“KRIPTOGRAFIYA 2” FANIDAN LOYIHA ISHI TOPSHIRIQLARI
1.
Quyidagi shifr mattni rasshifrovkalang (to‘rt kishi uchun).
N:
161576568432146305407822605195988788423367831773489290174076332113
521363679607546240195027460240509513858989808742833775844501328148
896686607335571077186467172699191870655807123126697642718467380022
525453169592854127254638514649573642026181569381054458981110496782
935446149117820012609966190965416354266154169940483964403517744509
298895261491842431708238017438381902558507620664199347932657618079
354432119435701891621511300974265440859708372450816921618200844969
391722749781316544437220151754178898992546171106782568194794747100
139084377474644269973938692328580102268545122126101079883764692809
227755619814566292469180303288004049276244256149776068993360178140
161708660059348212746565539084136115402589067975751406045610310419
925591716467816197273585893946479096044834598894148149905024867312
865650805528503709002643968384726653628316014207164301543481347346
346973311218232867870670211605403661827750699766653456784676393869
233506995575524443841537793344002949837895535587750274321530576881
4857864433151287
e: 3
ciphertext
(c):
122001231858887188613252475789888442217453455805559371330908830491
027399107355473265997713398068537089925785012197081240570079371054
667406215423754484017761674680566866631748114087260565376848486729
213813994907610290739983199882756764523098634545591569286309436479
752649730208273495590375505063815520289059980815455803470776737752
450030275445980792333181058517301097765798206988899694583078909252
693236465845903414545650505746911303613455974565907923646611951500
464818927822777755041502184014014731906147018384021403441791716194
037935127339421202284703769626553296868459235494147979947394135771
595320448723688871264249487754520100580777635485439035801573349533
110107785113248998366593964318806498644688359523984262144091845620
178716823498841065915321927732942623013649909609807268193949184091
396129053685121767704356574364446986299231024156389146422593561567
624208465861793122561853717368955941960768890514368360300748799642
256043026975030507928281897655728578625302577488315812597816487824
5223052992502106
2.
Quyidagi shifr mattni rasshifrovkalang (to‘rt kishi uchun).
N:
2933192249979498578273597604559116493668305938055895038656016
0105740343201513369939006307531165922708949619162698623675349
0304308595478257089947083218037053094594380993404277705800644
0091143185665690198278994828530995611184868690615266447335094
0486507451771223435835260168971210087470894448460745593956840
5865305279158025414500929465746948095848808966013175197944428
6297747112931978131316184205650171504055596401189958900286373
0868679527184420789010551475067862907739054966183120621407246
3985180989811064312192076978702934121764404829001835504673751
9023989845520117083141046048382944860347736130583874385275693
8687673
e: 3
ciphertext
(c):
2205316413931134031074603746928247799030155221252519872650080
5192637550753558252433275152114797475366975176884680953255172
0991168868430989490099289970750408764757599784771718076637783
2435022794675332132906451858990782325436498952049751141
3.
Quyidagi shifr mattni rasshifrovkalang (to‘rt kishi uchun).
c
=
1946990934266279413917993964398835970466043305162067149027416
2053008128349816033069247844970800863920955661712219866590366
9590908102703307250646216395967327417349581565673419272952219
8445430968778476727718084419319887006224079130320273805411547
7138201099392091672934450078062438509784548210810385639731494
7527381565904894782921374911825451607
n
=
3521965691682498349894156297871351510003392837134480208222044
4753957752560806765915557404471872854222134926440656261548716
5251895509139075621688493447805478090729096096693340629223579
5059531555148850225316762953811321830535065868925602454064437
6849173366230815779694365269990430445181132796913932904941185
478344783047790284034086549612594449881
e = 65537
4.
Faktorizatsiyalas (to‘rt kishi uchun).
Quyidagi RSA modullarini faktorlarga ajratib ko'ring va asosiy sonlarni olish
uchun zarur bo'lgan vaqtni yozib oling. RSA modullari qanchalik xavfsiz:
N=2416300553330965013
N=612499708047648285159224809949393
N=19108177365161625412461084593954711324687644626704669476977
8643815378126051777
Eng uzun murakkab sonni yozish o'rniga, nusxa ko'chirish va joylashtirish
uchun composite.txt faylidan foydalaning.
Odatda, modullar kattaligi 24096 yoki undan ham katta bo‘lishi mumkin.
Faktorizatsiya uchun qayd etilgan vaqtlarga asoslanib, 24096 qiymatli modulni
faktorlarga ajratish uchun zarur bo'lgan vaqtni taxminiy hisoblang. E'tibor bering,
yerning yoshi 4,6 109 yil.
5. RSA (to‘rt kishi uchun).
 Shifrlamoqchi bo'lgan xabarni yozing. Xabarni turli uzunlikdagi bir
nechta tub sonlar bilan shifrlang. Modul o'lchami N oshganda
shifrlangan matn uzunligi oshadimi?
 Ruxsat etilgan modul N uchun xabar blokining m eng katta hajmi
qancha?
 Faraz qilaylik, 𝑞 = 8191 va 𝑟 = 8111 ikkita tub son va ochiq kalit
𝑒 = 2047. Maxfiy kalitlarni hisoblang. Nima uchun ochiq kalitni
tanlash e=2048 ishlamayotganligini tushuntiring.
6. Diskret logarifm, Diffie Hellman kalit almashinuvi, ElGamal sxemasi
(to‘rt kishi uchun).
 Baby Step, Giant Step yordamida 3𝑥 = 5 𝑚𝑜𝑑 7 ning 𝑥 ni toping.
 𝑝 = 47 va 𝑔 = 5 bo'lsin. Har qanday ikkita raqamni ikkita aloqa
qiluvchi tomonning shaxsiy kalitlari sifatida oling va ular umumiy
kalitni umumiy maxfiy kalit sifatida qanday olishlarini ko'rsating.
 Xabar sifatida o‘nli sanoq tizimida raqamni tanlang va aloqa
tomonlari umumiy kalit yordamida xabarlarni qanday
shifrlashi/shifrini ochishi mumkinligini ko'rsating.
7. 𝑃 = 2256 − 2224 + 2192 + 296 − 1 va 𝑔 = 2 asosini hisobga olgan
holda, 𝑚𝑜𝑑 𝑝 da 123456789 ning diskret logarifmini toping (ikki kishi
uchun).
8. Diskret logarifmlash muammosi (to‘rt kishi uchun)
 𝐸1048583 (436743, 67111) bilan berilgan elliptik egri chiziqda
chekli maydon 𝐸 da nuqtalar soni 1049580 ga teng. Pohlig Hellman
algoritmi yordamida P = (169541 ∶ 556330) va 𝑄 =
(858751 ∶ 762468) uchun diskret logarifm muammosini log 𝑃 (𝑄)
yeching.
 𝐸1048583 (900410, 465299) bilan berilgan elliptik egri chiziqda
chekli maydon 𝐸 da nuqtalar soni 1049623 ga teng. Qo’pol kuch
hujumidan tashqari, ixtiyoriy algoritm yordamida P = (815314 ∶
582035) va 𝑄 = (67861 ∶ 1005415) uchun diskret logarifm
muammosini log 𝑃 (𝑄) yeching.
 Ikkala masalani hal qilish uchun zarur bo'lgan elliptik egri chiziq
operatsiyalar sonini va vaqtini solishtiring.
9. Diskret lograifmlash muammosi (to‘rt kishi uchun)
 Indeks hisoblash usuli yordamida 𝑥 ni toping:
a) 55𝑥 ≡ 444 (𝑚𝑜𝑑 569)
b) 7𝑥 ≡ 92 (𝑚𝑜𝑑 1433)
 Diffie-Hellman muammosi (DHP) va El Gamal muammosi
(ELGAMAL) hisoblash nuqtai nazaridan ekvivalent ekanligini
ko'rsating.
 Aytaylik, Elis El Gamal ochiq kalit kriptotizimida foydalanish
uchun kalitni (1237, 34, 383) nashr etdi.
o Siz Elisga m = 14 xabarini yubormoqchisiz. Siz aslida nimani
uzatasiz?
o Siz Elisga (94, 225) uzatishni kuzatdingiz.
 Elisning maxfiy kalitini 𝑎 topish uchun Diskret
logarifmlash muammosini hal qilish uchun indekslarni
hisoblash usulidan foydalaning?
 Asl xabar nima edi?
10. Elliptik egri chiziqga asoslangan DSA algoritmi (to‘rt kishi uchun).
ECDSA algoritmi uchun chekli 𝐸751 (−1; −563) F751 maydonini oling,
elliptik egri chiziq tayanch nuqtasi 𝑃 = (2, 373), tartibi 727 va kofaktor 1. Har bir
kichik harf 1 dan 26 gacha (shu jumladan) va har bir bosh harf uchun 27 dan 52
gacha bo'lgan qiymatni belgilaydi. Xesh funksiyani 727 modul bo‘yicha xabar
yig'indisi sifatida aniqlang. Sizning shaxsiy kalitingiz d = 113.
 “Men zo'rman” xabari uchun imzo yaratish uchun ECDSA dan
foydalaning, k uchun tasodifiy qiymat sifatida k = 235 dan foydalaning.
(ECDSA ning har bir bosqichini ko'rsating va kP ni hisoblash uchun
nuqtalarni qo‘shish hamda ikkilantirish ketma-ketligini ko'rsating.);
 Ochiq kalit Q = 113P yordamida imzongiz to‘g‘ri ekanligini tekshiring
(avval buni hisoblashingiz kerak);
 Ushbu xabarni imzolash orqali siz "imzolagan" boshqa xabarni (bu
shunchaki axlat emas) toping.
11. Diffi-Hellman kalitlarni taqsimlash algoritmi va El-Gamal elektron
raqamli imzo algoritmi (besh kishi uchun).
 Sizning o'qituvchingiz DHKE algoritmida birlamchi ildiz 𝑟 = 103
bo'lgan hamda chekli maydon qiymati 𝑝 = 11717 dan foydalanadi.
Bundan tashqari, o'qituvchingiz ElGamal ochiq kalitini (𝑝, 𝑟, 𝑎) =
(11717,103,1020) to'ldirish uchun 𝑎 = 1020 qiymatini hisoblab
chiqdi. Ushbu ochiq kalitdan foydalanib, siz 42 raqamidan iborat
bo'lishi kerak bo'lgan o'qituvchingiz xabar yubormoqchisiz (bu
"hayot, koinot va hamma narsa" ga javobdir). Siz qanday
shifrlangan matnni yuborasiz? Ishingizni ko'rsating!
 Sizning o'qituvchingiz sizga elektron pochta orqali yaqinda
o'tkazilgan testda sizning baholaringizni yubormoqchi bo'lsa va bu
elektron xat aslida o'qituvchingizdan kelib chiqqanligini isbotlash
uchun elektron pochtada 97 ball qiymati va " “Ushbu ball qiymati
mening ochiq kalitim yordamida ElGamal raqamli imzosi bilan
imzolangan ((p,r,a)=(11717,103,1020)); imzo qiymatiga ega
(6220,10407)" qo'shimchasini o'z ichiga oladi. Siz buni haqiqatan
ham o'qituvchingizdan kelgan deb qabul qilasizmi? Ishingizni
ko'rsating!
12. Quyidagi qitmatni “kvadrat elak” usuli yordamida faktorlarga
ajrating (to‘rt kishi uchun).
𝑁
= 2799783391122132787082946763872260162107044674070975356536
948079411322889349850197156207726828824297798741533122674663
24664879544163509079879.
13. Quyidagi qitmatni “brent” usuli yordamida faktorlarga ajrating (to‘rt
kishi uchun).
𝑁
= 21103175723325770325822795411414573403619421415019877707386860
4030636584321757797167901611506977156335718398322841012483665288
9322747516628218630786886185480849506355788733345084562838177755
6548679338293100762367188895036288525143912088800884584687522905
7882804989071193473026147578126398674375180316408645773612053041
0131350699232328851062144562699590996891630507177104956346150578
7351169282998393180211851214819438493842546081531033265487305719
8852626817812204833636223865524309115957670580695329250092000862
4096184069979286095181534551918182825906724092051973577500240522
2225074761142292844852151014620747227154458222332840787817112327
407303967988599944726539391
14. Quyidagi qitmatni “p-1 Pollard” usuli yordamida faktorlarga
ajrating (to‘rt kishi uchun).
𝑁
= 2522450745615916315657587779602845647366236973690487810093306
088788724421612026028169532886173905629427064309779431730897675
995836743754924480886377290668093952107048958264749843824491519
970546839581757560659781401956291370756230639716876456768678729
3933393417031620511244511892437239390863502308155450078419924080
2631029786638975323653639817197620817505526271972119454861414422
7452874632235452265190290890340695006009434372266898812931803072
9213169059762443357730983377434699924583492845212206319390320703
6228664770087110136440108865558203677349965120614325293916934211
90426715896428082690439073605633199475981032823