Non-Boolean almost perfect nonlinear functions on non-Abelian groups

The purpose of this paper is to present the extended definitions and characterizations of the classical notions of APN and maximum nonlinear Boolean functions to deal with the case of mappings from a finite group K to another one N with the possibility that one or both groups are non-Abelian.Comment: 17 page

Semifields, relative difference sets, and bent functions

Recently, the interest in semifields has increased due to the discovery of several new families and progress in the classification problem. Commutative semifields play an important role since they are equivalent to certain planar functions (in the case of odd characteristic) and to modified planar functions in even characteristic. Similarly, commutative semifields are equivalent to relative difference sets. The goal of this survey is to describe the connection between these concepts. Moreover, we shall discuss power mappings that are planar and consider component functions of planar mappings, which may be also viewed as projections of relative difference sets. It turns out that the component functions in the even characteristic case are related to negabent functions as well as to $\mathbb{Z}_4$-valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite fields", 09-13 December 2013, Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the "Radon Series on Computational and Applied Mathematics" by DeGruyte

On the Derivative Imbalance and Ambiguity of Functions

In 2007, Carlet and Ding introduced two parameters, denoted by $Nb_F$ and $NB_F$, quantifying respectively the balancedness of general functions $F$ between finite Abelian groups and the (global) balancedness of their derivatives $D_a F(x)=F(x+a)-F(x)$, $a\in G\setminus\{0\}$ (providing an indicator of the nonlinearity of the functions). These authors studied the properties and cryptographic significance of these two measures. They provided for S-boxes inequalities relating the nonlinearity $\mathcal{NL}(F)$ to $NB_F$, and obtained in particular an upper bound on the nonlinearity which unifies Sidelnikov-Chabaud-Vaudenay's bound and the covering radius bound. At the Workshop WCC 2009 and in its postproceedings in 2011, a further study of these parameters was made; in particular, the first parameter was applied to the functions $F+L$ where $L$ is affine, providing more nonlinearity parameters. In 2010, motivated by the study of Costas arrays, two parameters called ambiguity and deficiency were introduced by Panario \emph{et al.} for permutations over finite Abelian groups to measure the injectivity and surjectivity of the derivatives respectively. These authors also studied some fundamental properties and cryptographic significance of these two measures. Further studies followed without that the second pair of parameters be compared to the first one. In the present paper, we observe that ambiguity is the same parameter as $NB_F$, up to additive and multiplicative constants (i.e. up to rescaling). We make the necessary work of comparison and unification of the results on $NB_F$, respectively on ambiguity, which have been obtained in the five papers devoted to these parameters. We generalize some known results to any Abelian groups and we more importantly derive many new results on these parameters

(2^n,2^n,2^n,1)-relative difference sets and their representations

We show that every $(2^n,2^n,2^n,1)$-relative difference set $D$ in $\Z_4^n$ relative to $\Z_2^n$ can be represented by a polynomial f(x)\in \F_{2^n}[x], where $f(x+a)+f(x)+xa$ is a permutation for each nonzero $a$. We call such an $f$ a planar function on \F_{2^n}. The projective plane $\Pi$ obtained from $D$ in the way of Ganley and Spence \cite{ganley_relative_1975} is coordinatized, and we obtain necessary and sufficient conditions of $\Pi$ to be a presemifield plane. We also prove that a function $f$ on \F_{2^n} with exactly two elements in its image set and $f(0)=0$ is planar, if and only if, $f(x+y)=f(x)+f(y)$ for any x,y\in\F_{2^n}

Generalized binary arrays from quasi-orthogonal cocycles

Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is 2 or a multiple of 4. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by 4, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of 2-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length.Junta de Andalucía FQM-01

Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements

We introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases, being equivalent whenever the design is composed of d+1 bases in dimension d. We show that, for the purpose of quantum state determination, these designs specify an optimal collection of orthogonal measurements. Using highly nonlinear functions on abelian groups, we construct explicit examples from d+2 orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10, and 12, for example, where no complete sets of mutually unbiased bases have thus far been found.Comment: 28 pages, to appear in J. Math. Phy