Related papers: Faster integer and polynomial multiplication using…
The $N$th power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in~$N$. When Fast Fourier Transform (FFT) is available, the resulting complexity is…
We present efficient algorithms for counting points on a smooth plane quartic curve $X$ modulo a prime $p$. We address both the case where $X$ is defined over $\mathbb F_p$ and the case where $X$ is defined over $\mathbb Q$ and $p$ is a…
We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two $n \cross n$ matrices over finite fields. In particular we obtain the following results: 1. We show that the number of…
We show that any $n$-variate polynomial computable by a syntactically multilinear circuit of size $\operatorname{poly}(n)$ can be computed by a depth-$4$ syntactically multilinear ($\Sigma\Pi\Sigma\Pi$) circuit of size at most…
For $0\le k\le n$, write $\binom nk=uv$ where the primes dividing $u$ are at most $k$ and the primes dividing $v$ exceed $k$, and let $f(n)$ be the least $k$ with $u>n^2$; Erd\H{o}s problem 684 asks for bounds on $f(n)$. We resolve the…
Let p be prime and Zpn the degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast algorithms for common operations in Zpn modulo p^N. Combining existing methods with recent work…
We show that all perfect odd integer squares not divisible by 3, can be usefully written as sqrt(N) = a + 18p, where the constant a is determined by the basic properties of N. The equation can be solved deterministically by an efficient…
We present optimized quantum circuits for GF$(2^m)$ multiplication and division operations, which are essential computing primitives in various quantum algorithms. Our ancilla-free GF multiplication circuit has the gate count complexity of…
We address the question of computing one selected term of an algebraic power series. In characteristic zero, the best algorithm currently known for computing the $N$th coefficient of an algebraic series uses differential equations and has…
In this paper we construct quantum algorithms for matrix products over several algebraic structures called semirings, including the (max,min)-matrix product, the distance matrix product and the Boolean matrix product. In particular, we…
Polynomial-time quantum Turing machines are provably superior to their classical counterparts within a common space bound in $o(\log \log n)$. For $\Omega(\log \log n)$ space, the only known quantum advantage result has been the fact…
This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial…
Decimal multiplication is the task of multiplying two numbers in base $10^N.$ Specifically, we focus on the number-theoretic transform (NTT) family of algorithms. Using only portable techniques, we achieve a 3x-5x speedup over the mpdecimal…
We study the problem of multiway number partition optimization, which has a myriad of applications in the decision, learning and optimization literature. Even though the original multiway partitioning problem is NP-hard and requires…
In this paper, two approximation algorithms are given. Let N be an odd composite number. The algorithms give new directions regarding primality test of given N. The first algorithm is given using a new method called digital coding method.…
We develop a fast algorithm for computing the bound of an Ore polynomial over a skew field, under mild conditions. As an application, we state a criterion for deciding whether a bounded Ore polynomial is irreducible, and we discuss a…
Hittmeir recently presented a deterministic algorithm that provably computes the prime factorisation of a positive integer $N$ in $N^{2/9+o(1)}$ bit operations. Prior to this breakthrough, the best known complexity bound for this problem…
We show how to compute the edit distance between two strings of length n up to a factor of 2^{\~O(sqrt(log n))} in n^(1+o(1)) time. This is the first sub-polynomial approximation algorithm for this problem that runs in near-linear time,…
We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of nonrecursive programs.…
A method for computing the n'th decimal digit of pi in O(n^3 log(n)^3) time and with very little memory is presented here. The computation is based on the recently discovered Bailey-Borwein-Plouffe algorithm and the use of a new algorithm…