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Effectivity and Limitations of Homomorphic Secret Sharing Schemes
Jančová, Ľubica ; Hubáček, Pavel (advisor) ; Holub, Štěpán (referee)
This thesis focuses on constructions of Homomorphic Secret Sharing (HSS) based on assumptions not known to imply fully homomorphic encryption. The efficiency of these constructions depends on the complexity of the Distributed Discrete Logarithm (DDLog) problem in the corresponding groups. We describe this problem in detail, focusing on the possibility of leveraging preprocessing in prime order groups, and deriving upper bounds on the success probability for the DDLog problem with preprocessing in the generic group model. Further, we present a new HSS construction. We base our construction on the Joye-Libert encryption scheme which we adapt to support an efficient distributed discrete logarithm protocol. Our modified Joye-Libert scheme requires a new set of security assumptions, which we introduce, proving the IND-CPA security of our scheme given these assumptions. 1
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Alexander polynomial
Jančová, Ľubica ; Stanovský, David (advisor) ; Peksová, Lada (referee)
Title: Alexander polynomial Author: Ľubica Jančová Department: Department of Algebra Supervisor: doc. RNDr. David Stanovský, Ph.D., Department of Algebra Abstract: The subject of interest of this thesis is the Alexander polynomial in the knot theory as a knot invariant and various methods of its computa- tion. The thesis focuses on the description of the computation of the Alexander polynomial using four different methods, namely: colouring regions of the knot diagram, colouring arcs of the knot diagram, Seifert's method and the method using the Conway polynomial. In the first chapter we introduce basic notions of the knot theory. In the following chapters we describe methods of computa- tion of the Alexander polynomial. The final chapter deals with the possibility of using the Conway polynomial to show that all of the mentioned methods result in the same polynomial. The main result of this thesis are proofs that might lead to the complete proof of equivalence of algorithms of computation of the Alexander polynomial. Keywords: knot theory, Alexander polynomial, knot invariant
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