Ing. Ivo Petr, Ph.D.

Ing. Ivo Petr, Ph.D.

Ing. Karel Klouda, Ph.D.

Department of Computer Systems

Many of today's cryptosystems are based on the discrete logarithm problem. There is no known algorithm that is able to solve it quickly in an arbitrary group. Often a group of points of an elliptic curve is used to implement the problem in. Although it works only for anomalous elliptic curves, Smart's attack solves the problem with linear time complexity. In this thesis we describe how Smart's attack works and we implement it in SAGE software. In the end we present a measurement comparing its time complexity to time complexity of Baby-step giant-step and Pollard's Rho - algorithms that can solve the discrete logarithm problem in an arbitrary group.

Thesis on DSpace

Ing. Ivo Petr, Ph.D.

Mgr. Martin Jureček, Ph.D.

Department of Computer Systems

In my thesis, I focus on the issue of secure key distribution between communicating parties using the laws of quantum mechanics. This approach allows parties in addition to key distribution, the methods to detect eavesdropping by a third party on a quantum channel. In my thesis, I present the basic concepts of quantum computation and the BB84, B92 and E91 protocols that enable quantum key distribution. In the created solution, I provide a way to deal with the noisy quantum channel using linear codes. For each protocol, I list the method of possible eavesdropping and the means by which eavesdropping can be detected. Based on the collected data, I create a comparison of protocols on communication and security level. At the end of the thesis, I confirm these comparisons using the results from protocols simulation.

Thesis on DSpace

Ing. Ivo Petr, Ph.D.

prof. Ing. Pavel Tvrdík, CSc.

Department of Computer Systems

In this thesis we study the underlying mathematical principles that are funda- mental for lattice-based cryptosystem, namely NTRU. We have benchmarked algorithms proposed by Nicolas Gama, et al. on a low-dimensional NTRU matrices.

Thesis on DSpace

Ing. Ivo Petr, Ph.D.

Ing. Daniel Vašata, Ph.D.

Department of Theoretical Computer Science

Machine learning field has shown incredible impact on many kinds of optimization problems. Recently the power of machine learning was applied to speed up the quantum states preparation. Although approximation with quantum generative adversarial networks is one of the fastest ways to prepare a generic quantum state, training time for such models is still significant and can easily impair quantum advantage. This thesis explores incremental learning of quantum generative adversarial networks for the quantum states preparation problem and introduces learning use cases reducing the training time.

Thesis on DSpace

Ing. Ivo Petr, Ph.D.

Mgr. Martin Jureček

Department of Information Security

The elliptic curve discrete logarithm problem (ECDLP) is one of the most important problems in asymmetric cryptography. In recent years, several papers were concerned with the use of summation polynomials for solving the ECDLP efficiently. In this thesis, we summarize the state-of-the-art algorithms based on summation polynomials, and use these algorithms to solve the ECDLP over prime fields. A detailed complexity analysis of said algorithms is presented and verified by extensive tests. After a comparison of the presented algorithms with the well-known Pollard's $\rho$-algorithm we have come to a conclusion; the algorithms presented in this thesis are not yet practical and more research needs to be done.

Thesis on DSpace

Ing. Ivo Petr, Ph.D.

Mgr. Martin Jureček

Department of Computer Systems

One of the standard methods of solution of the discrete logarithm problem (DLP) in a group of points of an elliptic curve is the Baby-Step Giant-Step algorithm (BSGS). There are many variants of this algorithm and this work aims to investigate the state-of-the-art of the algorithm. I generated random elliptic curves in Weierstrass, Edwards and Montgomery form and compared their performance over different variants of BSGS. I came to a conclusion that some properties of elliptic curves and the form of the elliptic curve got a huge impact on the performance of the algorithms.

Thesis on DSpace