Related papers: Fast square-free decomposition of integers using c…
We introduce an efficient scheme for the construction of quadrature rules for bandlimited functions. While the scheme is predominantly based on well-known facts about prolate spheroidal wave functions of order zero, it has the asymptotic…
An \emph{indexing} of a finite set $S$ is a bijection $D : \{1,...,|S|\} \rightarrow S$. We present an indexing for the set of quadratic residues modulo $N$ that is decodable in polynomial time on the size of $N$, given the factorization of…
Symmetric tensor decomposition is an important problem with applications in several areas for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous…
In this paper we consider the problem of computing the longest common abelian factor (LCAF) between two given strings. We present a simple $O(\sigma~ n^2)$ time algorithm, where $n$ is the length of the strings and $\sigma$ is the alphabet…
We report a quantum-classical hybrid scheme for factorization of bi-prime numbers (which are odd and square-free) using IBM's quantum processors. The hybrid scheme proposed here involves both classical optimization techniques and adiabatic…
Fermat's well-known factorization algorithm is based on finding a representation of natural numbers $N$ as the difference of squares. In 1895, Lawrence generalized this idea and applied it to multiples $kN$ of the original number. A…
Stencil computations are widely used to simulate the change of state of physical systems across a multidimensional grid over multiple timesteps. The state-of-the-art techniques in this area fall into three groups: cache-aware tiled looping…
A square-root-free matrix QR decomposition (QRD) scheme was rederived in [1] based on [2] to simplify computations when solving least-squares (LS) problems on embedded systems. The scheme of [1] aims at eliminating both the square-root and…
We propose a decoder for the correction of erasures with hypergraph product codes, which form one of the most popular families of quantum LDPC codes. Our numerical simulations show that this decoder provides a close approximation of the…
We present an algorithm that computes the composition factors of the n-th tensor power of the free associative algebra on a vector space. The composition factors admit a description in terms of certain coefficients $c_{\lambda\mu}$…
In this work, we introduce bidirectional collision detection --- a new algorithmic tool that applies to the collision problems that arise in many isomorphism problems. For the group isomorphism problem, we show that bidirectional collision…
This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+\epsilon})$. Strassen's well known factoring approach is…
We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The algorithm…
We study a multiplicative function analogue of Linnik's problem on the least prime in an arithmetic progression. Let $h\colon \mathbb{N}\to\mathbb{R}\setminus\{0\}$ be a multiplicative function, and let $a \pmod q$ be a reduced residue…
In this paper we make a step towards a time and space efficient algorithm for the hidden shift problem for groups of the form $\mathbb{Z}_k^n$. We give a solution to the case when $k$ is a power of 2, which has polynomial running time in…
A polynomial-time algorithm is produced which, given generators for a group of permutations on a finite set, returns a direct product decomposition of the group into directly indecomposable subgroups. The process uses bilinear maps and…
We propose a novel algorithm for finding square roots modulo p. Although there exists a direct formula to calculate square root of an element modulo prime (3 mod 4), but calculating square root modulo prime (1 mod 4) is non trivial.…
This paper shows how to decode errors and erasures with Gabidulin codes in sub-quadratic time in the code length, improving previous algorithms which had at least quadratic complexity. The complexity reduction is achieved by accelerating…
We try to minimize the number of qubits needed to factor an integer of n bits using Shor's algorithm on a quantum computer. We introduce a circuit which uses 2n+3 qubits and O(n^3 lg(n)) elementary quantum gates in a depth of O(n^3) to…
The discriminant of a trinomial of the form $x^n \pm x^m \pm 1$ has the form $\pm n^n \pm (n-m)^{n-m} m^m$ if $n$ and $m$ are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various…