Inner Automorphisms of Finite Semifields

EnUnlike finite fields, finite semifields possess inner automorphisms. A further surprise is that even noncommutative semifields possess inner automorphisms. We compute inner automorphisms and automorphism groups for semifields quadratic over the nucleus, the Hughes-Kleinfeld semifields and the Dickson commutative semifields

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

Non-minimum tensor rank Gabidulin codes

The tensor rank of some Gabidulin codes of small dimension is investigated. In particular, we determine the tensor rank of any rank metric code equivalent to an 8-dimensional Fq-linear generalized Gabidulin code in Fq4×4. This shows that such a code is never minimum tensor rank. In this way, we detect the first infinite family of Gabidulin codes which are not minimum tensor rank

Finite semifields and nonsingular tensors

In this article, we give an overview of the classification results in the theory of finite semifields (note that this is not intended as a survey of finite semifields including a complete state of the art (see also Remark 1.10)) and elaborate on the approach using nonsingular tensors based on Liebler (Geom Dedicata 11(4):455-464, 1981)

Classification and computational search for planar functions in characteristic 3

Masteroppgave i informatikkINF399MAMN-PROGMAMN-IN

The semiring-theoretic approach to MV-algebras: a survey

In this paper we review some of the main achievements of the semiring-theoretic approach to MV-algebras initiated and pursued mainly by the present authors and their collaborators. The survey focuses mainly on the connections between MV-algebras and other theories that such a semiringbased approach enabled, and on an application of such a framework to Digital Image Processing. We also give some suggestions for further developments by stating several open problems and possible research lines.Comment: Published versio

Presemifields, bundles and polynomials over GF (pn)

The content of this thesis is first and foremost about presemifields and the equivalence classes they may be categorized by. This equivalence has been termed “bundle equivalence'' by Horadam. Bundle equivalence is inherited from multiplicative orthogonal cocycles, and the final Chapter is devoted entirely to coboundaries and cocycles. In this thesis we provide a complete computational classification of the bundles of presemifields in all presemifield isotopism classes of order p n , provide a formula for the number of bundles in the presemifields isotopism class of GF (p 2 ) and give a representative of each bundle, for any prime p . We provide computational classification of the bundles of presemifields in the isotopism class of GF (p 3 )  for the cases  p =3,5,7,11 and give representatives, give formulae for two of the three possible size bundles in the presemifield isotopism class of  GF (p 3 )   which we call the minimum and the mid-size bundles. We provide a Conjecture which states the total number of mid-size bundles in the isotopism class of  GF (p 3 ) and give a computational classification of the bundles of presemifields in the isotopism class of  GF (2 5 ) and  GF (3 4 ) . We provide a measurement of the differential uniformity of functions derived from the diagonal map of presemifield multiplications with order p n < 16 and derive bivariate polynomial formulae for cocycles and coboundaries in We produce a basis for the ( p n - 1 - n ) - dimensional -space of coboundaries. When p = 2 we give a recursive definition of the basis coboundaries. We use the Kronecker product to explain the self-similarity of the binomial coefficients modulo a prime and use the Kronecker product to define recursively the basis coboundaries for p odd, and we demonstrate this holds for the case p = 2. We show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form when  p = 2.  The results of this thesis have been published in the Proceedings of the International Workshop on Coding and Cryptography, Designs, Codes and Cryptography and the Proceedings of IEEE International Symposium on Information Theory and will appear in the Journal of the Australian Mathematical Society