A note on semi-bent functions with multiple trace terms and hyperelliptic curves

Semi-bent functions with even number of variables are a class of important Boolean functions whose Hadamard transform takes three values. In this note we are interested in the property of semi-bentness of Boolean functions defined on the Galois field $F_{2^n}$ (n even) with multiple trace terms obtained via Niho functions and two Dillon-like functions (the first one has been studied by Mesnager and the second one have been studied very recently by Wang, Tang, Qi, Yang and Xu). We subsequently give a connection between the property of semi-bentness and the number of rational points on some associated hyperelliptic curves. We use the hyperelliptic curve formalism to reduce the computational complexity in order to provide a polynomial time and space test leading to an efficient characterization of semi-bentness of such functions (which includes an efficient characterization of the hyperbent functions proposed by Wang et al.). The idea of this approach goes back to the recent work of Lisonek on the hyperbent functions studied by Charpin and Gong

A new class of hyper-bent functions and Kloosterman sums

This paper is devoted to the characterization of hyper-bent functions. Several classes of hyper-bent functions have been studied, such as Charpin and Gong\u27s $\sum\limits_{r\in R}\mathrm{Tr}_{1}^{n} (a_{r}x^{r(2^m-1)})$ and Mesnager\u27s $\sum\limits_{r\in R}\mathrm{Tr}_{1}^{n}(a_{r}x^{r(2^m-1)}) +\mathrm{Tr}_{1}^{2}(bx^{\frac{2^n-1}{3}})$, where $R$ is a set of representations of the cyclotomic cosets modulo $2^m+1$ of full size $n$ and $a_{r}\in \mathbb{F}_{2^m}$. In this paper, we generalize their results and consider a class of Boolean functions of the form $\sum_{r\in R}\sum_{i=0}^{2}Tr^n_1(a_{r,i}x^{r(2^m-1)+\frac{2^n-1}{3}i}) +Tr^2_1(bx^{\frac{2^n-1}{3}})$, where $n=2m$, $m$ is odd, $b\in\mathbb{F}_4$, and $a_{r,i}\in \mathbb{F}_{2^n}$. With the restriction of $a_{r,i}\in \mathbb{F}_{2^m}$, we present the characterization of hyper-bentness of these functions with character sums. Further, we reformulate this characterization in terms of the number of points on hyper-elliptic curves. For some special cases, with the help of Kloosterman sums and cubic sums, we determine the characterization for some hyper-bent functions including functions with four, six and ten traces terms. Evaluations of Kloosterman sums at three general points are used in the characterization. Actually, our results can generalized to the general case: $a_{r,i}\in \mathbb{F}_{2^n}$. And we explain this for characterizing binomial, trinomial and quadrinomial hyper-bent functions

A note on constructions of bent functions from involutions

Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied rstly by Dillon and next by many researchers for four decades. Since the complete classication of bent functions seems elusive, many researchers turn to design constructions of bent functions. In this note, we show that linear involutions (which are an important class of permutations) over nite elds give rise to bent functions in bivariate representations. In particular, we exhibit new constructions of bent functions involving binomial linear involutions whose dual functions are directly obtained without computation

On the Primary Constructions of Vectorial Boolean Bent Functions

Vectorial Boolean bent functions, which possess the maximal nonlinearity and the minimum differential uniformity, contribute to optimum resistance against linear cryptanalysis and differential cryptanalysis for the cryptographic algorithms that adopt them as nonlinear components. This paper is devoted to the new primary constructions of vectorial Boolean bent functions, including four types: vectorial monomial bent functions, vectorial Boolean bent functions with multiple trace terms, $\mathcal{H}$ vectorial functions and $\mathcal{H}$-like vectorial functions. For vectorial monomial bent functions, this paper answers one open problem proposed by E. Pasalic et al. and characterizes the vectorial monomial bent functions corresponding to the five known classes of bent exponents. For the vectorial Boolean bent functions with multiple trace terms, this paper answers one open problem proposed by A. Muratović-Ribić et al., presents six new infinite classes of explicit constructions and shows the nonexistence of the vectorial Boolean bent functions from $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{k}}$ of the form $\sum_{i=1}^{2^{k-2}}Tr^{n}_{k}(ax^{(2i-1)(2^{k}-1)})$ with $n=2k$ and $a\in\mathbb{F}_{2^{k}}^{*}$. Moreover, $\mathcal{H}$ vectorial functions are further characterized. In addition, a new infinite class of vectorial Boolean bent function named as $\mathcal{H}$-like vectorial functions are derived, which includes $\mathcal{H}$ vectorial functions as a subclass

Exploiting phonological constraints for handshape recognition in sign language video

The ability to recognize handshapes in signing video is essential in algorithms for sign recognition and retrieval. Handshape recognition from isolated images is, however, an insufficiently constrained problem. Many handshapes share similar 3D configurations and are indistinguishable for some hand orientations in 2D image projections. Additionally, significant differences in handshape appearance are induced by the articulated structure of the hand and variants produced by different signers. Linguistic rules involved in the production of signs impose strong constraints on the articulations of the hands, yet, little attention has been paid towards exploiting these constraints in previous works on sign recognition. Among the different classes of signs in any signed language, lexical signs constitute the prevalent class. Morphemes (or, meaningful units) for signs in this class involve a combination of particular handshapes, palm orientations, locations for articulation, and movement type. These are thus analyzed by many sign linguists as analogues of phonemes in spoken languages. Phonological constraints govern the ways in which phonemes combine in American Sign Language (ASL), as in other signed and spoken languages; utilizing these constraints for handshape recognition in ASL is the focus of the proposed thesis. Handshapes in monomorphemic lexical signs are specified at the start and end of the sign. The handshape transition within a sign are constrained to involve either closing or opening of the hand (i.e., constrained to exclusively use either folding or unfolding of the palm and one or more fingers). Furthermore, akin to allophonic variations in spoken languages, both inter- and intra- signer variations in the production of specific handshapes are observed. We propose a Bayesian network formulation to exploit handshape co-occurrence constraints also utilizing information about allophonic variations to aid in handshape recognition. We propose a fast non-rigid image alignment method to gain improved robustness to handshape appearance variations during computation of observation likelihoods in the Bayesian network. We evaluate our handshape recognition approach on a large dataset of monomorphemic lexical signs. We demonstrate that leveraging linguistic constraints on handshapes results in improved handshape recognition accuracy. As part of the overall project, we are collecting and preparing for dissemination a large corpus (three thousand signs from three native signers) of ASL video annotated with linguistic information such as glosses, morphological properties and variations, and start/end handshapes associated with each ASL sign

Innovative approaches for Monte Carlo simulations of orientational effects in crystals and their experimental verification

Interaction of either charged or neutral particles with crystals is an area of science under development. Coherent effects of ultra‐relativistic particles in crystals allow manipulating particle trajectories thanks to the strong electrical field generated between atomic planes and axes. Coherent effects for interaction of particles with aligned structures always exploited opportunity furnished by the most advanced calculators and calculation methods of the current period. In this thesis two Monte Carlo codes were developed for the simulation of coherent interactions between charged particles and crystals. The Monte Carlo codes were tested for comparison with the experimental results of various experiments on channeling and related topics. The first code, named DYNECHARM++, is completely object‐oriented and deals with numerical integration of the equation of motion to determine the trajectory of a particle in straight and bent complex crystalline structures. The second code addresses the implementation of coherent effects, such as planar channeling and volume reflection to Geant4, which is a widespread used toolkit for the simulation of the passage of particles through matter. Experiments on coherent interactions were carried out at the H8 and H4 external lines of the SPS at CERN and at the MAMI of the Johannes Gutenberg University of Mainz. At the H8 line experiments of coherent interaction in "exotic" atomic structure and crystal configuration were worked out. Within the UA9 experiment, a procedure for the on‐beam characterization of the strips for the SPS crystal collimation experiment was developed at the H8 line. At the H4 line and at MAMI the interaction of negative particles with bent crystals was studie