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Computational methods for finding cryptographic functions
Kroutil, Jaroslav ; Göloglu, Faruk (advisor) ; Růžička, Pavel (referee)
Recent studies have demonstrated several methods on different approaches to classi- fication of vectorial Boolean functions up to certain equivalence relation and to finding new quadratic Almost Perfect Nonlinear (APN) functions. In this work we explore these classification methods of vectorial Boolean functions, in particular those that minimise the search space up to EA-equivalence or linear-equivalence. We also investigate various strategies for finding quadratic APN functions. These methods are rooted in various aspects of algebraic theory. We explore the mathematical theory in more detail, and provide a guide to practical application of the theory. We also provide implementations of these methods and illustrate them in the context of the presented theory. 1
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Preprocessing Techniques in Algebraic Cryptanalysis
Mašková, Kristýna ; Hubáček, Pavel (advisor) ; Göloglu, Faruk (referee)
Algebraic cryptanalysis is a standard set of techniques for analyzing and attacking practical symmetric cryptographic primitives. It involves representing the relationship between a pair of plaintext, ciphertext and the key as a system of polynomial equations and then solving the system using Gröbner bases. When the equations depend only on the key, we can generate multiple systems of equations. This thesis examines preprocessing techniques in algebraic cryptanalysis, reducing large systems of equations to improve the performance of practical solving algorithms. Concentrating on a technique that aims to increase the sparsity of the polynomials, we lay the theoretical foundations for two methods. The first method of exhaustively going over all pairs and the second method of finding candidates for similar pairs using Locality- Sensitive Hashing. Finally, we improve on the latter method by targeting the leading monomials. 1
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Semifields and skew polynomial rings
Liška, Jakub ; Göloglu, Faruk (advisor) ; Pavlů, Jiří (referee)
In this thesis we give constructions of semifields, often characterized as not necessarily associative division algebras, from skew polynomial rings which are rings of polynomials over a field where multiplication is not commutative. These constructions are crucial for their use as maximum rank-distance codes, a family of self-correcting codes with a rank distance metric. We explore various connections between these structures through isotopy, isomorphy and equality. We also make the effort to prove as much of the fun- damental theory as possible since it is often regarded as obvious by the experts of the field. 1
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Security of Trapdoor Permutations under Preimage Leakage
Sedláček, Petr ; Hubáček, Pavel (advisor) ; Göloglu, Faruk (referee)
This thesis explores preimage leakage-resilient trapdoor permutations (PLR-TDPs) and their applications in proofs of storage replication and incompressible encodings. The thesis consists of three chapters covering the trapdoor permutations, formal definition of PLR-TDPs, and analysis of security properties of PLR-TDPs. The first chapter provides an overview of trapdoor permutations (TDPs), their def- initions, and applications in proofs of storage replication. Our results are presented in the second and third chapters. The second chapter formally defines PLR-TDPs and demonstrates their use by constructing a simple incompressible encoding in the random oracle model. The third chapter focuses on the existence of PLR-TDPs. It demonstrates the strong preimage leakage-resilience of fully random TDPs in an idealized model. We are the first to provide a partial formal justification for the conjecture of the preimage leakage-resilience of practical TDPs, such as RSA or Rabin permutations.
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Cryptosystems based on coding theory
Parýzková, Zuzana ; Žemlička, Jan (advisor) ; Göloglu, Faruk (referee)
Nowadays public-key cryptosystems such as RSA are threatened by quantum comput- ing. Therefore, a post-quantum standardization process was initiated by NIST in 2017. As of today, several cryptosystems have been selected for standardization and several still remain in the process. A cryptosystem based on coding theory - Classic MeEliece - is one of the cryptosystems that might be standardized. This thesis covers McEliece and Niederreiter cryptosystems as well as their rank-metric variants (GGPT cryptosystem). Sidelnikov-Shestakov's attack is explained in detail and an example of the attack is given. Stern's and Overbeck's attacks are discussed as well. Furthermore, a new polynomial-time attack against GGPT without distortion matrix X is given. 1
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Rank Two Commutative Semifields
Tittl, Ondřej ; Göloglu, Faruk (advisor) ; Růžička, Pavel (referee)
In this thesis we will explain what are semifields and what interesting properties these algebraic objects possesses. In the first chapter we will go over some basics and preliminaries to understand what semifields are. In the second chapter we will prove some useful lemmata for either commutative and non-commutative case of semifields and provide some examples. At last we will try to do some research by ourselves, where we will try to find some examples of semifields. 1
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Kryptografická primitiva v liché charakteristice
Richterová, Berenika ; Göloglu, Faruk (advisor) ; Pavlů, Jiří (referee)
In the thesis, we focus on the cryptanalytic techniques that can be used against the cryptographic primitives operating over odd characteristic finite fields. We introduce the GMiMC permutation, which belongs to such primitives. We describe an integral attack and generalize a certain type of this attack to finite fields of any characteristic. We also explain the notion of an impossible differential attack and then apply both the integral and the impossible differential attack to the GMiMC permutation. 1
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Slide Attacks
Zenknerová, Karolína ; Göloglu, Faruk (advisor) ; El Bashir, Robert (referee)
Slide attack is an attack against block ciphers which have all rounds the same. The success and the complexity of the attack is independent on the number of rounds. The original slide attack was mainly used on a Feistel structure, but very rarely on SPN networks, because in general, SPN networks have the last round diferent. This property does not allow to use normal slide attack. In the paper New slide attacks on almost self-similar ciphers by Orr Dunkelman, Nathan Keller, Noam Lasry, and Adi Shamir are introduced new slide attacks (four of them) which focus on SPN networks and they overcome a problem of the last round. In this thesis we explain main idea of the original slide attack and the main idea of two new slide attacks - a slid sets attack and a slide attack using a hypercube of slid pairs. In both these attacks we create and use special structures of plaintexts and ciphertexts to get more pairs of plaintexts which we call slid pairs. Moreover, we explain some selected parts of two new slide attacks and we compute the complexity. 1
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Boomerang attacks
Procházka, Martin ; Göloglu, Faruk (advisor) ; Středa, Adolf (referee)
In 2020 Dunkelman et al. introduced the Retracing Boomerang Attack as an extension of the Boomerang attack introduced in 1999 by Wagner. Both of these attacks are presented in this thesis. In the thesis we also study in detail the independences of the differential characteristics of the Boomerang attack as an element affecting the success probability of the attack. All this, together with background information and detail explanation of the differential cryptanalysis is presented in the well understandable form even for reader with just an elementary cryptography experience. Finally, in the last chapter we show idea and theoretical description of the best currently known differential attack on 5-round AES, which is based on the Retracing Boomerang Attack. 1
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Rank Two Commutative Semifields
Tittl, Ondřej ; Göloglu, Faruk (advisor) ; Růžička, Pavel (referee)
In this thesis we will explain what are semifields and what interesting properties these algebraic objects possesses. In the first chapter we will go over some basics and preliminaries to understand what semifields are. In the second chapter we will prove some useful lemmata for either commutative and non-commutative case of semifields and provide some examples. At last we will try to do some research by ourselves, where we will try to find some examples of semifields. 1
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