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Solution of a Weakly Delayed Difference System
Šafařík, Jan
The paper solves a weakly delayed difference systém x(k+1) = Ax(k)+Bx(k-1) where k = 0;1; : : : , A = (ai j)3 i; j=1, B = (bi j)3i ; j=1 are constant matrices. An explicit solution is given with a discussion on the number of independent initial data.
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Weakly Delayed Systems of Linear Discrete Equations in R^3
Šafařík, Jan ; Khusainov, Denys (oponent) ; Růžičková, Miroslava (oponent) ; Diblík, Josef (vedoucí práce)
The present thesis deals with the construction of a general solution of weakly delayed systems of linear discrete equations in ${\mathbb R}^3$ of the form \begin{equation*} x(k+1)=Ax(k)+Bx(k-m) \end{equation*} where $m>0$ is a positive integer, $x\colon \bZ_{-m}^{\infty}\to\bR^3$, $\bZ_{-m}^{\infty} := \{-m, -m+1, \dots, \infty\}$, $k\in\bZ_0^{\infty}$, $A=(a_{ij})$ and $B=(b_{ij})$ are constant $3\times 3$ matrices. The characteristic equations of weakly delayed systems are identical with those of the same systems but without delayed terms. The criteria ensuring that a system is weakly delayed are developed and then specified for every possible case of the Jordan form of matrix $A$. The system is solved by transforming it into a higher-dimensional system but without delays \begin{equation*} y(k+1)=\mathcal{A}y(k), \end{equation*} where ${\mathrm{dim}}\ y = 3(m+1)$. Using methods of linear algebra, it is possible to find the Jordan forms of $\mathcal{A}$ depending on the eigenvalues of matrices $A$ and $B$. Therefore, general the solution of the new system can be found and, consequently, the general solution of the initial system deduced.
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Digital Forensics: The Acceleration of Password Cracking
Hranický, Radek ; Hudec,, Ladislav (oponent) ; Rowe, Neil (oponent) ; Šafařík, Jiří (oponent) ; Ryšavý, Ondřej (vedoucí práce)
Cryptographic protection of sensitive data is one of the biggest challenges in digital forensics. A password is both a traditional way of authentication and a pivotal input for creating encryption keys. Therefore, they frequently protect devices, systems, documents, and disks. Forensic experts know that a single password may notably complicate the entire investigation. With suspects unwilling to comply, the only way the investigators can break the protection is password cracking. While its basic principle is relatively simple, the complexity of a single cracking session may be enormous. Serious tasks require to verify billions of candidate passwords and may take days and months to solve. The purpose of the thesis is thereby to explore how to accelerate the cracking process. I studied methods of distributing the workload across multiple nodes. This way, if done correctly, one can achieve higher cracking performance and shorten the time necessary to resolve a task. To answer what "correctly" means, I analyzed the aspects that influence the actual acceleration of cracking sessions. My research revealed that a distributed attack's efficiency relies upon the attack mode - i.e., how we guess the passwords, cryptographic algorithms involved, concrete technology, and distribution strategy. Therefore, the thesis compares available frameworks for distributed processing and possible schemes of assigning work. For different attack modes, it discusses potential distribution strategies and suggests the most convenient one. I demonstrate the proposed techniques on a proof-of-concept password cracking system, the Fitcrack - built upon the BOINC framework, and using the hashcat tool as a "cracking engine." A series of experiments aim to study the time, performance, and efficiency properties of distributed attacks with Fitcrack. Moreover, they compare the solution with an existing hashcat-based distributed tool - the Hashtopolis. Another way to accelerate the cracking process is by reducing the number of candidate passwords. Since users prefer strings that are easy to remember, they unwittingly follow a series of common password-creation patterns. Automated processing of leaked user credentials can create a mathematical model of these patterns. Forensic investigators may use such a model to guess passwords more precisely and limit tested candidates' set to the most probable ones. Cracking with probabilistic context-free grammars represents a smart alternative to traditional brute-force and dictionary password guessing. The thesis contributes with a series of enhancements to grammar-based cracking, including the proposal of a novelty parallel and distributed solution. The idea is to distribute sentential forms of partially-generated passwords, which reduces the amount of data necessary to transfer through the network. Solving tasks is thus more efficient and takes less amount of time. A proof-of-concept implementation and a series of practical experiments demonstrate the usability of the proposed techniques.
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Digital Forensics: The Acceleration of Password Cracking
Hranický, Radek ; Hudec,, Ladislav (oponent) ; Rowe, Neil (oponent) ; Šafařík, Jiří (oponent) ; Ryšavý, Ondřej (vedoucí práce)
Cryptographic protection of sensitive data is one of the biggest challenges in digital forensics. A password is both a traditional way of authentication and a pivotal input for creating encryption keys. Therefore, they frequently protect devices, systems, documents, and disks. Forensic experts know that a single password may notably complicate the entire investigation. With suspects unwilling to comply, the only way the investigators can break the protection is password cracking. While its basic principle is relatively simple, the complexity of a single cracking session may be enormous. Serious tasks require to verify billions of candidate passwords and may take days and months to solve. The purpose of the thesis is thereby to explore how to accelerate the cracking process. I studied methods of distributing the workload across multiple nodes. This way, if done correctly, one can achieve higher cracking performance and shorten the time necessary to resolve a task. To answer what "correctly" means, I analyzed the aspects that influence the actual acceleration of cracking sessions. My research revealed that a distributed attack's efficiency relies upon the attack mode - i.e., how we guess the passwords, cryptographic algorithms involved, concrete technology, and distribution strategy. Therefore, the thesis compares available frameworks for distributed processing and possible schemes of assigning work. For different attack modes, it discusses potential distribution strategies and suggests the most convenient one. I demonstrate the proposed techniques on a proof-of-concept password cracking system, the Fitcrack - built upon the BOINC framework, and using the hashcat tool as a "cracking engine." A series of experiments aim to study the time, performance, and efficiency properties of distributed attacks with Fitcrack. Moreover, they compare the solution with an existing hashcat-based distributed tool - the Hashtopolis. Another way to accelerate the cracking process is by reducing the number of candidate passwords. Since users prefer strings that are easy to remember, they unwittingly follow a series of common password-creation patterns. Automated processing of leaked user credentials can create a mathematical model of these patterns. Forensic investigators may use such a model to guess passwords more precisely and limit tested candidates' set to the most probable ones. Cracking with probabilistic context-free grammars represents a smart alternative to traditional brute-force and dictionary password guessing. The thesis contributes with a series of enhancements to grammar-based cracking, including the proposal of a novelty parallel and distributed solution. The idea is to distribute sentential forms of partially-generated passwords, which reduces the amount of data necessary to transfer through the network. Solving tasks is thus more efficient and takes less amount of time. A proof-of-concept implementation and a series of practical experiments demonstrate the usability of the proposed techniques.
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Weakly Delayed Systems In ℝ3
Šafařík, Jan
The paper is concerned with a linear discrete system with delay x(k+1) = Ax(k)+Bx(k-m); k = 0,1,…, in R3. It is assumed that the system is weakly delayed. For one of the possible Jordan forms solution of an arbitrary initial problem is given.
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Solution of a Weakly Delayed Difference System
Šafařík, Jan
The paper solves a weakly delayed difference systém x(k+1) = Ax(k)+Bx(k-1) where k = 0;1; : : : , A = (ai j)3 i; j=1, B = (bi j)3i ; j=1 are constant matrices. An explicit solution is given with a discussion on the number of independent initial data.
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Weakly Delayed Systems of Linear Discrete Equations in R^3
Šafařík, Jan ; Khusainov, Denys (oponent) ; Růžičková, Miroslava (oponent) ; Diblík, Josef (vedoucí práce)
The present thesis deals with the construction of a general solution of weakly delayed systems of linear discrete equations in ${\mathbb R}^3$ of the form \begin{equation*} x(k+1)=Ax(k)+Bx(k-m) \end{equation*} where $m>0$ is a positive integer, $x\colon \bZ_{-m}^{\infty}\to\bR^3$, $\bZ_{-m}^{\infty} := \{-m, -m+1, \dots, \infty\}$, $k\in\bZ_0^{\infty}$, $A=(a_{ij})$ and $B=(b_{ij})$ are constant $3\times 3$ matrices. The characteristic equations of weakly delayed systems are identical with those of the same systems but without delayed terms. The criteria ensuring that a system is weakly delayed are developed and then specified for every possible case of the Jordan form of matrix $A$. The system is solved by transforming it into a higher-dimensional system but without delays \begin{equation*} y(k+1)=\mathcal{A}y(k), \end{equation*} where ${\mathrm{dim}}\ y = 3(m+1)$. Using methods of linear algebra, it is possible to find the Jordan forms of $\mathcal{A}$ depending on the eigenvalues of matrices $A$ and $B$. Therefore, general the solution of the new system can be found and, consequently, the general solution of the initial system deduced.
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Weakly delayed planar linear discrete systems and conditional stability
Šafařík, J.
A discrete planar system x(k+1) = Ax(k)+B1x(k−m1)+B2x(k−m2), k ≥ 0 is analysed, where m1, m2 are constant integer delays, 0 < m1 < m2, A,B1,B2 are constant 2 × 2 matrices, A = (aij), Bl = (blij), i, j = 1,2, l = 1,2 and x: {−m2,−m2 +1,...} → R2. We get new results on conditional stability and asymptotic conditional stabilit.
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