
Multiplication in a finite field of characteristic 2 and XORmetrics
Carulkov, Nikita Edward ; Žemlička, Jan (advisor) ; Göloglu, Faruk (referee)
XORcounts measure the efficiency of multiplication in finite fields of characteristic 2. In the first chapter we define two XORcounts (the direct XORcount and the sequential XORcount) and present detailed proofs of some propositions from the paper from Lukas Kolsch about the XORcounts of inverse matrices and permutation similar matrices. It seems that the case when the direct XORcount is lower than the sequential XORcount is rare. We will explore those cases in the second chapter. Some of them were already described in the paper from Lukas Kolsch and we prove that they occur only for matrices with order higher or equal to six. 1


A study on ``A New PublicKey Cryptosystem via Mersenne Numbers''
Richter, Filip ; Göloglu, Faruk (advisor) ; El Bashir, Robert (referee)
In 2016 NIST announced a start of a process of development and standardiza tion of a postquantum publickey encryption scheme. Mersenne756839 was one of the proposals. This proposal is described in this thesis, as well as the known attacks against it. The description and the theoretical background behind these attacks are presented in a rigorous way and are accessible to the reader without any previous knowledge about the postquantum cryptography. New additional ideas for the implementation of the attacks are also presented. Finally, these attacks are implemented and attached to the thesis. 1


APN functions with nonclassical Walsh spectra
Maršálek, Michal ; Göloglu, Faruk (advisor) ; Drápal, Aleš (referee)
An interesting class of Boolean functions are APN functions  these func tions are "as far" from linear functions as possible. Most of the quadratic APN functions have the same (=classical) Walsh spectrum  a sort of footprint of the function. The aim of this thesis is to describe a method which might lead to a generalisation of a sporadic example of a quadratic APN function with nonclassical Walsh spectrum. Up until recently, it was believed that no such function exists. This was proven to be false in 2009, as an example of such func tion in dimension 6 was introduced. In this thesis, we describe the construction and then deduce necessary conditions for some free coefficients in order to reduce the search space to a level which enables a computer search. 1


Hledání APN permutací ve známých APN funkcích
Pavlů, Jiří ; Göloglu, Faruk (advisor) ; Drápal, Aleš (referee)
In the thesis a new way of checking whether a function is CCZequivalent to a permutation is given. The results for known families of almost perfect nonlinear (APN) functions are presented for functions defined over GF(2n ), for even n ≤ 12. The ways how to reduce the number of polynomials from each family are studied. For functions of the form x3 + a1 tr1(a3 x9 ) it is shown, that they cannot be CCZequivalent to a permutation on fields GF(24n ) for n ∈ ℕ .


Constructions of APN permutations
Krasnayová, Dáša ; Göloglu, Faruk (advisor) ; Lisoněk, Petr (referee)
In this thesis, we examine a family of vectorial boolean functions on F22m inspired by Kim function, in order to find new APN permutations on F22m for m > 2. The functions of this family are defined as F(X) = X3 + bX3q + cX2q+1 + dXq+2 , where parameters b, c and d are from F2m . Necessary and sufficient conditions for this functions to be APN or equivalent to a permutation are presented in this thesis. To find conditions for being APN, Trace0/Trace1 decomposition method is used. A method using exponential sums is used to deduce which functions of this family is CCZequivalent to a certain type of permutation. These results were then used to search for APN permutations on F26 and F210 . 1


Relationship between higher order attacks and CCZequivalence
Deptová, Lucie ; Göloglu, Faruk (advisor) ; Hojsík, Michal (referee)
In this thesis, we explain the term CCZequivalence in more detail to gether with an analysis of a special type of matrices of this equivalence. We also clarify the higher order differential cryptanalysis and its generalized ver sion. To demonstrate this method we present several attacks on a simple five round Feistel cipher, two of these attacks are our own. We have implemented the most important attacks and results of these experiments can be found in the text. We also explore how to use a decomposition F = F2 ◦F−1 1 (where F1 and F2 are permutations) to construct a generalized higher order differential attack to a block cipher with an Sbox F. This construction may be used while searching for an attack to F using the CCZequivalence which is gener ally a hard question. The result of our research is a theorem which presents a necessary condition on a degree of F which is needed for an existence of such a decomposition. 1


Algebraicdifferential analysis of Keccak
Seidlová, Monika ; Göloglu, Faruk (advisor) ; Hojsík, Michal (referee)
In this thesis, we analyze the cryptographic sponge function family Keccak  the winner of the SHA3 Cryptographic Hash Standard competition. Firstly, we explore how higher order differentials can be used to forge a tag in a parallelizable MAC function. We introduce new terms and theory studying what affine spaces remain affine after one round of Keccak's underlying permutation Keccakf. This allows us to improve the forgery. Secondly, collisions in Keccak could be generated from pairs of values, that follow particular differential trails in Keccakf. We tested finding pairs for a given differential trail in reducedround Keccakf using algebraic techniques with the mathematics software SAGE. We found a pair in a 4round trail in Keccakf[50] in under 5 minutes and a 3round trail in Keccakf[100] in 80 seconds on a regular PC. Powered by TCPDF (www.tcpdf.org)


Links Between Differential and Linear Cryptanalysis
Töpfer, Jakub ; Hojsík, Michal (advisor) ; Göloglu, Faruk (referee)
This thesis concerns the relations between correlation matrix, difference propagation matrix and other matrices used in the block cipher cryptanalysis. We show that some relations between these matrices can be seen just as a change of basis provided by the discrete Fourier transform. This can be used for an easier proof of a wellknown theorem. We also study properties of difference propagation matrix, describe a class of vectorial Boolean functions which have the same difference propagation matrix and state a numerically justified hypothesis that this class contains all such functions.
