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Security of cryptographic schemes for contact tracing
Pastyřík, Martin ; Hubáček, Pavel (advisor) ; Göloglu, Faruk (referee)
Due to the Covid-19 pandemic in 2020 there was a big development of contact tracing schemes and applications. In this thesis, we describe the DP3T scheme and some possible attacks against it mainly the replay and relay attacks. In order to resist these attacks, we formally define and construct Pietrzak's Delay-MAC (INDOCRYPT 2020). Using this construction and the definition of a DCT scheme by Danz et al. (IACR Cryptol. ePrint Arch. 2020: 1309), we formally define Pietrzak's (INDOCRYPT 2020) contact tracing scheme, which we call DP4T. Using the security model presented in (IACR Cryptol. ePrint Arch. 2020: 1309), we prove that DP4T is resistant to replay attacks and discuss if the improvement of DP4T presented in (INDOCRYPT 2020) is resilient to relay attacks. Using definitions and properties from (IACR Cryptol. ePrint Arch. 2020: 1309) we discuss privacy of DP4T. We then present two new attacks on DCT schemes and prove that other schemes from literature are not resistant to them. We prove that DP4T is resilient against one of those attacks and discuss the importance of this result to the improvement of DP4T resistant to relay attacks. 1
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On search complexity of discrete logarithm
Václavek, Jan ; Hubáček, Pavel (advisor) ; Koucký, Michal (referee)
In this thesis, we study the discrete logarithm problem in the context of TFNP - the complexity class of search problems with a syntactically guaranteed existence of a solution for all instances. Our main results show that suitable variants of the discrete logarithm problem, which we call Index and DLog, are complete for the classes PPP and PWPP, respectively. Additionally, our reductions provide new structural insights into PWPP by establishing two new PWPP-complete problems. First, the problem Dove, a relaxation of the PPP-complete problem Pigeon. Dove is the first PWPP-complete problem not defined in terms of an explicitly shrinking function. Second, the problem Claw, a total search problem capturing the computational complexity of breaking claw-free permuta- tions. In the context of TFNP, the PWPP-completeness of Claw matches the known intrinsic relationship between collision-resistant hash functions and claw-free permuta- tions established in the cryptographic literature. 1
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Modern applications of zero-knowledge protocols
Krňák, Tomáš ; Hubáček, Pavel (advisor) ; Mareš, Martin (referee)
zk-SNARK is a cryptographic protocol, which enables transformation of an arbitrary computation into short effectively verifiable argument of correctness of this computation. Further more, it enables a prover to decide exactly, which inputs of the computation will be public and which inputs will stay private. The goal of this work is to present features, construction and applications of modern zk-SNARKs. In a construction part of this work we describe construction based on linear PCP and Paillier cryptosystem. In an application part we explain principles of anonymous cryptocurrency Zcash and we describe a completely new application of zk-SNARKs in networks of trust. 1
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Gröbnerovy báze v kryptografii
Hubáček, Pavel ; Stanovský, David (advisor) ; Šťovíček, Jan (referee)
The thesis focuses on the use of GrÄobner bases in cryptography and especially on applications in cryptanalysis of block ciphers. Some elementary concepts from the theory of GrÄobner bases are introduced together with Buchberger's algorithm, a method for constructing such bases. The principle of solving of poly nomial systems using suitable GrÄobner bases is explained. This is followed by pre sentation of modern algorithms that improve the Buchberger's algorithm. In the last part of the thesis present results achieved by GrÄobner bases are summarised and the notion of algebraic cryptanalysis is introduced. In algebraic cryptanalysis we transform breaking of given cryptosystem into a problem of solving polynomial equations over some nite eld. Examples of polynomial descriptions of block ciphers are provided together with some experimental result on arising polynomial systems.
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