On values of vectorial Boolean functions and related problems in APN functions

In this paper we prove that there are only differential 4-uniform functions which are on distance 1 from an APN function. Also we prove that there are no APN functions of distance 1 from another APN functions up to dimension 5. We determine some properties of the set of values of an arbitrary vectorial Boolean function from F_n^2 to F_n^2 in connection to the set of values of its derivatives. These results are connected to several open question concerning metric properties of APN functions

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

Analysis and design of some cryptographic Boolean functions

Boolean functions are vital components of symmetric-key ciphers such as block ciphers, stream ciphers and hash functions. When used in cipher systems, Boolean functions should satisfy several cryptographic properties such as balance, high nonlinearity, resiliency and high algebraic degree. Bent functions achieve the maximum possible nonlinearity and hence they provide optimal resistance to several cryptographic attacks such as linear and differential cryptanalysis. We present some simple constructions for binary bent functions of length 2 2 k using a known bent function of length 2 2 k -2 . Adams and Tavares introduced two classes of bent functions: bent-based bent functions and linear-based bent functions. In this thesis we explore different bent-based constructions. In particular, we show that all nonlinear resilient functions with maximum order resiliency are either bent-based or linear-based. We provide an explicit count for the number of such resilient functions that belong to both classes. We also provide a simple proof that all symmetric functions that achieve the maximum possible nonlinearity are bent-based. In particular, for n even, we have 4 bent-based bent functions. For n odd, we also have 4 bent-based functions. We also prove that there is no bent-based homogeneous functions with algebraic degree >2. Almost all cryptographic properties of Boolean functions can be determined efficiently from its Walsh transform. In this thesis, we present some restrictions on the partial sum of the Walsh transform of binary functions. In several parts of the thesis, we extend the obtained results to functions defined over GF(p

On Equivalence of Known Families of APN Functions in Small Dimensions

In this extended abstract, we computationally check and list the CCZ-inequivalent APN functions from infinite families on $\mathbb{F}_2^n$ for n from 6 to 11. These functions are selected with simplest coefficients from CCZ-inequivalent classes. This work can simplify checking CCZ-equivalence between any APN function and infinite APN families.Comment: This paper is already in "PROCEEDING OF THE 20TH CONFERENCE OF FRUCT ASSOCIATION