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This work aims to introduce the framework of polynomial optimization theory to solve fractional polynomial problems (FPPs). Unlike other widely used optimization frameworks, the proposed one applies to a larger class of FPPs, not…
In distributed matrix multiplication, a common scenario is to assign each worker a fraction of the multiplication task, by partitioning the input matrices into smaller submatrices. In particular, by dividing two input matrices into…
Nowadays there are several classes of constrained codes intended for different applications. The following two large classes can be distinguished. The first class contains codes with local constraints; for example, the source data must be…
The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable…
SMT solvers use sophisticated techniques for polynomial (linear or non-linear) integer arithmetic. In contrast, non-polynomial integer arithmetic has mostly been neglected so far. However, in the context of program verification, polynomials…
We consider the problem of secure distributed matrix multiplication in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. We show how to construct polynomial schemes for the outer…
The problem of verifying multi-threaded execution against the memory consistency model of a processor is known to be an NP hard problem. However polynomial time algorithms exist that detect almost all failures in such execution. These are…
In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly…
The Android operating system is pervasively adopted as the operating system platform of choice for smart devices. However, the strong adoption has also resulted in exponential growth in the number of Android based malicious software or…
Signature-based algorithms have become a standard approach for Gr\"obner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this…
Probabilistic computing has emerged as a viable approach to treat optimization problems. To achieve superior computing performance, the key aspect during computation is massive sampling and tuning on the probability states of each…
The BMR16 circuit garbling scheme introduces gadgets that allow for ciphertext-free modular addition, while the multiplication of private inputs modulo a prime p can be done with 2(p - 1) ciphertexts as described in Malkin, Pastro, and…
Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of…
Cover-free families are set systems used as solutions for a large variety of problems, and in particular, problems where we deal with $n$ elements and want to identify $d$ invalid ones among them by performing only $t$ tests ($t \leq n$).…
We consider polynomial equations, or systems of polynomial equations, with integer coefficients, modulo prime numbers $p$. We offer an elementary approach based on a counting method. The outcome is a weak form of the Lang-Weil lower bound…
We investigate the use of invariant polynomials in the construction of data-driven interatomic potentials for material systems. The "atomic body-ordered permutation-invariant polynomials" (aPIPs) comprise a systematic basis and are…
Artificial neural networks (ANNs) have achieved significant success in tackling classical and modern machine learning problems. As learning problems grow in scale and complexity, and expand into multi-disciplinary territory, a more modular…
There has been significant work recently on integer programs (IPs) $\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\}$ with a constraint marix $A$ with bounded subdeterminants. This is motivated by a well-known conjecture claiming that,…
The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far…
Current trends in technology, such as cloud computing, allow outsourcing the storage, backup, and archiving of data. This provides efficiency and flexibility, but also poses new risks for data security. It in particular became crucial to…