Overview. Block Cipher Cryptanalysis - Part 1. Block Cipher Cryptanalysis. Block Cipher Cryptanalysis

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1 Ovrvw Blck Cphr Crpaalss - Par 1 lssar Bargh Dparm f Elcrcs, Ifrma a Bgrg (DEIB) Plcc Mla alssar.bargh - a - plm. Lss cs Bascs f Blck Cphr Crpaalss lgbrac w Plax ack f Blck Cphrs w/ Sbxs Th Trval cphr lgbrac Crpaalss f Trval Sascal w Plax ack: Lar Crpaalss Th Smpl cphr Lar Crpaalss f Smpl G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 1 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 2 / 37 Blck Cphr Crpaalss Blck Cphr Crpaalss pprachs Mr blck cphr crpaalss ams a hr rcvrg h k wh a ffr smallr ha brfrc w-cphrx-l crpaalss s a smpm f a pl flaw cphr (Casar, Vgér, Plafar al.) W wll ackl chqs rqrg hr a kw plax r a rla (chs) plax assmp Blck cphr crpaalss ar sall spl w famls: algbrac a sascal lgbrac alss cphr ca b s as a larg s f Bla smlas qas vlvg k bs (varabls), px a cx bs (kw rms). Ehr r bh f h fllwg fgs shl b ba ffcl hav a sccssfl crpaalss Fg a sl ( cls frm), csrg h k bs as kws, ls h k frm (r mr) px/cx pars Fg ha l a sbs f h k bs ar vlv h smlas qas rvals a flaw Ol h vlv k bs ar ffcvl s b h cphr crp G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 3 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 4 / 37

2 Blck Cphr Crpaalss Blck Cphr Crpaalss Sascal alss prfc cphr s b abl prc a cx p frm h px Ol h OTP s, all h hrs prc a cx whr h prbabl ha a b s hr r zr s bas (= 1 2 ± ɛ) b px val Ths, a rla r bw px a cx bs ma hl wh bas ɛ 0 Sampl r: hr s a v mbrs f 1s h plax hr s a v mbr f 1s h cphrx Gal: f sch rlas bw px a cx, a xpl h sascal bass xrac a pr f h k Sarg Frm w, lar wll b rfrrg lar w.r.. h a m2,.., Xclsv-O (xr) call ha h mlplca m 2 s h Bla a pra a ha x 2 m 2 = x Frs sp: bsrv ha a lar cphr s rval crpaalz Obag h k wh a kw plax aack = slvg a s f smlas lar qas Frs xampl: h Trval cphr, l has ffs a k a G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 5 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 6 / 37 Blck Cphr Crpaalss Blck Cphr Crpaalss Trval Cphr P 0 k 0 k 0 k 0 C 0 Trval Crpaalss: lgbrac pprach wr h rlas bw px, cx a k bs as qas wh bar cffcs: P 1 P 2 k 1 k 2 k 3 k 1 k 2 k 3 k 1 k 2 k 3 C 1 C 2 C 0 = P 2 k 2 k 3 k 0 C 1 = P 3 k 3 k 0 k 1 C 2 = P 0 k 0 k 1 k 2 C 3 = P 1 k 1 k 2 k 3 P 2 C 0 = k 2 k 3 k 0 P 3 C 1 = k 3 k 0 k 1 P 0 C 2 = k 0 k 1 k 2 P 1 C 3 = k 1 k 2 k 3 P 3 4-b blck, 4-b k schl: a rp f h k C 3 Ths s a lar smlas q. s, whr h pxs a cxs ar kw: js slv fr k 0, k 1, k 2, k 3 cphr ma l wh a lar k a a a lar ffs lar ca b brk wh h sam chq G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 7 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 8 / 37

3 Blck Cphr Crpaalss Trval Crpaalss: Sascal pprach Sc Trval C 0 = P 2 k 2 k 1 k 0 : If h rla r : C 0 = P 2 hls, h hr: 1 k 0 = k 2 = k 1 = 0 2 k 0 = k 1 = 1, k 2 = 0 3 k 0 = k 2 = 1, k 1 = 0 4 k 1 = k 2 = 1, k 0 = 0 If h rla r : C 0 P 2 hls, h hr: 1 k 0 = k 2 = k 1 = 1 2 k 0 = k 1 = 0, k 2 = 1 3 k 0 = k 2 = 0, k 1 = 1 4 k 1 = k 2 = 0, k 0 = 1 I bh cass, lkg a h val f a px-cx b par, f 2 3 = 8 pssbl vals fr k 0, k 1, k 2 l 4 pssbls ar lf Ths s qval a kspac rc b b, a ca alwas b sc hr r r r wll hl wh Pr(r) = 1. Nlar las Er h S-Bxs S cphrs hav lar cmps (S-Bxs) prvg ffc cls frm sl f smlas qas Slvg smlas qas wh g. > 1 Z p s NP-Cmpl Wh S-Bxs, h lar rlas bw px a cx bs alwas hl (.., wh prbabl =1) Wh S-Bxs, lar rlas ma sll hl wh a prbabl 1 2 Ia f Lar Crpaalss f Blck Cphrs 1 Expl hs rlas hlg wh a bas (.., hlg wh a prbabl 1 2 ) prv a k-p lar apprxma f a cphr pr 2 Us h apprxma shrc h cphr h las r a rcvr h las r k G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 9 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 10 / 37 S-Bx Dscrp Lar Crpaalss Small, rblsm W wll ackl a 4 4-b S-Bx, crrspg a {0, 1} 4 {0, 1} 4 lar map I s ba csrg h frs S-Bx f DES a sg 00 h w lfms p bs Th cmpl scrp s prv blw (p h p rw, X 0 X 1 X 2 X 3, cmal; p bbl, Y 0 Y 1 Y 2 Y 3, h bm ) S-bx pprxma W chck hw wll lar rlas f h S-Bx Hw ma l. rl. bw S-Bx ps a ps ca b wr? Lar rla plmals wh gr 1 (.g.: X 1 X 3 X 4 ) Th rlas ca b wr as X = j Y j c, c {0, 1} Thr ar 8 varabls vlv (4 ps, 4 ps) = 2 9 pssbl rlas If a gv rla hls wh prbabl 1 2 +ɛ wh c=0, h h sg c=1 ls hlg wh bas ɛ Sc w ar rs h mag f ɛ, w wll arbrarl pck c=0 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 11 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 12 / 37

4 Lar Crpaalss Bas cmpas fr h 4--4 S-bx lar p-p rla S-bx s: a 0 X 0 a 1 X 1 a 2 X 2 a 3 X 3 = b 0 Y 0 b 1 Y 1 b 2 Y 2 b 3 Y 3 ; a, b {0, 1} Fr ach chc f h bar cffcs a, b f (2 4 ) 2 =256 pssbl chcs: { 0000, 0000,..., 0010, 1001,...}, w slc a rla Ialz w vals: cr 0, cs Sbx sz= Fr ach f h S-bx ps (x 0, x 1, x 2, x 3 ) {0000, 0001,...} Cmp h p vals ( 0, 1, 2, 3 ) = Sbx(x 0, x 1, x 2, x 3 ) If h rla fr h gv chc f a, b hls crm cr Sr h bas val ɛ=pr 1 2 = cr 1 2 fr h slc rla as Sbx sz cr 2 ( maag grs sa f fracs) Obvsl, h rla gv b (a 0, a 1, a 2, a 3 ), (b 0, b 1, b 2, b 3 ) = 0000, 0000 wll alwas b r (sm f p bs = sm f p bs).., Pr(r)=1 r bas ɛ= 1 2 (maxmm) r gr bas= Sbx sz 2 (maxmm) Bas abl fr h S-bx a=(a 0a 1a 2a 3) : X (X ) b=(b 0b 1b 2b 3) : Y(Y ) {}}{{}}{ a 0 X 0 a 1 X 1 a 2 X 2 a 3 X 3 = b 0 Y 0 b 1 Y 1 b 2 Y 2 b 3 Y 3 ; a, b {0, 1} a :X (X ) b :Y(Y ) B C D E F 0 : : X : X : X2 X : X : X1 X : X1 X : X1 X2 X : X : X0 X : X0 X B : X0 X2 X C : X0 X D : X0 X1 X E : X0 X1 X F : X0... X G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 13 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 14 / 37 Lar Crpaalss Hw ra a bas abl Th prvs bas abl ca b ra as fllws: Chs p bs cmp h wghs (a 0, a 1, a 2, a 3 ) ra hm as a rw x val Chs p bs cmp h wghs (b 0, b 1, b 2, b 3 ) ra hm as a clm x val Lk p h val v. If sgs ar s, h a Clr-Cg s ap: r s psv, whras bl s gav Cmp h rla bas as ɛ = v (Sbx sz) Exampl: Chs p bs X 0, X 1, X 2, X 3 (0, 0, 0, 1) rw 1 Chs p bs Y 0, Y 1, Y 2, Y 3 (0, 1, 1, 1) clm 7 Th val v s +6, hs h rla r : X 3 =Y 1 Y 2 Y 3 hls wh prbabl Pr(r)= (+6) = = 7 8 Wh h sa bs f h cphr ar pass hrgh h S-Bx h p-p apprxma rla r : X 3 =Y 1 Y 2 Y 3 hls 7 ms f 8 Tacklg a N s Trval Cphr cap w: w hav a s f apprxma lar rlas whch ca rplac h S-bx wh a kw prbabl w: h lar pr f h cphr s scrb b lar rlas Gal-1: Oba a lar apprxma f a cphr p rgh bfr h las k-a Gal-2: Us h apprxma rv h fal r-k hrgh slvg h q. ssm G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 15 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 / 37

5 Targ cphr Targ cphr h Smpl cphr -b blck, 4 rs, 80 b k N kschl, -b s ach r SPN sg: [, SBx, Prm] 3, [, SBx] 1, pls a fal Th S-bxs ar all h sam, a mach h w js aals Prma: -h b f j-h bx gs h j-h b f -h Sbx N prma h las r: rval lma, acs bws Th r k a s prfrm va bws xr h Smpl cphr 1 S 11 S 12 S 13 S 14 2 S 21 S 22 S 23 S 24 3 S 31 S 32 S 33 S 34 4 S 41 S 42 S 43 S 44 5 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 17 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 18 / 37 Chag pprxmas Hw bl a whl cphr apprxma W w kw hw apprxma a sgl S-bx a wr a lar rla fr a ffs sag T ba a fll cphr apprxma w sll : 1 F a wa cmp h bas f w s mr ha apprxma (.., h cphr hr ar ma S-Bx lars) 2 G r f h rma k as W wll sar wh h sascal l whch allws s cmb apprxmas: h Pl-p Lmma [1] Lar Crpaalss Pl-p Lmma [1] - Tlbx Csr w ram varabls Z 1, Z 2 vr {0, 1}, wh hr rspcv srbs { { p1 f z Pr(Z 1 =z 1 )= 1 = 0 p2 f z Pr(Z 1 p 1 f z 1 = 1 2 =z 2 )= 2 = 0 1 p 2 f z 2 = 1 If w ak Z 1 b p frm Z 2, w ba: p 1 p 2 z 1 = 0, z 2 = 0 p Pr(Z 1 = z 1, Z 2 = z 2 ) = 1 (1 p 2 ) z 1 = 0, z 2 = 1 (1 p 1 )p 2 z 1 = 1, z 2 = 0 (1 p 1 )(1 p 2 ) z 1 = 1, z 2 = 1 W ca w rv Pr(Z 1 Z 2 =0) as Pr(Z 1 = 0, Z 2 = 0) + Pr(Z 1 = 1, Z 2 = 1)=p 1 p 2 +(1 p 1 )(1 p 2 ) G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 19 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 20 / 37

6 Lar Crpaalss Pl-p Lmma - Tlbx = Fr h sak f clar, xplc ɛ 1, ɛ 2 rwrg: p 1 = 1 2 +ɛ 1, p 2 = 1 2 +ɛ 2 Th bass ɛ 1, ɛ 2, ar h rag 1 2 ɛ 1, ɛ a hghlgh hw mch h prbabls va frm a hs c ss W ca w ba h fllwg f fr Pr(Z 1 Z 2 =0): Pr(Z 1 Z 2 = 0) = p 1 p 2 + (1 p 1 )(1 p 2 ) ɛ 1ɛ 2 Ths, h bas assca h r : Z 1 Z 2 =0 as 2ɛ 1 ɛ 2 =ɛ 1,2 Lar Crpaalss Pl-p Lmma [1] Fllwg h prvs rsls, w ca gralz 2 p ram varabls, b c: Pr(Z 1 Z 2... Z = 0) = Eqvall, b a shrr frm: ɛ 1,..., = 2 1 Prf: ssm val f h Lmma fr 1 varabls: Pr( 1 =1 Z = 0)= 1 2 +ɛ 1,..., 1 whr ɛ 1,..., 1 =2 2 1 =1 ɛ Df h bar varabl W = 1 =1 Z. Th Pr(W Z =0)=...= ɛ 1,..., 1 ɛ, ɛ 1,..., = 2 1 =1 ɛ =1 =1 ɛ ɛ G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 21 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 22 / 37 Pl-p Lmma a Ipc Isss wh varabls ar p L Z 1, Z 2, Z 3 b p bar ram varabl wh bass ɛ 1 =ɛ 2 =ɛ 3 = 1 4. W ca cmp rcl Z 12 = Z 1 Z 2 wh bas ɛ 12 = 2ɛ 1 ɛ 2 = 1 8 Z 23 = Z 2 Z 3 wh bas ɛ 23 = 2ɛ 2 ɛ 3 = 1 8 Z 13 = Z 1 Z 3 wh bas ɛ 13 = 2ɛ 1 ɛ 3 = 1 8 N ha Z 12 a Z 23 ca b p. If h wr p, h b h Pl-p lmma h bas f Z 13 =Z 12 Z 23 wl b qal = 1 32, whch s h cas Us f h Pl-p lmma fr cmbg rlas Exampl Nw w assm rlas amg p (X ) a p (Z ) vals as qals 0, hs ach f hm ca b rpr as a ram varabl. Th gal s hav a rla bw X a Z wh a hr rma varabl X 0 X 1 X 2 X 3 S Y 0 Y 1 Y 2 Y 3 P r 1 r 2 r 3 r 1 : X 3 Y 3 Y 2 Y 1 = 0 ɛ 1 = + 6 r 2 : (Y 0 Y 3 0 ) (Y 1 Y 2 1 ) (Y 2 Y 1 2 ) = 0 ɛ 3 = r 3 : Y 2 Y 1 Y 0 Z 2 Z 1 = 0 ɛ 2 = 4 (r 2 s alwas r as prma a k a ar lar) Y 3 Y 2 Y 1 Y 0 Y' 0 Y' 1 Y' 2 Y' 3 S Z 0 Z 1 Z 2 Z 3 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 23 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 24 / 37

7 Us f h Pl-p lmma fr cmbg rlas Exampl X 0 X 1 X 2 X 3 S Y 0 Y 1 Y 2 Y 3 P Y 3 Y 2 Y 1 Y 0 Y' 0 Y' 1 Y' 2 Y' 3 S r 1 r 2 r 3 Cmb r 2, r 3 r 23 ag rlas mmbrws a cmp ɛ 23 r 23 : Y 3 Y 2 Y Z 2 Z 1 = 0 ɛ 23 = ɛ 2 = 4 r 1 : X 3 Y 3 Y 2 Y 1 = 0 ɛ 1 = + 6 Cmb r 23 a r 1 (s pl-p lmma, assmg r 1, r 23 as p hs s ssbl ms cass [1]) r 123 : X 3 Z 2 Z = 0 ɛ 123 = 2ɛ 1 ɛ 23 = 3 Z 0 Z 1 Z 2 Z 3 Lar Crpaalss Whl cphr apprxma W ar clsg r arg: ba a apprxma f a par f h cphr p bfr h las k-a W sll w hgs 1 Chs whch bass s fr h S-Bxs a whch S-Bxs apprxma 2 Dal wh h fr r-ks bg a bw W wll sar chsg whch bas rlas s a whch S-Bxs apprxma Th rlas ar chs wh w (pssbl crasg) gals: 1 Mmz h mbr f p bs s mmz h mbr f acv bxs (r h bas wll h w...) 2 Empl SBx lar apprxmas wh h hghs pssbl bass G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 25 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 26 / 37 Lar Crpaalss Lar Pah Na: U,j rprss h j-h p b (cg frm 1, bm p r pcr) h -h r S-Bxs V,j rprss h j-h p b (cg frm 1, bm p r pcr) h -h r S-Bxs,j rprss h j-h b (cg frm 1, bm p r pcr) f h -h r k W sar b csrg h p h frs r S-bxs: j [1, ] U 1,j = P j 1,j as h frs acs all bs W w chs h bas fr r frs sp: r 1 : X 0 X 2 X 3 = Y 1 S-Bx S 13 Pr(r 1 ) = 12/, ɛ r1 = 1 4 Lar Crpaalss Lar Pah pplg r 1 : X 0 X 2 X 3 = Y 1 (Pr(r 1 ) = 12/, ɛ r1 = 1 4 ) S-bx S 13, allws s sa ha V 1,6 = U 1,5 U 1,7 U 1,8 = (P 5 1,5 ) (P 7 1,7 ) (P 8 1,8 ) wh prbabl Pr = ɛ r 1 = O bx w, hr g :-) Sc h V 1,6 b gs hrgh h sc, U 2,6 = V 1,6 2,6 Fllwg h wrs, w ba ha r x rla b csr s vlv l U 2,6 amg h p bs W chs a hgh-bas rla vlvg X 1 (h sc Sbx p crrsp. V 1,6 ); l s ak r 2 : X 1 = Y 1 Y 3 wh bas ɛ r2 = 1 4 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 27 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 28 / 37

8 Lar Crpaalss Lar Pah pplg r 1 : X 1 = Y 1 Y 3 (ɛ r2 = 1 4 ) SBx S 23, allws s sa ha V 2,6 V 2,8 = U 2,6 = V 1,6 2,6 wh prbabl Pr = ɛ r 2 = I s m mrg h w rlas! pplg h Pl-p lmma, w sbs V 1,6 wh P 5 1,5 (P 7 1,7 ) P 8 1,8, bag r 12 : V 2,6 V 2,8 = P 5 1,5 (P 7 1,7 ) P 8 1,8 2,6 wh a bas f ɛ r1,r 2 = 2ɛ r1 ɛ r2 = 1 4 ( 1 4 ) = Tw bxs w: w hav a lar apprxma h plax (a k bs) p afr h sc S-bx lar hlg wh Pr(r 1, r 2 ) = = 3 8 Lar Crpaalss Lar Pah s h prvs rla vlvs wrs blgg w bs f h p lar, w w cmb w rlas apprxma S 31 a S 33 Emplg h sam r 2 as bfr bh bxs w ba ha V 3,6 V 3,8 = V 2,6 3,6 V 3,14 V 3, = V 2,8 3,14 bh hlg wh prbabl 1 4, ach Plg hm p w ba ha r 3 : V 3,6 V 3,8 V 2,6 3,6 V 3,14 V 3, V 2,8 3,14 = 0 hlg wh a (pl-p) bas f 2( 1 4 )( 1 4 ) = 1 8 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 29 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 30 / 37 Lar Crpaalss Lar Pah Cmbg r apprxma f h hr lar f bxs r 3 : V 3,6 V 3,8 V 2,6 3,6 V 3,14 V 3, V 2,8 3,14 = 0 (Pr(r 3 ) = ) wh h prvs fr h frs w r 12 : V 2,6 V 2,8 = P 5 1,5 (P 7 1,7 ) P 8 1,8 2,6 (Pr(r 12 ) = ) r 123 : V 3,6 V 3,8 V 3,14 V 3, P 5 P 7 P 8 Σ k = 0 hlg wh bas 2( 1 8 )( 1 8 ) = 1 32, whr Σ k s Lar Crpaalss Drppg h k bs g h frh r k, a packg all h k bs Σ k w g r 4 : U 4,6 U 4,8 U 4,14 U 4, P 5 P 7 P 8 Σ k = 0 hlg wh a bas f 1 32 W w csr h sm f h vlv k bs as a sgl val Σ k : hs ca l assm 0 r 1 as s val s h k s fx, all h k bs hav (a ms) h ffc f flppg h sg f h rla bas whch s hs ± 1 32 Thr w g :) W hav a lar apprxma, p h k bs, frm h plax h p f h frh SBx lar! Σ k = 3,14 3,6 1,5 1,7 1,8 2,6 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 31 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 32 / 37

9 Lar Crpaalss Lar Crpaalss Cmpl lar apprxma pah S 11 S 12 S 21 S 22 S 31 S 32 S 41 S 42 Exracg h k bs fr h las r Gv a lar pah apprxma f h cphr, r, vlvg sm px bs a sm bs cmg frm h plma k-a lar, hlg wh bas ɛ Call paral sbks h k bs f h las r-k flc b h p bs f r apprxma Cllc N 1 ɛ 2 px/cx pars [1] 1 S 13 S 14 2 S 23 S 24 3 S 33 S 34 4 S 43 S Fr ach gv paral sbk val, sk, c hw ma px/cx pars mak h rla r hl (sa N sk ) 2 Cmp h sampl prbabls p = N sk N fr ach sk val 3 Chck whch val p = 1 2 ± ɛ sk s a caa sbk! Lahr, rs, rpa wh hr lar apprxmas xrac h hr paral sbks G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 33 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 34 / 37 Lar Crpaalss Frhr rags Smmar 1 alz h cphr Sbxs, cmpg h lar bas abl 2 Chs vabl rlas fr a S-Bx 3 Prpaga h bass p h las-b- mplg h Pl-p lmma 4 Cllc 1 ɛ 2 px/cx pars 5 Chck fr all h pssbl paral sbks vlv h lar apprxma whch maks hl wh h crrc bas 6 pa fr hr sbks, vall rmvg rs l all h k maral s f Pracc a ral wrl xampls C mplma f h whl aack a a cmpr fr h lar bass s avalabl h wbs ral wrl cphr kw b rrbl vlrabl lar crpaalss s FEL [2]: s l 4 rs, s ca r a crack Fr h mr rm: ca ra [1] xplr h fll DES lar crpaalss G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 35 / 37 G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 36 / 37

10 Bblgraph I Msr Mas. Lar crpaalss mh fr DES cphr. I Wrkshp h hr a applca f crpgraphc chqs vacs crplg, EUOCYPT 93, pags , Msr Mas a shr Yamagsh. w mh fr kw plax aack f fal cphr. I ar. ppl, r, vacs Crplg EUOCYPT 92, vlm 658 f Lcr Ns Cmpr Scc, pags Sprgr Brl Hlbrg, G. Pls,. Bargh (DEIB) Blck Cphr Crpaalss - Par 1 37 / 37

Mathematical Statistics. Chapter VIII Sampling Distributions and the Central Limit Theorem

Mathematical Statistics. Chapter VIII Sampling Distributions and the Central Limit Theorem Mahmacal ascs 8 Chapr VIII amplg Dsrbos ad h Cral Lm Thorm Fcos of radom arabls ar sall of rs sascal applcao Cosdr a s of obsrabl radom arabls L For ampl sppos h arabls ar a radom sampl of s from a poplao