A new construction of the d-dimensional Buratti–Del Fra dual hyperoval

AbstractThe Buratti–Del Fra dual hyperoval Dd(F2) is one of the four known infinite families of simply connected d-dimensional dual hyperovals over F2 with ambient space of vector dimension (d+1)(d+2)/2 (Buratti and Del Fra (2003) [1]). A criterion (Proposition 1) is given for a d-dimensional dual hyperoval over F2 to be covered by Dd(F2) in terms of the addition formula. Using it, we provide a simpler model of Dd(F2) (Proposition 3). We also give conditions (Lemma 4) for a collection S[B] of (d+1)-dimensional subspaces of K⊕K constructed from a symmetric bilinear form B on K≅F2d+1 to be a quotient of Dd(F2). For when d is even, an explicit form B satisfying these conditions is given. We also provide a proof for the fact that the affine expansion of Dd(F2) is covered by the halved hypercube (Proposition 10)

Flag-transitive L_h.L*-geometries

The classification of finite flag-transitive linear spaces, obtained by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl [20] at the end of the eighties, gave new impulse to the program of classifying various classes of locally finite flag-transitive geometries belonging to diagrams obtained from a Coxeter diagram by putting a label L or L ∗ on some (possibly, all) of the singlebond strokes for projective planes

On the dual of the dual hyperoval from APN function f(x)=x3+Tr(x9)

AbstractUsing a quadratic APN function f on GF(2d+1), Yoshiara (2009) [15] constructed a d-dimensional dual hyperoval Sf in PG(2d+1,2). In Taniguchi and Yoshiara (2005) [13], we prove that the dual of Sf, which we denote by Sf⊥, is also a d-dimensional dual hyperoval if and only if d is even. In this note, for a quadratic APN function f(x)=x3+Tr(x9) on GF(2d+1) by Budaghyan, Carlet and Leander (2009) [2], we show that the dual Sf⊥ and the transpose of the dual Sf⊥T are not isomorphic to the known bilinear dual hyperovals if d is even and d⩾6

Almost perfect nonlinear functions and related combinatorial structures

A map f(x) from the finite field Fpn to itself is said to be differentially k-uniform if k is the maximum number of solutions of the equation f(x + a) - f(x) = b where a, b [is in] Fpn , a ≠ 0. In particular, 2-uniform maps over F2n are called almost perfect nonlinear (APN) maps. These maps are of interest in cryptography because they offer optimum resistance to linear and differential attacks on certain cryptosystems. They can also be used to construct several combinatorial structures of interest.;In this dissertation, we characterize and classify all known power maps f(x) = xd over F2n , which are APN or of low uniformity. We discuss some basic properties of APN maps, collect all known APN power maps, and give a classification of APN power maps up to equivalence. We also give some insight regarding efforts to find other APN functions or prove that others do not exist and classify all power maps according to their degree of uniformity for n up to 13.;In the latter part of this dissertation, through the introduction of an incidence structure, we study how these functions can be used to construct semi-biplanes utilizing the method of Robert S. Coulter and Marie Henderson. We then consider a particular class of APN functions, from which we construct symmetric association schemes of class two and three. Using the result of E. R. van Dam and D. Fon-Der-Flaass, we can see that the relation graphs of some of these association schemes are distance-regular graphs. We discuss the local structure of these distance-regular graphs and characterize them

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