Mật mã học và xác nhận chữ ký điện tử.doc

7/30/2019 Mt m hc v xc nhn ch k in t.doc

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TRNG I HC S PHM H NIKHOA CNG NGH THNG TIN

------------ ------------

NGHIN CU KHOA HC ti:

TM HIU MT M HC V NG DNGTRONG XC THC CH K IN T

Gio vin hng dn:PGS.TS.V nh HaSinh vin thc hin:Trnh Mai Hng

H ni ,2008


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Mc lcLi ni u .............................................................................................................. 4Chng 1.Tng quan v mt m hc ....................................................................5

1.1.Lch s pht trin ca mt m ........................................................................ 51.1.1.Mt m hc c in ........................................................................................................51.1.2.Thi trung c ................................................................................................................. 61.1.4.Mt m hc trong Th chin II ......................................................................................81.1.5.Mt m hc hin i .................................................................................................... 11

1.2.Mt s thut ng s dng trong h mt m ........................................................................ 161.3.nh ngha mt m hc .......................................................................................................191.4.Phn loi h mt m hc .....................................................................................................21

1.4.1.Mt m c in (ci ny ngy nay vn hay dng trong tr chi tm mt th).Da vo kiu ca php bin i trong h mt m c in, ngi ta chia h mt m

lm 2 nhm: m thay th (substitution cipher) v m hon v (permutation/ transposition

cipher)................................................................................................................................... 211.4.2.Mt m hin i ........................................................................................................... 23

Chng 2.H mt m c in ..............................................................................282.1.H m Caesar ......................................................................................................................282.2.H m Affinne .................................................................................................................... 292.3.H m Vigenre .................................................................................................................. 312.4.H mt Hill ......................................................................................................................... 332.5. H mt Playfair ..................................................................................................................34

Chng 3. Mt s cng c h tr cho thuyt mt m .......................................363.1.L thuyt s ........................................................................................................................ 36

3.1.1.Kin thc ng d thc ............................................................................................... 363.1.2.Mt s nh l s dng trong thut m ha cng khai ................................................ 38

3.2.L thuyt phc tp .........................................................................................................44Chng 4. H mt m cng khai .........................................................................47

4.1.Gii thiu mt m vi kha cng khai ................................................................................474.1.1.Lch s ......................................................................................................................... 474.1.2.L thuyt mt m cng khai ........................................................................................ 494.1.3.Nhng yu im, hn ch ca mt m vi kha cng khai ......................................... 514.1.4.ng dng ca mt m ..................................................................................................52

4.2.H mt RSA ........................................................................................................................ 544.2.1.Lch s ......................................................................................................................... 544.2.2.M t thut ton ...........................................................................................................55b. M ha .............................................................................................................................. 57c. Gii m ..............................................................................................................................57V d ..................................................................................................................................... 584.2.3.Tc m ha RSA .....................................................................................................594.2.4. an ton ca RSA .................................................................................................... 604.2.5.S che du thng tin trong h thng RSA ...................................................................63

4.3.H mt Rabin ...................................................................................................................... 664.3.1.M t gii thut Rabin ................................................................................................. 664.3.2.nh gi hiu qu ........................................................................................................ 68


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4.4.Ch k in t .................................................................................................................... 684.4.1.nh ngha ................................................................................................................... 704.4.2.Hm bm ......................................................................................................................714.4.3.Mt s s ch k in t ........................................................................................ 75

Chng 5. Xy dng phn mm ng dng ........................................................81

5.1.nh ngha bi ton .............................................................................................................815.2.Phn tch v thit k ............................................................................................................825.2.1. Qu trnh k trong Message ........................................................................................835.2.2. Qu trnh kim tra xc nhn ch k trn ti liu........................................................ 84

5.3.Chng trnh ci t ........................................................................................................... 87Chng trnh chy trn hu ht cc h iu hnh ca windows. Ci t bng ngn ng C#trn mi trng Visual Studio 2005. ....................................................................................... 87


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Li ni uHin nay , cng ngh thng tin, cng ngh Internet, cng ngh E-mail, E-

business pht trin nh v bo.Vit Nam , ang tng bc p dng cng ngh

mi tin hc ha x hi tc l a tin hc vo cc lnh vc ca x hi ci

thin hot ng th cng trc y.Tin hc ha gii phng sc lao ng ca

con ngi bng cch sng ch my ht bi, my git , my ra bt, cc con robot

lm vic trong hm m-ni rt nguy him v c hi cho sc khe ca con

ngi

Ngoi ra,Tin hc cn c a vo qun l hnh chnh Nh nc.Trong giai

on 2001-2005, Th tng Phan Vn Khi ph duyt nhiu n tin hc ha

qun l hnh chnh Nh nc vi mc tiu quyt tm xy dng mt Chnh ph

in t Vit Nam.Nu n ny thnh cng th ngi dn c th tm hiu thng

tin cn thit vn mang tnh giy t nh giy khai sinh, khai t, ng k lp hc,

xin thnh lp doanh nghip,xin cp h chiu, xin bo h tc quyn hay quyn s

hu cng nghipthng qua a ch mng m khng cn phi n c quan hnh

chnh.Nh vy chng ta c th trao i mi thng tin qua mng.Thng tin m

chng ta gi i c th l thng tin qun s, ti chnh, kinh doanh hoc n gin l

mt thng tin no mang tnh ring tiu ny dn ti mt vn xy ra l

Internet l mi trng khng an ton, y ri ro v nguy him, khng c g mbo rng thng tin m chng ta truyn i khng b c trm trn ng truyn. Do

, mt bin php c a ra nhm gip chng ta t bo v chnh mnh cng

nh nhng thng tin m chng ta gi i l cn phi m ha thng tin.Ngy nay

bin php ny c nhiu ni s dng nh l cng c bo v an ton cho bn

thn.Mt v d in hnh cc ngn hng li dng tnh nng ca m ha tch hp

cng ngh ch k s vo cc giao dch thng mi in t trc tuyn, m bo

tnh ton vn ca d liu, tnh b mt, tnh chng chi b giao dch (bng chng)trong cc giao dch thng mi in t online

V l mc ch chnh ca lun vn l tm hiu l thuyt mt m a l

thuyt ng dng vo thc t.


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Chng 1.Tng quan v mt m hc

1.1.Lch s pht trin ca mt m

Mt m hc l mt ngnh c lch s t hng nghn nm nay. Trong phn ln

thi gian pht trin ca mnh (ngoi tr vi thp k tr li y), lch s mt m

hc chnh l lch s ca nhng phng php mt m hc c in - cc phng

php mt m ha vi bt v giy, i khi c h tr t nhng dng c c kh n

gin. Vo u th k XX, s xut hin ca cc c cu c kh v in c, chng hn

nh my Enigma, cung cp nhng c ch phc tp v hiu qu hn cho vic

mt m ha. S ra i v pht trin mnh m ca ngnh in t v my tnh trong

nhng thp k gn y to iu kin mt m hc pht trin nhy vt ln mt

tm cao mi.

S pht trin ca mt m hc lun lun i km vi s pht trin ca cc k

thut ph m (hay thm m). Cc pht hin v ng dng ca cc k thut ph m

trong mt s trng hp c nh hng ng k n cc s kin lch s. Mt vi

s kin ng ghi nh bao gm vic pht hin ra bc in Zimmermann khin Hoa

K tham gia Th chin 1 v vic ph m thnh cng h thng mt m ca c

Quc x gp phn lm y nhanh thi im kt thc th chin II.

Cho ti u thp k 1970, cc k thut lin quan ti mt m hc hu nh

ch nm trong tay cc chnh ph. Hai s kin khin cho mt m hc tr nn

thch hp cho mi ngi, l: s xut hin ca tiu chun mt m ha DES v

s ra i ca cc k thut mt m ha kha cng khai.

1.1.1.Mt m hc c in

Nhng bng chng sm nht v s dng mt m hc l cc ch tng hnhkhng tiu chun tm thy trn cc bc tng Ai Cp c i (cch y khong

4500). Nhng k hiu t ra khng phi phc v mc ch truyn thng tin b

mt m c v nh l nhm mc ch gi nn nhng iu thn b, tr t m hoc

thm ch to s thch th cho ngi xem. Ngoi ra cn rt nhiu v d khc v

nhng ng dng ca mt m hc hoc l nhng iu tng t. Mun hn, cc hc


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gi v ting Hebrew c s dng mt phng php m ha thay th bng ch ci

n gin chng hn nh mt m ha Atbash (khong nm 500 n nm 600). Mt

m hc t lu c s dng trong cc tc phm tn gio che giu thng tin

vi chnh quyn hoc nn vn ha thng tr. V d tiu biu nht l "s ch k th

ca Cha" (ting Anh:Number of the Beast) xut hin trong kinh Tn c ca C

c gio. y, s 666 c th l cch m ha ch n ch La M hoc l

n hong Nero ca ch ny. Vic khng cp trc tip s gy rc ri

khi cun sch b chnh quyn ch . i vi C c gio chnh thng th vic che

du ny kt thc khi Constantine ci o v chp nhn o C c l tn gio

chnh thng ca ch.

Ngi Hy Lp c i cng c bit n l s dng cc k thut mt m(chng hn nh mt m scytale). Cng c nhng bng chng r rng chng t

ngi La M nm c cc k thut mt m (mt m Caesar v cc bin th).

Thm ch c nhng cp n mt cun sch ni v mt m trong qun i La

M; tuy nhin cun sch ny tht truyn.

Ti n , mt m hc cng kh ni ting. Trong cun sch Kama Sutra,

mt m hc c xem l cch nhng ngi yu nhau trao i thng tin m khng

b pht hin.

1.1.2.Thi trung c

Nguyn do xut pht c th l t vic phn tch bn kinh Quran, do nhu

cu tn gio, m k thut phn tch tn sut c pht minh ph v cc h

thng mt m n k t vo khong nm 1000. y chnh l k thut ph m c

bn nht c s dng, mi cho ti tn thi im ca th chin th II. V nguyn

tc, mi k thut mt m u khng chng li c k thut phn tch m(cryptanalytic technique) ny cho ti khi k thut mt m a k t c Alberti

sng to (nm 1465).

Mt m hc ngy cng tr nn quan trng di tc ng ca nhng thay

i, cnh tranh trong chnh tr v tn gio. Chng hn ti chu u, trong v sau


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thi k Phc hng, cc cng dn ca cc thnh bang thuc , gm c cc thnh

bang thuc gio phn v Cng gio La M, s dng v pht trin rng ri cc

k thut mt m. Tuy nhin rt t trong s ny tip thu c cng trnh ca Alberti

(cc cng trnh ca h khng phn nh s hiu bit hoc tri thc v k thut tn

tin ca Alberti) v do hu nh tt c nhng ngi pht trin v s dng cc h

thng ny u qu lc quan v an ton. iu ny hu nh vn cn ng cho ti

tn hin nay, nhiu nh pht trin khng xc nh c im yu ca h thng. Do

thiu hiu bit cho nn cc nh gi da trn suy on v hy vng l ph bin.

Mt m hc, phn tch m hc v s phn bi ca nhn vin tnh bo, ca

ngi a th, u xut hin trong m mu Babington din ra di triu i ca

n hong Elizabeth I dn n kt cc x t n hong Mary I ca Scotland. Mtthng ip c m ha t thi "ngi di mt n st" (Man in the Iron Mask)

(c gii m vo khong 1900 bi tienne Bazeries) cho bit mt s thng tin v

s phn ca t nhn ny (ng tic thay l nhng thng tin ny cng cha c r

rng cho lm). Mt m hc, v nhng lm dng ca n, cng l nhng phn t lin

quan n mu dn ti vic x t Mata Hari v m mu qu quyt dn n tr

h trong vic kt n Dreyfus v b t hai ngi u th k 20. May mn thay,

nhng nh mt m hc (cryptographer) cng nhng tay vo vic phi by mu

dn n cc khc mc ca Dreyfus; Mata Hari, ngc li, b bn cht.

Ngoi cc nc Trung ng v chu u, mt m hc hu nh khng

c pht trin. Ti Nht Bn, mi cho ti 1510, mt m hc vn cha c s

dng v cc k thut tin tin ch c bit n sau khi nc ny m ca vi

phng Ty (thp k 1860).

1.1.3.Mt m hc t nm 1800 n Th chin II

Tuy mt m hc c mt lch s di v phc tp, mi cho n th k 19 n

mi c pht trin mt cch c h thng, khng ch cn l nhng tip cn nht

thi, v t chc. Nhng v d v phn tch m bao gm cng trnh ca Charles

Babbage trong k nguyn ca Chin tranh Krim (Crimean War) v ton phn tch

mt m n k t. Cng trnh ca ng, tuy hi mun mng, c Friedrich


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Kasiski, ngi Ph, khi phc v cng b. Ti thi im ny, hiu c mt

m hc, ngi ta thng phi da vo nhng kinh nghim tng tri ( rules of

thumb); xin xem thm cc bi vit v mt m hc ca Auguste Kerckhoffs cui

th k 19. Trong thp nin 1840, Edgar Allan Poe xy dng mt s phng

php c h thng gii mt m. C th l, ng by t kh nng ca mnh

trong t bo hng tun Alexander's Weekly (Express) Messenger Philadelphia,

mi mi ngi trnh cc phng php m ha ca h, v ng l ngi ng ra

gii. S thnh cng ca ng gy chn ng vi cng chng trong vi thng. Sau

ny ng c vit mt lun vn v cc phng php mt m ha v chng tr thnh

nhng cng c rt c li, c p dng vo vic gii m ca c trong Th chin

II.

Trong thi gian trc v ti thi im ca Th chin II, nhiu phng php

ton hc hnh thnh (ng ch l ng dng ca William F. Friedman dng k

thut thng k phn tch v kin to mt m, v thnh cng bc u ca

Marian Rejewski trong vic b gy mt m ca h thng Enigma ca Qun i

c). Sau Th chin II tr i, c hai ngnh, mt m hc v phn tch m, ngy

cng s dng nhiu cc c s ton hc. Tuy th, ch n khi my tnh v cc

phng tin truyn thng Internet tr nn ph bin, ngi ta mi c th mang tnh

hu dng ca mt m hc vo trong nhng thi quen s dng hng ngy ca mi

ngi, thay v ch c dng bi cc chnh quyn quc gia hay cc hot ng kinh

doanh ln trc .

1.1.4.Mt m hc trong Th chin II

Trong th chin II, cc h thng mt m c kh v c in t c s dng rng

ri mc d cc h thng th cng vn c dng ti nhng ni khng iu

kin. Cc k thut phn tch mt m c nhng t ph trong thi k ny, tt cu din ra trong b mt. Cho n gn y, cc thng tin ny mi dn c tit l

do thi k gi b mt 50 nm ca chnh ph Anh kt thc, cc bn lu ca Hoa

K dn c cng b cng vi s xut hin ca cc bi bo v hi k c lin

quan.


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Ngi c s dng rng ri mt h thng my rto c in t, di

nhiu hnh thc khc nhau, c tn gi l my Enigma. Vo thng 12 nm 1932,

Marian Rejewski, mt nh ton hc ti Cc mt m Ba Lan (ting Ba Lan: Biuro

Szyfrw), dng li h thng ny da trn ton hc v mt s thng tin c c

t cc ti liu do i y Gustave Bertrand ca tnh bo qun s Php cung cp.

y c th coi l t ph ln nht trong lch s phn tch mt m trong sut mt

nghn nm tr li. Rejewski cng vi cc ng s ca mnh l Jerzy Rycki v

Henryk Zygalski tip tc nghin cu v bt nhp vi nhng tin ha trong cc

thnh phn ca h thng cng nh cc th tc mt m ha. Cng vi nhng tin

trin ca tnh hnh chnh tr, ngun ti chnh ca Ba Lan tr nn cn kit v nguy

c ca cuc chin tranh tr nn gn k, vo ngy 25 thng 7 nm 1939 ti

Warszawa, cc mt m Ba Lan, di ch o ca b tham mu, trao cho idin tnh bo Php v Anh nhng thng tin b mt v h thng Enigma.

Ngay sau khi Th chin II bt u (ngy 1 thng 9 nm 1939), cc thnh

vin ch cht ca cc mt m Ba Lan c s tn v pha ty nam; v n ngy 17

thng 9, khi qun i Lin X tin vo Ba Lan, th h li c chuyn sang

Romania. T y, h ti Paris (Php). Ti PC Bruno, gn Paris, h tip tc phn

tch Enigma v hp tc vi cc nh mt m hc ca Anh ti Bletchley Park lc

ny tin b kp thi. Nhng ngi Anh, trong bao gm nhng tn tui ln

ca ngnh mt m hc nh Gordon Welchaman v Alan Turing, ngi sng lp

khi nim khoa hc in ton hin i, gp cng ln trong vic pht trin cc

k thut ph m h thng my Enigma.

Ngy 19 thng 4 nm 1945, cc tng lnh cp cao ca Anh c ch th

khng c tit l tin tc rng m Enigma b ph, bi v nh vy n s to iu

kin cho k th b nh bi c s ni rng h "khng b nh bi mt cchsng phng" (were not well and fairly beaten).

Cc nh mt m hc ca Hi qun M (vi s hp tc ca cc nh mt m

hc Anh v H Lan sau 1940) xm nhp c vo mt s h thng mt m ca

Hi qun Nht. Vic xm nhp vo h thng JN-25 trong s chng mang li

chin thng v vang cho M trong trn Midway. SIS, mt nhm trong qun i


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M, thnh cng trong vic xm nhp h thng mt m ngoi giao ti mt ca

Nht (mt my c in dng "b chuyn mch dch bc" (stepping switch) c

ngi M gi l Purple) ngay c trc khi th chin II bt u. Ngi M t tn

cho nhng b mt m hc tm c t vic thm m, c th c bit l t vic ph

m my Purple, vi ci tn "Magic". Ngi Anh sau ny t tn cho nhng b mt

m h tm ra trong vic thm m, c bit l t lung thng ip c m ha bi

cc my Enigma, l "Ultra". Ci tn Anh trc ca Ultra lBoniface.

Qun i c cng cho trin khai mt s th nghim c hc s dng thut

ton mt m dng mt ln (one-time pad). Bletchley Park gi chng l m Fish, v

ng Max Newman cng ng nghip ca mnh thit k ra mt my tnh in t

s kh lp trnh (programmable digital electronic computer) u tin l myColossus gip vic thm m ca h. B ngoi giao c bt u s dng thut

ton mt m dng mt ln vo nm 1919; mt s lung giao thng ca n b

ngi ta c c trong Th chin II, mt phn do kt qu ca vic khm ph ra

mt s ti liu ch cht ti Nam M, do s bt cn ca nhng ngi a th ca

c khng hy thng ip mt cch cn thn.

B ngoi giao ca Nht cng cc b xy dng mt h thng da trn

nguyn l ca "b in c chuyn mch dch bc" (c M gi l Purple), vng thi cng s dng mt s my tng t trang b cho mt s ta i s

Nht Bn. Mt trong s chng c ngi M gi l "My-M" (M-machine), v

mt ci na c gi l "Red". Tt c nhng my ny u t nhiu b pha

ng Minh ph m.

SIGABA c miu t trong Bng sng ch ca M 6.175.625, trnh

nm 1944 song mi n nm 2001 mi c pht hnh


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Cc my mt m m phe ng minh s dng trong th chin II, bao gm c

my TypeX ca Anh v my SIGABA ca M, u l nhng thit k c in dngrto trn tinh thn tng t nh my Enigma, song vi nhiu nng cp ln. Khng

c h thng no b ph m trong qu trnh ca cuc chin tranh. Ngi Ba Lan s

dng my Lacida, song do tnh thiu an ninh, my khng tip tc c dng. Cc

phn i trn mt trn ch s dng my M-209 v cc my thuc h M-94 t bo

an hn. u tin, cc nhn vin mt v trong C quan c v ca Anh (Special

Operations Executive - SOE) s dng "mt m th" (cc bi th m h ghi nh l

nhng cha kha), song nhng thi k sau trong cuc chin, h bt u chuyn

sang dng cc hnh thc ca mt m dng mt ln (one-time pad).

1.1.5.Mt m hc hin i

Nhiu ngi cho rng k nguyn ca mt m hc hin i c bt u vi

Claude Shannon, ngi c coi l cha ca mt m ton hc. Nm 1949 ng

cng b bi L thuyt v truyn thng trong cc h thng bo mt

(Communication Theory of Secrecy Systems) trn tp san Bell System Technical

Journal- Tp san k thut ca h thng Bell - v mt thi gian ngn sau , trongcun Mathematical Theory of Communication - L thuyt ton hc trong truyn

thng - cng vi tc gi Warren Weaver. Nhng cng trnh ny, cng vi nhng

cng trnh nghin cu khc ca ng v l thuyt v tin hc v truyn thng

(information and communication theory), thit lp mt nn tng l thuyt c

bn cho mt m hc v thm m hc. Vi nh hng , mt m hc hu nh b
http://vi.wikipedia.org/wiki/H%C3%ACnh:SIGABA-patent.png


12/89

thu tm bi cc c quan truyn thng mt ca chnh ph, chng hn nh NSA, v

bin mt khi tm hiu bit ca cng chng. Rt t cc cng trnh c tip tc

cng b, cho n thi k gia thp nin 1970, khi mi s c thay i.

Thi k gia thp nin k 1970 c chng kin hai tin b cng chnh ln(cng khai). u tin l s cng b xut Tiu chun mt m ha d liu (Data

Encryption Standard) trong "Cng bo Lin bang" (Federal Register) nc M

vo ngy 17 thng 3 nm 1975. Vi c ca Cc Tiu chun Quc gia (National

Bureau of Standards - NBS) (hin l NIST), bn xut DES c cng ty IBM

(International Business Machines) trnh tr thnh mt trong nhng c gng

trong vic xy dng cc cng c tin ch cho thng mi, nh cho cc nh bng v

cho cc t chc ti chnh ln. Sau nhng ch o v thay i ca NSA, vo nm1977, n c chp thun v c pht hnh di ci tn Bn Cng b v Tiu

chun X l Thng tin ca Lin bang (Federal Information Processing Standard

Publication - FIPS) (phin bn hin nay l FIPS 46-3). DES l phng thc mt

m cng khai u tin c mt c quan quc gia nh NSA "tn sng". S pht

hnh bn c t ca n bi NBS khuyn khch s quan tm ch ca cng

chng cng nh ca cc t chc nghin cu v mt m hc.

Nm 2001, DES chnh thc c thay th bi AES (vit tt caAdvanced Encryption Standard- Tiu chun m ha tin tin) khi NIST cng b

phin bn FIPS 197. Sau mt cuc thi t chc cng khai, NIST chn Rijndael,

do hai nh mt m ngi B trnh, v n tr thnh AES. Hin nay DES v mt

s bin th ca n (nh Tam phn DES (Triple DES); xin xem thm trong phin

bn FIPS 46-3), vn cn c s dng, do trc y n c gn lin vi

nhiu tiu chun ca quc gia v ca cc t chc. Vi chiu di kho ch l 56-bit,

n c chng minh l khng sc chng li nhng tn cng kiu vt cn(brute force attack - tn cng dng bo lc). Mt trong nhng cuc tn cng kiu

ny c thc hin bi nhm "nhn quyn cyber" (cyber civil-rights group) tn l

T chc tin tuyn in t (Electronic Frontier Foundation) vo nm 1997, v

ph m thnh cng trong 56 ting ng h -- cu chuyn ny c nhc n trong

cun Cracking DES(Ph v DES), c xut bn bi "O'Reilly and Associates".


13/89

Do kt qu ny m hin nay vic s dng phng php mt m ha DES nguyn

dng, c th c khng nh mt cch khng nghi ng, l mt vic lm mo

him, khng an ton, v nhng thng ip di s bo v ca nhng h thng

m ha trc y dng DES, cng nh tt c cc thng ip c truyn gi t

nm 1976 tr i s dng DES, u trong tnh trng rt ng lo ngi. Bt chp

cht lng vn c ca n, mt s s kin xy ra trong nm 1976, c bit l s

kin cng khai nht ca Whitfield Diffie, ch ra rng chiu di kha m DES s

dng (56-bit) l mt kha qu nh. c mt s nghi ng xut hin ni rng mt

s cc t chc ca chnh ph, ngay ti thi im hi by gi, cng c cng

sut my tnh ph m cc thng ip dng DES; r rng l nhng c quan khc

cng c kh nng thc hin vic ny ri.

Tin trin th hai, vo nm 1976, c l cn t ph hn na, v tin trin

ny thay i nn tng c bn trong cch lm vic ca cc h thng mt m ha.

chnh l cng b ca bi vit phng hng mi trong mt m hc (New

Directions in Cryptography) ca Whitfield Diffie v Martin Hellman. Bi vit gii

thiu mt phng php hon ton mi v cch thc phn phi cc kha mt m.

y l mt bc tin kh xa trong vic gii quyt mt vn c bn trong mt m

hc, vn phn phi kha, v n c gi l trao i kha Diffie-Hellman

(Diffie-Hellman key exchange). Bi vit cn kch thch s pht trin gn nh tc

thi ca mt lp cc thut ton mt m ha mi, cc thut ton cha kha bt i

xng (asymmetric key algorithms).

Trc thi k ny, hu ht cc thut ton mt m ha hin i u l nhng

thut ton kha i xng (symmetric key algorithms), trong c ngi gi v

ngi nhn phi dng chung mt kha, tc kha dng trong thut ton mt m, v

c hai ngi u phi gi b mt v kha ny. Tt c cc my in c dng trongth chin II, k c m Caesar v m Atbash, v v bn cht m ni, k c hu ht

cc h thng m c dng trong sut qu trnh lch s na u thuc v loi ny.

ng nhin, kha ca mt m chnh l sch m (codebook), v l ci cng phi

c phn phi v gi gn mt cch b mt tng t.


14/89

Do nhu cu an ninh, kha cho mi mt h thng nh vy nht thit phi

c trao i gia cc bn giao thng lin lc bng mt phng thc an ton no

y, trc khi h s dng h thng (thut ng thng c dng l 'thng qua mt

knh an ton'), v d nh bng vic s dng mt ngi a th ng tin cy vi

mt cp ti liu c kha vo c tay bng mt cp kha tay, hoc bng cuc gp

g mt i mt, hay bng mt con chim b cu a th trung thnh... Vn ny

cha bao gi c xem l d thc hin, v n nhanh chng tr nn mt vic gn

nh khng th qun l c khi s lng ngi tham gia tng ln, hay khi ngi

ta khng cn cc knh an ton trao i kha na, hoc lc h phi lin tc thay

i cc cha kha - mt thi quen nn thc hin trong khi lm vic vi mt m. C

th l mi mt cp truyn thng cn phi c mt kha ring nu, theo nh thit k

ca h thng mt m, khng mt ngi th ba no, k c khi ngi y l mtngi dng, c php gii m cc thng ip. Mt h thng thuc loi ny c

gi l mt h thng dng cha kha mt, hoc mt h thng mt m ha dng kha

i xng. H thng trao i kha Diffie-Hellman (cng nhng phin bn c

nng cp k tip hay cc bin th ca n) to iu kin cho cc hot ng ny

trong cc h thng tr nn d dng hn rt nhiu, ng thi cng an ton hn, hn

tt c nhng g c th lm trc y.

Ngc li, i vi mt m ha dng kha bt i xng, ngi ta phi c

mt cp kha c quan h ton hc dng trong thut ton, mt dng m ha

v mt dng gii m. Mt s nhng thut ton ny, song khng phi tt c, c

thm c tnh l mt trong cc kha c th c cng b cng khai trong khi ci

kia khng th no (t nht bng nhng phng php hin c) c suy ra t kha

'cng khai'. Trong cc h thng ny, kha cn li phi c gi b mt v n

thng c gi bng mt ci tn, hi c v ln xn, l kha 'c nhn' (private

key) hay kha b mt. Mt thut ton thuc loi ny c gi l mt h thng'kha cng khai' hay h thng kha bt i xng. i vi nhng h thng dng

cc thut ton ny, mi ngi nhn ch cn c mt cp cha kha m thi (bt chp

s ngi gi l bao nhiu i chng na). Trong 2 kha, mt kha lun c gi b

mt v mt c cng b cng khai nn khng cn phi dng n mt knh an

ton trao i kha. Ch cn m bo kha b mt khng b l th an ninh ca h


15/89

thng vn c m bo v c th s dng cp kha trong mt thi gian di. c

tnh ng ngc nhin ny ca cc thut ton to kh nng, cng nh tnh kh thi,

cho php vic trin khai cc h thng mt m c cht lng cao mt cch rng ri,

v ai cng c th s dng chng c.

Cc thut ton mt m kha bt i xng da trn mt lp cc bi ton gi

l hm mt chiu (one-way functions). Cc hm ny c c tnh l rt d dng

thc hin theo chiu xui nhng li rt kh (v khi lng tnh ton) thc hin

theo chiu ngc li. Mt v d kinh in cho lp bi ton ny l hm nhn hai s

nguyn t rt ln. Ta c th tnh tch s ca 2 s nguyn t ny mt cch kh d

dng nhng nu ch cho bit tch s th rt kh tm ra 2 tha s ban u. Do

nhng c tnh ca hm mt chiu, hu ht cc kha c th li l nhng kha yuv ch cn li mt phn nh c th dng lm kha. V th, cc thut ton kha

bt i xng i hi di kha ln hn rt nhiu so vi cc thut ton kha i

xng t c an ton tng ng. Ngoi ra, vic thc hin thut ton

kha bt i xng i hi khi lng tnh ton ln hn nhiu ln so vi thut ton

kha i xng. Bn cnh , i vi cc h thng kha i xng, vic to ra mt

kha ngu nhin lm kha phin ch dng trong mt phin giao dch l kh d

dng. V th, trong thc t ngi ta thng dng kt hp: h thng mt m kha

bt i xng c dng trao i kha phin cn h thng mt m kha i

xng dng kha phin c c trao i cc bn tin thc s.

Mt m hc dng kha bt i xng, tc trao i kha Diffie-Hellman, v

nhng thut ton ni ting dng kha cng khai / kha b mt (v d nh ci m

ngi ta vn thng gi l thut ton RSA), tt c hnh nh c xy dng mt

cch c lp ti mt c quan tnh bo ca Anh, trc thi im cng b ca Diffie

and Hellman vo nm 1976. S ch huy giao thng lin lc ca chnh ph(Government Communications Headquarters - GCHQ) - C quan tnh bo Anh

Quc - c xut bn mt s ti liu qu quyt rng chnh h xy dng mt m

hc dng kha cng khai, trc khi bi vit ca Diffie v Hellman c cng b.

Nhiu ti liu mt do GCHQ vit trong qu trnh nhng nm 1960 v 1970, l

nhng bi cui cng cng dn n mt s k hoch i b phn tng t nh


16/89

phng php mt m ha RSA v phng php trao i cha kha Diffie-Hellman

vo nm 1973 v 1974. Mt s ti liu ny hin c pht hnh, v nhng nh

sng ch (James H. Ellis, Clifford Cocks, v Malcolm Williamson) cng cho

cng b (mt s) cng trnh ca h.

1.2.Mt s thut ng s dng trong h mt m

Sender/Receiver: Ngi gi/Ngi nhn d liu.

Vn bn (Plaintext -Cleartext): Thng tin trc khi c m ho. y l d liu

ban u dng r. Thng tin gc c ghi bng hnh nh m thanh, ch s, ch

vitmi tn hiu u c th c s ha thnh cc xu k t s

Ciphertext: Thng tin, d liu c m ho dng m Kha (key): Thnh phn quan trng trong vic m ho v gii m. Kha l ilng b mt, bin thin trong mt h mt. Kha nht nh phi l b mt. Kha

nht nh phi l i lng bin thin. Tuy nhin, c th c trng hp i lng

bin thin trong h mt khng phi l kha. V d: vector khi to (IV = Initial

Vector) ch CBC, OFB v CFB ca m khi.

CryptoGraphic Algorithm: L cc thut ton c s dng trong vic m ho hoc

gii m thng tin

H m (CryptoSystem hay cn gi l h thng m): H thng m ho bao gm

thut ton m ho, kho, Plaintext,Ciphertext

K thut mt m (cryptology) l mn khoa hc bao gm hai lnh vc: mt m

(crytography) v m thm (cryptoanalysis).

Mt m (cryptography) l lnh vc khoa hc v cc phng php bin i thng

tin nhm mc ch bo v thng tin khi s truy cp ca nhng ngi khng cthm quyn.

M thm (cryptoanalysis) l lnh vc khoa hc chuyn nghin cu, tm kim yu

im ca cc h mt t a ra phng php tn cng cc h mt . Mt

m v m thm l hai lnh vc i lp nhau nhng gn b mt thit vi nhau.

Khng th xy dng mt h mt tt nu khng hiu bit su v m thm. M thm


17/89

ch ra yu im ca h mt. Yu im ny c th c s dng tn cng h mt

ny nhng cng c th c s dng ci tin h mt cho tt hn. Nu ngi

xy dng h mt khng c hiu bit rng v m thm, khng kim tra an ton

ca h mt trc cc phng php tn cng th h mt ca anh ta c th t ra km

an ton trc mt phng php tn cng no m anh ta cha bit. Tuy nhin,

khng ai c th khng nh l c nhng phng php thm m no c bit

n. c nhim ca cc nc lun gi b mt nhng kt qu thu c trong lnh

vc m thm: k c phng php thm m v kt qa ca vic thm m.

S mt m l tp hp cc thut ton m ha, gi m, kim tra s ton vn v

cc chc nng khc ca mt h mt.

Giao thc mt m l tp hp cc quy tc, th tc quy nh cch thc s dng s

mt m trong mt h m. C th thy rng "giao thc mt m" v "s mt m"khng i lin vi nhau. C th c nhiu giao thc khc mt m khc nhau quy

nh cc cch thc s dng khc nhau ca cng mt s mt m no .

Lp m (Encrypt) l vic bin vn bn ngun thnh vn bn m

Gii m (Decrypt) l vic a vn bn m ha tr thnh dng vn bn ngun.

nh m (encode/decode) l vic xc nh ra php tng ng gia cc ch v s

- Tc m c c trng bi s lng php tnh (N) cn thc hin m ha

(gii m) mt n v thng tin. Cn hiu rng tc m ch ph thuc vo bnthn h m ch khng ph thuc vo c tnh ca thit b trin trin khai n (tc

my tnh, my m...).

an ton ca h m c trng cho kh nng ca h m chng li s thm

m; n c o bng s lng php tnh n gin cn thc hin thm h m

trong iu kin s dng thut ton (phng php) thm tt nht. Cn phi ni

thm rng c th xy dng nhng h mt vi an tan bng v cng (tc l

khng th thm c v mt l thuyt). Tuy nhin cc h mt ny khng thun

tin cho vic s dng, i hi chi ph cao. V th, trn thc t, ngi ta s dng

nhng h mt c gii hn i vi an tan. Do bt k h mt no cng c th

b thm trong thi gian no (v d nh sau... 500 nm chng hn).

Kh nng chng nhiu ca m l kh nng chng li s pht tn li trong bn tin

sau khi gii m, nu trc xy ra li vi bn m trong qu trnh bn m c

truyn t ngi gi n ngi nhn. C 3 loi li l:


18/89

li thay th k t: mt k t b thay i thnh mt k t khc.

V d: abcd atcd

li chn k t: mt k t c chn vo chui k t c truyn i.

V d: abcd azbcd

li mt k t: mt k t trong chui b mt.

V d: abcd abd.

Nh vy khi nim kh nng chng nhiu trong mt m c hiu khc

hn so vi khi nim ny trong lnh vc truyn tin. Trong truyn tin kh nng

chng nhiu l mt trong nhng c trng ca m chng nhiu (noise

combating code) - kh nng pht hin v sa li ca m chng nhiu. V d: m

(7,4) ca Hemming c th pht hin 2 li v sa 1 li trong khi 7 bits (4 bitsthng tin c ch v 3 bits dng kim tra v sa li).

M dng (Stream cipher) l vic tin hnh m ha lin tc trn tng k t hay

tng bit.

M khi (Block cipher) l vic tin hnh m trn tng khi vn bn.

Mc ch ca m ha l che du thng tin trc khi truyn trn knh truyn.

C nhiu phng php mt m khc nhau, tuy vy tt c chng c hai php ton

thc hin trong mt m l php m ha v gii m. C th biu th php mha v php ton gii m nh cc hm ca hai bin s, hoc c th nh mt thut

ton, c ngha l mt th tc i xng tnh kt qu khi gi tr cc tham s

cho.

Bn tin r y l tp hp cc d liu trc khi thc hin m ha. Kt qu

ca php m ha l bn tin c m ha. Vic gii m bn tin c m ha

s thu c bn tin r ban u. C biu thc bn tin r v bn tin m ha

u c lin quan n mt mt m c th. Cc ch ci vit hoa D (Decipherment)v E (Encipherment) l k hiu cho cc hm gii m v m ha tng ng. K

hiu x l l bn tin v y l bn tin m ha th biu thc ton hc ca php m

ha l:

y= Ek(x)

v ca php gii m l:


19/89

x=Dk(y)

Trong tham s ph k l kha m

Kha m l mt c tnh quan trng ca thut ton mt m.V nguyn l nu hm

y=E(x) khng c mt kha m no, th cng c th che du c gi tr ca x

Tp hp cc gi tr ca kho k c gi l khng gian cc kha. Trong mt mt

m no , nu kha m c 20 s thp phn s cho khn gian cc kha l 10 20 .

Nu kha no c 50 s nh phn th khng gian cc kha s l 250. Nu kha l

mt hon v ca 26 ch ci A,B,CZ th khng gian cc kha s l 26!

K hiu chung: P l thng tin ban u, trc khi m ho. E() l thut ton m ho.

D() l thut ton gii m. C l thng tin m ho. K l kho. Chng ta biu din qu

trnh m ho v gii m nh sau:

Qu trnh m ho c m t bng cng thc: Ek(P)=CQu trnh gii m c m t bng cng thc: Dk(C)=P

1.3.nh ngha mt m hci tng c bn ca mt m l to ra kh nng lin lc trn mt knh

khng mt cho hai ngi s dng (tm gi l Alice v Bob) sao cho i phng

(Oscar) khng th hiu c thng tin truyn i. Knh ny c th l mt ng

dy in thoi hoc mt mng my tnh. Thng tin m Alice mun gi cho Bob

(bn r) c th l bn ting anh, cc d liu bng s hoc bt k ti liu no c cu

trc ty . Alice s m ha bn r bng mt kha c xc nh trc v gi

bn m kt qu trn knh. Osar c bn m thu trm c trn knh song khng th

xc nh ni dung ca bn r, nhng Bob (ngi bit kha m) c th gii m

v thu c bn r.

Ta s m t hnh thc ha ni dung bng cch dng khi nim ton hc nh

sauMt h mt m l mt b 5 thnh phn (P,C,K,E,D) tha mn cc tnh cht sau:

1.Pl mt tp hu hn cc bn r c th

2.Cl mt tp hu hn cc bn m c th

3.K(khng gian kha) l tp hu hn cc kha c th


20/89

4.i vi mi kK c mt quy tc m ek: PC v mt quy tc gii m

tng ng dkD. Mi ek:PCv dk:CPl nhng hm

Dk(ek(x))=x vi mi bn r xP

Trong tnh cht 4 l tnh cht ch yu y. Ni dung ca n l nu mtbn r x c m ha bng ek v bn m nhn c sau c gii m bng d k

th ta phi thu c bn r ban u x. Alice v Bob s p dng th tc sau khi

dng h mt kha ring. Trc tin h chn mt kha ngu nhin kK. iu ny

c thc hin khi h cng mt ch v khng b Oscar theo di hoc h c mt

knh mt trong trng hp h xa nhau. Sau gi s Alice mun gi mt thng

bo cho Bob trn mt knh khng mt v ta xem thng bo ny l mt chui:

x = x1,x2 ,. . .,xn

vi s nguyn n1 no . y mi k hiu ca mi bn r xiP, 1 i n. Mi

xi s c m ha bng quy tc m ek vi kha k xc nh trc .Bi vy Alice

s tnh yi =ek(xi), 1 i n v chui bn nhn c

y = y1,y2 ,. . .,yn

s c gi trn knh. Khi Bob nhn c y = y1,y2 ,. . .,yn anh ta s gii m bng

hm gii m dk v thu c bn r gc x1,x2 ,. . .,xn. Hnh 1.1. l mt v d v mt

knh lin lc


21/89

R rng trong trng hp ny hm m ho phi l hm n nh (tc l nh x 1-

1), nu khng vic giai rmax s khng thc hin c mt cch tng minh. V d

y= ek(x1)=ek(x2)

trong x1 x2, th Bob s khng c cch no bit liu s phi gii m thnh x1 hay

x2. Ch rng nu P = Cth mi hm m ha ize=2. Bn quyn Cng ty Pht

tp cc bn m v tp cc bn r l ng nht th mi mt hm m s l mt s spxp li (hay hon v) cc phn t ca tp ny

1.4.Phn loi h mt m hcLch s ca mt m hc chnh l lch s ca phng php mt m hc c

in- phng php m ha bt v giy. Sau ny da trn nn tng ca mt m hc

c in xut hin phng php m ha mi. Chnh v vy mt m hc c

phn chia thnh mt m hc c in v mt m hc hin i

1.4.1.Mt m c in (ci ny ngy nay vn hay dng trong tr chi tm mtth).

Da vo kiu ca php bin i trong h mt m c in, ngita chia h mt m lm 2 nhm: m thay th (substitution cipher) v mhon v (permutation/ transposition cipher).

Oscar

B gii mB m ha BobAlice

Knh anton

Ngun kha


22/89

Substitution: thay th phng php m ha trong tng k t (hoc tng

nhm k t) ca vn bn ban u(bn r - Plaintext) c thay th bng mt (hay

mt nhm) k t khc to ra bn m (Ciphertext). Bn nhn ch cn o ngc

trnh t thay th trn Ciphertext c c Plaintext ban u.

Mt v d v m thay th thun ty l m bng t in. Ngi lm cng

tc mt m c mt quyn t in. m ha mt bn tin (dng vn bn), anh ta

tm t hoc cm t ca bn tin trong t in v thay bng mt nhm ch s tng

ng. N ging nh tra t in Vit-XXX, trong XXX l th ngn ng m ch

bao gm cc ch s, ng thi cc t lun c di c nh (thng l 4-5 ch

s). Sau khi dch t ting Vit sang ting XXX, ngi ta s cng tng t trong

ca vn bn (trong ting XXX) vi kha theo module no . Kha cng l mtt ngu nhin trong ting XXX.

Mt v d n gin na minh ha m thay th: cho mt vn bn ch gm

cc k t latin, tm trong cc nguyn m (a,e,i,o,u) v bin i chng theo quy tc

a thay bi e, e thay bi i,.... , u thay bi a.

V d 2: Vit trn mt dng cc k t trong bng ch ci theo ng th t.Trn

dng th hai, cng vit ra cc k t ca bn ch ci nhng khng bt u bng

ch a m bng ch f chng hn. m ha mt k t ca bn r , hy tm n

trn dng th nht , thay n bi k t nm trn dng th hai (ngay di n).

Thay th n tr v thay th a tr l hai trng hp ring ca m thay

th.Tr li vi v d v m t in, vi ngn ng XXX nu trn.Nu nh trong

t in, 1 t Ting Vit tng ng vi 1 v ch 1 t ting XXX th l m thay

th n tr.Cn nu mt t Ting Vit tng ng vi 2 hay nhiu hn 2 t trong

ting XXX (tc l nhiu t trong ting XXX c cng mt ngha trong Ting Vit)

th l m thay th a tr.

Tuy khng cn c s dng nhng tng ca phng php ny vn c tiptc trong nhng thut ton hin i

Transposition: hon v

Bn cnh phng php m ho thay th th trong m ho c in c mt

phng php khc na cng ni ting khng km, chnh l m ho hon v.


23/89

Nu nh trong phng php m ho thay th, cc k t trong Plaintext c thay

th hon ton bng cc k t trong Ciphertext, th trong phng php m ho hon

v, cc k t trong Plaintext vn c gi nguyn, chng ch c sp xp li v tr

to ra Ciphertext. Tc l cc k t trong Plaintext hon ton khng b thay i

bng k t khc.

C th phng php hon v l phng php m ha trong cc k t

trong vn bn ban u ch thay i v tr cho nhau cn bn thn cc k t khng h

b bin i.

V d n gin nht: m ha bn r bng cch o ngc th t cc k t

ca n. Gi s bn r ca bn c di N k t. Bn s hon i v tr k t th 1

v k t N, k t 2 v k t N-1,Phc tp hn mt cht, hon v khng phiton b bn r m chia nios ra cc on vi di L v thc hin php hon v

theo tng on.Khi L s l kha ca bn! Mt khc L c th nhn gi tr tuyt

i (2,3,4) hoc gi tr tng i (1/2,1/3,1/4ca N).

Vo khong th k V-IV trc Cng nguyn, ngi ta ngh ra thit b

m ha. l mt ng hnh tr vi bn hnh R. m ha, ngi ta qun bng

giy (nh, di nh giy dng trong in tn) quanh ng hnh tr ny v vit ni

dung cn m ha ln giy theo chiu dc ca ng. Sau khi g bng giy khi ng

th ni dung s c che du. Muoons gii m th phi cun bng giy ln ng

cng c bn knh R.Bn knh R chnh l kha trong h mt ny.

1.4.2.Mt m hin i

a. Symmetric cryptography: m ha i xng, tc l c hai qu trnh m ha v

gii m u dng mt cha kha. m bo tnh an ton, cha kha ny phi

c gi b mt. V th cc thut ton loi ny cn c tn gi khc l secret key

cryptography (hay private key cryptography), tc l thut ton m ha dng chakha ring (hay b mt). Cc thut ton loi ny l tng cho mc ch m ha d

liu ca c nhn hay t chc n l nhng bc l hn ch khi thng tin phi

c chia s vi mt bn th hai.


24/89

Gi s nu Alice ch gi thng ip m ha cho Bob m khng h bo

trc v thut ton s dng, Bob s chng hiu Alice mun ni g. V th bt buc

Alice phi thng bo cho Bob v cha kha v thut ton s dng ti mt thi

im no trc y. Alice c th lm iu ny mt cch trc tip (mt i mt)

hay gin tip (gi qua email, tin nhn...). iu ny dn ti kh nng b ngi th

ba xem trm cha kha v c th gii m c thng ip Alice m ha gi cho

Bob.

Hnh 1.Thut ton m ha i xng

Bob v Alice c cng mt kha KA-B. Kha ny c xy dng sao cho:

m = KA-B(KA-B(m)).

Trn thc t, i vi cc h mt i xng, kho K lun chu s bin i

trc mi pha m ha v gii m. Kt qu ca s bin i ny pha gii m Kd s

khc vi kt qu bin i pha m ha Ke.Nu coi Ke v Kd ln lt l kha m

ha v kha gii m th s c kha gii m khng trng vi kha m ha. Tuy

nhin nu bit c kha Ke th c th d dng tnh c Kd v ngc li. Vy nn

c mt nh ngha rng hn cho m i xng l: M i xng l nhm m trong

kha dng gii m Kd c th d dng tnh c t kha dng m ha Ke.

Trong h thng m ho i xng, trc khi truyn d liu, 2 bn gi v

nhn phi tho thun v kho dng chung cho qu trnh m ho v gii m. Sau, bn gi s m ho bn r (Plaintext) bng cch s dng kho b mt ny v gi

thng ip m ho cho bn nhn. Bn nhn sau khi nhn c thng ip

m ho s s dng chnh kho b mt m hai bn tho thun gii m v ly li

bn r (Plaintext). Trong qu trnh tin hnh trao i thng tin gia bn gi v bn

nhn thng qua vic s dng phng php m ho i xng, th thnh phn quan


25/89

trng nht cn phi c gi b mt chnh l kho. Vic trao i, tho thun v

thut ton c s dng trong vic m ho c th tin hnh mt cch cng khai,

nhng bc tho thun v kho trong vic m ho v gii m phi tin hnh b

mt. Chng ta c th thy rng thut ton m ho i xng s rt c li khi c

p dng trong cc c quan hay t chc n l. Nhng nu cn phi trao i thng

tin vi mt bn th ba th vic m bo tnh b mt ca kho phi c t ln

hng u.

M ha i xng c th phn thnh hai nhm ph:

- Block ciphers: thut ton khi trong tng khi d liu trong vn bn

ban u c thay th bng mt khi d liu khc c cng di. di

mi khi gi l block size, thng c tnh bng n v bit. V d thut

ton 3-Way c kch thc khi bng 96 bit. Mt s thut ton khi thngdng l:DES, 3DES, RC5, RC6, 3-Way, CAST, Camelia, Blowfish, MARS,

Serpent, Twofish, GOST...

- Stream ciphers: thut ton dng trong d liu u vo c m ha

tng bit mt. Cc thut ton dng c tc nhanh hn cc thut ton khi,

c dng khi khi lng d liu cn m ha cha c bit trc, v d

trong kt ni khng dy. C th coi thut ton dng l thut ton khi vi

kch thc mi khi l 1 bit. Mt s thut ton dng thng dng: RC4,

A5/1, A5/2, Chameleon

b. Asymmetric cryptography: m ha bt i xng, s dng mt cp cha kha c

lin quan vi nhau v mt ton hc, mt cha cng khai dng m ho (public

key) v mt cha b mt dng gii m (private key). Mt thng ip sau khi

c m ha bi cha cng khai s ch c th c gii m vi cha b mt tng

ng. Do cc thut ton loi ny s dng mt cha kha cng khai (khng b mt)nn cn c tn gi khc lpublic-key cryptography (thut ton m ha dng cha

kha cng khai). Mt s thut ton bt i xng thng dng l : RSA, Elliptic

Curve, ElGamal, Diffie Hellman...

Quay li vi Alice v Bob, nu Alice mun gi mt thng ip b mt ti

Bob, c ta s tm cha cng khai ca Bob. Sau khi kim tra chc chn cha kha


26/89

chnh l ca Bob ch khng ca ai khc (thng qua chng ch in t digital

certificate), Alice dng n m ha thng ip ca mnh v gi ti Bob. Khi

Bob nhn c bc thng ip m ha anh ta s dng cha b mt ca mnh

gii m n. Nu gii m thnh cng th bc thng ip ng l dnh cho Bob.

Alice v Bob trong trng hp ny c th l hai ngi cha tng quen bit. Mt

h thng nh vy cho php hai ngi thc hin c giao dch trong khi khng

chia s trc mt thng tin b mt no c.

Hnh 2.Thut ton m ha bt i xng

Trong v d trn ta thy kha public v kha private phi p ngv t kha public ngi ta khng th tm ra c kha

private.

M ho kho cng khai ra i gii quyt vn v qun l v phn phi kho

ca cc phng php m ho i xng. Qu trnh truyn v s dng m ho kho

cng khai c thc hin nh sau:

- Bn gi yu cu cung cp hoc t tm kho cng khai ca bn nhn trn

mt server chu trch nhim qun l kho.

- Sau hai bn thng nht thut ton dng m ho d liu, bn gi s

dng kho cng khai ca bn nhn cng vi thut ton thng nht m

ho thng tin c gi i.

- Khi nhn c thng tin m ho, bn nhn s dng kho b mt ca

mnh gii m v ly ra thng tin ban u.


27/89

Vy l vi s ra i ca M ho cng khai th kho c qun l mt cch linh

hot v hiu qu hn. Ngi s dng ch cn bo v Private key. Tuy nhin nhc

im ca M ho kho cng khai nm tc thc hin, n chm hn rt nhiu

so vi m ho i xng. Do , ngi ta thng kt hp hai h thng m ho kho

i xng v cng khai li vi nhau v c gi l Hybrid Cryptosystems. Mt s

thut ton m ho cng khai ni ting: Diffle-Hellman, RSA,

Trn thc t h thng m ho kho cng khai c hn ch v tc chm nn cha

th thay th h thng m ho kho b mt c, n t c s dng m ho d

liu m thng dng m ho kho. H thng m ho kho lai ra i l s kt

hp gia tc v tnh an ton ca hai h thng m ho trn. V vy ngi ta

thng s dng mt h thng lai tp trong d liu c m ha bi mt thut

ton i xng, ch c cha dng thc hin vic m ha ny mi c m habng thut ton bt i xng. Hay ni mt cch khc l ngi ta dng thut ton

bt i xng chia s cha kha b mt ri sau dng thut ton i xng vi

cha kha b mt trn truyn thng tin.

Chng ta c th hnh dung c hot ng ca h thng m ho ny nh

sau:

- Bn gi to ra mt kho b mt dng m ho d liu. Kho ny cn

c gi l Session Key.- Sau , Session Key ny li c m ho bng kho cng khai ca bn

nhn d liu.

- Tip theo d liu m ho cng vi Session Key m ho c gi i ti

bn nhn.

- Lc ny bn nhn dng kho ring gii m Session Key v c c

Session Key ban u.

- Dng Session Key sau khi gii m gii m d liu.

Nh vy, h thng m ho kho lai tn dng tt c cc im mnh ca hai h

thng m ho trn l: tc v tnh an ton. iu ny s lm hn ch bt kh

nng gii m ca tin tc.

Cn lu rng trn y, chng ta nhc n hai khi nim c tnh cht

tng i l d v kh. Ngi ta quy c rng nu thut ton c phc tp


28/89

khng vt qu phc tp a thc th bi ton c coi l d; cn ln hn th

bi ton c coi l kh.

Chng 2.H mt m c in

2.1.H m CaesarH m Caesar c xc nh trn Z26 (do c 26 ch ci trn bng ch ci

ting Anh) mc d c th xc nh n trn Zm vi modulus m ty .D dng thy

rng , m dch vng s to nn mt h mt nh xc nh trn, tc l Dk(Ek(x))

= x vi xZ26.

nh ngha:

Mt h mt gm b 5 (P,C,K,E,D). Gi s P = C = K = Z26 vi 0 k 25,

nh ngha:Ek(x)=x+k mod 26

V Dk(x)=y-k mod 26 (x,y Z26)

Nhn xt:Trong trng hp k=3, h mt thng c gi l m Caesar tng

c Julius Caesar s dng

Ta s s dng m dch vng (vi modulo 26) m ha mt vn bn ting Anh

thng thng bng cch thit lp s tng ng gia cc k t v cc thng d theo

modulo 26 nh sau: A0, B1,.,Z25.

A B C D E F G H I J K L M0 1 2 3 4 5 6 7 8 9 1

0

1

1

12

N O P Q R S T U V W X Y Z1 1 1 1 1 1 1 2 2 2 2 2 25


29/89

3 4 5 6 7 8 9 0 1 2 3 4

V d

Gi s kha cho m dch vng k=11 v bn r l: wewillmeetatmidnight

Trc tin bin i bn r thnh dy cc s nguyn nh dng php tng ngtrn.Ta c:

22 4 22 8 11 11 12 4 4 190 19 12 8 3 13 8 6 7 19

Sau cng 11 vo mi gi tr ri rt gn tng theo modulo 26

7 15 7 19 22 22 23 15 15 411 4 23 19 14 24 19 17 18 4

Cui cng bin i dy s nguyn ny thnh cc k t thu c bn m sau

HPHTWWXPPELEXTOYTRSE

gi m bn m ny, trc tin, Bob s bin i bn m thnh dy cc s

nguyn ri tr i gi tr cho 11 (rt gn modulo 26) v cui cng bin i li dy

ny thnh cc k t

2.2.H m Affinnenh ngha: M tuyn tnh Affinne l b 5 (P,C,K,E,D) tha mn:

1.Cho P=C=Z26 v gi s P={(a,b) Z26 x Z26:UCLN(a,26)=1}

2.Vi k=(a,b) K, ta nh ngha:

Ek(x)=ax+bmod26

V Dk(y)=a-1(y-b)mod26, x,yZ26

vic gii m thc hin c, yu cu cn thit l hm Affine phi l nnh.Ni cch khc, vi bt k yZ26, ta mun c ng nht thc sau:

ax+b y(mod26)

phi c nghim x duy nht.ng d thc ny tng ng vi

ax y-b(mod 26)


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v y thay i trn Z26 nn y-b cng thay i trn Z26.Bi vy, ta ch cn nghin cu

phng trnh ng d:

ax y(mod 26) (yZ26)

ta bit rng phng trnh ny c mt nghim duy nht i vi mi y khi v ch khi

UCLN(a,26)=1.

Chng minh:Trc tin ta gi s rng, UCLN(a,26)=d>1. Khi , ng d thc

ax 0(mod26) s c t nht hai nghim phn bit trong Z26 l x=0 v x=26/d.

Trong trng hp ny, E(x)=ax+b(mod 26) khng pahir l mt hm n nh v

bi vy n khng th l hm m ha hp l.

V d do UCLN(4,26)=2 nn 4x+7 khng l hm m ha hp l: x v x+13 s m

ha thnh cng mt gi tr i vi bt k xZ26.

Ta gi thit UCLN(a,26)=1.Gi s vi x1 v x2 no tha mn:

ax1 ax2(mod 26)

Khi

a(x1 x2) 0 (mod 26)

bi vy 26| a(x1 x2)

By gi ta s s dng mt tnh cht ca php chia sau: Nu UCLN(a,b)=1 v a | bc

th a |c. V 26 | a(x1 x2) v UCLN(a,26)=1 nn ta c:26 |(x1 x2)

Tc l

x1 x2 (mod 26)

Ti y ta chng t rng, nu UCLN(a,26)=1 th mt ng d thc dng ax y

(mod 26) ch c nhiu nht mt nghim trong Z26.D , nu ta cho x thay i trn

Z26 th ax mod 26 s nhn c 26 gi tr khc nhau theo modulo 26 v ng d

thc ax y(mod 26) ch c nghim duy nht.V d:

Gi s k=(7,3).Ta c 7-1 mod 26= 15.Hm m ha l:

Ek(x)=7x+3

V hm gii m tng ng l

Dk(x)=15(y-3) mod 26=15y-19


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y tt c cc php ton u thc hin trn Z26. Ta s kim tra liu Dk(Ek(x))=x

vi xZ26 khng? Dng cc tnh ton trn Z26, ta c

Dk(Ek(x))= Dk(7x+3)

= 15(7x+3)-19

=x+45-19

=x

minh ha, ta hy m ha bn r hot. Trc tin bin i cc ch h,o,t thnh

cc thng d theo modulo 26. Ta c cc s tng ng l: 7, 14 v 19.By gi

m ha:

7 7 +3 mod 26 = 52 mod 26 = 0

7 14 + 3 mod 26 = 101 mod 26 =23

7 19 +3 mod 26 = 136 mod 26 = 6

By gi 3 k t ca bn m l 0, 23 v 6 tng ng vi xu k t AXG.

Gii m: t xu k t ca bn m chuyn thnh s nguyn trong bng ch ci

ting Anh (26 ch ci), ta c cc s tng ng 0, 23, 6

Dk(0)=15 0- 19 mod 26 =7

Dk(23)=15 23- 19 mod 26 =14

Dk(6)=15 6- 19 mod 26 =19

By gi 3 k t ca bn r: h, o, t.

2.3.H m VigenreTrong c hai h m dch chuyn v m tuyn tnh(mt khi kha c chn )

mi k t s c nh x vo mt k t duy nht. V l do , cc h mt cn li

c gi l h thay th n biu. By gi ti s trnh by mt h mt khng phi

l b ch n, l h m Vigenre ni ting. Mt m ny ly tn ca Blaise de

Vigenre sng vo th k XVI.

S dng php tng ng A 0, B 1, .,Z25 m t trn, ta c th gn cho

mi kha k vi mt chui k t c di m c gi l t kha.Mt m V s m

ha ng thi m k t: mi phn t ca bn r tng ng vi m k t

V d


32/89

Gi s m=6 v t kha l CIPHER. T kha ny tng ng vi dy s

k=(2,8,15,4,17).Gi s bn r l xu

thiscryptosystemisnotsecure

nh ngha:

Cho m l mt s dng c nh no . Cho P=C=K=(Z26)m. Vi kha K=(k1, k2 ,

,km) ta xc nh:

EK(x1, x2, . . . ,xm) = (x1+k1, x2+k2, . . . , xm+km)

v

DK(y1, y2, . . . ,ym) = (y1-k1, y2-k2, . . . , ym-km)

Trong tt c cc php ton c thc hin trong Z26

Ta s bin i cc phn t ca bn r thnh cc thng d theo modulo 26,

vit chng thnh cc nhm 6 ri cng vi t kha theo modulo nh sau19 7 8 18 2 17 24 15 19 14 18 24

2 8 15 7 4 17 2 8 15 7 4 17

21 15 23 25 6 8 0 23 8 21 22 15

18 19 4 12 8 18 13 14 19 18 4 2

2 8 15 7 4 17 2 8 15 7 4 17

20 1 19 19 12 9 15 22 8 15 8 19

20 17 4

2 8 15

22 25 19

Bi vy, dy k t tng ng ca xu bn m s l:

V P X Z G I A X I V W P U B T T M J P W I Z I T W Z T

gii m ta c th dng cng t kha nhng thay cho cng, ta tr n theo

modulo 26


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Ta thy rng cc t kha c th vi s di m trong mt m Vigenre l

26m, bi vy, thm ch vi cc gi tr m kh nh, phng php tm kim vt cn

cng yu cu thi gian kh ln. V d, nu m=5 th khn gian kha cng c kch

thc ln hn 1,1 107. Lng kha ny ln ngn nga vic tm kha bng

tay

Trong h mt Vigenre c t kha di m, mi k t c th c nh x

vo trong m k t c th c (gi s rng t kha cha m k t phn bit).Mt h

mt nh vy c gi l h mt thay th a kiu (poly alphabetic). Ni chung,

vic thm m h thay th a kiu s kh khn hn so vic thm m h n kiu.

2.4.H mt HillTrong phn ny s m t mt h mt thay th a kiu khc c gi l mt

m Hill. Mt m ny do Lester S.Hill a ra nm 1929. Gi s m l mt s

nguyn, tP = C = (Z26)m . tng y l ly t hp tuyn tnh ca m k t

trong mt phn t ca bn r to ra m k t mt phn t ca bn m.

nh ngha: Mt m Hill l b 5(P, C, K, E, D). Cho m l mt s nguyn dng c

nh. ChoP = C = (Z26)m v cho

K={cc ma trn kh nghch cp m m trn Z26}

Vi mt kha KK ta xc nh

EK(x) = xK

v DK(y) = yK-1

tt c cc php ton c thc hin trong Z26

V d

Gi s kha

T cc tnh ton trn ta c


34/89

Gi s cn m ha bn r July. Ta c hai phn t ca bn r m ha:(9,20)

(ng vi Ju) v (11,24)(ng vi ly). Ta tnh nh sau:

V

Bi vy bn m ca july l DELW. gii m Bob s tnh

V

Nh vy Bob nhn c bn ng

Cho ti lc ny ta ch ra rng c th thc hin php gii m nu K c mt

nghch o. Trn thc t, php gii m l c th thc hin c, iu kin cn

l K phi c nghch o. (iu ny d dng rt ra t i s tuyn tnh s cp).

2.5. H mt PlayfairPhp thay th n-gram:thay v thay th i vi cc k t, ngi ta c th thay

th cho tng cm 2 k t (gi l digram) hoc cho tng cm 3 k t (gi l trigram)

v tng qut cho tng cm n k t (gi l n-gram). Nu bng ch ci gm 26 k

t ting Anh th php thay th n-gram s c kho l mt hon v ca 26n n-gram

khc nhau. Trong trng hp digram th hon v gm 262 digram v c th biu

din tt nht bng mt dy 2 chiu 26 26 trong cc hng biu din k hiu

u tin, cc ct biu din k hiu th hai, ni dung ca cc biu din chui thayth. V d bng 2 chiu sau biu th AA c thay bng EG, AB c thay bng

RS, BA c thay bng BO, BB c thay bng SC,

A B A EG RSB BO SC


35/89

y l mt s da trn s thay th digram trong kho l mt hnh vung

kch thc 5 5 cha mt s sp xp no ca 25 k t ca bng ch ci (khng

tnh k t J v s xut hin t ca n v c th thay n bng I). Gi s chng ta c

ma trn kho nh sau

B Y D G Z

W S F U P

L A R K X

C O I V E

Q N M H T

S thay th s c thc hin nh sau. Chng hn nu digram cn thay th

l AV th trong hnh ch nht c A, V l hai nh cho nhau thay A bng nh k

ca n theo ng thng ng chnh l O v tng t thay V bng nh k ca ntheo ng thng ng chnh l K.

Tng t nu digram cn thay th l VN th chui thay th l HO. Nu cc k t

ca digram nm trn hng ngang th chui thay th l cc k t bn phi ca

chng. Chng hn nu digram l WU th chui thay th l SP, nu digram l FP th

chui thay th l UW, nu digram l XR th chui thay th l LK. Tng t nu

cc k t ca digram nm trn hng dc th chui thay th l cc k t bn di ca

chng. Chng hn nu digram l SO th chui thay th l AN, nu digram l MRth chui thay th l DI, nu digram l GH th chui thay th l UG. Trong trng

hp digram l mt cp k t ging nhau chng hn OO hoc l mt k t c i

km mt khong trng chng hn B th c nhiu cch x l, cch n gin nht

l gi nguyn khng bin i digram ny.


36/89

Chng 3. Mt s cng c h tr cho thuyt mt m

3.1.L thuyt s3.1.1.Kin thc ng d thc

a. nh ngha: Cho l s nguyn dng. Hai s nguyn v c gi l ng

d vi nhau theo module m nu hiu a

K hiu a b(mod m) c gi l mt ng d thc. Nu khng chia ht

cho , ta vit

V d 3 -1 (mod 4)

5 17 (mod 6)

18 0 (mod 6)

iu kin a 0(mod m) ngha l a

b. Tnh cht v cc h qu

Tnh cht 1:

Vi mi s nguyn , ta c: a a (mod m)

Tnh cht 2:

a b (mod m) b a (mod m)

Tnh cht 3

a b (mod m), b c (mod m) a c (mod m)

Chng minh:


37/89

a b (mod m) m | (a - b)

b c(mod m) m | (b- c

v a c = (a b) + (b c ) m | (a - c

Tnh cht 4

Chng minh:

Tnh cht 5

Chng minh:

Theo tnh cht 4 ta c:

Nhn tng v hai T ta c:

Nhn xt:

1, Nu a 1(mod 2) v b 1(mod 2) th a + b 2(mod 2), v 2 0 (mod 2)

suy ra: a + b 0(mod 2), cn a.b 1(mod 2)

iu ny c ngha : Tng ca hai s l l mt s chn; Tch ca hai s l l mt s

l

2,Nu a 3(mod 7) a2 9 (mod 7) 2(mod 7)

C ngha: Nu mt s chia cho 7 d 3 th bnh phng s chia 7 d 2.

Cc h qu ca tnh cht 4 v 5:

3. , vi

Ch :


38/89

1_Chia hai v cho mt ng thc, ni chung l khng c.

nhng

2 nhng ab c th ng d vi 0 theo module m. Chng

hn : nhng 2.5=10 0(mod 10)

3.1.2.Mt s nh l s dng trong thut m ha cng khai

a.Thut gii Euclid- Tm UCLN ca hai s nguyn

Gii thut Euclid hay thut ton Euclid, l mt gii thut gip tnh c s

chung ln nht (SCLN) ca hai s mt cch hiu qu. Gii thut ny c

bit n t khong nm 300 trc Cng Nguyn. Nh ton hc Hy Lp c Euclid

vit gii thut ny trong cun sch ton ni tingElements.

Gi s a = bq + r, vi a, b, q, rl cc s nguyn, ta c:

Gii thut

Input: hai s nguyn khng m a v b, b>0

Output: UCLN ca a, b.

(1) While b 0 do

r= a mod b, a= b, b=r

(2) Return (a)

b.Gii thut Euclid m rng

Gii thut Euclid m rng s dng gii phng trnh v nh nguyn (cn c

gi l phng trnh i--phng)

a*x+b*y=c,trong a, b,c l cc h s nguyn, x, y l cc n nhn gi tr nguyn. iu kin

cn v phng trnh ny c nghim (nguyn) l UCLN(a,b) l c ca c.

Khng nh ny da trn mt mnh sau:

Trong s hc bit rng nu d=UCLN(a,b) th tn ti cc s nguyn x, y

sao cho


39/89

a*x+b*y = d

Gii thut

Input: hai s nguyn khng m a v b , a>b

Output: d= UCLN(a,b) v cc s nguyn x v y tha mn ax + by = d

(1) Nu b = 0 th t d =a, y = 0, v return (d,x,y)

(2) Khai bo 5 bin trung gian x1, x2, y1, y2 v q

(3) t x2 = 1, x1 = 0, y2 = 0, y1 = 1

(4) While b > 0 do

(4.1) q = [a/b], r = a qb, x = x2 qx1, y = y2 qy1

(4.2) a = b, b = r, x2 = x1 , x1 = x, y2 = y1, y1 = y

(5) t d = a, x = x2, y = y2 v return (d,x,y).

nh gi phc tp: Thut ton Euclid m rng c phc tp v thi gian lO((lg n)2).

V d: Xt v d vi a=4864 v b=3458.

q r x y a b x2 x1 y2 y1 4864 3458 1 0 0 11 1406 1 -1 3458 1406 0 1 1 -12 646 -2 3 1406 646 1 -2 -1 32 114 5 -7 646 114 -2 5 3 -75 76 -27 38 114 76 5 -27 -7 381 38 32 -45 76 38 -27 32 38 -452 0 -91 128 38 0 32 -91 45 128

ng dng thut ton Euclid m rng tm phn t nghch o

Thut ton Euclid m rng c s dng rt thng xuyn trong mt m

vi kha cng khai tm phn t nghch o. Xt mt trng hp ring khi vn

dng thut ton Euclid m rng:

Cho hai s nguyn dng nguyn t cng nhau a, n: n>a, (a,n)=1. Cn tms nguyn dng b nh nht sao cho ab 1 (mod n). S b nh th c gi l

"nghch o" ca a theo module n (v ngc li, a l "nghch o" ca b theo

module n).

p dng thut ton Euclid m rng cho cp s (n,a) ta tm c b 3 s

(d,x,y) tha mn d=(n,a) v nx+ay=d. Bi v a v n nguyn t cng nhau nn d=1


40/89

v nx+ay=1. V nx lun chia ht cho n nn t ng thc cui cng ta suy ra c

ay 1 (mod n).

i chiu vi yu cu ca bi ton, ta c b = y + zn. Trong z l s

nguyn nh nht tha mn b > 0. Dng rt gn ca thut ton Euclid m rng.

Bi v bi tan tm "phn t nghch o" l trng hp ring ca thut ton Euclid

m rng, li c dng rt thng xuyn trong mt m vi kha cng khai nn

ngi ta xy dng thut ton n gin hn gii bi ton ny. Thut ton c

th hin bng di y:

I ui vi qi1 0 n2 1 a [n/a]

3 u1-q2.u2 v1-q2.v2 [v2/v3]... ... ... ...K uk-2-qk-1.uk-1 vk-2-qk-1.vk-1 [vk-1/vk]... ... ... ...? y 1I ui vi qi1 0 232 1 5 4

3 -4 3 14 5 2 15 -9 1

Bc 1:

1. u := 0;

2. v := n; (v d: n=23)

3. Chuyn n bc 2

Bc 2:

1. u := 1;

2. v := a; (v d: a=5)

3. Nu v=1 th chuyn n bc 5.

4. q = n/a


41/89

5. Chuyn n bc 3

Bc 3:

1. uk := uk-2-qk-1.uk-1;2. vk := vk-2-qk-1.vk-1;

3. Nu vk=1 th chuyn n bc 5.

4. qk := [vk-1/vk];

5. Chuyn n bc 4

Bc 4: Tr li bc 3.

Bc 5: n y ta thu c gi tr v = y. S b cn tm c xc nh bi b = y +

zn. Trong , z l s nguyn nh nht tha mn b > 0. v d trn y, i vin=23 v a=5 ta tm c y = -9 nn b = 14 (vi z=1).

c.nh l phn d Trung Hoa

nh l phn d Trung Hoa, hay bi ton Hn Tn im binh, l mt nh l

ni v nghim ca h phng trnh ng d bc nht.

Ni dung

Cho tp cc s nguyn t cng nhau tng i mt :m1, m2, , mk. Vi mi b snguyn bt k a1, a2, , ak. H phng trnh ng d:

Lun c nghim duy nht theo moun M = m1.m2...mkl:

trong

M1 = M / m1, M2 = M / m2,..., Mk = M / mk

y1 = (M1) 1(mod m1), y2 = (M2) 1(mod m2),..., yk = (Mk) 1(mod mk)


42/89

d.Thut gii Rabin Miller (1980)

Cho n 3 l, thut ton sau y xc nh rng n l mt hp s hoc in ra thng

bao sn l s nguyn t

(1) Write n 1 = 2km, where m is old

(2) Chose a random integer, 1 a n 1

(3) Compute b = am mod n

(4) If b=1 (mod n) then anwer n is prime and quit

(5) For i =0 to k 1 do

If b = -1 (mod n) then anwer n is prime and quit

else b = b2 (mod n)

(6) Anwser n is composite

f. Thut gii tnh xp mod m

Cho x Zm v mt s nguyn p N* c biu din nh phn

p = pi2i(i = 0, 1). Vic tnh gi tr y = xp mod m c gi l php ly tha mod

Input: x Zm, p = pi2i(i = 0, 1)

Output: y = xp mod m

(1) y = 1

(2) for i = 1 downto 0 do

y = y2 mod m

if pi = 1 then y = (y*x) mod m

(3) return y


43/89

g. nh l Ferma

Nu p l mt s nguyn t cn a l mt s nguyn th ap a(mod p).

Nu p khng chia ht cho a (tc l a(mod p) 0) th ap-1 1(mod p)(nh lFerma nh )

D nhn thy rng nh l Fermat nh l trng hp ring ca nh l Euler khi n

l s nguyn t.

h. nh l Euler

nh ngha hm Euler: Cho n l mt s nguyn dng. Hm Euler ca n c khiu l (n) v c xc nh bi cng sut ca tp hp M cc s nguyn dng

nh hn n v nguyn t cng nhau vi n.

Gii thch:

Cho trc s nguyn dng n

Xc nh tp hp M (di vi s n cho): s x thuc tp hp M khi v ch

khi tha mn cc iu kin sau:

1. x N2. 0 < x < n

3. (x,n) = 1

Hm Euler ca n c gi tr bng s phn t ca tp hp M: (n) = #M

Quy tc tnh gi tr ca hm Euler:

1. (p) = p 1, nu p l s nguyn t;

2. (pi) = (pi 1), trong pi l cc s nguyn t khc nhau;3. (piki) = (pi(pi 1)ki), trong pi l cc s nguyn t khc nhau;

4. (mn) = (m)(n), nu (m,n)=1.

nh l Euler:Cho a v n l 2 s nguyn dng, nguyn t cng nhau: (a,n)=1.

nh l Euler khng nh: a(n) 1 (mod n), trong (n) l hm Euler ca n.


44/89

3.2.L thuyt phc tp

Mt chng trnh my tnh thng c ci t da trn mt thut ton ng

gii quyt bi ton hay vn . Tuy nhin, ngay c khi thut ton ng, chng

trnh vn c th khng s dng c i vi mt d liu u vo no v thi

gian cho ra kt qu l qu lu hoc s dng qu nhiu b nh (vt qu kh

nng p ng ca my tnh).

Khi tin hnhphn tch thut ton ngha l chng ta tm ra mt nh gi v thi

gian v "khng gian" cn thit thc hin thut ton. Khng gian y c

hiu l cc yu cu v b nh, thit b lu tr, ... ca my tnh thut ton c thlm vic. Vic xem xt v khng gian ca thut ton ph thuc phn ln vo cch

t chc d liu ca thut ton. Trong phn ny, khi ni n phc tp ca thut

ton, chng ta ch cp n nhng nh gi v mt thi gian m thi.

Phn tch thut ton l mt cng vic rt kh khn, i hi phi c nhng hiu

bit su sc v thut ton v nhiu kin thc ton hc khc. y l cng vic m

khng phi bt c ngi no cng lm c. Rt may mn l cc nh ton hc

phn tch cho chng ta phc tp ca hu ht cc thut ton c s (sp xp, tm

kim, cc thut ton s hc, ...). Chnh v vy, nhim v cn li ca chng ta l

hiu c cc khi nim lin quan n phc tp ca thut ton.

nh gi v thi gian ca thut ton khng phi l xc nh thi gian tuyt i

(chy thut ton mt bao nhiu giy, bao nhiu pht,...) thc hin thut ton m

l xc nh mi lin quan gia d liu u vo (input) ca thut ton v chi ph (s

thao tc, s php tnh cng,tr, nhn, chia, rt cn,...) thc hin thut ton. Sd ngi ta khng quan tm n thi gian tuyt i ca thut ton v yu t ny

ph thuc vo tc ca my tnh, m cc my tnh khc nhau th c tc rt

khc nhau. Mt cch tng qut, chi ph thc hin thut ton l mt hm s ph

thuc vo d liu u vo :

T = f(input)


45/89

Tuy vy, khi phn tch thut ton, ngi ta thng ch ch n mi lin quan

gia ln ca d liu u vo v chi ph. Trong cc thut ton, ln ca d

liu u vothng c th hin bng mt con s nguyn n. Chng hn :sp xp

n con s nguyn, tm con s ln nht trong n s, tnh im trung bnh ca n hc

sinh, ... Lc ny, ngi ta th hin chi ph thc hin thut ton bng mt hm s

ph thuc vo n :

T = f(n)

Vic xy dng mt hm T tng qut nh trn trong mi trng hp ca thut

ton l mt vic rt kh khn, nhiu lc khng th thc hin c. Chnh v vy

m ngi ta ch xy dng hm T cho mt s trng hp ng ch nht ca thut

ton, thng l trng hp tt nhtv xu nht. nh gi trng hp tt nht

v xu nht ngi ta da vo nh ngha sau:

Cho hai hm f v g c min xc nh trong tp s t nhin . Ta vit

f(n) = O(g(n)) v ni f(n) c cp cao nht l g(n) khi tn ti hng s C v k sao

cho | f(n) | C.g(n) vi mi n > k

Tuy chi ph ca thut ton trong trng hp tt nht v xu nht c th ni lnnhiu iu nhng vn cha a ra c mt hnh dung tt nht v phc tp ca

thut ton. c th hnh dung chnh xc v phc tp ca thut ton, ta xt n

mt yu t khc l tngca chi ph khi ln n ca d liu u vo tng.

Mt cch tng qut, nu hm chi ph ca thut ton (xt trong mt trng hp

no ) b chn bi O(f(n)) th ta ni rng thut ton c phc tp l O(f(n))

trong trng hp .

Nh vy, thut ton tm s ln nht c phc tp trong trng hp tt nht v

xu nht u l O(n). Ngi ta gi cc thut ton c phc tp O(n) l cc thut

ton c phc tp tuyn tnh.


46/89

Sau y l mt s "thc o" phc tp ca thut ton c s dng rng ri.

Cc phc tp c sp xp theo th t tng dn. Ngha l mt bi ton c

phc tp O(nk) s phc tp hn bi ton c phc tp O(n) hoc O(logn).


47/89

Chng 4. H mt m cng khai

4.1.Gii thiu mt m vi kha cng khai

4.1.1.Lch s

Mt m ha kha cng khai l mt dng mt m ha cho php ngi s

dng trao i cc thng tin mt m khng cn phi trao i cc kha chung b mt

trc . iu ny c thc hin bng cch s dng mt cp kha c quan h

ton hc vi nhau l kha cng khai v kha c nhn (hay kha b mt).

Thut ng mt m ha kha bt i xng thng c dng ng ngha vi

mt m ha kha cng khai mc d hai khi nim khng hon ton tng ng.

C nhng thut ton mt m kha bt i xng khng c tnh cht kha cng khaiv b mt nh cp trn m c hai kha (cho m ha v gii m) u cn phi

gi b mt.

Trong mt m ha kha cng khai, kha c nhn phi c gi b mt trong

khi kha cng khai c ph bin cng khai. Trong 2 kha, mt dng m ha

v kha cn li dng gii m. iu quan trng i vi h thng l khng th

tm ra kha b mt nu ch bit kha cng khai.

H thng mt m ha kha cng khai c th s dng vi cc mc ch:

M ha: gi b mt thng tin v ch c ngi c kha b mt mi gii m

c.

To ch k s: cho php kim tra mt vn bn c phi c to vi mt

kha b mt no hay khng.

Tha thun kha: cho php thit lp kha dng trao i thng tin mt

gia 2 bn.

Thng thng, cc k thut mt m ha kha cng khai i hi khi lng tnh

ton nhiu hn cc k thut m ha kha i xng nhng nhng li im m

chng mang li khin cho chng c p dng trong nhiu ng dng.


48/89

Trong hu ht lch s mt m hc, kha dng trong cc qu trnh m ha v

gii m phi c gi b mt v cn c trao i bng mt phng php an ton

khc (khng dng mt m) nh gp nhau trc tip hay thng qua mt ngi a

th tin cy. V vy qu trnh phn phi kha trong thc t gp rt nhiu kh khn,

c bit l khi s lng ngi s dng rt ln. Mt m ha kha cng khai gii

quyt c vn ny v n cho php ngi dng gi thng tin mt trn ng

truyn khng an ton m khng cn tha thun kha t trc.

Nm 1874, William Stanley Jevons xut bn mt cun sch m t mi quan

h gia cc hm mt chiu vi mt m hc ng thi i su vo bi ton phn tch

ra tha s nguyn t (s dng trong thut ton RSA). Thng 7 nm 1996, mt nh

nghin cu

bnh lun v cun sch trn nh sau:

Trong cun The Principles of Science: A Treatise on Logic and Scientific

Methodc xut bn nm 1890, William S. Jevons pht hin nhiu php ton

rt d thc hin theo mt chiu nhng rt kh theo chiu ngc li. Mt v d

chng t m ha rt d dng trong khi gii m th khng. Vn trong phn ni trn

chng 7 (Gii thiu v php tnh ngc) tc gi cp n nguyn l: ta c th

d dng nhn cc s t nhin nhng phn tch kt qu ra tha s nguyn t th

khng h n gin. y chnh l nguyn tc c bn ca thut ton mt m hakha cng khai RSA mc d tc gi khng phi l ngi pht minh ra mt m

ha kha cng khai.

Thut ton mt m ha kha cng khai c thit k u tin bi James H.

Ellis, Clifford Cocks, v Malcolm Williamson ti GCHQ (Anh) vo u thp k

1970. Thut ton sau ny c pht trin v bit n di tn Diffie-Hellman, v

l mt trng hp c bit ca RSA. Tuy nhin nhng thng tin ny ch c tit

l vo nm 1997.

Nm 1976, Whitfield Diffie v Martin Hellman cng b mt h thng mt

m ha kha bt i xng trong nu ra phng php trao i kha cng khai.

Cng trnh ny chu s nh hng t xut bn trc ca Ralph Merkle v phn

phi kha cng khai. Trao i kha Diffie-Hellman l phng php c th p


49/89

dng trn thc t u tin phn phi kha b mt thng qua mt knh thng tin

khng an ton. K thut tha thun kha ca Merkle c tn l h thng cu

Merkle.

Thut ton u tin cng c Rivest, Shamir v Adleman tm ra vo nm1977 ti MIT. Cng trnh ny c cng b vo nm 1978 v thut ton c t

tn l RSA. RSA s dng php ton tnh hm m mun (mun c tnh bng

tch s ca 2 s nguyn t ln) m ha v gii m cng nh to [ch k s]. An

ton ca thut ton c m bo vi iu kin l khng tn ti k thut hiu qu

phn tch mt s rt ln thnh tha s nguyn t.

K t thp k 1970, c rt nhiu thut ton m ha, to ch k s, tha

thun kha.. c pht trin. Cc thut ton nh ElGamal (mt m) do Netscape

pht trin hay DSA do NSA v NIST cng da trn cc bi ton lgarit ri rc

tng t nh RSA. Vo gia thp k 1980, Neal Koblitz bt u cho mt dng

thut ton mi: mt m ng cong elliptic v cng to ra nhiu thut ton tng

t. Mc d c s ton hc ca dng thut ton ny phc tp hn nhng li gip

lm gim khi lng tnh ton c bit khi kha c di ln.

4.1.2.L thuyt mt m cng khai

Khi nim v mt m kha cng khai to ra s c gng gii quyt hai vn

kh khn nht trong mt m kha quy c, l s phn b kha v ch k s:

- Trong m quy c s phn b kha yu cu hoc l hai ngi truyn thng

cng tham gia mt kha m bng cch no c phn b ti h hoc

s dng chung mt trung tm phn b kha.

- Nu vic s dng mt m tr nn ph bin, khng ch trong qun i mcn trong thng mi v nhng mc ch c nhn th nhng on tin v ti

liu in t s cn nhng ch k tng ng s dng trong cc ti liu

giy. Tc l, mt phng php c th c ngh ra c quy nh lm hi lng

tt c nhng ngi tham gia khi m mt on tin s c gi bi mt c

nhn c bit hay khng


50/89

Trong s m ha quy c, cc kha c dng cho m ha v gii m mt

on tin l ging nhau. y l mt iu kin khng cn thit, n c th pht

trin gii thut m ha da trn mt kha cho m ha v mt kha khc cho

gii m

Cc bc cn thit trong qu trnh m ha cng khai

- Mi h thng cui trong mng to ra mt cp kha dng cho m ha v

gii m on tin m n s nhn

- Mi h thng cng b rng ri kha m ha bng cch t kha vo mt

thanh ghi hay mt file cng khai, kha cn li c gi ring

- Nu A mun gi mt on tin ti B th A m ha on tin bng kha cng

khai ca B

- Khi B nhn on tin m ha, n c th gii m bng kha b mt ca mnh.

Khng mt ngi no khc c th gii m oan tin ny bi v ch c mnh B

bit kha b mt thi .

Vic cc tip cn ny, tt c nhng ngi tham gia c th truy xut kha cng

khai. Kha b mt c to bi tng c nhn, v vy khng bao gi c phn

b. bt k thi im no, h thng cng c th chuyn i cp kha mbo tnh b mt.

Bng sau tm tt mt s kha cnh quan trng v m ha quy c v m ha

cng khai : phn bit c hai loi chng ta tng qut ha lin h kha s

dng trong m ha quy c l kha b mt, hai kha s dng trong m ha

cng khai l kha cng khai v kha b mt.

M ha quy c M ha cng khai* Yu cu

- Thut gii tng t cho m ha v

gii m.

- Ngi gi v ngi nhn phi tham

* Yu cu

- Mt thut gii cho m ha v mt

thut gii cho gii m

- Ngi gi v ngi nhn, mi


51/89

gia cng thut gii v cng kha

* Tnh bo mt

- Kha phi c b mt

- Khng th hay t nht khng c tnh

thc t gii m on tin nu thng

tin khc c sn

- Kin thc v thut gii cng vi

cc mu v mt m khng xc

nh kha

ngi phi c cp kha ring ca

mnh

* Tnh bo mt

- Mt trong hai kha phi c gi

b mt

- Khng th hay t nht khng c tnh

thc t gii m on tn nu thng

tin khc khng c sn

- Kin thc v thut gii cng vi

mt trong cc kha, cng vi cc

mu v mt m khng xc nh

kha

4.1.3.Nhng yu im, hn ch ca mt m vi kha cng khai

Tn ti kh nng mt ngi no c th tm ra c kha b mt. Khng ging

vi h thng mt m s dng mt ln (one-time pad) hoc tng ng, cha cthut ton m ha kha bt i xng no c chng minh l an ton trc cc

tn cng da trn bn cht ton hc ca thut ton. Kh nng mt mi quan h no

gia 2 kha hay im yu ca thut ton dn ti cho php gii m khng cn

ti kha hay ch cn kha m ha vn cha c loi tr. An ton ca cc thut

ton ny u da trn cc c lng v khi lng tnh ton gii cc bi ton

gn vi chng. Cc c lng ny li lun thay i ty thuc kh nng ca my

tnh v cc pht hin ton hc mi.

Mc d vy, an ton ca cc thut ton mt m ha kha cng khai cng tng

i m bo. Nu thi gian ph mt m (bng phng php duyt ton b)

c c lng l 1000 nm th thut ton ny hon ton c th dng m ha

cc thng tin v th tn dng - R rng l thi gian ph m ln hn nhiu ln thi

gian tn ti ca th (vi nm).


52/89

Nhiu im yu ca mt s thut ton mt m ha kha bt i xng c tm

ra trong qu kh. Thut ton ng gi ba l l mt v d. N ch c xem l

khng an ton khi mt dng tn cng khng lng trc b pht hin. Gn y,

mt s dng tn cng n gin ha vic tm kha gii m da trn vic o c

chnh xc thi gian m mt h thng phn cng thc hin m ha. V vy, vic s

dng m ha kha bt i xng khng th m bo an ton tuyt i. y l mt

lnh vc ang c tch cc nghin cu tm ra nhng dng tn cng mi.

Mt im yu tim tng trong vic s dng kha bt i xng l kh nng b tn

cng dng k tn cng ng gia (man in the middle attack): k tn cng li dng

vic phn phi kha cng khai thay i kha cng khai. Sau khi gi mo

c kha cng khai, k tn cng ng gia 2 bn nhn cc gi tin, gii mri li m ha vi kha ng v gi n ni nhn trnh b pht hin. Dng tn

cng kiu ny c th phng nga bng cc phng php trao i kha an ton

nhm m bo nhn thc ngi gi v ton vn thng tin. Mt iu cn lu l

khi cc chnh ph quan tm n dng tn cng ny: h c th thuyt phc (hay bt

buc) nh cung cp chng thc s xc nhn mt kha gi mo v c th c cc

thng tin m ha.

4.1.4.ng dng ca mt m

a.Bo mt

ng dng r rng nht ca mt m ha kha cng khai l bo mt: mt vn

bn c m ha bng kha cng khai ca mt ngi s dng th ch c th gii

m vi kha b mt ca ngi .

Phn mm PGP min ph ch c s dng cho ngi dng c nhn vimc ch phi thng mi, c th ti v ti a ch :

http://www.pgp.com/products/freeware.html

b.Chng thc
http://www.pgp.com/products/freeware.htmlhttp://www.pgp.com/products/freeware.html


53/89

Cc thut ton to ch k s kha cng khai c th dng nhn thc. Mt

ngi s dng c th m ha vn bn vi kha b mt ca mnh. Nu mt ngi

khc c th gii m vi kha cng khai ca ngi gi th c th tin rng vn bn

thc s xut pht t ngi gn vi kha cng khai . Dng ch k s cho email

v m ha email khi gi i thng qua nh cung cp chng ch s lm trng ti iu

khin

Nh chng ch s ca nh cung cp Thawte(www.thawte.com) cho php

bn c th ng k cho mnh mt ti khon Personal Email Certificate haonf ton

min ph ti y thc hin giao dch khi gi v nhn mail

(http://www.thawte.com/secure-email/personal-email-certificates/index.htm)

c.ng dng trong thng mi in t

Nhiu n v, t chc Vit Nam ang xy dng mng my tnh c quy

m ln phc v cho cng vic kinh doanh ca mnh: mng chng khon, mng

ngn hng, mng bn v tu xe, k khai v np thu qua mng.

Cng ty phn mm v Truyn thng VASC chnh thc k kt hp ng

ng dng chng ch s trong giao dch ngn hng in t vi ngn hng c phn

thng mi Chu (ACB) t ngy 30/9/2003, cho php khch hng ACB s giao

dch trc tuyn trn mng vi ch k in t do VASC cp.

Mng giao dch chng khon VCBS (http://www.vebs.vn) : m ti khon

ngn hng cho php giao dch trc tip qua sn, bo gi c phiu, cho php t

lnh mua bn c phn ch bng thao tc click chut.

Mng ngn hng VCB, EAB (http://www.vietcombank.com.vn,

http://ebanking.dongabank.com.vn) cho php xem s d, chuyn khon cho ti

khon khc cng h thng t 20-500 triu ng mi ngy, bn k chi tit gaio dch

ca ti khon trn Internet.
http://www.thawte.com/http://www.vebs.vn/http://www.vietcombank.com.vn/http://ebanking.dongabank.com.vn/http://www.thawte.com/http://www.vebs.vn/http://www.vietcombank.com.vn/http://ebanking.dongabank.com.vn/


54/89

H thng bn v qua mng ca ngnh hng khng

(http://www.pacificairline.com.vn), ng st (http://www.vr.com.vn) trin

khai 1/2007, mua bn trc tuyn (http://www.ebay.vn).

Chi cc thu thnh ph H Ch Minh (http://www.hcmtax.gov.vn) ang thnghim cho php doanh nghip ng k t in ha n theo mu, t k khai bo

co thu, khu tr thu qua mng

Nu nh c c mt c ch bo mt tt, m bo xc thc r rng gia cc

bn tham gia vo h thng th chc chn rng nhng vn lin quan n mng

my tnh nu trn ch cn l vn thi gian.

4.2.H mt RSA

Trong mt m hc, RSA l mt thut ton mt m ha kha cng khai. y

l thut ton u tin ph hp vi vic to ra ch k in t ng thi vi vic m

ha. N nh du mt s tin b vt bc ca lnh vc mt m hc trong vic s

dng kha cng cng. RSA ang c s dng ph bin trong thng mi in t

v c cho l m bo an ton vi iu kin di kha ln.

4.2.1.Lch s

Thut ton c Ron Rivest, Adi Shamir v Len Adleman m t ln u tin vo

nm 1977 ti Hc vin Cng ngh Massachusetts (MIT). Tn ca thut ton ly t

3 ch ci u ca tn 3 tc gi.

Trc , vo nm 1973, Clifford Cocks, mt nh ton hc ngi Anh lm vic

ti GCHQ, m t mt thut ton tng t. Vi kh nng tnh ton ti thi im

th thut ton ny khng kh thi v cha bao gi c thc nghim. Tuy nhin,pht minh ny ch c cng b vo nm 1997 v c xp vo loi tuyt mt.

Thut ton RSA c MIT ng k bng sng ch ti Hoa K vo nm 1983 (S

ng k 4,405,829). Bng sng ch ny ht hn vo ngy 21 thng 9 nm 2000.

Tuy nhin, do thut ton c cng b trc khi c ng k bo h nn s bo

h hu nh khng c gi tr bn ngoi Hoa K. Ngoi ra, nu nh cng trnh ca
http://www.pacificairline.com.vn/http://www.vr.com.vn/http://www.ebay.vn/http://www.hcmtax.gov.vn/http://www.pacificairline.com.vn/http://www.vr.com.vn/http://www.ebay.vn/http://www.hcmtax.gov.vn/


55/89

Clifford Cocks c cng b trc th bng sng ch RSA khng th

c ng k.

4.2.2.M t thut ton

Thut ton RSA c hai kha: kha cng khai (hay kha cng cng) v kha

b mt (hay kha c nhn). Mi kha l nhng s c nh s dng trong qu trnh

m ha v gii m. Kha cng khai c cng b rng ri cho mi ngi v c

dng m ha. Nhng thng tin c m ha bng kha cng khai ch c th

c gii m bng kha b mt tng ng. Ni cch khc, mi ngi u c th

m ha nhng ch c ngi bit kha c nhn (b mt) mi c th gii m c.

Ta c th m phng trc quan mt h mt m kho cng khai nh sau : Bobmun gi cho Alice mt thng tin mt m Bob mun duy nht Alice c th c

c. lm c iu ny, Alice gi cho Bob mt chic hp c kha m sn

v gi li cha kha. Bob nhn chic hp, cho vo mt t giy vit th bnh

thng v kha li (nh loi kho thng thng ch cn sp cht li, sau khi sp

cht kha ngay c Bob cng khng th m li c-khng c li hay sa thng

tin trong th c na). Sau Bob gi chic hp li cho Alice. Alice m hp vi

cha kha ca mnh v c thng tin trong th. Trong v d ny, chic hp vi

kha m ng vai tr kha cng khai, chic cha kha chnh l kha b mt.

a. To kha

Gi s Alice v Bob cn trao i thng tin b mt thng qua mt knh khng an

ton (v d nh Internet). Vi thut ton RSA, Alice u tin cn to ra cho mnh

cp kha gm kha cng khai v kha b mt theo cc bc sau:

1. Chn 2 s nguyn t ln p v q vi pq, la chn ngu nhin v c lp.

2. Tnh: n= pq

3. Tnh: gi tr hm s le (n)= (p-1)(q-1).

4. Chn mt s t nhin e sao cho 1< e< (n) v l s nguyn t cng nhau

vi (n) .


56/89

5. Tnh: dsao cho de 1 (mod (n).

Mt s lu :

Cc s nguyn t thng c chn bng phng php th xc sut. Cc bc 4 v 5 c th c thc hin bng gii thut Euclid m rng (xem

thm: s hc mun ).

Bc 5 c th vit cch khc: Tm s t nhin sao cho

cng l s t nhin. Khi s dng gi tr

.

T bc 3, PKCS#1 v2.1 s dng thay cho

).

Kha cng khai bao gm:

n, mun

e, s m cng khai (cng gi ls m m ha).

Kha b mt bao gm:

n, mun, xut hin c trong kha cng khai v kha b mt, v

d, s m b mt (cng gi ls m gii m).

Mt dng khc ca kha b mt bao gm:

p and q, hai s nguyn t chn ban u,

d mod (p-1) v d mod (q-1) (thng c gi l dmp1 v dmq1), (1/q) mod p (thng c gi l iqmp)

Dng ny cho php thc hin gii m v k nhanh hn vi vic s dng nh l s

d Trung Quc (ting Anh: Chinese Remainder Theorem - CRT). dng ny, tt

c thnh phn ca kha b mt phi c gi b mt.


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Alice gi kha cng khai cho Bob, v gi b mt kha c nhn ca mnh. y,p

v q gi vai tr rt quan trng. Chng l cc phn t ca n v cho php tnh dkhi

bit e. Nu khng s dng dng sau ca kha b mt (dng CRT) th p v q s

c xa ngay sau khi thc hin xong qu trnh to kha.

b. M ha

Gi s Bob mun gi on thng tinMcho Alice. u tin Bob chuynMthnh

mt s m < n theo mt hm c th o ngc (t m c th xc nh li M) c

tha thun trc. Qu trnh ny c m t phn sau

Lc ny Bob c m v bit n cng nh e do Alice gi. Bob s tnh c l bn m ha

ca m theo cng thc:

Hm trn c th tnh d dng s dng phng php tnh hm m (theo mun)

bng thut ton bnh phng v nhn. Cui cng Bob gi c cho Alice.

c. Gii m

Alice nhn c t Bob v bit kha b mt d. Alice c th tm c m t c theo cng

thc sau:

Bit m, Alice tm liMtheo phng php tha thun trc. Qu trnh gii m

hot ng v ta c

.

Do ed 1 (modp-1) v ed 1 (mod q-1), (theo nh l Fermat nh) nn:

v


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Dop v q l hai s nguyn t cng nhau, p dng nh l s d Trung Quc, ta c:

.

hay:

.

V d

Sau y l mt v d vi nhng s c th. y chng ta s dng nhng s nh tin tnh ton cn trong thc t phi dng cc s c gi tr ln.

Ly:

p = 61 s nguyn t th nht (gi b mt hoc hy sau khi to kha)q = 53 s nguyn t th hai (gi b mt hoc hy sau khi to kha)n = pq =

3233 mun (cng b cng khai)

e = 17 s m cng khaid= 2753 s m b mt

Kha cng khai l cp (e, n). Kha b mt l d. Hm m ha l:

encrypt(m) = me mod n = m17 mod 3233

vi m l vn bn r. Hm gii m l:

decrypt(c) = cd mod n = c2753 mod 3233

vi c l vn bn m.

m ha vn bn c gi tr 123, ta thc hin php tnh:

encrypt(123) = 12317 mod 3233 = 855


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gii m vn bn c gi tr 855, ta thc hin php tnh:

decrypt(855) = 8552753 mod 3233 = 123

C hai php tnh trn u c th c thc hin hiu qu nh gii thut bnhphng v nhn.

4.2.3.Tc m ha RSA

Tc v hiu qu ca nhiu phn mm thng mi c sn v cng c phn cng

ca RSA ang gia tng mt cch nhanh chng. Vic Pentium 90Mhz, b toolkit

BSAFE 3.0 ca c quan bo mt d liu RSA t tc tnh kha b mt l 21,6

Kbps vi kha 512 bit v 7,4 Kbps vi kha 1024 bit. Phn cng RSA nhanh nht

y 300 Kbps vi kha 512 bit, nu c x l song song th t 600 Kbps vi

kha 512 bit v 185 Kbps vi kha 970 bit.

So snh vi gii thut DES v cc gii thut m khi khc th RSA chm hn: v

phn mm DES nhanh hn RSA 100 ln, v phn cng DES nhanh hn RSA t

1000 ti 10000 ln ty thuc cng c (implementation) s dng (thng tin ny

c ly t http://www.rsa.com)

Kch thc ca kha trong RSA:

Hiu qu ca mt h thng mt m kha bt i xng ph thuc vo kh (l

thuyt hoc tnh ton) ca mt vn ton hc no chng hn nh bi ton

phn tch ra tha s nguyn t. Gii cc bi ton ny thng mt nhiu thi gian

nhng thng thng vn nhanh hn l th ln lt tng kha theo kiu duyt ton

b. V th, kha dng trong cc h thng ny cn phi di hn trong cc h thng

mt m kha i xng. Ti thi im nm 2002, di 1024 bt c xem l gitr ti thiu cho h thng s dng thut ton RSA.

Nm 2003, cng ty RSA Security cho rng kha RSA 1024 bt c an ton

tng ng vi kha 80 bt, kha RSA 2048 bt tng ng vi kha 112 bt v

kha RSA 3072 bt tng ng vi kha 128 bt ca h thng mt m kha i
http://www.rsa.com/http://www.rsa.com/


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xng. H cng nh gi rng, kha 1024 bt c th b ph v trong khong t

2006 ti 2010 v kha 2048 bt s an ton ti 2030. Cc kha 3072 bt cn c

s dng trong trng hp thng tin cn gi b mt sau 2030. Cc hng dn v

qun l kha ca NIST cng gi rng kha RSA 15360 bt c an ton tng

ng vi kha i xng 256 bt.

Mt dng khc ca thut ton mt m ha kha bt i xng, mt m ng

cong elliptic (ECC), t ra an ton vi kha ngn hn kh nhiu so vi cc thut

ton khc. Hng dn ca NIST cho rng kha ca ECC ch cn di gp i kha

ca h thng kha i xng. Gi nh ny ng trong trng hp khng c nhng

t ph trong vic gii cc bi ton m ECC ang s dng. Mt vn bn m ha

bng ECC vi kha 109 bt b ph v bng cch tn cng duyt ton b.

Ty thuc vo kch thc bo mt ca mi ngi v thi gian sng ca kha m

kha c chiu di thch hp

- loi Export 512 bit

- loi Person 768 bit

- loi Commercial 1024 bit

- loi Militery 2048 bit

Chu k sng ca kha ph thuc vo

- vic ng k v to kha

- vic phn b kha

- vic kch hot v khng kch hot kha

- vic thay th hoc cp nht kha

- vic hy b kha- vic kt thc kha bao gm s ph hoi hoc s lu tr

4.2.4. an ton ca RSA

an ton ca h thng RSA da trn 2 vn ca ton hc: bi ton phn

tch ra tha s nguyn t cc s nguyn ln v bi ton RSA. Nu 2 bi ton trn


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l kh (khng tm c thut ton hiu qu gii chng) th khng th thc hin

c vic ph m ton b i vi RSA. Ph m mt phn phi c ngn chn

bng cc phng php chuyn i bn r an ton.

Bi ton RSA l bi ton tnh cn bc e mun n (vi n l hp s): tm s msao cho me=c mod n, trong (e, n) chnh l kha cng khai v c l bn m. Hin

nay phng php trin vng nht gii bi ton ny l phn tc

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