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A fundamental problem in computer science is to find all the common zeroes of $m$ quadratic polynomials in $n$ unknowns over $\mathbb{F}_2$. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity…
In this paper we present a finite element method for the direct transcription of constrained non-linear optimal control problems. We prove that our method converges of high order under mild assumptions. Our analysis uses a regularized…
Description Logics (DLs) are used in knowledge-based systems to represent and reason about terminological knowledge of the application domain in a semantically well-defined manner. In this thesis, we establish a number of novel complexity…
We define a reduction mechanism for LP and SDP formulations that degrades approximation factors in a controlled fashion. Our reduction mechanism is a minor restriction of classical reductions establishing inapproximability in the context of…
Deep neural networks (DNNs) are valuable assets, yet their public accessibility raises security concerns about parameter extraction by malicious actors. Recent work by Carlini et al. (crypto'20) and Canales-Mart\'inez et al. (eurocrypt'24)…
In this note, we consider the problem of differentially privately (DP) computing an anonymized histogram, which is defined as the multiset of counts of the input dataset (without bucket labels). In the low-privacy regime $\epsilon \geq 1$,…
We describe an efficient quantum algorithm for computing discrete logarithms in semigroups using Shor's algorithms for period finding and discrete log as subroutines. Thus proposed cryptosystems based on the presumed hardness of discrete…
Low probability of detection (or covert) communication refers to the scenario where information must be sent reliably to a receiver, but with low probability of detection by an adversary. Recent works on the fundamental limits of this…
We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially…
The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that…
Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we…
We propose a general framework for neural network compression that is motivated by the Minimum Description Length (MDL) principle. For that we first derive an expression for the entropy of a neural network, which measures its complexity…
We show how to reduce a general, strictly-feasible LP problem, into a min-max problem, which can be solved by the algorithm from the third section of my thesis.
We study the problem of solving linear programs of the form $Ax\le b$, $x\ge0$ with differential privacy. For homogeneous LPs $Ax\ge0$, we give an efficient $(\epsilon,\delta)$-differentially private algorithm which with probability at…
The System of Linear Equations Problem (SLEP) is specified by a complex invertible matrix $A$, the condition number $\kappa$ of $A$, a vector $b$, a Hermitian matrix $M$ and an accuracy $\epsilon$, and the task is to estimate $x^\dagger…
The constrained linear representability problem (CLRP) for polymatroids determines whether there exists a polymatroid that is linear over a specified field while satisfying a collection of constraints on the rank function. Using a computer…
We present efficient and practical algorithms for a large, distributed system of processors to achieve reliable computations in a secure manner. Specifically, we address the problem of computing a general function of several private inputs…
This paper considers a large class of linear operator equations, including linear boundary value problems for partial differential equations, and treats them as linear recovery problems for objects from their data. Well-posedness of the…
We have observed an interesting, yet unexplained, phenomenon: Semidefinite programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend to be tight in recovery problems with noisy data, even when MLE cannot exactly…
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent…