mật mã học và xác nhận chữ ký điện tử.doc
TRANSCRIPT
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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
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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".
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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.
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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
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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
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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
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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:
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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:
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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
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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
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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
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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.
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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.
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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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:
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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 :
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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
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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
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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
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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)
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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
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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.
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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)
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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.
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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).
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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.
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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
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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
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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
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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).
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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 -
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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/ -
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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/ -
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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) .
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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
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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