Analysis, classification and construction of optimal cryptographic Boolean functions

Modern cryptography is deeply founded on mathematical theory and vectorial Boolean functions play an important role in it. In this context, some cryptographic properties of Boolean functions are defined. In simple terms, these properties evaluate the quality of the cryptographic algorithm in which the functions are implemented. One cryptographic property is the differential uniformity, introduced by Nyberg in 1993. This property is related to the differential attack, introduced by Biham and Shamir in 1990. The corresponding optimal functions are called Almost Perfect Nonlinear functions, shortly APN. APN functions have been constructed, studied and classified up to equivalence relations. Very important is their classification in infinite families, i.e. constructing APN functions that are defined for infinitely many dimensions. In spite of an intensive study of these maps, many fundamental problems related to APN functions are still open and relatively few infinite families are known so far. In this thesis we present some constructions of APN functions and study some of their properties. Specifically, we consider a known construction, L1(x^3)+L2(x^9) with L1 and L2 linear maps, and we introduce two new constructions, the isotopic shift and the generalised isotopic shift. In particular, using the two isotopic shift constructing techniques, in dimensions 8 and 9 we obtain new APN functions and we cover many unclassified cases of APN maps. Here new stands for inequivalent (in respect to the so-called CCZ-equivalence) to already known ones. Afterwards, we study two infinite families of APN functions and their generalisations. We show that all these families are equivalent to each other and they are included in another known family. For many years it was not known whether all the constructed infinite families of APN maps were pairwise inequivalent. With our work, we reduce the list to those inequivalent to each other. Furthermore, we consider optimal functions with respect to the differential uniformity in fields of odd characteristic. These functions, called planar, have been valuable for the construction of new commutative semifields. Planar functions present often a close connection with APN maps. Indeed, the idea behind the isotopic shift construction comes from the study of isotopic equivalence, which is defined for quadratic planar functions. We completely characterise the mentioned equivalence by means of the isotopic shift and the extended affine equivalence. We show that the isotopic shift construction leads also to inequivalent planar functions and we analyse some particular cases of this construction. Finally, we study another cryptographic property, the boomerang uniformity, introduced by Cid et al. in 2018. This property is related to the boomerang attack, presented by Wagner in 1999. Here, we study the boomerang uniformity for some known classes of permutation polynomials.Doktorgradsavhandlin

Design and Analysis of Cryptographic Hash Functions

Wydział Matematyki i InformatykiKryptograficzne funkcje haszujące stanowią element składowy wielu algorytmów kryptograficznych. Przykładowymi zastosowaniami kryptograficznych funkcji haszujących są podpisy cyfrowe oraz kody uwierzytelniania wiadomości. Ich własności kryptograficzne mają znaczący wpływ na poziom bezpieczeństwa systemów kryptograficznych wykorzystujących haszowanie. W dysertacji analizowane są kryptograficzne funkcje haszujące oraz omówione główne zasady tworzenia bezpiecznych kryptograficznych funkcji haszujących. Analizujemy bezpieczeństwo dedykowanych funkcji haszujących (BMW, Shabal, SIMD, BLAKE2, Skein) oraz funkcji haszujących zbudowanych z szyfrów blokowych (Crypton, Hierocrypt-3, IDEA, SAFER++, Square). Głównymi metodami kryptoanalizy użytymi są skrócona analiza różnicowa, analiza rotacyjna i przesuwna. Uzyskane wyniki pokazują słabości analizowanych konstrukcji.Cryptographic Hash Functions (CHFs) are building blocks of many cryptographic algorithms. For instance, they are indispensable tools for efficient digital signature and authentication tags. Their security properties have tremendous impact on the security level of systems, which use cryptographic hashing. This thesis analyzes CHFs and studies the design principles for construction of secure and efficient CHFs. The dissertation investigates security of both dedicated hash functions (BMW, Shabal, SIMD, BLAKE2, Skein) and hash functions based on block ciphers (Crypton, Hierocrypt-3, IDEA, SAFER++, Square). The main cryptographic tools applied are truncated differentials, rotational and shift analysis. The findings show weaknesses in the designs

Denial-of-Service Resistance in Key Establishment

Denial of Service (DoS) attacks are an increasing problem for network connected systems. Key establishment protocols are applications that are particularly vulnerable to DoS attack as they are typically required to perform computationally expensive cryptographic operations in order to authenticate the protocol initiator and to generate the cryptographic keying material that will subsequently be used to secure the communications between initiator and responder. The goal of DoS resistance in key establishment protocols is to ensure that attackers cannot prevent a legitimate initiator and responder deriving cryptographic keys without expending resources beyond a responder-determined threshold. In this work we review the strategies and techniques used to improve resistance to DoS attacks. Three key establishment protocols implementing DoS resistance techniques are critically reviewed and the impact of misapplication of the techniques on DoS resistance is discussed. Recommendations on effectively applying resistance techniques to key establishment protocols are made

Using Simon's Algorithm to Attack Symmetric-Key Cryptographic Primitives

We present new connections between quantum information and the field of classical cryptography. In particular, we provide examples where Simon's algorithm can be used to show insecurity of commonly used cryptographic symmetric-key primitives. Specifically, these examples consist of a quantum distinguisher for the 3-round Feistel network and a forgery attack on CBC-MAC which forges a tag for a chosen-prefix message querying only other messages (of the same length). We assume that an adversary has quantum-oracle access to the respective classical primitives. Similar results have been achieved recently in independent work by Kaplan et al. Our findings shed new light on the post-quantum security of cryptographic schemes and underline that classical security proofs of cryptographic constructions need to be revisited in light of quantum attackers.Comment: 14 pages, 2 figures. v3: final polished version, more formal definitions adde