1.1.5 Mt m Hill Mt m ny c pht minh vo nm 1929 bi Lester S. Hill. Cho mt s nguyn dng m v nh ngha P = C = (Z26)m. tng ca thut ton l ly m t hp tuyn tnh ca m k t ch ci trong mt phn t vn bn gc , theo sn xut m k t ch ci trong mt phn t vn bn m.

Hnh 1.6 Mt m HillCho m l mt s nguyn dng cho trc. Cho P = C = (Z26)m v cho K = {cc ma trn m m c nghch o trn Z26 }Cho mt kha K, chng ta nh ngha eK(x) = xKv dK(y) = yK-1 , vi K-1 l ma trn nghch o ca K, y tt c cc php ton c thc hin trong Z26

nh ngha 1.5nh thc ca ma trn 2 2 A = (ai,j) l gi tr det A = a1,1a2,2 a1.2a2,1Nhn xt: nh thc ca mt ma trn vung m m c th c tnh bi cc php ton c bn, xem trong cc sch i s tuyn tnh.Hai c tnh quan trng ca nh thc l det Im = 1 v qui tc nhn det(AB) = det A det B.

V d 1.5Gi s kha l K= T vic tnh ton ta thu cK-1 =

Gi s chng ta mun m ha vn bn july. Chng ta c 2 phn ca vn bn m l: (9,20) (tng ng ju) v (11,24) (tng ng ly). Chng ta tnh nh sau:(9,20) = (99 + 60, 72 + 140) = (3,4)v (11,24) = (121 + 72, 88 + 168) = (11,22).

V d: nu m = 2 chng ta c th vit mt phn t vn bn l x = (x1,x2) v mt phn t mt m l y = (y1,y2), y y1, y2 l mt t hp tuyn tnh ca x1 v x2. Chng ta c th c:y1 = 11x1 + 3x2y2 = 8x1 + 7x2Tt nhin ta cng c th vit di dng ma trn nh sau:(y1,y2) = (x1, x2)

Trong trng hp tng qut, chng ta s ly ma trn K m m l kha. Nu u vo hng i v ct j ca K l ki,j th chng ta vit K=(ki,j). Cho x = (x1,xm) P v K K, chng ta tnh y = eK(x) = (y1, .ym) nh sau:

(y1,y2, ,.ym) = (x1,x2,.,xm)

Cch vit khc y = xK

Chng ta ni rng vn bn m thu c t vn bn gc bng php bin i tuyn tnh. Chng ta phi xem xt vic gii m s c thc hin nh th no, lm th no tnh x t y. Nhng ngi hc i s tuyn tnh s nhn ra rng chng ta s dng ma trn nghch o K-1 gii m. Vn bn m c gii m s dng cng thc x = yK-1 trong mod 26.

Do , m ha ca july l DELW . gii m, Bob s tnh ton nh sau:

(3,4) = (9,20)

V (11,22) = (11,24).

Do , vn bn thu c l ng.

Mt ma trn s thc K c nghch o nu v ch nu nh thc ca n l khc khng. Tuy nhin, mt iu quan trng cn nh rng chng ta ang lm vic vt qu Z26. Kt qu lin quan ti mc ch ca chng ta l mt ma trn K c nghch o moldulo 26 nu v ch nu gcd(det K, 26) =1.

1.1.6 Mt m hon vTt c h thng mt m chng ta tho lun v su xa n bao hm s thay th: vn bn gc c thay th bi vn bn m khc. tng ca mt m hon v l gi nguyn vn bn gc nhng thay i v tr ca chng bng cch sp xp li chng. Mt m hon v (cn c gi l mt m chuyn i v tr) c s dng trong hng trm nm. Trong thc t, s khc bit gia mt m hon v v mt m thay th c ch rt sm t nm 1563 bi Giovanni Porta. Mt nh ngha hnh thc c cho trong hnh 1.7

Hnh 1.7 Mt m hon vCho m l mt s nguyn dng cho trc. Cho P = C = (Z26)m v cho K gm tt c cc hon v ca {1,,m}. Cho mt kha (ngha l mt hon v) chng ta nh ngha

V

y l hon v nghch o t .

V d 1.6Cho m = 6 v kha l hon v c cho nh sau:

Khi ta c hon v nghch o -1 l

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Gi s chng ta c vn bn Shesellsseashellsbytheseashoretrc tin chng ta nhm vn bn cho thnh cc nhm, mi nhm 6 ch ciShesel | lsseas | hellsb | ythese | ashore

By gi mi nhm gm 6 ch ci l sp xp ty hon v kt qu nh sau:ELSEHS | SSLASE | LBHSEL | HEYSTE | HEARSOV th vn bn m ha l:ELSEHS | SSLASE | LBHSEL | HEYSTE | HEARSO

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Vn bn m ha c th c gii m tng t nh cch m ha, s dng hon v nghch o -1Trn thc t, mt m hon v l trng hp c bit ca mt m Hill. Cho mt hon v ca ca tp hp {1,,m}, chng ta c th nh ngha ma trn hon v (kt hp) m m, K = (ki,j) vi

k j,i =

(mt ma trn hon v l mt ma trn m mi hng v ct u cha chnh xc mt gi tr 1 v cc v tr khc u cha gi tr 0. Mt ma trn hon v c th thu c t mt ma trn ng nht bng cch hon v cc hng v ct.)

1.1.7 Mt m dngTrong h thng mt m chng ta tm hiu vn ny, vn bn vi cc phn t k tip l mt m s dng kha K. vn bn m xu y thu c nh sau: y = y1y2=eK(x1)eK(x2)H thng mt m kiu ny thng c gi l mt m khi Mt cch tip cn khc l s dng ci c gi l mt m dng. tng c bn l sn sinh mt kha dng z = z1z2., v s dng n m ha xu gc x = x1x2ty theo qui tc y = y1y2=ez1(x1)ez2(x2)

hot ng ca mt m dng l nh sau: cho K K l kha v x1x2 l xu gc. Hm fi c s dng sn sinh zi (phn t th i ca kha dng), y fi l hm ca kha K v i-1 k t u ca xu gc:zi = fi (K, x1,,xi-1).phn t kha dng zi s dng m ha xi, kt qu yi = ezi(xi). v th m ha xu gc x1x2.....chng ta s tnh z1, y1, z2, y2Gii m xu m ha y1y2 c th c hon thnh bi vic tnh z1, x1, z2, x2.

V d:

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nh ngha 1.6Mt m dng l mt b (P,C,K,L,F, , D) tha mn cc iu kin sau:P l tp hp hu hn ca cc vn bn gcC l tp hp hu hn ca cc vn bn mK l tp hp hu hn ca cc kha L l tp hp hu hn gi l bng ch ci khaF = ( f1, f2..) l hm to kha. Vi i 1fi:K Pi-1 LVi mi z L, c mt qui tc m ha ez v tng ng c mt qui tc gii m dz Lz ez : P C v dz : C P l cc hm sao cho dz(ez(x)) = x vi mi vn bn gc x P.

Chng ta c th coi mt m dch chuyn l trng hp c bit ca mt m dng khi kha dng l hng zi = k vi mi i 1.

Mt v d ca mt m dng khng ng b c bit n l mt m kha t ng c cho trong hnh 1.9. nhn b ngoi ging vi mt m Vegenre.L do dng thut ng kha t ng l vn bn gc c s dng kha (ngoi tr kha ban u K). Hnh 1.9 Mt m kha t ngCho P = C = K = L = Z26, cho z1 = K v zi = xi-1 (i 2). Cho 0 z 25, nh nghaez(x) = x + z mod 26vdz(y) = y z mod 26(x,y thuc Z26).

y l v d minh ha:V d 1.8Gi s kha K = 8, v vn bn l rendezvous.Trc ht ta chuyn vn bn gc thnh mt dy s nguyn:17 4 13 3 4 25 21 14 20 18Kha dng l:8 17 4 13 3 4 25 21 14 20By gi chng ta cng cc phn t tng ng, qui v modulo 2625 21 17 16 7 3 20 9 8 12

i chiu trong bng ch ci ta c vn bn m l:ZVRQHDUJIMBy gi hy nhn xem Alice gii m nh th no. Trc ht c s chuyn xu ch ci thnh dy s nguyn25 21 17 16 7 3 20 9 8 12Sau c y tnhx1 = d8(25) = 25 8 mod 26 =17.tip theox2 = d17(21) = 21 17 mod 26 = 4. v tip tc nh vy. Mi ln c y nhn c ch ci vn bn gc khc nhau. C cng s dng n l phn t kha dng tip theo