3 research outputs found

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

    Get PDF
    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)

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

    Get PDF
    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 dâ©Ÿ6

    Analysis, classification and construction of optimal cryptographic Boolean functions

    Get PDF
    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
    corecore