Publications

2022

• Impact de perturbations internes sur l'entraînement de réseaux profonds pour la détection d'évènements sonores
  
  • Perera David
  • Essid Slim
  • Richard Gael
  , 2022. L'apprentissage d'invariants est une méthode d'entraînement prometteuse pour les réseaux de neurones profonds, puisqu'elle permet à la fois de pallier le manque de diversité des bases de données disponibles, et de rendre les modèles entraînés plus interprétables. En pratique, l'apprentissage d'invariants passe souvent par l'utilisation d'augmentations de données et de coûts de consistance pénalisant la sensibilité d'un modèle à ces augmentations. Il n'existe cependant pas de consensus concernant la sélection de ces augmentations pour une tâche cible. Cet article étudie l'impact de plusieurs types d'augmentations sur l'entraînement d'un modèle de l'état de l'art, dans le cadre de la détection et de la classification d'évènements sonores. Nous montrons en particulier que la perturbation des représentations internes d'un réseau de neurones profond est bénéfique pour cette tâche.
• Future Trends and Recommendations
  
  • Sibille Alain
  • Lau Buon Kiong
  • Oestges Claude
  • Burr Alister
  • Ruiz Silvia
  • Jamsa Tommi
  , 2022, pp.469-485. (10.13052/rp-9788793379145)
  DOI : 10.13052/rp-9788793379145
• Apprentissage de bancs de filtres pour la séparation aveugle de sources sonores
  
  • Mathieu Félix
  • Courtat Thomas
  • Richard Gael
  • Peeters Geoffroy
  , 2022. L'utilisation d'encodeurs audio paramétrés s'est révélée être une piste encourageante pour améliorer l'interprétabilité et les performances des modèles de séparation de sources bout-à-bout. Nous présentons des propriétés d'intérêt nécessaires à l'apprentissage des filtres de ces encodeurs ; et proposons une paramétrisation pour contraindre ces filtres. Sur la base de la transformée de Hilbert et du théorème de Bedrosian, nous proposons de construire un ensemble de filtres déphasés en modulant des sinusoïdes à travers des filtres passe-bas appris librement. Ces filtres permettent d'obtenir des invariances pour des décalages temporels, des décalages de phases tout en évitant l'utilisation de réseaux de neurones complexes grâce à une astuce de sur-paramétrisation de la phase pour une forme d'onde donnée.
• Point gamma
  
  • Zayana Karim
  • Boyer Ivan
  • Rabiet Victor
  CultureMath, ENS, 2022.
• Prediction of RF-EMF Exposure by Outdoor Drive Test Measurements
  
  • Wang Shanshan
  • Mazloum Taghrid
  • Wiart Joe
  Telecom, MDPI, 2022, 3 (3), pp.396-406. In this paper, we exploit the artificial neural network (ANN) model for a spatial reconstruction of radio-frequency (RF) electromagnetic field (EMF) exposure in an outdoor urban environment. To this end, we have carried out a drive test measurement campaign covering a large part of Paris, along a route of approximately 65 Km. The electric (E) field strength has been recorded over a wide band ranging from 700 to 2700 MHz. From these measurement data, the E-field strength is extracted and computed for each frequency band of each telecommunication operator. First, the correlation between the E-fields at different frequency bands is computed and analyzed. The results show that a strong correlation of E-field levels is observed for bands belonging to the same operator. Then, we build ANN models with input data encompassing information related to distances to N neighboring base stations (BS), receiver location and time variation. We consider two different models. The first one is a fully connected ANN model, where we take into account the N nearest BSs ignoring the corresponding operator. The second one is a hybrid model, where we consider locally connected blocks with the N nearest BSs for each operator, followed by fully connected layers. The results show that the hybrid model achieves better performance than the fully connected one. Among N∈{3,5,7}, we found out that with N=3, the proposed hybrid model allows a good prediction of the exposure level while the maintaining acceptable complexity of the model. (10.3390/telecom3030021)
  DOI : 10.3390/telecom3030021
• 3D mesh cutting for high quality atlas packing
  
  • Wang Shiyi
  • Chen Jiong
  • Gao Xifeng
  • Bao Hujun
  • Huang Jin
  Computer Aided Geometric Design, Elsevier, 2022, 99, pp.102149. An efficiently packed, low-distortion parameterization with a short boundary can save a great amount of memory and improve both quality and efficiency of rendering. Existing packing methods begin with an input atlas (or parameterization), but the cuts in the input atlas may be not suitable for a high quality result. We propose a simple yet effective approach to cut an input surface and generate an atlas that comprehensively considers the packing efficiency, the mapping distortion and the boundary length. Viewing the desired cuts on the input mesh as the pullback of a low-distortion mapping from a polysquare boundary, we notice that the above three objectives actually imply a small number of cone singularities with angle deficit of pi/2 k, k in Z, and orthogonally intersected short cuts passing through all singularities. Therefore, we first leverage a cross frame-field to identify a set of singularities and cancel some of them to balance their amount and atlas distortion. Then, the singularities remained are heuristically connected by short cuts which intersect with each other nearly orthogonally. Results show that our method produces a low-distortion and polysquare-like atlas with controllable number of singularities. Comparing with other atlas generation methods only focusing on distortion, taking our atlas as the input for the subsequent packing algorithm (Liu et al., 2019) is better than using previous cutting strategies on a benchmark containing 5519 cases, because the conflicts among those desired objectives for packing are alleviated at an early stage. (10.1016/j.cagd.2022.102149)
  DOI : 10.1016/j.cagd.2022.102149
• Débruitage multi-temporel d'images radar à synthèse d'ouverture par apprentissage profond auto-supervisé
  
  • Meraoumia Inès
  • Dalsasso Emanuele
  • Denis Loïc
  • Tupin Florence
  , 2022. Les satellites imageurs radar (SAR) représentent une modalité très utilisée pour l'observation de la terre, fournissant à chaque revisite une nouvelle image de la zone d'intérêt. L'interprétation des images SAR est cependant difficile à cause du phénomène de chatoiement: des fluctuations apparaissent dans l'image, d'autant plus fortes là où la réflectivité radar est élevée. Un grand nombre de méthodes de réduction du chatoiement ont donc été développées. Nous proposons ici une approche d'apprentissage profond présentant deux originalités: 1) l'exploitation d'une série temporelle d'images afin d'améliorer la restauration d'une image d'intérêt et 2) l'entraînement sans référence du réseau de neurones.
• Ensembles bien équilibrés
  
  • Zayana Karim
  • Michalak Pierre
  • Felloneau Claude
  CultureMath, ENS, 2022.
• Be my guess: Guessing entropy vs. success rate for evaluating side-channel attacks of secure chips
  
  • Béguinot Julien
  • Cheng Wei
  • Guilley Sylvain
  • Rioul Olivier
  , 2022. In a theoretical context of side-channel attacks, optimal bounds between success rate and guessing entropy are derived with a simple majorization (Schur-concavity) argument. They are further theoretically refined for different versions of the classical Hamming weight leakage model, in particular assuming apriori equiprobable secret keys and additive white Gaussian measurement noise. Closed-form expressions and numerical computation are given. A study of the impact of the choice of the substitution box with respect to side-channel resistance reveals that its nonlinearity tends to homogenize the expressivity of success rate and guessing entropy. The intriguing approximate relation GE = 1/SR is observed in the case of 8-bit bytes and low noise.
• Morphisms and minimisation of weighted automata
  
  • Lombardy Sylvain
  • Sakarovitch Jacques
  Fundamenta Informaticae, Polskie Towarzystwo Matematyczne, 2022, 186 (1-4), pp.195-218. This paper studies the algorithms for the minimisation of weighted automata. It starts with the definition of morphisms-which generalises and unifies the notion of bisimulation to the whole class of weighted automata-and the unicity of a minimal quotient for every automaton, obtained by partition refinement. From a general scheme for the refinement of partitions, two strategies are considered for the computation of the minimal quotient: the Domain Split and the Predecesor Class Split algorithms. They correspond respectivly to the classical Moore and Hopcroft algorithms for the computation of the minimal quotient of deterministic Boolean automata. We show that these two strategies yield algorithms with the same quadratic complexity and we study the cases when the second one can be improved in order to achieve a complexity similar to the one of Hopcroft algorithm. (10.3233/FI-222126)
  DOI : 10.3233/FI-222126
• Elliptically Contoured Alpha-Stable Representation for MUSIC-Based Sound Source Localization
  
  • Fontaine Mathieu
  • Di Carlo Diego
  • Sekiguchi Kouhei
  • Nugraha Aditya Arie
  • Bando Yoshiaki
  • Yoshii Kazuyoshi
  , 2022, pp.26-30. This paper introduces a theoretically-rigorous sound source localization (SSL) method based on a robust extension of the classical multiple signal classification (MUSIC) algorithm. The original SSL method estimates the noise eigenvectors and the MUSIC spectrum by computing the spatial covariance matrix of the observed multichannel signal and then detects the peaks from the spectrum. In this work, the covariance matrix is replaced with the positive definite shape matrix originating from the elliptically contoured α-stable model, which is more suitable under real noisy high-reverberant conditions. Evaluation on synthetic data shows that the proposed method outperforms baseline methods under such adverse conditions, while it is comparable on real data recorded in a mild acoustic condition. (10.23919/EUSIPCO55093.2022.9909944)
  DOI : 10.23919/EUSIPCO55093.2022.9909944
• Preface to the special issue on dynamic recommender systems and user models
  
  • Vinagre João
  • Jorge Alípio Mário
  • Al-Ghossein Marie
  • Bifet Albert
  • Cremonesi Paolo
  User Modeling and User-Adapted Interaction, Springer Verlag, 2022, 32 (4), pp.503--507. The ever-growing and dynamic nature of user-generated data in online systems poses obvious challenges on how we learn from such data. The underlying problem is how to adapt, in real time, to multiple simultaneous changes involving individual users, user contexts and the system as a whole. Many algorithms are able to adjust their output to some of these changes in real time; however, this requires that the model has been previously trained on data with very similar phenomena. To adapt to new trends, preferences and other unpredictable phenomena, algorithms must be able to update the underlying model itself, which should preferably happen online, incrementally and in real time. This motivates the research on adaptive methods able to maintain and evolve predictive models over time. Incremental learning algorithms and data stream mining have gained maturity in recent years. However, this body of knowledge has not been applied to predictive user modeling, and although the potential to solve relevant problems is high, advances in this direction are far from trivial, calling for further research in this direction. This special issue provides contributions on the above challenges, explored and discussed in the Online Recommender Systems and User Modeling (ORSUM) workshop series that have taken place since 2018 in The Web Conference 2018 (Jorge et al. 2018) and the ACM Conference on Recommender Systems between 2019 and 2022 (Vinagre et al. 2019, 2020, 2021). (10.1007/S11257-022-09341-Y)
  DOI : 10.1007/S11257-022-09341-Y
• Foreground Static Error Calibration for Current Sources Using Backgate Body Biasing
  
  • Beauquier Clément
  • David Duperray
  • Jabbour Chadi
  • Desgreys Patricia
  • Frappé Antoine
  • Kaiser Andreas
  , 2022. This work presents a detection and calibration circuit for current sources static mismatch introduced by the process of fabrication. The current is corrected through backgate body bias voltage control, which has the benefit of reduced additional parasitic elements, compared to more classic amplitude calibration or sort-and-map solutions. The main application are high speed and high resolution current steering Digital to Analog Converters (DAC). The calibration circuit is applied on a 2 timeinterleaved (TI) DAC, with 12 bits of resolution and sampled at a frequency of 16 GHz. Its main requirement is to be able to generate signals up to the Nyquist Band (8 GHz) with Spurious Free Dynamic Range (SFDR) of at least 70 dBFS. We validate the method with a schematic 28 nm FDSOI CMOS transistor level testbench, Montecarlo simulations and temperature variations from 27 °C to 125 °C.
• The Role of Causality in a Formal Definition of Timing Anomalies
  
  • Binder Benjamin
  • Asavoae Mihail
  • Brandner Florian
  • Hedia Belgacem Ben
  • Jan Mathieu
  , 2022, pp.91-102. Intuitively, a counter-intuitive timing anomaly manifests when a locally faster execution becomes globally slower. While the presence of such timing anomalies threatens the soundness and/or scalability of timing analyses, tools to systematically detect them do not exist. The main reason lies in the absence of a definition of counter-intuitive timing anomalies that establishes relations between local and global timing effects. In this paper, we address these relations through an important concept, that of causality, which we further use to revise the formalization of counter-intuitive timing anomalies. We also propose a specialized instance of the notions to implement a detection procedure for out-of-order pipelines. (10.1109/RTCSA55878.2022.00016)
  DOI : 10.1109/RTCSA55878.2022.00016
• Weighted Counting of Matchings in Unbounded-Treewidth Graph Families
  
  • Amarilli Antoine
  • Monet Mikaël
  , 2022. We consider a weighted counting problem on matchings, denoted $\textrm{PrMatching}(\mathcal{G})$, on an arbitrary fixed graph family $\mathcal{G}$. The input consists of a graph $G\in \mathcal{G}$ and of rational probabilities of existence on every edge of $G$, assuming independence. The output is the probability of obtaining a matching of $G$ in the resulting distribution, i.e., a set of edges that are pairwise disjoint. It is known that, if $\mathcal{G}$ has bounded treewidth, then $\textrm{PrMatching}(\mathcal{G})$ can be solved in polynomial time. In this paper we show that, under some assumptions, bounded treewidth in fact characterizes the tractable graph families for this problem. More precisely, we show intractability for all graph families $\mathcal{G}$ satisfying the following treewidth-constructibility requirement: given an integer $k$ in unary, we can construct in polynomial time a graph $G \in \mathcal{G}$ with treewidth at least $k$. Our hardness result is then the following: for any treewidth-constructible graph family $\mathcal{G}$, the problem $\textrm{PrMatching}(\mathcal{G})$ is intractable. This generalizes known hardness results for weighted matching counting under some restrictions that do not bound treewidth, e.g., being planar, 3-regular, or bipartite; it also answers a question left open in Amarilli, Bourhis and Senellart (PODS'16). We also obtain a similar lower bound for the weighted counting of edge covers. (10.4230/LIPIcs.MFCS.2022.9)
  DOI : 10.4230/LIPIcs.MFCS.2022.9
• Bridging the 100 GHz-10 THz domain with unipolar quantum optoelectronics
  
  • Grillot Frédéric
  • Didier Pierre
  • Spitz Olivier
  • del Balzo Livia
  • Kim Hyunah
  • Dely Hamza
  • Bonazzi Thomas
  • Rodriguez Etienne
  • Gacemi Djamal
  • Vasanelli Angela
  • Sirtori Carlo
  , 2022, pp.10. The challenge of Unipolar Quantum Optoelectronics (UQO) is to bring reliable technology in the mid-infrared and terahertz domains with dozens of GHz bandwidth and room-temperature operation. The semiconductor devices based on this novel technology rely on two-dimensional electronic states localized in the conduction band, which implies that electrons are the only charge carriers involved. Though UQO technology has been proven useful for emission (quantum cascade lasers) and detection (quantum cascade detectors), it is still underdeveloped for other applications, like high-speed modulation. In this paper, we will review our recent results with a full transmission system UQO in the 8 to 14 µm atmospheric window, composed of a quantum cascade (QC) laser, an external modulator and a QC detector, all optimized for operation at 33 THz optical wavelength. Dynamics down to a few dozens of picoseconds are observed, which allow us demonstrating data rate transmission of 10 Gbps without any signal processing. In addition, the paper aims at discussing further applications of UQO in particular for radio over free-space. The basic principle for producing microwave carriers is based on an optical heterodyne beating technique taking advantage of the high-bandwidth potential of QC detectors. Then, the microwave signal is transmitted through a point-to-point wireless link by using radiofrequency antennas. With UQO, microwave signals of dozens of GHz can be achieved. To sum, this paper highlights the importance of using UQO devices operating at a few dozens of THz optical wavelength for both free-space optics and microwave photonics targeting 100 GHz radiofrequencies. (10.1117/12.2631999)
  DOI : 10.1117/12.2631999
• An Exploratory Study on Group Potency Classification from Non-verbal Social Behaviours
  
  • Corbellini Nicola
  • Ceccaldi Eleonora
  • Varni Giovanna
  • Volpe Gualtiero
  , 2022.
• Computational Multimodal Models of Users’ Interactional Trust in Multiparty Human-Robot Interaction
  
  • Hulcelle Marc
  • Varni Giovanna
  • Rollet Nicolas
  • Clavel Chloé
  , 2023, 13643, pp.225-239. In this paper, we present multimodal computational models of interactional trust in a humans-robot interaction scenario. We address trust modeling as a binary as well as a multi-class classification problem. We also investigate how early- and late-fusion of modalities impact trust modeling. Our results indicate that early-fusion performs better in both the binary and multi-class formulations, meaning that modalities have co-dependencies when studying trust. We also run a SHapley Additive exPlanation (SHAP) values analysis for a Random Forest in the binary classification problem, as it is the model with the best results, to explore which multimodal features are the most relevant to detect trust or mistrust (10.1007/978-3-031-37660-3_16)
  DOI : 10.1007/978-3-031-37660-3_16
• A statistically constrained internal method for single image super-resolution
  
  • Chatillon Pierrick
  • Gousseau Yann
  • Lefebvre Sidonie
  , 2022, pp.1322-1328. Deep learning based methods for single-image superresolution (SR) have drawn a lot of attention lately. In particular, various papers have shown that the learning stage can be performed on a single image, resulting in the so-called internal approaches. The SinGAN method is one of these contributions, where the distribution of image patches is learnt on the image at hand and propagated at finer scales. Now, there are situations where some statistical a priori can be assumed for the final image. In particular, many natural phenomena yield images having power law Fourier spectrum, such as clouds and other texture images. In this work, we show how such a priori information can be integrated into an internal super-resolution approach, by constraining the learned up-sampling procedure of SinGAN. We consider various types of constraints, related to the Fourier power spectrum, the color histograms and the consistency of the upsampling scheme. We demonstrate on various experiments that these constraints are indeed satisfied, but also that some perceptual quality measures can be improved by the proposed approach. (10.1109/ICPR56361.2022.9956498)
  DOI : 10.1109/ICPR56361.2022.9956498
• Partial Key Exposure Attacks on BIKE, Rainbow and NTRU
  
  • Esser Andre
  • May Alexander
  • Verbel Javier A.
  • Wen Weiqiang
  , 2022, 13509, pp.3-31. In a so-called partial key exposure attack one obtains some information about the secret key, e.g. via some side-channel leakage. This information might be a certain fraction of the secret key bits (erasure model) or some erroneous version of the secret key (error model). The goal is to recover the secret key from the leaked information. There is a common belief that, as opposed to e.g. the RSA cryptosystem, most post-quantum cryptosystems are usually resistant against partial key exposure attacks. We strongly question this belief by constructing partial key exposure attacks on code-based, multivariate, and latticebased schemes (BIKE, Rainbow and NTRU). Our attacks exploit the redundancy that modern PQ cryptosystems inherently use for efficiency reasons. The application and development of techniques from information set decoding plays a crucial role for achieving our results. On the theoretical side, we show non-trivial information leakage bounds that allow for a polynomial time key recovery attack. As an example, for all schemes the knowledge of a constant fraction of the secret key bits suffices to reconstruct the full key in polynomial time. Even if we no longer insist on polynomial time attacks, most of our attacks extend well and remain feasible up to large erasure and error rates. In the case of BIKE for example we obtain attack complexities around 60 bits when half of the secret key bits are erased, or a quarter of the secret key bits are faulty. Our results show that even highly error-prone key leakage of modern PQ cryptosystems may lead to full secret key recoveries. (10.1007/978-3-031-15982-4_1)
  DOI : 10.1007/978-3-031-15982-4_1
• Online Clustering: Algorithms, Evaluation, Metrics, Applications and Benchmarking
  
  • Montiel Jacob
  • Ngo Hoang-Anh
  • Nguyen Minh-Huong Le
  • Bifet Albert
  , 2022, pp.4808--4809. Online clustering algorithms play a critical role in data science, especially with the advantages regarding time, memory usage and complexity, while maintaining a high performance compared to traditional clustering methods. This tutorial serves, first, as a survey on online machine learning and, in particular, data stream clustering methods. During this tutorial, state-of-the-art algorithms and the associated core research threads will be presented by identifying different categories based on distance, density grids and hidden statistical models. Clustering validity indices, an important part of the clustering process which are usually neglected or replaced with classification metrics, resulting in misleading interpretation of final results, will also be deeply investigated. Then, this introduction will be put into the context with River, a go-to Python library merged between Creme and scikit-multiflow. It is also the first open-source project to include an online clustering module that can facilitate reproducibility and allow direct further improvements. From this, we propose methods of clustering configuration, applications and settings for benchmarking, using real-world problems and datasets. (10.1145/3534678.3542600)
  DOI : 10.1145/3534678.3542600
• SOKNL: A novel way of integrating K-nearest neighbours with adaptive random forest regression for data streams
  
  • Sun Yibin
  • Pfahringer Bernhard
  • Gomes Heitor Murilo
  • Bifet Albert
  Data Mining and Knowledge Discovery, Springer, 2022, 36 (5), pp.2006--2032. Most research in machine learning for data streams has focused on classification algorithms, whereas regression methods have received a lot less attention. This paper proposes Self-Optimising K-Nearest Leaves (SOKNL), a novel forest-based algorithm for streaming regression problems. Specifically, the Adaptive Random Forest Regression, a state-of-the-art online regression algorithm is extended like this: in each leaf, a representative data point – also called centroid – is generated by compressing the information from all instances in that leaf. During the prediction step, instead of letting all trees in the forest participate, the distances between the input instance and all centroids from relevant leaves are calculated, only k trees that possess the smallest distances are utilised for the prediction. Furthermore, we simplify the algorithm by introducing a mechanism for tuning the k values, which is dynamically and automatically optimised based on historical information. This new algorithm produces promising predictive results and achieves a superior ranking according to statistical testing when compared with several standard stream regression methods over typical benchmark datasets. This improvement incurs only a small increase in runtime and memory consumption over the basic Adaptive Random Forest Regressor. (10.1007/S10618-022-00858-9)
  DOI : 10.1007/S10618-022-00858-9
• Go Green: General Regularized Green’s Functions for Elasticity
  
  • Chen Jiong
  • Desbrun Mathieu
  , 2022 (6), pp.1-8. The fundamental solutions (Green's functions) of linear elasticity for an infinite and isotropic media are ubiquitous in interactive graphics applications that cannot afford the computational costs of volumetric meshing and finite-element simulation. For instance, the recent work of de Goes and James [2017] leveraged these Green's functions to formulate sculpting tools capturing in real-time broad and physically-plausible deformations more intuitively and realistically than traditional editing brushes. In this paper, we extend this family of Green's functions by exploiting the anisotropic behavior of general linear elastic materials, where the relationship between stress and strain in the material depends on its orientation. While this more general framework prevents the existence of analytical expressions for its fundamental solutions, we show that a finite sum of spherical harmonics can be used to decompose a Green's function, which can be further factorized into directional, radial, and material-dependent terms. From such a decoupling, we show how to numerically derive sculpting brushes to generate anisotropic deformation and finely control their falloff profiles in real-time. (10.1145/3528233.3530726)
  DOI : 10.1145/3528233.3530726
• System for reducing the reflectivity of an incident electromagnetic wave on a surface and device implementing said system
  
  • Soiron Michel
  • Barka André
  • Lepage Anne Claire
  • Rance Olivier
  • Begaud Xavier
  • Parneix Patrick
  • Laybros Sarah
  , 2022.
• Concrete constructions of weightwise perfectly balanced (2-rotation symmetric) functions with optimal algebraic immunity and high weightwise nonlinearity
  
  • Mesnager Sihem
  • Su Sihong
  • Li Jingjing
  • Zhu Linya
  Cryptography and Communications - Discrete Structures, Boolean Functions and Sequences, Springer, 2022, 14 (6), pp.1371-1389. Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the well-known FLIP stream cipher proposed by Méaux et al. at the conference Eurocrypt 2016. After providing a security analysis on the FLIP cipher, those functions were nicely-investigated firstly by Carlet et al. in 2017 before taking a high interest by the community. Handling such Boolean functions and designing those with optimal characteristic cryptographic properties is no easy assignment. This article attempts to broaden the range of choices for these functions by offering two new concrete constructions of weightwise perfectly balanced (WPB) functions on variables (where m is a positive integer) with optimal algebraic immunity. It is worth noting that the second class of WPB functions can be linearly transformed to be 2-rotation symmetric. Simultaneously, the k-weight nonlinearities of these newly constructed WPB functions on 2m variables are discussed for small values of m. Lastly, comparisons of the k-weight nonlinearities of all the known WPB functions are given, including the known results from computer investigations. The comparison to the current literature shows that despite its simplicity (an advantage from the implementation point of view), the WPB functions presented in this paper are the best in behavior from the algebraic immunity and the k-weight nonlinearities. Specifically, the even-weight nonlinearities of our second class of WPB functions are much higher than all the known WPB functions in the literature. (10.1007/s12095-022-00603-5)
  DOI : 10.1007/s12095-022-00603-5