Related papers: Computing $\varphi(N)$ for an RSA module with a si…

We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-$2$ polynomial threshold function. Given a degree-2 input polynomial $p(x_1,\dots,x_n)$ and a parameter $\eps >…

In this paper, we consider the ``Shortest Superstring Problem''(SSP) or the ``Shortest Common Superstring Problem''(SCS). The problem is as follows. For a positive integer $n$, a sequence of n strings $S=(s^1,\dots,s^n)$ is given. We should…

We report a proof-of-concept demonstration of a quantum order-finding algorithm for factoring the integer 21. Our demonstration involves the use of a compiled version of the quantum phase estimation routine, and builds upon a previous…

Many variants of RSA cryptosystem exist in the literature. One of them is RSA over polynomials based on Galois approach. In standard RSA modulus is product of two large primes whereas in the Galois approach author considered the modulus as…

We present a new FPTAS for the Subset Sum Ratio problem, which, given a set of integers, asks for two disjoint subsets such that the ratio of their sums is as close to $1$ as possible. Our scheme makes use of exact and approximate…

We exhibit a randomized algorithm which given a matrix $A\in \mathbb{C}^{n\times n}$ with $\|A\|\le 1$ and $\delta>0$, computes with high probability an invertible $V$ and diagonal $D$ such that $\|A-VDV^{-1}\|\le \delta$ using…

An algorithm is devised for computing $\Phi(n) = \phi(1) + \phi(2) + \cdots + \phi(n)$ in time $\widetilde{\Theta}(n^{2/3})$ and space $\widetilde{\Theta}(n^{1/3})$. The starting point is an existing algorithm based on the Dirichlet…

We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. The algorithms use the \emph{black-box} model which is suitable for sparse and structured matrices. The algorithms depend on a number of…

The algorithm and complexity of approximating the permanent of a matrix is an extensively studied topic. Recently, its connection with quantum supremacy and more specifically BosonSampling draws special attention to the average-case…

We prove lower bounds on the error probability of a quantum algorithm for searching through an unordered list of N items, as a function of the number T of queries it makes. In particular, if T=O(sqrt{N}) then the error is lower bounded by a…

We study the complexity of estimating the partition function $\mathsf{Z}(\beta)=\sum_{x\in\chi} e^{-\beta H(x)}$ for a Gibbs distribution characterized by the Hamiltonian $H(x)$. We provide a simple and natural lower bound for quantum…

Many combinatorial problems involve determining whether a universe of $n$ elements contains a witness consisting of $k$ elements which have some specified property. In this paper we investigate the relationship between the decision and…

This paper describes an algorithm (thus far referred to as the "Dragonfly Algorithm") by which the subset-sum problem can be solved in $O(n^{11}\log(n))$ time complexity. The paper will first detail the generalized "product-derivative"…

Integer factorization is a fundamental problem in algorithmic number theory and computer science. It is considered as a one way or trapdoor function in the (RSA) cryptosystem. To date, from elementary trial division to sophisticated methods…

We present a polynomial time algorithm, which solves a nonstandard Variation of the well-known PARTITION-problem: Given positive integers $n, k$ and $t$ such that $t \geq n$ and $k \cdot t = {n+1 \choose 2}$, the algorithm partitions the…

We say that a function is rare-case hard against a given class of algorithms (the adversary) if all algorithms in the class can compute the function only on an $o(1)$-fraction of instances of size $n$ for large enough $n$. Starting from any…

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this…

Given $\kappa$ databases of unstructured entries, we propose a quantum algorithm to find the common entries between those databases. The proposed algorithm requires $\mathcal{O}(\kappa \sqrt{N})$ queries to find the common entries, where…

In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, the algorithm can only compare pairs of…

Analyzing large sparse electrical networks is a fundamental task in physics, electrical engineering and computer science. We propose two classes of quantum algorithms for this task. The first class is based on solving linear systems, and…