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RSA in number fields and on lattices
Kucka, Filip Miroslav ; Kala, Vítězslav (advisor) ; Šůstek Vyhnalová, Sára (referee)
This thesis is focused on the RSA algorithm in number fields and on lattices. Specif- ically, we extend the work the authors Zheng and Liu in their article High Dimensional RSA. In the thesis we precisely describe all the theory required theory with theorems and examples using mostly Algebraic number theory and lattice theory. In the second chapter, we create the RSA only in number fields, we discuss its problems and the ne- cessity of lattices. In the third chapter, we precisely describe and prove properties of ideal matrices, we define the vector multiplication in Rn and at the end ve prove the ring isomorphism K ≃ Qn ≃ M∗ Q. In the fourth chapter, we prove the ring isomorphism Z[x]/(mθ(x)) ≃ OK ≃ Zn ≃ M∗ Z, we define ideal lattices and we create all the required theory over lattices for RSA. The last chapter consists of the complete RSA algorithm in number fields and on lattices and example. 1
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Ideal lattices in cryptography
Vyhnalová, Sára ; Kala, Vítězslav (advisor) ; Příhoda, Pavel (referee)
The thesis is focused on the theory of special lattices that are important in cryptography, namely ideal, cyclic and NTRU lattices. Specifically, we expand and generalise the work of Ding and Lindner on Identifying Ideal Lattices. The algorithm for identifying ideal lattices is included, along with illustrative examples and more detailed proofs of propositions on which the algorithm is based. In the section about Lattice Isomorphism there is also included a generalised theorem from the paper. We extend the result of identifying the NTRU lattices and supplement it with several examples. The thesis also contains Chapter Applications in Cryptography where we describe a cryptographic hash function based on ideal lattices. And finally, we provide a brief overview of the cryptographic algorithms using NTRU lattices.
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Solovay-Strassen primality test
Vyhnalová, Sára ; Kala, Vítězslav (advisor) ; Vávra, Tomáš (referee)
This thesis studies the Solovay-Strassen test for primality of an integer n, which is based on the Jacobi symbol. After formulating the basic algorithm, we compute the probability that the number n being tested is really a prime number if the Solovay-Strassen test declared it so. We further improve the computation of the probability under the assumption that n is not divisible by specific small primes, which can be easily verified. Finally, we construct a new test, as an analogy of the Solovay-Strassen test, based on the quartic residue symbol. 1
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