b Fv(\036)p Ft(\()p Fv(n)p Ft(\))22 b FC(bli)n(v)o(er)d(s\345:)i Fv(\036)p Ft(\()p Fv(n)p Ft(\))j(=)f(\()p Fv(p)18 b Fm(\000)g Ft(1\)\()p Fv(q)k Fm(\000)c Ft(1\))23 b(=)f(36)c Fm(\001)g Ft(88)k(=)h(3168)308 977 y FC(3.)40 b(T)-7 b(allet)24 b Fv(e)f FC(v\346lges)g(til)g(at)g(v\346re)g Ft(25)f FC(\226)h(at)g Ft(\()p Fv(e;\036)p Ft(\()p Fv(n)p Ft(\)\))29 b(=)e(1)c FC(er)g(opfyldt)e(kan)h(f.eks.)g(testes)i(med)e(en)h (ber\370mt)e(algoritme)411 1077 y(af)g(Euklid)e([Beutelspacher,)f (1994,)h(s.)i(108])e(som)h(gi)n(v)o(er)f(den)g(st\370rste)i(f\346lles)g (di)n(visor)-5 b(.)308 1243 y(4.)40 b(T)m(il)21 b(sidst)g(\002ndes)f (tallet)h Fv(d)i Ft(=)g(2281)c FC(som)h(opfylder)e(at)j Fv(ed)14 b Ft(\(mo)r(d)g Fv(\036)p Ft(\()p Fv(n)p Ft(\)\))24 b(=)e(1)p FC(.)328 1425 y(V)-5 b(i)35 b(har)g(nu)f(bestemt)g(den)g(pri) n(v)n(ate)f(n\370gle)h(til)h Ft(\()p Fv(n;d)p Ft(\))51 b(=)e(\(3293)p Fv(;)p Ft(2281\))32 b FC(og)i(den)g(of)n(fentlige)f (n\370gle)g(til)i Ft(\()p Fv(n;e)p Ft(\))50 b(=)204 1525 y(\(3293)p Fv(;)p Ft(25\))p FC(.)30 b(Bem\346rk)i(at)g(primtallene)f Fv(p)h FC(og)f Fv(q)36 b FC(samt)c Fv(\036)p Ft(\()p Fv(n)p Ft(\))h FC(ikk)o(e)f(bruges)f(til)i(noget)d(efter)i (n\370gles\346ttet)g(er)g(bestemt.)204 1624 y(Man)21 b(skal)g(f)o(aktisk)h(sikre)f(sig)h(at)f(de)h(er)f(slettet,)h(idet)f (hele)h(sikk)o(erheden)d(i)j(RSA)g(hviler)f(p\345)g(at)h(det)f(er)g (sv\346rt)h(at)g(bestemme)204 1724 y Fv(p)e FC(og)g Fv(q)k FC(ud)c(fra)g Fv(n)p FC(.)328 1824 y(Med)33 b(n\370gles\346ttet)f(i)h (h\345nden)e(kan)h(vi)h(nu)f(kryptere)e(klarteksten)k(E)t FD(N)j(K)t(RY)t(P)t(T)t(E)t(R)t(E)t(T)f(B)t(E)t(S)t(K)t(E)t(D)r FC(.)d(Hvis)g(vi)g(tildeler)204 1923 y(bogsta)n(v)o(erne)21 b(v\346rdier)i(efter)g Fv(A)29 b Ft(=)g(01)p FC(,)23 b Fv(B)33 b Ft(=)c(02)23 b FC(osv)-5 b(.)23 b(helt)h(op)f(til)h Fw(\305)29 b Ft(=)g(29)23 b FC(\(mellemrum)e(f\345r)j(s\345)g Ft(30)p FC(\).)e(F)o(or)h(klarteksten)204 2023 y(skal)c(der)f(g\346lde) h Ft(0)k Fv(<)f(m)h(<)g(n)c FC(s\345)h(besk)o(eden)d(skal)i(deles)g(op) g(i)g(blokk)o(e)f(\226)h(med)f(den)g(v)n(algte)h(v\346rdi)f(af)h Fv(n)g FC(kan)g(man)f(kryptere)204 2123 y(2)i(bogsta)n(v)o(er)f(af)h (gangen:)204 2305 y Fc(E)49 b(N)199 b(K)100 b(R)50 b(Y)99 b(P)50 b(T)99 b(E)50 b(R)99 b(E)50 b(T)199 b(B)99 b(E)50 b(S)99 b(K)50 b(E)99 b(D)204 2405 y(0514)49 b(3011)g(1825)g(1620)g (0518)g(0520)g(3002)g(0519)g(1105)g(0430)328 2588 y FC(Allerede)21 b(v)o(ed)f(en)h(s\345)g(lille)h Fv(n)p FC(-v\346rdi,)e(st\370der)g(man) h(p\345)f(det)h(praktisk)f(problem)f(at)j(bere)o(gne)d Ft(0514)3157 2557 y Fo(25)3250 2588 y Ft(=)37 b(\(mo)r(d)15 b(3293\))p FC(.)204 2687 y(Bare)k(den)f(f\370rste)g(del)g Ft(0514)1015 2657 y Fo(25)1107 2687 y Fm(\031)k Ft(10)1278 2657 y Fo(67)1367 2687 y FC(er)c(s\345)i(stor)e(at)h(det)g (besv\346rligt)f(at)h(re)o(gne)e(med.)h(Men)g(eftersom)g Ft(514)3318 2657 y Fo(25)3406 2687 y FC(kan)g(skri)n(v)o(es)204 2787 y(som)1181 2969 y Ft(514)1307 2935 y Fo(25)1399 2969 y Ft(=)23 b(514)17 b Fm(\001)h Ft(\(514\))1861 2935 y Fo(24)1954 2969 y Ft(=)23 b(514)16 b Fm(\001)j Ft(\(514)2384 2935 y Fo(2)2420 2969 y Ft(\))2452 2935 y Fo(12)1399 3104 y Ft(=)k(514)17 b Fm(\001)h Ft(\(\(514)1861 3070 y Fo(2)1898 3104 y Ft(\))1930 3070 y Fo(2)1967 3104 y Ft(\))1999 3070 y Fo(6)2060 3104 y Ft(=)k(514)17 b Fm(\001)i Ft(\(\(\(514)2554 3070 y Fo(2)2591 3104 y Ft(\))2623 3070 y Fo(2)2660 3104 y Ft(\))2692 3070 y Fo(2)2730 3104 y Ft(\))2762 3070 y Fo(3)1399 3239 y Ft(=)k(514)17 b Fm(\001)h Ft(\(\(\(514)1893 3205 y Fo(2)1930 3239 y Ft(\))1962 3205 y Fo(2)2000 3239 y Ft(\))2032 3205 y Fo(2)2069 3239 y Ft(\))2101 3205 y Fo(2)2157 3239 y Fm(\001)h Ft(\(\(514)2389 3205 y Fo(2)2425 3239 y Ft(\))2457 3205 y Fo(2)2494 3239 y Ft(\))2526 3205 y Fo(2)204 3422 y FC(kan)g(man)h(vha.)g(en)g(af)g (modulore)o(gnere)o(glerne)14 b(\(s\346tning)20 b(B.2.2\))f(undg\345)g (de)h(store)g(tal.)1949 6016 y(16)p eop %%Page: 17 19 17 18 bop 204 800 a Fs(Kapitel)43 b(6)204 1215 y FE(Det)51 b(fr)l(emtidige)h(perspekti)n(v)204 1696 y Fr(6.1)119 b(Hvilk)o(e)30 b(algoritmer)g(har)g(en)g(fr)n(emtid?)204 1882 y FC(Man)e(skulle)h(jo)f(tro)h(at)g(hele)g(v)o(erden)d(straks)j (ville)g(g\345)g(o)o(v)o(er)e(til)i(RSA)h(da)f(den)f(har)g(et)h (smartere)f(n\370glesystem)f(end)h(de)204 1981 y(symmetrisk)o(e)19 b(algoritmer)-5 b(.)19 b(Men)h(RSA)i(er)e(omkring)e(100)i(gange)f (langsommere)f(end)i(DES,)g(og)g(derfor)f(bruges)g(DES)i(og)204 2081 y(specielt)26 b(T)m(ripleDES)f(stadig)h(i)h(praksis.)e(F)-7 b(.eks.)27 b(v)o(ed)e(client/serv)o(er)n(-programmel)c(kan)26 b(dette)g(ha)n(v)o(e)f(stor)h(betydning)e(for)204 2181 y(hastigheden)18 b(af)i(programmet.)328 2280 y(En)28 b(anden)e(fordel)h(v)o(ed)g(symmetrisk)f(kryptering)f(er)j(at)g(den)g (kan)f(g\370res)g(st\346rk)o(ere)g(i)i(takt)e(med)h(computere)d(bli)n (v)o(er)204 2380 y(st\346rk)o(ere,)g(uden)f(det)i(har)f(stor)g (betydning)f(for)g(krypteringens)f(hastighed.)h(Asymmetrisk)o(e)g(kan)h (ogs\345)g(g\370res)g(st\346rk)o(ere)204 2480 y(v)o(ed)19 b(at)i(v\346lge)f(st\370rre)g(primtal,)f(men)h(dette)g(kr\346v)o(er)f (\003ere)h(bere)o(gninger)d(og)j(g\345r)g(dermed)f(ud)g(o)o(v)o(er)g (hastigheden.)328 2579 y(Men)e(asymmetrisk)o(e)e(krypteringsalgoritmer) d(er)17 b(uundv\346rlige)d(v)o(ed)i(k)o(ommunikation)d(o)o(v)o(er)i (Internettet,)g(hv)n(or)h(de)g(to)204 2679 y(parter)k(m\345sk)o(e)g (sj\346ldent)h(er)g(i)g(fysisk)f(k)o(ontakt)g(med)g(hinanden.)e(Her)j (har)f(bl.a.)g(RSA)i(gjort)e(det)g(muligt)g(at)h(k)o(ommunik)o(ere)204 2779 y(krypteret)d(uden)h(forudg\345ende)e(at)k(k)o(ende)e(en)h (hemmelig)f(n\370gle.)204 3016 y Fq(6.1.1)99 b(Hybridl\370sninger)204 3172 y FC(En)31 b(l\370sning)g(der)h(er)f(ble)n(v)o(et)g(mere)h(og)f (mere)g(popul\346r)m(,)f(er)i(hv)n(ad)f(man)g(kunne)g(kalde)g(en)g (hybridl\370sning)e(der)i(bruger)204 3271 y(fordelene)18 b(fra)i(be)o(gge)f(typer)-5 b(.)328 3371 y(Hvis)26 b Fv(A)g FC(skal)f(snakk)o(e)f(med)h Fv(B)t FC(,)g(starter)g Fv(A)h FC(med)e(at)i(bede)e Fv(B)30 b FC(om)25 b(en)g(n\370gle.)e (Derefter)i(bruger)e Fv(B)29 b FC(en)c(asymmetrisk)204 3471 y(kryptering)13 b(til)k(at)f(sende)f(en)h(krypteringsn\370gle)c Fv(K)1678 3483 y Fu(e)1730 3471 y FC(til)k Fv(A)p FC(.)h Fv(A)f FC(har)g(nu)f(en)h(sikk)o(er)f(kanal,)g(som)h(han)f(v)o(ed)g(at) i(kun)e Fv(B)20 b FC(kan)15 b(l\346se.)204 3570 y(Herefter)i(sender)h (af)h(en)f(tilf\346ldig)g(dannet)g(n\370gle)f(til)i(k)o(ommunikation)c (som)k(han)f(krypterer)e(med)i Fv(K)3145 3582 y Fu(e)3180 3570 y FC(.)h Fv(B)k FC(dekrypterer)16 b(s\345)204 3670 y(denne)i(n\370gle)h(og)g(de)g(kan)g(nu)g(bruge)g(denne)f(n\370gle)g (til)j(k)o(ommunikation)16 b(med)j(f.eks.)g(T)m(ripleDES.)g(P\345)h (samme)f(m\345de)g(kan)204 3770 y(dette)h(k)o(ombineres)e(med)i (signering)f(s\345)i Fv(B)k FC(er)20 b(sikk)o(er)g(p\345)g(at)g(han)g (ikk)o(e)g(f\345r)g(en)g(n\370gle)f(fra)h(en)g(anden.)204 4007 y Fq(6.1.2)99 b(Nutidige)25 b(an)l(v)o(endelser)204 4163 y FC(I)18 b(det)h(f\370lgende)d(vil)j(vi)g(k)o(ort)f(beskri)n(v)o (e)f(to)h(systemer)g(baseret)g(p\345)h(hybridl\370sningen)14 b(som)k(er)h(uundv\346rlige)d(for)i(mennesk)o(er)204 4263 y(i)g(dag.)e(Det)i(f\370rste)f(er)g(SSH)h(som)f(er)h(et)g(program) d(med)i(en)g(protok)o(ol)e(til)j(sikk)o(ert)f(at)h(udf\370re)d(k)o (ommandoer)g(o)o(v)o(er)g(et)j(netv\346rk)204 4362 y(p\345)i(en)g (anden)f(maskine,)g(det)h(andet,)g(PGP)-9 b(,)20 b(bruges)f(til)i(at)g (signere)f(og)f(kryptere)g(email.)204 4583 y Fx(Secur)o(e)h(Shell)204 4739 y FC(Den)f(f\370rste)g(v)o(ersion)f(af)h(Secure)g(Shell)g(ble)n(v) g(udviklet)f(i)i(1995)e(af)h(en)g(uni)n(v)o(ersitetsstuderende)d(fra)j (Helsinki)g(Uni)n(v)o(ersitet,)204 4838 y(fordi)k(han)h(v)n(ar)f (tr\346t)i(af)g(at)g(f\345)f(snif)n(fet)g(sine)g(adgangsk)o(oder)d ([Barrett)j(og)g(Silv)o(erman,)f(2001].)f(I)i(dag)g(ansl\345s)h(der)f (at)g(v\346re)204 4938 y(omkring)18 b(2)i(mill.)h(SSH-brugere)d ([Acheson,)g(2001].)328 5038 y(SSH)j(bruger)d(et)i(kryptosystem)e(med)i (of)n(fentlige)e(n\370gler)m(,)g(som)i(f.eks.)f(RSA,)i(til)f (identi\002kation)e(og)i(sikring)f(af)h(at)g(det)204 5137 y(er)k(den)g(rigtige)g(maskine)g(man)g(snakk)o(er)g(med.)g (Herefter)f(fore)o(g\345r)g(al)i(kryptering)d(med)i(et)h(symmetrisk)f (kryptosystem,)204 5237 y(f.eks.)19 b(T)m(riple)h(DES)g(eller)g(et)h(v) o(ed)e(na)n(vn)g(Blo)n(w\002sh.)h(Herv)o(ed)f(fungerer)f(SSH)i(som)g (en)g(erstatning)f(for)g(tidligere)h(l\370sninger)204 5337 y(som)26 b(telnet)f(eller)h(rlogin)f([Myers,)g(2000].)e(Med)j(til) h(SSH)f(h\370rer)f(ogs\345)g(krypterede)e(\002lo)o(v)o(erf\370rsler)h (s\345)i(man)f(kan)h(sende)204 5436 y(sine)20 b(\002ler)h(uden)e(risik) o(o)h(for)g(opsnapning.)328 5536 y(En)26 b(ting)g(i)h(forbindelse)d (med)h(SSH)i(er)f(at)h(\003ere)f(af)g(krypteringsalgoritmerne)21 b(er)26 b(besk)o(yttede)e(af)i(patenter)m(,)f(hvilk)o(et)204 5635 y(is\346r)h(gi)n(v)o(er)e(problemer)f(for)i(de)g(frie,)g(\345bne)g (udga)n(v)o(er)e(der)i(er)g(i)h(oml\370b)m(.)d(Enten)i(m\345)g(man)g (s\345)h(undv\346re)e(visse)i(algoritmer)204 5735 y(eller)20 b(ogs\345)g(m\345)g(man)g(fors\370ge)f(at)h(arbejde)f(sig)i(rundt)e(om) h(det)g(idet)g(ikk)o(e)g(alle)h(lande)f(tillader)g(denne)f(form)g(for)g (patenter)-5 b(.)1949 6016 y(17)p eop %%Page: 18 20 18 19 bop 2204 -143 a FA(6.2.)40 b(KR)-5 b(YPT)o(OSYSTEMERNES)19 b(SIKKERHED)p 204 -110 3573 4 v 204 177 a Fx(Pr)o(etty)f(Good)i(Pri)o (v)o(acy)204 333 y FC(Pretty)32 b(Good)e(Pri)n(v)n(ac)o(y)h(ble)n(v)g (startet)i(af)f(Phil)g(R.)g(Zimmermann,)d(og)j(programmets)d(omtumlede) h(historie)i(afspejler)204 432 y(me)o(get)24 b(godt)g(USA)-9 b(')k(s)26 b(holdning)c(til)k(kryptering)d(\226)i(den)f(viser)h(NSA)-9 b(')k(s)26 b(og)e(den)g(amerikansk)o(e)g(re)o(geringens)e(fors\370g)i (p\345)h(at)204 532 y(hindre)d(udbredelsen)f(af)j(st\346rk)f (kryptering)e(\(hele)i(historien)g(kan)g(l\346ses)i(her)e([Back,)g (2001]\).)e(F)o(or)i(at)h(k)o(omme)e(uden)g(de)204 632 y(amerikansk)o(e)15 b(eksportrestriktioner)e(ble)n(v)j(kildek)o(oden)f (en)h(o)o(v)o(er)o(gang)d(udgi)n(v)o(et)i(som)i(en)f(bog)g(som)g(man)h (s\345)g(kunne)e(skanne)204 731 y(direkte)k(ind;)h(dermed)f(ble)n(v)g (der)h(jo)h(ikk)o(e)e(eksporteret)g(noget)g(i)i(elektronisk)e(form)g Fv(:)14 b(:)g(:)328 831 y FC(En)22 b(foruds\346tning)d(for)i (signeringen)f(og)h(krypteringen)d(af)k(email)g(er)g(at)g(man)f(f\345r) g(PGP)i(til)f(at)g(generere)e(en)i(pri)n(v)n(at)f(og)204 930 y(of)n(fentlig)d(n\370gle.)h(Den)g(of)n(fentlige)f(n\370gle)h(kan)h (derefter)e(sendes)i(til)h(en)e(n\370gle-serv)o(er)e(s\345)k(andre)e (har)g(let)i(adgang)c(til)k(den)e(\226)204 1030 y(der)i(er)h(et)g (indviklet)e(system)i(hv)n(or)f(folk)f(siger)i(god)f(for)g(hinandens)e (n\370gler)i(til)h(at)g(sikre)g(at)g(de)f(f)o(aktisk)h(er)f(\346gte)h ([Ashle)o(y,)204 1130 y(1999].)328 1229 y(Krypteringen)16 b(af)i(selv)o(e)g(teksten)f(sk)o(er)h(p\345)g(samme)g(m\345de)g(som)f (beskre)n(v)o(et)g(i)i(fore)o(g\345ende)c(afsnit)j(med)f(pri)n(v)n(ate) g(n\370gler)m(,)204 1329 y(simpelthen)i(fordi)g(det)h(er)h(hurtigere.) 204 1609 y Fr(6.2)119 b(Kryptosystemer)n(nes)28 b(sikk)o(erhed)204 1795 y FC(Et)h(hyppigt)f(an)m(v)o(endt)f(m\345l)i(for)g(sikk)o(erheden) e(i)j(kryptosystemer)d(er)i(antallet)g(af)h(k)o(ombinationsmuligheder) 24 b(der)29 b(skal)204 1894 y(afpr\370)o(v)o(es)17 b(v)o(ed)h(en)i (udt\370mmende)c(s\370gning.)h(Men)i(den)g(udt\370mmende)d(s\370gning)i (er)h(blot)g(et)h(udtryk)d(for)i(arbejdsm\346ngden)204 1994 y(i)25 b(det)f(v\346rst)h(t\346nk)o(elige)f(tilf\346lde)h(\226)g (f.eks.)f(kan)g(simple)g(bogsta)n(v)o(erstatningssystemer)d(hurtigt)j (brydes)f(v)o(ed)h(at)h(udn)o(ytte)204 2094 y(anden)f(k)o(endt)g(viden) h(om)g(klarteksten,)f(nemlig)g(at)i(de)f(forsk)o(ellige)g(bogsta)n(v)o (er)e(i)j(et)g(almindeligt)e(sprog)h(optr\346der)f(med)204 2193 y(bestemte)c(hyppigheder)-5 b(.)328 2293 y(Man)27 b(kan)f(ikk)o(e)g(afvise)g(at)i(lignende)d(analyser)g(eller)i(andre)f (smarte)g(metoder)g(der)g(udn)o(ytter)e(ekstra)j(viden)e(om)i(si-)204 2392 y(tuationen,)h(kan)g(bruges)h(til)h(at)g(sk\346re)g(ned)e(p\345)i (antallet)f(af)h(muligheder)d(v)o(ed)h(de)i(systemer)f(som)g(bruges)g (i)h(dag,)e(som)204 2492 y(DES)e(og)f(RSA,)h(og)f(derv)o(ed)f(rokk)o(e) h(v)o(ed)f(deres)i(sikk)o(erhed.)d(En)j(s\345dan)f(afvisning)f(ville)i (skulle)f(bygge)f(p\345)i(matematisk)204 2592 y(stringente)c(be)n (viser)m(,)g(og)g(dem)h(har)g(man)f(ikk)o(e)h([Landrock)e(og)h(Nissen,) i(1997,)d(s.)j(120].)d(Kryptering)g(med)i(de)g(of)n(fentlige)204 2691 y(n\370glesystemer)18 b(er)i(i)h(denne)e(sammenh\346ng)g(specielt) h(s\345rbare)f(fordi)h(den)f(samme)h(n\370gle)f(bruges)h(o)o(v)o(er)e (lange)i(perioder)-5 b(.)328 2791 y(Og)29 b(man)f(skal)h(ogs\345)f (husk)o(e)g(p\345)g(at)h(et)g(kryptosystem)e(er)h(t\346t)i(forb)n (undet)c(med)i(den)g(tid)h(det)g(er)f(opfundet)e(i,)j(ingen)204 2891 y(af)g(de)h(hidtidige)e(systemer)h(er)g(rigtigt)g(fremtidssikrede) f(selv)n(om)h(de)g(m\345sk)o(e)g(kan)g(udvides)f(i)i(takt)g(med)f (tidens)g(gang,)204 2990 y(f.eks.)21 b(i)h(form)e(af)i(l\346ngere)e (n\370gler)-5 b(.)21 b(Lidt)g(ekstrapolering)e(p\345)i(den)g(hidtidige) g(eksponentionelle)d(v\346kst)k(i)g(re)o(gnekraft)d(for)204 3090 y(computere)i(gi)n(v)o(er)g(at)j(de)f(n\370gler)e(man)i(ben)o (ytter)e(i)j(dag)e(til)i(RSA)g(om)e(blot)h(et)g(par)g(\345rtier)g (sandsynligvis)e(v\346re)i(inden)f(for)204 3189 y(r\346kk)o(e)n(vidden) 27 b(af)i(hv)n(ad)f(man)g(kan)h(n\345)g(at)h(re)o(gne)e(sig)h(igennem)f (selv)h(med)g(en)g(udt\370mmende)e(s\370gning)g([Landrock)g(og)204 3289 y(Nissen,)20 b(1997,)f(s.)i(120].)328 3389 y(Men)j(DES)h(og)f(RSA) h(har)f(trods)g(alt)h(efterh\345nden)d(v\346ret)i(brugt)f(i)i(mange)f (\345r)g(uden)f(at)i(der)f(er)g(ble)n(v)o(et)g(fundet)f(nogen)204 3488 y(alv)n(orlige)c(fejl)h(v)o(ed)g(dem.)204 3769 y Fr(6.3)119 b(Inter)n(essek)n(on\003ikter)31 b(og)e(krypteringslo)o(vgi) o(vning)204 3954 y FC(Efterh\345nden)21 b(som)j(krypteringsmetoderne)19 b(bli)n(v)o(er)k(bedre)g(og)g(sikre,)h(bli)n(v)o(er)e(det)i(n\370dv)o (endigt)d(at)k(tage)e(politisk)h(stilling)204 4054 y(til)d(kryptering.) c(F)o(orsk)o(ellige)i(parter)h(har)g(forsk)o(ellige)f(interesser:)328 4236 y Fm(\017)41 b FC(Den)18 b(almindelige)f(bor)o(ger)f(er)i (interesseret)g(i)g(at)h(besk)o(ytte)e(sine)i(pri)n(v)n(ate)e (oplysninger)e(og)j(at)h(kunne)d(k)o(ommunik)o(ere)411 4336 y(via)32 b(internettet)f(uden)g(andre)f(kan)h(f\370lge)g(med)g (\(besk)o(yttelse)g(af)h(pri)n(v)n(atli)n(v)o(ets)e(fred)h(er)h(en)f (mennesk)o(erettighed)411 4436 y([Mennesk)o(erettighederne,)16 b(1948,)i(art.)j(12]\).)328 4602 y Fm(\017)41 b FC(V)-5 b(irksomheder)15 b(vil)i(gerne)g(kunne)e(besk)o(ytte)h(fortrolige)g (data)h(for)f(at)i(forhindre)d(industrispionage)f(\226)j(spionagenet-) 411 4701 y(v\346rk)o(et)j(Echelon)f(menes)h(at)h(ha)n(v)o(e)e(k)o (ostet)i(europ\346isk)o(e)e(virksomheder)e(milliarder)i(af)h(kroner)f ([Langv)n(ad,)e(2001].)328 4867 y Fm(\017)41 b FC(T)-6 b(errorister)20 b(og)h(andre)g(forbrydere)d(\370nsk)o(er)i(af)h (indlysende)f(grunde)f(at)j(kunne)e(k)o(ommunik)o(ere)e(uden)j (politiet)g(kan)411 4967 y(lytte)g(med.)328 5133 y Fm(\017)41 b FC(Staten)30 b(har)f(interesse)g(i)h(at)f(kunne)f(f\345)h (efterretninger)e(om)i(trusler)g(mod)f(landet,)h(b\345de)f(indre)h(og)f (ydre.)h(Det)g(er)411 5233 y(derfor)22 b(i)i (inderrigsefterretningstjenestens)19 b(interesse)24 b(at)g(bor)o(gerne) c Fw(ikk)o(e)k FC(kan)f(skjule)g(noget,)g(og)g(det)g(er)h(i)g(uden-)411 5332 y(rigsefterretningstjenestens)d(interesse)j(at)h(andre)e(lande)g (ikk)o(e)h(bli)n(v)o(er)f(i)h(stand)g(til)h(holde)e(deres)h(k)o (ommunikation)411 5432 y(hemmelig.)19 b(Under)g(en)h(krig)g(er)g(dette) g(vitalt)h(for)e(milit\346ret,)i(som)f(det)g(s\345s)h(under)e(sidste)i (v)o(erdenskrig.)328 5615 y(Den)g(enk)o(elte)g(stats)i(interesser)e(er) g(alts\345)h(i)g(k)o(on\003ikt)e(med)h(bor)o(gernes)e(og)i(andre)f (staters)i(interesser)-5 b(.)21 b(Dette)g(udm\370nter)204 5714 y(sig)c(i)h(tiltag)f(til)h(at)f(lo)o(vgi)n(v)o(e)e(om)i(f.eks.)f (tvungne)f(bagd\370re)g(eller)i(sv)n(agheder)e(i)j(kryptosystemerne)13 b(eller)18 b(forb)n(ud)d(mod)h(st\346rk)1949 6016 y(18)p eop %%Page: 19 21 19 20 bop 1406 -143 a FA(6.3.)40 b(INTERESSEK)n(ONFLIKTER)18 b(OG)j(KR)-5 b(YPTERINGSLO)l(V)o(GIVNING)p 204 -110 3573 4 v 204 177 a FC(kryptering)20 b(\(dvs.)j(kryptering)d(der)j(rent)g(f)o (aktisk)g(kan)f(holde)h(noget)f(hemmeligt\).)f(Eller)i(restriktioner)f (p\345)h(eksporten)e(af)204 277 y(krypteringsalgoritmer)-5 b(.)328 376 y(Internt)17 b(i)i(et)g(land)e(er)h(lo)o(vgi)n(vningen)d (klart)j(pr\346get)g(af)g(etisk)o(e)g(o)o(v)o(erv)o(ejelser)-5 b(.)16 b(\330nsk)o(er)i(man)f(en)h(o)o(v)o(erv\345gningsstat)d(eller) 204 476 y(er)23 b(retten)g(til)h(bre)n(vhemmelighed)19 b(vigtigere)j(end)h(politiets)h(muligheder)d(for)h(at)i(opklare)e (kriminelle)g(handlinger?)f(Pro-)204 576 y(blemet)f(er)i(med)e(det)h(n) o(ylige)f(terroristangreb)f(mod)h(W)-7 b(orld)21 b(T)m(rade)f(Center)h (ble)n(v)o(et)f(sat)i(yderligere)e(p\345)h(spidsen.)f(Eksternt)204 675 y(er)g(det)g(nok)g(mere)f(et)i(sp\370r)o(gsm\345l)e(om)g(det)i(er)f (muligt)f(at)i(h\345ndh\346v)o(e)d(en)i(gi)n(v)o(en)f(lo)o(v)h(i)h (praksis,)f(pga.)e(f.eks.)i(internettet.)328 775 y(Et)26 b(af)g(de)f(mest)h(akti)n(v)o(e)e(lande)h(n\345r)g(det)g(handler)f(om)h (lo)o(vgi)n(vning)e(mod)h(kryptering)f(er)i(USA)h(\226)g(det)f (f\370rn\346vnte)f(ek-)204 874 y(sempel)f(med)h(PGP)g(viser)g(ogs\345)f (dette,)h(Phil)g(Zimmerman)e(skaf)n(fede)g(sig)i(f)o(aktisk)g(en)g (retssag)g(p\345)f(halsen)h([Back,)f(2001])204 974 y(som)d(dog)f (senere)h(ble)n(v)g(opgi)n(v)o(et.)204 1212 y Fq(6.3.1)99 b(W)-6 b(assenaar)l(-aftalen)204 1367 y FC(W)f(assenaar)n(-aftalen)32 b(er)h(et)g(fors\370g)e(fra)h(USA)i(p\345)e(at)h(f\345)g(andre)f(lande) g(til)h(at)g(indf\370re)e(samme)h(lo)o(vgi)n(vning)e(som)j(dem)204 1467 y([K)m(oops,)16 b(2001].)g(Aftalen)h(fors\370ger)f(at)i(la)n(v)o (e)g(eksportrestriktioner)d(p\345)i(st\346rk)o(e)h (krypteringsalgoritmer)m(,)13 b(dvs.)k(o)o(v)o(er)g(56)g(bit)204 1567 y(v)o(ed)i(symmetrisk)g(kryptering)f(og)i(512)f(bit)i(v)o(ed)e (asymmetrisk)g(kryptering.)328 1666 y(Mange)f(lande,)g(inklusi)n(v)g (Danmark,)f(har)h(underskre)n(v)o(et)e(aftalen,)i(men)g(landene)f(skal) i(selv)g(indarbejde)e(aftalen)h(i)i(de-)204 1766 y(res)e(lo)o(vgi)n (vning,)c(og)j(4)g(dansk)o(e)g(ministre)g(har)g(senere,)g(i)h(et)g(bre) n(v)f(til)h(IT)-8 b(-sikk)o(erhedsudv)n(alget,)13 b(of)n(fentliggjort)i (re)o(geringens)204 1865 y(holdning)23 b(p\345)i(omr\345det.)e(De)j(4)f (ministre)g(siger)g(at)h(kryptering)c(skal)k(kunne)d(bruges)h(og)h (spredes)g(frit)g(inden)f(for)g(EU)i(og)204 1965 y(W)-7 b(assenaar)n(-aftalens)23 b(lande,)f(men)h(man)f(skal)i(stadig)f (beholde)f(k)o(ontrollen)f(o)o(v)o(er)h(spredning)f(af)i (\223f\370lsomme)f(produkter)204 2065 y(til)f(f\370lsomme)e (brugere\224)f([W)-7 b(eiss)21 b(et)g(al.,)f(2000].)328 2164 y(Men)i(hv)n(ordan)e(dette)h(skulle)h(kunne)e(lade)i(sig)g (g\370re,)f(og)g(hv)o(em)g(der)g(skal)h(g\370re)f(det)h(n\346vner)f (bre)n(v)o(et)g(ikk)o(e)g(noget)g(om,)204 2264 y(og)j(det)h(lader)f (ikk)o(e)h(til)h(at)f(der)f(er)h(sk)o(et)h(noget)e(k)o(onkret.)f (M\345sk)o(e)i(er)g(dette)f(en)h(erk)o(endelse)f(af)h(at)g(den)f (aftale)h(som)g(de)g(har)204 2364 y(skre)n(v)o(et)19 b(under)g(p\345)h(ikk)o(e)g(kan)f(bruges)g(i)i(praksis.)f(USA)g(har)g (ikk)o(e)g(selv)g(v\346ret)g(i)h(stand)f(til)h(at)g(forhindre)c (distrib)n(ution)i(af)h(de)204 2463 y(algoritmer)e(der)i(er)g(ble)n(v)o (et)g(la)n(v)o(et)g(i)h(landet.)1949 6016 y(19)p eop %%Page: 20 22 20 21 bop 204 800 a Fs(Kapitel)43 b(7)204 1215 y FE(K)-5 b(onklusion)204 1646 y FC(Kryptering)18 b(g\345r)i(ud)f(p\345)i(at)f (g\370re)g(data)g(uforst\345elig)e(s\345)j(man)f(kan)g(undg\345)e(at)j (uv)o(edk)o(ommende)16 b(kan)k(drage)f(n)o(ytte)g(af)i(dem.)328 1746 y(Historisk)33 b(set)g(har)f(kryptologi)d(specielt)k(under)e (krige)g(spillet)i(en)g(stor)f(rolle)g(fordi)f(det)i(kan)e(betyde)h (forsk)o(ellen)204 1846 y(mellem)16 b(li)n(v)h(og)f(d\370d)g(om)h(man)f (kan)g(holde)g(k)o(ommunikationen)d(hemmelig)i(\226)i(is\346r)h(med)e (indf\370relsen)f(af)i(letopsnappelige)204 1945 y(radiosignaler)-5 b(.)328 2045 y(Eksempler)21 b(p\345)i(klassisk)o(e)g(kryptosystemer)d (er)i Fv(R)q(O)r(T)1930 2057 y Fo(13)2000 2045 y FC(,)h(XOR)h(og)e(DES) g(som)h(bygger)d(p\345)j(pri)n(v)n(ate)e(n\370gler)m(,)g(hvilk)o(et)204 2145 y(betyder)e(at)h(man)g(bruger)e(samme)i(n\370gle)g(til)h(at)f (kryptere)f(og)g(dekryptere)f(med.)328 2244 y(RSA)33 b(er)g(et)f(eksempel)f(p\345)i(et)f(moderne)e(kryptosystem)g(der)i (bygger)e(p\345)i(of)n(fentlige)f(n\370gler)-5 b(.)31 b(Of)n(fentlige)g(n\370gler)204 2344 y(er)e(v)o(ele)o(gnede)d(til)k (internettet)e(fordi)g(man)g(ikk)o(e)h(l\370ber)f(nogen)f(risik)o(o)i 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y(eller)g (forhindre)e(forbrydere)f(i)k(at)g(kryptere)d(data)i(s\345)h(sikk)o (ert)f(at)h(politiet)f(ikk)o(e)g(kan)f(bryde)g(det.)h(De)h(fors\370g)e (p\345)h(eksport-)204 3141 y(restriktioner)j(der)g(har)h(v\346ret,)h (er)f(en)g(total)h(\002ask)o(o)f(som)h(ikk)o(e)f(har)g(haft)g(nogen)e (reel)j(virkning.)d(PGP)j(har)f(p\345)g(trods)g(af)204 3241 y(eksportrestriktioner)17 b(under)i(hele)h(programmets)d (eksistens)k(v\346ret)f(lettilg\346ngelig)g(for)f(hele)h(v)o(erdenen.) 328 3340 y(V)-5 b(i)30 b(mener)e(at)h(man)g(b\370r)f(s\370ge)h(n)o(ye)f (v)o(eje)h(for)f(at)i(sikre)f(sig)g(be)n(vismateriale)f(n\345r)g(den)h (v)o(ej)f(der)h(har)g(v\346ret)g(tr\345dt)f(de)204 3440 y(seneste)20 b(\345r)h(nu)e(ikk)o(e)h(har)g(virk)o(et.)f(M\345sk)o(e)h (kunne)f(man)h(forestille)g(sig)g(en)g(l\370sning)g(hv)n(or)f(man)g (efter)h(en)g(dommerk)o(endelse)204 3539 y(ha)n(vde)f(pligt)h(til)h(at) g(udle)n(v)o(ere)d(sine)j(pri)n(v)n(ate)e(n\370gler)m(,)f(p\345)i 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(fremg\345r)h(af)h(f\370lgende)e(s\346tning:)206 4886 y(S)t FD(\306)t(T)t(N)t(I)t(N)t(G)25 b FC(B)t(.)t(2)t(.)t(1)43 b FA(F)o(or)20 b(alle)h Fv(m;)14 b(k)25 b Fm(2)f Fg(Z)14 b FA(og)20 b Fv(n)j Fm(2)g Fg(N)31 b FA(g\346lder)308 5052 y(1.)40 b Fv(n)p Fm(j)p Fv(m)23 b Fm(,)g Fv(m)14 b Ft(\(mo)r(d)g Fv(n)p Ft(\))23 b(=)g Fv(n)308 5218 y FA(2.)40 b Ft(0)23 b Fm(\024)g Fv(m)g(<)f(n)h Fm(\))g Fv(m)14 b Ft(\(mo)r(d)g Fv(n)p Ft(\))23 b(=)g Fv(m)308 5384 y FA(3.)40 b Ft(\()p Fv(m)14 b Ft(\(mo)r(d)h Fv(n)p Ft(\)\))f(\(mo)r(d)g Fv(n)p Ft(\))23 b(=)g Fv(m)14 b Ft(\(mo)r(d)g Fv(n)p Ft(\))308 5550 y FA(4.)40 b Ft(\()p Fv(m)19 b Ft(+)f Fv(k)s(n)p Ft(\))c(\(mo)r(d)g Fv(n)p Ft(\))23 b(=)g Fv(m)14 b Ft(\(mo)r(d)g Fv(n)p Ft(\))328 5716 y FC(Endelig)19 b(er)i(der)e(er)i(en)f(r\346kk)o(e)g(re)o(gnere)o (gler)d(for)i(omformning)e(af)k(udtryk)d(med)i(modulo:)1949 6016 y(23)p eop %%Page: 24 26 24 25 bop 3240 -143 a FA(B.3.)41 b(PRIMT)-8 b(AL)p 204 -110 3573 4 v 206 177 a FC(S)t FD(\306)t(T)t(N)t(I)t(N)t(G)25 b FC(B)t(.)t(2)t(.)t(2)43 b FA(F)o(or)20 b(alle)h Fv(a;)14 b(b)22 b Fm(2)i Fg(Z)14 b FA(og)20 b Fv(n;)14 b(t)22 b Fm(2)i Fg(N)30 b FA(g\346lder)308 319 y(1.)40 b Ft(\()p Fv(a)19 b Ft(+)f Fv(b)p Ft(\))c(\(mo)r(d)g Fv(n)p Ft(\))23 b(=)g(\()p Fv(a)14 b Ft(\(mo)r(d)g Fv(n)p Ft(\))19 b(+)f Fv(b)c Ft(\(mo)r(d)f Fv(n)p Ft(\)\))h(\(mo)r(d)h Fv(n)p Ft(\))308 472 y FA(2.)40 b Ft(\()p Fv(a)19 b Fm(\001)f Fv(b)p Ft(\))c(\(mo)r(d)g Fv(n)p Ft(\))24 b(=)e(\()p Fv(a)14 b Ft(\(mo)r(d)h Fv(n)p Ft(\))j Fm(\001)h Fv(b)14 b Ft(\(mo)r(d)f Fv(n)p Ft(\)\))h(\(mo)r(d)h Fv(n)p Ft(\))308 626 y FA(3.)40 b Fv(a)455 596 y Fu(t)498 626 y Ft(\(mo)r(d)15 b Fv(n)p Ft(\))23 b(=)f(\()p Fv(a)14 b Ft(\(mo)r(d)h Fv(n)p Ft(\)\))1306 596 y Fu(t)1349 626 y Ft(\(mo)r(d)f Fv(n)p Ft(\))204 901 y Fr(B.3)119 b(Primtal)204 1086 y FC(F)o(or)20 b(at)g(klarl\346gge)g(hv)n(ad)f(vi)h(mener)f(med)h(et)h (primtal,)e(f)o(astl\346gges)h(det)h(i)f(f\370lgende)f(de\002nition:) 206 1267 y(D)t FD(E)t(FI)t(N)t(I)t(T)t(I)t(O)t(N)25 b FC(B)t(.)t(3)t(.)t(1)43 b FA(Et)20 b(helt)h(tal)f Fv(p)j(>)g Ft(1)d FA(kaldes)g(et)29 b FC(primtal)19 b FA(hvis)i Fv(p)f FA(kun)f(har)h(de)g(tri)n(vielle)g(di)n(visorer)f Ft(1)h FA(og)g Fv(p)p FA(.)328 1424 y FC(Positi)n(v)o(e)25 b(hele)f(tal)h(der)f(ikk)o(e)g(er)h(primtal,)e(kaldes)i Fw(sammensatte)f(tal)p FC(.)g(If\370lge)f(en)i(s\346tning)f(v)o(ed)g (na)n(vn)g(aritmetikk)o(ens)204 1524 y(fundamentals\346tning)16 b(kan)j(de)g(kan)g(alle)g Fw(primfaktoriser)m(es)h FC(entydigt,)d(dvs.) i(skri)n(v)o(es)g(som)g(et)h(entydigt)e(produkt)f(af)i(prim-)204 1623 y(tal;)j(f.eks.)g(er)g Ft(21)j(=)h(3)20 b Fm(\001)g Ft(7)p FC(.)h(S\346tningen)h(er)g(ikk)o(e)f(sv\346r)i(at)f(vise,)g(men) g(vil)g(ikk)o(e)g(bli)n(v)o(e)f(taget)h(op)g(her)f(\226)h(det)g (vigtige)f(er)i(blot)204 1723 y(at)d(primf)o(aktorisering)d(er)k (entydig.)328 1823 y(Hvis)g Fv(d)p Fm(j)p Fv(a)g FC(og)f Fv(d)p Fm(j)p Fv(b)p FC(,)g(siges)h Fv(d)g FC(at)g(v\346re)f(f\346lles) h(di)n(visor)f(i)g Fv(a)h FC(og)f Fv(b)p FC(,)g(og)g(den)g(st\370rste)g (f\346lles)h(di)n(visor)f(for)f Fv(a)i FC(og)f Fv(b)g FC(ben\346vnes)204 1922 y Ft(\()p Fv(a;)14 b(b)p Ft(\))p FC(.)20 b(Hvis)h Ft(\()p Fv(a;)14 b(b)p Ft(\))23 b(=)g(1)p FC(,)d(sv)n(arende)e(til)j(at)g(kun)e(1)i(g\345r)e(op)h(i)h(b\345de)e Fv(a)i FC(og)f Fv(b)p FC(,)g(siges)h Fv(a)f FC(og)g Fv(b)h FC(at)f(v\346re)g Fw(indbyr)m(des)g(primisk)o(e)p FC(.)204 2197 y Fr(B.4)119 b(Den)31 b(kinesisk)o(e)f(r)n(ests\346tning)204 2382 y FC(Den)18 b(kinesisk)o(e)g(rests\346tning)h(d\346kk)o(er)e(o)o (v)o(er)h(et)h(problem)d(mht.)i(at)h(\002nde)g(l\370sninger)e(til)i(et) g(ligningssystem)e(med)h(modulo-)204 2482 y(udtryk:)206 2663 y(S)t FD(\306)t(T)t(N)t(I)t(N)t(G)25 b FC(B)t(.)t(4)t(.)t(1)43 b FA(Lad)20 b Fv(a;)14 b(b)23 b Fm(2)g Fg(Z)14 b FA(og)20 b Fv(n;)14 b(m)23 b Fm(2)g Fg(N)t Fv(;)14 b Ft(\()p Fv(n;)g(m)p Ft(\))29 b(=)23 b(1)p FA(.)d(S\345)h(har)1121 2815 y Fv(x)14 b Ft(\(mo)r(d)g Fv(n)p Ft(\))23 b(=)g Fv(a)14 b Ft(\(mo)r(d)g Fv(n)p Ft(\))19 b Fm(^)f Fv(x)c Ft(\(mo)r(d)h Fv(m)p Ft(\))23 b(=)g Fv(b)14 b Ft(\(mo)r(d)g Fv(m)p Ft(\))204 2967 y FA(en)20 b(entydig)f(bestemt)h(l\370sning)f(modulo)f Fv(nm)j FA(gi)n(v)o(et)e(v)o(ed)1318 3119 y Fv(x)14 b Ft(\(mo)r(d)h Fv(nm)p Ft(\))23 b(=)g(\()p Fv(bsn)18 b Ft(+)g Fv(atm)p Ft(\))c(\(mo)r(d)g Fv(nm)p Ft(\))204 3271 y FA(hv)n(or)19 b Fv(s)i FA(og)f Fv(t)g FA(er)h(v)n(algt)e(s\345)i Fv(sn)d Ft(+)g Fv(tm)23 b Ft(=)g(1)p FA(.)204 3546 y Fr(B.5)119 b(Reducering)32 b(med)e Fh(\036)p Fa(\()p Fh(n)p Fa(\))204 3744 y FC(Man)24 b(kan)h(vise)g(at)g(hvis)g Ft(\()p Fv(a;)14 b(n)p Ft(\))32 b(=)f(1)25 b FC(s\345)g(er)g Fv(a)1553 3713 y Fu(\036)p Fo(\()p Fu(n)p Fo(\))1704 3744 y Ft(\(mo)r(d)14 b Fv(n)p Ft(\))32 b(=)f(1)14 b(\(mo)r(d)g Fv(n)p Ft(\))25 b FC(\(s\346tningen)f(kaldes)h(Eulers)f(s\346tning\).)g (En)204 3843 y(k)o(onsekv)o(ens)18 b(af)i(dette)g(er)h(f\370lgende)d (s\346tning:)206 4024 y(S)t FD(\306)t(T)t(N)t(I)t(N)t(G)25 b FC(B)t(.)t(5)t(.)t(1)43 b FA(F)o(or)20 b(alle)h Fv(a)i Fm(2)g Fg(Z)14 b FA(og)20 b Fv(n;)14 b(t)23 b Fm(2)g Fg(N)31 b FA(med)20 b Ft(\()p Fv(a;)14 b(n)p Ft(\))23 b(=)f(1)f FA(g\346lder)1394 4176 y Fv(a)1438 4142 y Fu(t)1481 4176 y Ft(\(mo)r(d)14 b Fv(n)p Ft(\))23 b(=)g Fv(a)1923 4142 y Fu(t)11 b Fo(\(mo)r(d)g Fu(\036)p Fo(\()p Fu(n)p Fo(\)\))2299 4176 y Ft(\(mo)r(d)k Fv(n)p Ft(\))204 4392 y Fx(Be)o(vis:)82 b FC(Der)31 b(\002ndes)g(if\370lge)f(\(B.1.1\))f (hele)i(tal)g Fv(q)j FC(og)d Fv(r)j FC(s\345dan)c(at)h Fv(s)43 b Ft(=)f Fv(q)s(\036)p Ft(\()p Fv(n)p Ft(\))27 b(+)f Fv(r)34 b FC(hv)n(or)29 b Fv(r)46 b Ft(=)c Fv(s)14 b Ft(\(mo)r(d)g Fv(\036)p Ft(\()p Fv(n)p Ft(\)\))p FC(.)32 b(Af)204 4491 y(re)o(gnere)o(glerne)16 b(for)k(modulo-udtryk)c(og)k (Eulers)g(s\346tning)g(f\345s)g(s\345)831 4643 y Fv(a)875 4609 y Fu(t)918 4643 y Ft(\(mo)r(d)15 b Fv(n)p Ft(\))23 b(=)f Fv(a)1360 4609 y Fu(q)r(\036)p Fo(\()p Fu(n)p Fo(\)+)p Fu(r)1627 4643 y Ft(\(mo)r(d)14 b Fv(n)p Ft(\))24 b(=)e(\()p Fv(a)2101 4609 y Fu(q)r(\036)p Fo(\()p Fu(n)p Fo(\))2271 4643 y Fv(a)2315 4609 y Fu(r)2352 4643 y Ft(\))14 b(\(mo)r(d)g Fv(n)p Ft(\))1229 4802 y(=)1316 4710 y Fp(\020)1366 4735 y(\000)1404 4802 y Ft(\()p Fv(a)1480 4768 y Fu(\036)p Fo(\()p Fu(n)p Fo(\))1631 4802 y Ft(\(mo)r(d)g Fv(n)p Ft(\)\))1950 4768 y Fu(q)2001 4802 y Ft(\(mo)r(d)g Fv(n)p Ft(\))2288 4735 y Fp(\001)q(\000)2365 4802 y Fv(a)2409 4768 y Fu(r)2459 4802 y Ft(\(mo)r(d)g Fv(n)p Ft(\))2746 4735 y Fp(\001)2785 4710 y(\021)2862 4802 y Ft(\(mo)r(d)g Fv(n)p Ft(\))1229 4960 y(=)1316 4892 y Fp(\000)1354 4960 y Ft(\(1)1428 4925 y Fu(q)1479 4960 y Ft(\(mo)r(d)g Fv(n)p Ft(\)\)\()p Fv(a)1874 4925 y Fu(r)1925 4960 y Ft(\(mo)r(d)g Fv(n)p Ft(\)\))2244 4892 y Fp(\001)2297 4960 y Ft(\(mo)r(d)g Fv(n)p Ft(\))1229 5101 y(=)22 b Fv(a)1360 5066 y Fu(r)1411 5101 y Ft(\(mo)r(d)14 b Fv(n)p Ft(\))23 b(=)g Fv(a)1853 5066 y Fu(s)11 b Fo(\(mo)r(d)g Fu(\036)p Fo(\()p Fu(n)p Fo(\)\))2236 5101 y Ft(\(mo)r(d)j Fv(n)p Ft(\))328 5253 y FC(Hvis)33 b Fv(p)f FC(er)g(et)h(primtal,)e(er)i Fv(\036)p Ft(\()p Fv(n)p Ft(\))46 b(=)e Fv(p)27 b Fm(\000)g Ft(1)32 b FC(da)g(alle)h(tal)g(fra)f(1)g(til)h Fv(p)f FC(er)g(indbyrdes)e (primisk)o(e)i(med)g Fv(p)p FC(.)g(Som)g(et)204 5352 y(specialtilf\346lde)20 b(af)g(Eulers)g(s\346tning)g(f\345s)h(da)206 5533 y(S)t FD(\306)t(T)t(N)t(I)t(N)t(G)k FC(B)t(.)t(5)t(.)t(2)g(\()t(F) t FD(E)t(R)t(M)t(A)m(T)t(S)f(L)t(I)t(L)t(L)t(E)g(S)t(\306)t(T)t(N)t(I)t (N)t(G)t FC(\))43 b FA(Lad)20 b Fv(a)j Fm(2)g Fg(Z)14 b FA(og)20 b Fv(p)h FA(v\346re)f(et)h(primtal)e(med)h Ft(\()p Fv(p;)14 b(a)p Ft(\))23 b(=)g(1)p FA(.)d(S\345)h(er)1537 5685 y Fv(a)1581 5651 y Fu(p)p Fd(\000)p Fo(1)1718 5685 y Ft(\(mo)r(d)15 b Fv(p)p Ft(\))23 b(=)f(1)14 b(\(mo)r(d)g Fv(p)p Ft(\))1949 6016 y FC(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF