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Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of $t$-weight linear codes over ${\mathbb F}_{q}$ are presented with the defining sets given by the intersection, difference and union of two certain sets, where $t=3,4,5,6$ and $q$ is an odd prime power. By using Weil sums and Gauss sums, the parameters and weight distributions of these codes are determined completely. Moreover, three classes of optimal codes meeting the Griesmer bound are obtained, and computer experiments show that many (almost) optimal codes can be derived from our constructions.

Dillon-like Boolean functions are known, in the literature, to be those trace polynomial functions from $\mathbb{F}_{2^{2n}}$ to $\mathbb{F}_{2}$, with all the exponents being multiples of $2^n-1$ often called Dillon-like exponents. This paper is devoted to bent functions in which we study the bentness of some classes of Dillon-like Boolean functions connected with rational trace functions. Specifically, we introduce a special infinite family of trace rational functions. We shall use these functions as building blocks and generalise notably a criterion due to Li et al. published in [IEEE Trans. Inf. Theory 59(3), pp. 1818-1831, 2013] on the bentness of Dillon-like functions in the binary case, we explicitly characterize three classes of bent functions. These characterizations are expressed in terms of the well-known binary Kloosterman sums. Furthermore, analysis and experiments indicate that new functions not EA-equivalent to all known classes of monomial functions are included in our classes., Comment: 28 pages

The concept of world models has garnered significant attention due to advancements in multimodal large language models such as GPT-4 and video generation models such as Sora, which are central to the pursuit of artificial general intelligence. This survey offers a comprehensive review of the literature on world models. Generally, world models are regarded as tools for either understanding the present state of the world or predicting its future dynamics. This review presents a systematic categorization of world models, emphasizing two primary functions: (1) constructing internal representations to understand the mechanisms of the world, and (2) predicting future states to simulate and guide decision-making. Initially, we examine the current progress in these two categories. We then explore the application of world models in key domains, including autonomous driving, robotics, and social simulacra, with a focus on how each domain utilizes these aspects. Finally, we outline key challenges and provide insights into potential future research directions.

Plasmonic nanomaterials offer a direct and effective approach to harnessing solar energy. Specifically, plasmonic semiconductors enable a highly efficient light-to-heat conversion process, outperforming noble metals in stability, cost-effectiveness, and accessibility. In this study, a composite 3D printing filament (T-TPU), composed of titanium nitride (TiN) and thermoplastic polyurethane (TPU), was synthesized using a combined extrusion process involving a twin-screw extruder and a single-screw extruder. The resulting T-TPU filament could be used with fused deposition modeling (FDM) 3D printing to produce custom-designed patterns. Notably, these printed patterns exhibited superior photothermal performance, with potential applications in photothermal de-icing and infrared labeling. Additionally, the wavelength-dependent plasmonic and photothermal responses of the printed patterns were experimentally investigated and supported by finite elemental method (FEM) simulations, revealing a temperature increase of approximately 2.5 under IR LED light when compared to commercial black thermoplastic polyurethane (C-TPU), that was more obvious than a difference less than 1 under UV or visible LED light sources. Finally, the mechanical properties of T-TPU, altered by the inclusion of TiN nanoparticles, were assessed, showing a slight enhancement in modulus and friction coefficient relative to neat TPU (N-TPU). Molecular dynamics (MD) simulations indicated that the TiN nanoparticles promoted strong interactions between polymer chains and TiN particles, enhancing the modulus of elasticity and contributing to the improved mechanical properties of T-TPU. These findings suggest improved abrasion resistance, demonstrating the stability and durability of the composite material.

Solar photothermal conversion is one of the most straightforward methods to utilize solar energy. In this manuscript, a novel double-layer structure constructed of graphene enhanced thermoplastic polyurethanes (G-TPU) and neat thermoplastic polyurethanes (N-TPU) was developed via fused deposition modelling (FDM) 3D printing process. The developed G-TPU-N-TPU double-layer structure exhibited anisotropic thermal conductivity that simultaneously satisfied high in-plane (IP) thermal conductivity and low through-plane (TP) thermal conductivity. The top G-TPU layer essentially offered a high IP thermal conductivity of 4.54 W(mK) that lead to overall structure anisotropic thermal conductivity ratio (TCIP-TCTP) of 8. And the low thermal conductivity in the TP direction led to the heat retention effects for thermal storage. Nonetheless, the exceptional photothermal conversion effect of graphene flakes guaranteed the superior photothermal performance that was promising in the photothermal de-icing and infrared labels applications. Finally, the graphene flake enhancement in the mechanical properties of the G-TPU-N-TPU double layer structure was also evaluated that contributed to excellent impact resistance with a puncture energy reaching 12.86 J, and extraordinary wear resistance with a small friction coefficient of 0.1 over 1000 cycles, which ensured the structure suitable for applications at harsh environment.

The Feistel Boomerang Connectivity Table ($\rm{FBCT}$), which is the Feistel version of the Boomerang Connectivity Table ($\rm{BCT}$), plays a vital role in analyzing block ciphers' ability to withstand strong attacks, such as boomerang attacks. However, as of now, only four classes of power functions are known to have explicit values for all entries in their $\rm{FBCT}$. In this paper, we focus on studying the FBCT of the power function $F(x)=x^{2^{n-2}-1}$ over $\mathbb{F}_{2^n}$, where $n$ is a positive integer. Through certain refined manipulations to solve specific equations over $\mathbb{F}_{2^n}$ and employing binary Kloosterman sums, we determine explicit values for all entries in the $\rm{FBCT}$ of $F(x)$ and further analyze its Feistel boomerang spectrum. Finally, we demonstrate that this power function exhibits the lowest Feistel boomerang uniformity.

As a novel and promising 2D material, bismuth oxyselenide (Bi$_2$O$_2$Se) has demonstrated significant potential to overcome existing technical barriers in various electronic device applications, due to its unique physical properties like high symmetry, adjustable electronic structure, ultra-high electron mobility. However, the rapid growth of Bi$_2$O$_2$Se films down to a few atomic layers with precise control remains a significant challenge. In this work, the growth of two-dimensional (2D) Bi$_2$O$_2$Se thin films by the pulsed laser deposition (PLD) method is systematically investigated. By controlling temperature, oxygen pressure, laser energy density and laser emission frequency, we successfully prepare atomically thin and flat Bi$_2$O$_2$Se (001) thin films on the (001) surface of SrTiO3. Importantly, we provide a fundamental and unique perspective toward understanding the growth process of atomically thin and flat Bi$_2$O$_2$Se films, and the growth process can be primarily summarized into four steps: i) anisotropic non-spontaneous nucleation preferentially along the step roots; ii) monolayer Bi$_2$O$_2$Se nanosheets expanding across the surrounding area, and eventually covering the entire STO substrate step; iii) vertical growth of Bi$_2$O$_2$Se monolayer in a 2D Frank-van der Merwe (FM) epitaxial growth, and iv) with a layer-by-layer 2D FM growth mode, ultimately producing an atomically flat and epitaxially aligned thin film. Moreover, the combined results of the crystallinity quality, surface morphology and the chemical states manifest the successful PLD-growth of high-quality Bi$_2$O$_2$Se films in a controllable and fast mode.

Recently, constructions of optimal linear codes from simplicial complexes have attracted much attention and some related nice works were presented. Let $q$ be a prime power. In this paper, by using the simplicial complexes of ${\mathbb F}_{q}^m$ with one single maximal element, we construct four families of linear codes over the ring ${\mathbb F}_{q}+u{\mathbb F}_{q}$ ($u^2=0$), which generalizes the results of [IEEE Trans. Inf. Theory 66(6):3657-3663, 2020]. The parameters and Lee weight distributions of these four families of codes are completely determined. Most notably, via the Gray map, we obtain several classes of optimal linear codes over ${\mathbb F}_{q}$, including (near) Griesmer codes and distance-optimal codes., Comment: 18 pages

In this paper, several infinite families of codes over the extension of non-unital non-commutative rings are constructed utilizing general simplicial complexes. Thanks to the special structure of the defining sets, the principal parameters of these codes are characterized. Specially, when the employed simplicial complexes are generated by a single maximal element, we determine their Lee weight distributions completely. Furthermore, by considering the Gray image codes and the corresponding subfield-like codes, numerous of linear codes over $\mathbb{F}_q$ are also obtained, where $q$ is a prime power. Certain conditions are given to ensure the above linear codes are (Hermitian) self-orthogonal in the case of $q=2,3,4$. It is noteworthy that most of the derived codes over $\mathbb{F}_q$ satisfy the Ashikhmin-Barg's condition for minimality. Besides, we obtain two infinite families of distance-optimal codes over $\mathbb{F}_q$ with respect to the Griesmer bound., Comment: 26 pages

Transformer-based architectures for speaker verification typically require more training data than ECAPA-TDNN. Therefore, recent work has generally been trained on VoxCeleb1&2. We propose a backbone network based on self-attention, which can achieve competitive results when trained on VoxCeleb2 alone. The network alternates between neighborhood attention and global attention to capture local and global features, then aggregates features of different hierarchical levels, and finally performs attentive statistics pooling. Additionally, we employ a progressive channel fusion strategy to expand the receptive field in the channel dimension as the network deepens. We trained the proposed PCF-NAT model on VoxCeleb2 and evaluated it on VoxCeleb1 and the validation sets of VoxSRC. The EER and minDCF of the shallow PCF-NAT are on average more than 20% lower than those of similarly sized ECAPA-TDNN. Deep PCF-NAT achieves an EER lower than 0.5% on VoxCeleb1-O. The code and models are publicly available at https://github.com/ChenNan1996/PCF-NAT., Comment: 8 pages, 2 figures, 3 tables; added github link

Recommender systems filter out information that meets user interests. However, users may be tired of the recommendations that are too similar to the content they have been exposed to in a short historical period, which is the so-called user fatigue. Despite the significance for a better user experience, user fatigue is seldom explored by existing recommenders. In fact, there are three main challenges to be addressed for modeling user fatigue, including what features support it, how it influences user interests, and how its explicit signals are obtained. In this paper, we propose to model user Fatigue in interest learning for sequential Recommendations (FRec). To address the first challenge, based on a multi-interest framework, we connect the target item with historical items and construct an interest-aware similarity matrix as features to support fatigue modeling. Regarding the second challenge, built upon feature cross, we propose a fatigue-enhanced multi-interest fusion to capture long-term interest. In addition, we develop a fatigue-gated recurrent unit for short-term interest learning, with temporal fatigue representations as important inputs for constructing update and reset gates. For the last challenge, we propose a novel sequence augmentation to obtain explicit fatigue signals for contrastive learning. We conduct extensive experiments on real-world datasets, including two public datasets and one large-scale industrial dataset. Experimental results show that FRec can improve AUC and GAUC up to 0.026 and 0.019 compared with state-of-the-art models, respectively. Moreover, large-scale online experiments demonstrate the effectiveness of FRec for fatigue reduction. Our codes are released at https://github.com/tsinghua-fib-lab/SIGIR24-FRec., Comment: SIGIR 2024

Substitution boxes (S-boxes) play a significant role in ensuring the resistance of block ciphers against various attacks. The Upper Boomerang Connectivity Table (UBCT), the Lower Boomerang Connectivity Table (LBCT) and the Double Boomerang Connectivity Table (DBCT) of a given S-box are crucial tools to analyze its security concerning specific attacks. However, there are currently no related results for this research. The inverse function is crucial for constructing S-boxes of block ciphers with good cryptographic properties in symmetric cryptography. Therefore, extensive research has been conducted on the inverse function, exploring various properties related to standard attacks. Thanks to the recent advancements in boomerang cryptanalysis, particularly the introduction of concepts such as UBCT, LBCT, and DBCT, this paper aims to further investigate the properties of the inverse function $F(x)=x^{2^n-2}$ over $\gf_{2^n}$ for arbitrary $n$. As a consequence, by carrying out certain finer manipulations of solving specific equations over $\gf_{2^n}$, we give all entries of the UBCT, LBCT of $F(x)$ over $\gf_{2^n}$ for arbitrary $n$. Besides, based on the results of the UBCT and LBCT for the inverse function, we determine that $F(x)$ is hard when $n$ is odd. Furthermore, we completely compute all entries of the DBCT of $F(x)$ over $\gf_{2^n}$ for arbitrary $n$. Additionally, we provide the precise number of elements with a given entry by means of the values of some Kloosterman sums. Further, we determine the double boomerang uniformity of $F(x)$ over $\gf_{2^n}$ for arbitrary $n$. Our in-depth analysis of the DBCT of $F(x)$ contributes to a better evaluation of the S-box's resistance against boomerang attacks., Comment: This manuscript was submitted to Finite Fields and Their Application on April 8, 2024. arXiv admin note: text overlap with arXiv:2309.01881

Filter bubbles have been studied extensively within the context of online content platforms due to their potential to cause undesirable outcomes such as user dissatisfaction or polarization. With the rise of short-video platforms, the filter bubble has been given extra attention because these platforms rely on an unprecedented use of the recommender system to provide relevant content. In our work, we investigate the deep filter bubble, which refers to the user being exposed to narrow content within their broad interests. We accomplish this using one-year interaction data from a top short-video platform in China, which includes hierarchical data with three levels of categories for each video. We formalize our definition of a "deep" filter bubble within this context, and then explore various correlations within the data: first understanding the evolution of the deep filter bubble over time, and later revealing some of the factors that give rise to this phenomenon, such as specific categories, user demographics, and feedback type. We observe that while the overall proportion of users in a filter bubble remains largely constant over time, the depth composition of their filter bubble changes. In addition, we find that some demographic groups that have a higher likelihood of seeing narrower content and implicit feedback signals can lead to less bubble formation. Finally, we propose some ways in which recommender systems can be designed to reduce the risk of a user getting caught in a bubble., Comment: accepted to WWW 2024

Heterogeneous information networks (HIN) have gained increasing popularity in recent years for capturing complex relations between diverse types of nodes. Meta-structures are proposed as a useful tool to identify the important patterns in HINs, but hand-crafted meta-structures pose significant challenges for scaling up, drawing wide research attention towards developing automatic search algorithms. Previous efforts primarily focused on searching for meta-structures with good empirical performance, overlooking the importance of human comprehensibility and generalizability. To address this challenge, we draw inspiration from the emergent reasoning abilities of large language models (LLMs). We propose ReStruct, a meta-structure search framework that integrates LLM reasoning into the evolutionary procedure. ReStruct uses a grammar translator to encode the meta-structures into natural language sentences, and leverages the reasoning power of LLMs to evaluate their semantic feasibility. Besides, ReStruct also employs performance-oriented evolutionary operations. These two competing forces allow ReStruct to jointly optimize the semantic explainability and empirical performance of meta-structures. Furthermore, ReStruct contains a differential LLM explainer to generate and refine natural language explanations for the discovered meta-structures by reasoning through the search history. Experiments on eight representative HIN datasets demonstrate that ReStruct achieves state-of-the-art performance in both recommendation and node classification tasks. Moreover, a survey study involving 73 graduate students shows that the discovered meta-structures and generated explanations by ReStruct are substantially more comprehensible. Our code and questionnaire are available at https://github.com/LinChen-65/ReStruct.

Agent-based modeling and simulation has evolved as a powerful tool for modeling complex systems, offering insights into emergent behaviors and interactions among diverse agents. Integrating large language models into agent-based modeling and simulation presents a promising avenue for enhancing simulation capabilities. This paper surveys the landscape of utilizing large language models in agent-based modeling and simulation, examining their challenges and promising future directions. In this survey, since this is an interdisciplinary field, we first introduce the background of agent-based modeling and simulation and large language model-empowered agents. We then discuss the motivation for applying large language models to agent-based simulation and systematically analyze the challenges in environment perception, human alignment, action generation, and evaluation. Most importantly, we provide a comprehensive overview of the recent works of large language model-empowered agent-based modeling and simulation in multiple scenarios, which can be divided into four domains: cyber, physical, social, and hybrid, covering simulation of both real-world and virtual environments. Finally, since this area is new and quickly evolving, we discuss the open problems and promising future directions., Comment: 37 pages

Boukerrou et al. (IACR Trans. Symmetric Cryptol. 2020(1), 331-362) introduced the notion of Feistel Boomerang Connectivity Table (FBCT), the Feistel counterpart of the Boomerang Connectivity Table (BCT), and the Feistel boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic). FBCT is a crucial table for the analysis of the resistance of block ciphers to power attacks such as differential and boomerang attacks. It is worth noting that the coefficients of FBCT are related to the second-order zero differential spectra of functions. In this paper, by carrying out certain finer manipulations of solving specific equations over the finite field $\mathbb{F}_{p^n}$, we explicitly determine the second-order zero differential spectra of some power functions with low differential uniformity, and show that our considered functions also have low second-order zero differential uniformity. Our study pushes further former investigations on second-order zero differential uniformity and Feistel boomerang differential uniformity for a power function $F$.

The advent of artificial intelligence has led to a growing emphasis on data-driven modeling in macroeconomics, with agent-based modeling (ABM) emerging as a prominent bottom-up simulation paradigm. In ABM, agents (e.g., households, firms) interact within a macroeconomic environment, collectively generating market dynamics. Existing agent modeling typically employs predetermined rules or learning-based neural networks for decision-making. However, customizing each agent presents significant challenges, complicating the modeling of agent heterogeneity. Additionally, the influence of multi-period market dynamics and multifaceted macroeconomic factors are often overlooked in decision-making processes. In this work, we introduce EconAgent, a large language model-empowered agent with human-like characteristics for macroeconomic simulation. We first construct a simulation environment that incorporates various market dynamics driven by agents' decisions regarding work and consumption. Through the perception module, we create heterogeneous agents with distinct decision-making mechanisms. Furthermore, we model the impact of macroeconomic trends using a memory module, which allows agents to reflect on past individual experiences and market dynamics. Simulation experiments show that EconAgent can make realistic decisions, leading to more reasonable macroeconomic phenomena compared to existing rule-based or learning-based agents. Our codes are released at https://github.com/tsinghua-fib-lab/ACL24-EconAgent., Comment: ACL 2024 (main conference)

We study a new method of constructing Boolean bent functions from cyclotomic mappings. Three generic constructions are obtained by considering different branch functions such as Dillon functions, Niho functions and Kasami functions over multiplicative cosets and additive cosets respectively. As a result, several new explicit infinite families of bent functions and their duals are derived. We demonstrate that some previous constructions are special cases of our simple constructions. In addition, by studying their polynomial forms, we observe that the last construction provides some examples which are EA-inequivalent to five classes of monomials, Dillon type and Niho type polynomials.

Substitution boxes (S-boxes) play a significant role in ensuring the resistance of block ciphers against various attacks. The Difference Distribution Table (DDT), the Feistel Boomerang Connectivity Table (FBCT), the Feistel Boomerang Difference Table (FBDT) and the Feistel Boomerang Extended Table (FBET) of a given S-box are crucial tools to analyze its security concerning specific attacks. However, the results on them are rare. In this paper, we investigate the properties of the power function $F(x):=x^{2^{m+1}-1}$ over the finite field $\gf_{2^n}$ of order $2^n$ where $n=2m$ or $n=2m+1$ ($m$ stands for a positive integer). As a consequence, by carrying out certain finer manipulations of solving specific equations over $\gf_{2^n}$, we give explicit values of all entries of the DDT, the FBCT, the FBDT and the FBET of the investigated power functions. From the theoretical point of view, our study pushes further former investigations on differential and Feistel boomerang differential uniformities for a novel power function $F$. From a cryptographic point of view, when considering Feistel block cipher involving $F$, our in-depth analysis helps select $F$ resistant to differential attacks, Feistel differential attacks and Feistel boomerang attacks, respectively.

Micro-videos platforms such as TikTok are extremely popular nowadays. One important feature is that users no longer select interested videos from a set, instead they either watch the recommended video or skip to the next one. As a result, the time length of users' watching behavior becomes the most important signal for identifying preferences. However, our empirical data analysis has shown a video-length effect that long videos are easier to receive a higher value of average view time, thus adopting such view-time labels for measuring user preferences can easily induce a biased model that favors the longer videos. In this paper, we propose a Video Length Debiasing Recommendation (VLDRec) method to alleviate such an effect for micro-video recommendation. VLDRec designs the data labeling approach and the sample generation module that better capture user preferences in a view-time oriented manner. It further leverages the multi-task learning technique to jointly optimize the above samples with original biased ones. Extensive experiments show that VLDRec can improve the users' view time by 1.81% and 11.32% on two real-world datasets, given a recommendation list of a fixed overall video length, compared with the best baseline method. Moreover, VLDRec is also more effective in matching users' interests in terms of the video content., Comment: Accept by TOIS

Short-video recommendation is one of the most important recommendation applications in today's industrial information systems. Compared with other recommendation tasks, the enormous amount of feedback is the most typical characteristic. Specifically, in short-video recommendation, the easiest-to-collect user feedback is the skipping behavior, which leads to two critical challenges for the recommendation model. First, the skipping behavior reflects implicit user preferences, and thus, it is challenging for interest extraction. Second, this kind of special feedback involves multiple objectives, such as total watching time and skipping rate, which is also very challenging. In this paper, we present our industrial solution in Kuaishou, which serves billion-level users every day. Specifically, we deploy a feedback-aware encoding module that extracts user preferences, taking the impact of context into consideration. We further design a multi-objective prediction module which well distinguishes the relation and differences among different model objectives in the short-video recommendation. We conduct extensive online A/B tests, along with detailed and careful analysis, which verify the effectiveness of our solution., Comment: Accepted by CIKM'23

In this paper, we construct a large family of projective linear codes over ${\mathbb F}_{q}$ from the general simplicial complexes of ${\mathbb F}_{q}^m$ via the defining-set construction, which generalizes the results of [IEEE Trans. Inf. Theory 66(11):6762-6773, 2020]. The parameters and weight distribution of this class of codes are completely determined. By using the Griesmer bound, we give a necessary and sufficient condition such that the codes are Griesmer codes and a sufficient condition such that the codes are distance-optimal. For a special case, we also present a necessary and sufficient condition for the codes to be near Griesmer codes. Moreover, by discussing the cases of simplicial complexes with one, two and three maximal elements respectively, the parameters and weight distributions of the codes are given more explicitly, which shows that the codes are at most $2$-weight, $5$-weight and $19$-weight respectively. By studying the optimality of the codes for the three cases in detail, many infinite families of optimal linear codes with few weights over ${\mathbb F}_{q}$ are obtained, including Griesmer codes, near Griesmer codes and distance-optimal codes., Comment: 19 pages

This paper deals with Niho functions which are one of the most important classes of functions thanks to their close connections with a wide variety of objects from mathematics, such as spreads and oval polynomials or from applied areas, such as symmetric cryptography, coding theory and sequences. In this paper, we investigate specifically the $c$-differential uniformity of the power function $F(x)=x^{s(2^m-1)+1}$ over the finite field $\mathbb{F}_{2^n}$, where $n=2m$, $m$ is odd and $s=(2^k+1)^{-1}$ is the multiplicative inverse of $2^k+1$ modulo $2^m+1$, and show that the $c$-differential uniformity of $F(x)$ is $2^{\gcd(k,m)}+1$ by carrying out some subtle manipulation of certain equations over $\mathbb{F}_{2^n}$. Notably, $F(x)$ has a very low $c$-differential uniformity equals $3$ when $k$ and $m$ are coprime.

Starting with the multiplication of elements in $\mathbb{F}_{q}^2$ which is consistent with that over $\mathbb{F}_{q^2}$, where $q$ is a prime power, via some identification of the two environments, we investigate the $c$-differential uniformity for bivariate functions $F(x,y)=(G(x,y),H(x,y))$. By carefully choosing the functions $G(x,y)$ and $H(x,y)$, we present several constructions of bivariate functions with low $c$-differential uniformity. Many P$c$N and AP$c$N functions can be produced from our constructions., Comment: Low $c$-differential uniformity, perfect and almost perfect $c$-nonlinearity, the bivariate function

The investigation of partially APN functions has attracted a lot of research interest recently. In this paper, we present several new infinite classes of 0-APN power functions over $\mathbb{F}_{2^n}$ by using the multivariate method and resultant elimination, and show that these 0-APN power functions are CCZ-inequivalent to the known ones., Comment: arXiv admin note: text overlap with arXiv:2210.02207, arXiv:2210.15103 by other authors

In this article, we focus on the concept of locally-APN-ness (``APN" is the abbreviation of the well-known notion of Almost Perfect Nonlinear) introduced by Blondeau, Canteaut, and Charpin, which makes the corpus of S-boxes somehow larger regarding their differential uniformity and, therefore, possibly, more suitable candidates against the differential attack (or their variants). Specifically, given two coprime positive integers $m$ and $k$ such that $\gcd(2^m+1,2^k+1)=1$, we investigate the locally-APN-ness property of an infinite family of Niho type power functions in the form $F(x)=x^{s(2^m-1)+1}$ over the finite field ${\mathbb F}_{2^{2m}}$ for $s=(2^k+1)^{-1}$, where $(2^k+1)^{-1}$ denotes the multiplicative inverse modulo $2^m+1$. By employing finer studies of the number of solutions of certain equations over finite fields (with even characteristic) as well as some subtle manipulations of solving some equations, we prove that $F(x)$ is locally APN and determine its differential spectrum. It is worth noting that computer experiments show that this class of locally-APN power functions covers all Niho type locally-APN power functions for $2\leq m\leq10$. In addition, we also determine the boomerang spectrum of $F(x)$ by using its differential spectrum, which particularly generalizes a recent result by Yan, Zhang, and Li.

In this paper, we study the boomerang spectrum of the power mapping $F(x)=x^{k(q-1)}$ over ${\mathbb F}_{q^2}$, where $q=p^m$, $p$ is a prime, $m$ is a positive integer and $\gcd(k,q+1)=1$. We first determine the differential spectrum of $F(x)$ and show that $F(x)$ is locally-APN. This extends a result of [IEEE Trans. Inf. Theory 57(12):8127-8137, 2011] from $(p,k)=(2,1)$ to general $(p,k)$. We then determine the boomerang spectrum of $F(x)$ by making use of its differential spectrum, which shows that the boomerang uniformity of $F(x)$ is 4 if $p=2$ and $m$ is odd and otherwise it is 2. Our results not only generalize the results in [Des. Codes Cryptogr. 89:2627-2636, 2021] and [arXiv:2203.00485, 2022] but also extend the example $x^{45}$ over ${\mathbb F}_{2^8}$ in [Des. Codes Cryptogr. 89:2627-2636, 2021] into an infinite class of power mappings with boomerang uniformity 2., Comment: 18 pages