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The breakthrough of achieving fully homomorphic encryption sparked enormous studies on where and how to apply homomorphic encryption schemes so that operations can be performed on encrypted data without the secret key while still obtaining…

The concept of \emph{data depth} in non-parametric multivariate descriptive statistics is the generalization of the univariate rank method to multivariate data. \emph{Halfspace depth} is a measure of data depth. Given a set $S$ of points…

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of…

Modern cryptographic methods for implementing privacy-preserving LLMs such as \gls{HE} require the LLMs to have a polynomial form. Forming such a representation is challenging because transformers include non-polynomial components, such as…

The convergence of fully homomorphic encryption (FHE) and machine learning offers unprecedented opportunities for private inference of sensitive data. FHE enables computation directly on encrypted data, safeguarding the entire machine…

For every constant $d$, we design a subexponential time deterministic algorithm that takes as input a multivariate polynomial $f$ given as a constant depth algebraic circuit over the field of rational numbers, and outputs all irreducible…

In the era of generative AI, ensuring the privacy of music data presents unique challenges: unlike static artworks such as images, music data is inherently temporal and multimodal, and it is sampled, transformed, and remixed at an…

This tutorial aims to establish connections between polynomial modular multiplication over a ring to circular convolution and discrete Fourier transform (DFT). The main goal is to extend the well-known theory of DFT in signal processing…

Fully homomorphic encryption (FHE) enables a simple, attractive framework for secure search. Compared to other secure search systems, no costly setup procedure is necessary; it is sufficient for the client merely to upload the encrypted…

As the demand for privacy-preserving computation continues to grow, fully homomorphic encryption (FHE)-which enables continuous computation on encrypted data-has become a critical solution. However, its adoption is hindered by significant…

Recent studies have explored the deployment of privacy-preserving deep neural networks utilizing homomorphic encryption (HE), especially for private inference (PI). Many works have attempted the approximation-aware training (AAT) approach…

New cryptographic techniques such as homomorphic encryption (HE) allow computations to be outsourced to and evaluated blindfolded in a resourceful cloud. These computations often require private data owned by multiple participants, engaging…

Fully Homomorphic Encryption (FHE) is a set of powerful cryptographic schemes that allows computation to be performed directly on encrypted data with an unlimited depth. Despite FHE's promising in privacy-preserving computing, yet in most…

Secure training, while protecting the confidentiality of both data and model weights, typically incurs significant training overhead. Traditional Fully Homomorphic Encryption (FHE)-based non-inter-active training models are heavily burdened…

High-dimensional planted problems, such as finding a hidden dense subgraph within a random graph, often exhibit a gap between statistical and computational feasibility. While recovering the hidden structure may be statistically possible, it…

Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate…

In this paper, we show that the classical discrete orthogonal univariate polynomials (namely, Hahn polynomials on an equidistant lattice with unit weights) of sufficiently high degrees have extremely small values near the endpoints (we call…

Homomorphic encryption (HE) is pivotal for secure computation on encrypted data, crucial in privacy-preserving data analysis. However, efficiently processing high-dimensional data in HE, especially for machine learning and statistical…

Incorporating fully homomorphic encryption (FHE) into the inference process of a convolutional neural network (CNN) draws enormous attention as a viable approach for achieving private inference (PI). FHE allows delegating the entire…

Given a weighted undirected graph $G=(V,E,w)$, a hopset $H$ of hopbound $\beta$ and stretch $(1+\epsilon)$ is a set of edges such that for any pair of nodes $u, v \in V$, there is a path in $G \cup H$ of at most $\beta$ hops, whose length…