Related papers: Generalized Implicit Factorization Problem
In binary polynomial optimization, the goal is to find a binary point maximizing a given polynomial function. In this paper, we propose a novel way of formulating this general optimization problem, which we call factorized binary polynomial…
Quantum computing devices are believed to be powerful in solving the prime factorization problem, which is at the heart of widely deployed public-key cryptographic tools. However, the implementation of Shor's quantum factorization algorithm…
We explore the computational implications of a superposition of spacetimes, a phenomenon hypothesized in quantum gravity theories. This was initiated by Shmueli (2024) where the author introduced the complexity class $\mathbf{BQP^{OI}}$…
Integer factorization is a famous computational problem unknown whether being solvable in the polynomial time. With the rise of deep neural networks, it is interesting whether they can facilitate faster factorization. We present an approach…
Prime factorization has been a buzzing topic in the field of number theory since time unknown. However, in recent years, alternative avenues to tackle this problem are being explored by researchers because of its direct application in the…
Lagarias and Odlyzko (J.~ACM~1985) proposed a polynomial time algorithm for solving ``\emph{almost all}'' instances of the Subset Sum problem with $n$ integers of size $\Omega(\Gamma_{\text{LO}})$, where $\log_2(\Gamma_{\text{LO}}) > n^2…
In this paper we study the Product Partition Problem (PPP), i.e. we are given a set of $n$ natural numbers represented on $m$ bits each and we are asked if a subset exists such that the product of the numbers in the subset equals the…
We point out critical deficiencies in lattice-based cryptanalysis of common prime RSA presented in ``Remarks on the cryptanalysis of common prime RSA for IoT constrained low power devices'' [Information Sciences, 538 (2020) 54--68]. To…
The integer factorization problem (IFP) underpins the security of RSA, yet becomes efficiently solvable on a quantum computer through Shor's algorithm. Regev's recent high-dimensional variant reduces the circuit size through lattice-based…
The splitting method is a powerful method for solving partial differential equations. Various splitting methods have been designed to separate different physics, nonlinearities, and so on. Recently, a new splitting approach has been…
In this paper, we present an approach to integer factorization using distributed representations formed with Vector Symbolic Architectures. The approach formulates integer factorization in a manner such that it can be solved using neural…
Our main result is a reduction from worst-case lattice problems such as GapSVP and SIVP to a certain learning problem. This learning problem is a natural extension of the `learning from parity with error' problem to higher moduli. It can…
Let $K$ be a number field, let $A$ be a finite-dimensional $K$-algebra, let $\mathrm{J}(A)$ denote the Jacobson radical of $A$, and let $\Lambda$ be an $\mathcal{O}_{K}$-order in $A$. Suppose that each simple component of the semisimple…
Implicit particle filters for data assimilation generate high-probability samples by representing each particle location as a separate function of a common reference variable. This representation requires that a certain underdetermined…
This article presents a validation of a recently proposed strongly polynomial-time algorithm for the general linear programming problem. The proposed algorithm is an implicit reduction procedure that combines primal and dual linear…
This article studies two notions of generalized matroid representations motivated by algorithmic information theory and cryptographic secret sharing. The first (entropic representability) involves discrete random variables, while the second…
The present lectures were prepared for the Faro International Summer School on Factorization and Integrable Systems in September 2000. They were intended for participants with the background in Analysis and Operator Theory but without…
We introduce a probabilistic model with implicit norm regularization for learning nonnegative matrix factorization (NMF) that is commonly used for predicting missing values and finding hidden patterns in the data, in which the matrix…
Numerous problems in signal processing and imaging, statistical learning and data mining, or computer vision can be formulated as optimization problems which consist in minimizing a sum of convex functions, not necessarily differentiable,…
Gradient descent for matrix factorization exhibits an implicit bias toward approximately low-rank solutions. While existing theories often assume the boundedness of iterates, empirically the bias persists even with unbounded sequences. This…