arXiv subjects

Tsz-Wo Sze

Publications and source records attributed to Tsz-Wo Sze.

5 recordsLinked to original sources

arXiv2019

A Fast Self-correcting $π$ Algorithm

We have rediscovered a simple algorithm to compute the mathematical constant \[ π=3.14159265\cdots. \] The algorithm had been known for a long time but it might not be recognized as a fast, practical algorithm. The time complexity of it can be proved to be \[ O(M(n)\log^2 n) \] bit operations for computing $π$ with error $O(2^{-n})$, where $M(n)$ is the time complexity to multiply two $n$-bit integers. We conjecture that the algorithm actually runs in \[ O(M(n)\log n). \] The algorithm is \emph{self-correcting} in the sense that, given an approximated value of $π$ as an input, it can compute a more accurate approximation of $π$ with cubic convergence.

math.NT↗

arXiv2011

Deterministic Primality Proving on Proth Numbers

We present an algorithm to decide the primality of Proth numbers, N=2^e t+1, without assuming any unproven hypothesis. The expected running time and the worst case running time of the algorithm are O ((t log t + log N)log N) and O ((t log t + log N) log^2 N) bit operations, respectively.

math.NT↗

arXiv2011

On Taking r-th Roots without r-th Nonresidues over Finite Fields and Its Applications

We first show a deterministic algorithm for taking $r$-th roots over $\F_q$ without being given any $r$-th nonresidue, where $\F_q$ is a finite field with $q$ elements and $r$ is a small prime such that $r^2$ divides of $q-1$. As applications, we illustrate deterministic algorithms over $\F_q$ for constructing $r$-th nonresidues, constructing primitive elements, solving polynomial equations and computing elliptic curve "$n$-th roots", and a deterministic primality test for the generalized Proth numbers. All algorithms are proved without assuming any unproven hypothesis. They are efficient only if all the factors of $q-1$ are small and some primitive roots of unity can be constructed efficiently over $\F_q$. In some cases, they are the fastest among the known deterministic algorithms.

math.NT↗

arXiv2010

The Two Quadrillionth Bit of Pi is 0! Distributed Computation of Pi with Apache Hadoop

We present a new record on computing specific bits of Pi, the mathematical constant, and discuss performing such computations on Apache Hadoop clusters. The specific bits represented in hexadecimal are 0E6C1294 AED40403 F56D2D76 4026265B CA98511D 0FCFFAA1 0F4D28B1 BB5392B8. These 256 bits end at the 2,000,000,000,000,252nd bit position, which doubles the position and quadruples the precision of the previous known record. The position of the first bit is 1,999,999,999,999,997 and the value of the two quadrillionth bit is 0. The computation is carried out by a MapReduce program called DistBbp. To effectively utilize available cluster resources without monopolizing the whole cluster, we develop an elastic computation framework that automatically schedules computation slices, each a DistBbp job, as either map-side or reduce-side computation based on changing cluster load condition. We have calculated Pi at varying bit positions and precisions, and one of the largest computations took 23 days of wall clock time and 503 years of CPU time on a 1000-node cluster.

cs.DC↗

arXiv2010

On Taking Square Roots without Quadratic Nonresidues over Finite Fields

We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in $\tilde{O}(\log^2 q)$ bit operations over finite fields with $q$ elements. As an application, we construct a deterministic primality proving algorithm, which runs in $\tilde{O}(\log^3 N)$ for some integers $N$.

math.NT↗