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Quasigroup based cryptography
Frisová, Andrea ; Drápal, Aleš (referee) ; Stanovský, David (advisor)
In this work, we study some properties of an in nite matrix, which consists of quasigroup elements. This matrix is generated from a certain sequence X using left iterated translations. We suppose that the sequence X is periodic and we examine how the periods of the rows of our matrix behave for various types of quasigroups. We show that for central quasigroups the periods increase at most linearly. Further, we try to apply our result to the stream cipher Edon-80.
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Quasigroups, one-way functions and hash mappings
Machek, Ivo ; Stanovský, David (referee) ; Drápal, Aleš (advisor)
In the rst part of this work we study the complexity of solving nonlinear quasigroup equations for di erent classes of quasigroups. In particular we study the application of principle of central quasigroups on the blocks of congruence. We show that these quasigroups can be shapeless and therefore we gain counterexample to the hypothesis which was stated by D. Gligoroski. In the second part of this work we apply previous results on the concrete quasigroups of the type Edon-R-I,II and we deduce the complexity of the corresponding algorithm for inverting the hash function Edon-R.
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Factorization of polynomials over finite fields
Straka, Milan ; Stanovský, David (referee) ; Žemlička, Jan (advisor)
Nazcv prace: Faktorizace polynoinu nad konccnynii telesy Autor: Milan Straka Katcdra (ustav): Katcdra algebry Vedouci bakalarske prace: Mgr. Jan Zcmlicka, Ph.D. E-mail vedouciho: Jan.Zemlicka((hnff. cuni.cz Abstrakt: Cilem prace je prozkoumat problem rozkladu polynomn nad konecnym telc- scm na soucin ircducibilnich polynoinu. PopHanim nekolika algoritmu hledaji- cich tento rozklad se ukaze, ze tento problem je vzdy fcsitclny v polynornialnim case vzhleclem kc stupni polynomu a poctu prvku konecneho telcsa. U jeduoho z algoritnm je po])sana implenientace s vclnii clobrou asymptotic- kou casovou slozito.sti O(nLylD log c/}, kdc i\. jc stupen rozkladaneho polynuinn nad telesem « q prvky. Program pouzivajiei jcdnodnssi, ale prakticky rychlcjsi variantu tohoto algoritnm jc soucasti ])racc. Klicova slova: faktorizace, kouecna telesa, polynoniy, algoritmns Title: Factoring polynomials over finite fields Author: Milan Straka Department: Department of Algebra Supervisor: Mgr. Jan Zemlicka, Ph.D. Supervisor's e-mail address: Jan. Zcirilicka@mJJ.cum.cz Abstract: The goal of this work is to present the problem of the decomposition of a polyno- mial over a finite field into a product of irreducible polynomials. By describing algorithms solving this problem, we show that the decomposition can always be found in...
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Identity checking
Polach, František ; Žemlička, Jan (referee) ; Stanovský, David (advisor)
Nazev prace: Testovdni identit Autor: FrantiSek Polnch Katedra (ustav): Katedra algebry Vedouci bakalafske prace: RNDr. David Stanovsky, Ph.D. e-mail vedouciho: stanovsk@karlin.mff.citni.cz Abstrakt: Na overeni, zda dana identita (napf. komlttativita, asociativita, apod.) plati v dane algebre (grupe, okrului,...), existuje ocividny algoritmtts, ktery ma exponencidlni slozitost vzh- ledem kdelce zadane identity (profixm'algebru)- Neni tezke nahlednout, ze tento problemje pro libovolnou algebra v I ride co-NP a ze existuji algebry, pro ktere je co-NP-uplny. Na druhou stranu, pro mnoho algeber (napr. pro abelovske grupy) existitje algoritmus polynomidlni. Ex- istuje mezindrodni projekt, jehoz cilein je charakterizovat ty algebry, pro ktere je tento prob- lem pollfnomidlni, rcsp. co-NP-iiplny. Cflem tcto prdceje shrnout nektere zndme vysledky o grupdch a okruzich. Konkretne ukdzeme polynomidlni algoritmy pro testovdni identit v nilpo- tentnich i dihedrdlnfch grupdch a nilpotentnich okruzich, a dokdzeme co-NP-iiplnost testovdni identit v nenilpotcntnfch okruzich. Klicova slova: testovdni identit, slozitost, grupy, okruhy Title: Identity checking Author: Franlisek Polach Department: Department of Algebra Supervisor: RNDr. David Stanovsky, Ph.D. Supervisor's e-mail address: stanovsk@karlin.inff.cuni.cz Abstract:...
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Binary equality words
Hadravová, Jana ; Stanovský, David (referee) ; Holub, Štěpán (advisor)
Binary equality language is a set consisting of all solutions of equation g(w) = h(w), where g, h are arbitrary binary morphisms. Recently, it has been prooved that equality set for each pair of morphisms g, h is generated by at most two words. Structure of binary equality language has been already known in the case that at least one of morphisms g, h is periodic or if their equality set is generated exactly by two words. The main objective of the paper was to find a structure of solutions for morphisms whose equality set is generated by one word. The problem in general case remains unsolved but special result for solutions consisting of just one block for marked morphisms was discovered. Using methods established in this paper (covering by the same pattern to find n-multiple p-overflows and working with the cyclic pair (e, f, z)) it is believed that some more results can be achieved in the near future.
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Interactive proofs
Kučerová, Jana ; Stanovský, David (referee) ; Tůma, Jiří (advisor)
Nazev prace: Interakt.ivui dukazy Aulor: .lana Kucerova Katedra. (uslav): Katedra algebry Vedonc] bakalafske praee: Dor.RNDr. Jin Tunia.DrSc. e-mail vedouei'ho: 1 uina'U'karliii.inll.rimi.rz Abstrakt: I'fedlo/ena pra.ce se veimje kryptografickym protokolum pro interak- tivm dukazy. Protoze tyto protokoly jsou jcduy zc slozitrjsicli, json v prvni casti pract' |>o]).s;'ina nrklrra. jtxbuKlussi kryijtograficka schrinata. kl.era jsou pozdrji vyuzita jako stavcljni prvkv ttrhtu protokohi. \ ttHn prari jsou intrraktivni diika.7.%' iR'jpi'vi^ (Irlinovany ja.kci vyporcl .s])ujcnr dvojict1 iiiU'raktivm'ch Turiii- i^ovych stroju. r>ozdt'ji ju uvctk'iia suuvislo.st taklo dcfinovaurho intcraktivni'ho duka/u s inl.fM-akl.ivniin dukazoTii /nalosli dnkax,o\'at,('luva lajcin.stvi v ideutili- kariiirh prolokulcrli. Xa pfikladu urknlika idrntifikarnirh protokulu je ukazaiio. jakyin /pusobcm l/x1 rozluxlnout. /da sc jediui o dukay, s nulo\'oii /ualosli imbo zda jc dauy prolokol prukazatrlm"' Ix'Zpcrny. Kli'cova iikjva: krypto.u.raficky protokol. Tiirin^uv stroj. proka/atrlua lK'Zp(.'cnost. diika./ s inilovoii /nalosti Tillc: lnUTarlivr ])roc)fs Author: Jana Kurt'rova, Depart incut: Df^jaitnuMil of Alp,cbra SuptTvLsor: Doc-.HNDv. Jiff Tuina,I)rSc. SujicTvisor's r-inail addi'rss: t uina "karlin.inll.cuiii.rz Abstract: Tlio main...
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