Related papers: Small primitive roots and malleability of RSA modu…

We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the $7$-modular and $11$-modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a…

Let $G$ be a connected reductive group over an algebraically closed field of characteristic $p>0$. Given an indecomposable G-module $M$, one can ask when it remains indecomposable upon restriction to the Frobenius kernel $G_r$, and when its…

Let $G$ be an abelian group of order $n$ and let $R$ be a commutative ring which admits a homomorphism ${\Bbb Z}[\zeta_{n}]\ra R$, where $\zeta_{n}$ is a (complex) primitive $n$-th root of unity. Given a finite $R[G\e]$-module $M$, we…

Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\phi$ to be weakly $S$-prime if…

The Rabin tree theorem yields an algorithm to solve the satisfiability problem for monadic second-order logic over infinite trees. Here we solve the probabilistic variant of this problem. Namely, we show how to compute the probability that…

Over the past years Robust PCA has been established as a standard tool for reliable low-rank approximation of matrices in the presence of outliers. Recently, the Robust PCA approach via nuclear norm minimization has been extended to…

The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the…

Engineering risk is concerned with the likelihood of failure and the scenarios when it occurs. The sensitivity of failure probability to change in system parameters is relevant to risk-informed decision making. Computing sensitivity is at…

Recently, there has been considerable progress on designing algorithms with provable guarantees -- typically using linear algebraic methods -- for parameter learning in latent variable models. But designing provable algorithms for inference…

Recent work has shown that state-of-the-art classifiers are quite brittle, in the sense that a small adversarial change of an originally with high confidence correctly classified input leads to a wrong classification again with high…

Greedy algorithms are widely used for problems in machine learning such as feature selection and set function optimization. Unfortunately, for large datasets, the running time of even greedy algorithms can be quite high. This is because for…

RSA is an incredibly successful and useful asymmetric encryption algorithm. One of the types of implementation flaws in RSA is low entropy of the key generation, specifically the prime number creation stage. This can occur due to flawed…

Despite their empirical success, neural networks remain vulnerable to small, adversarial perturbations. A longstanding hypothesis suggests that flat minima, regions of low curvature in the loss landscape, offer increased robustness. While…

Let $R$ be a commutative Noetherian ring. We give criteria for flatness of $R$-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if $R$ has…

In this set of three companion manuscripts/articles, we unveil our new results on primality testing and reveal new primality testing algorithms enabled by those results. The results have been classified (and referred to) as…

We demonstrate that a modification of the classical index calculus algorithm can be used to factor integers. More generally, we reduce the factoring problem to finding an overdetermined system of multiplicative relations in any factor base…

In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…

Much like admissibility is the key concept underlying preferred semantics, strong admissibility is the key concept underlying grounded semantics, as membership of a strongly admissible set is sufficient to show membership of the grounded…

In this article we investigate the coprimeness properties of one and two-dimensional discrete equations, in a situation where the equations are decomposable into several factors of polynomials. After experimenting on a simple equation, we…

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$…