A Randomized Sublinear Time Parallel GCD Algorithm for the EREW PRAM

We present a randomized parallel algorithm that computes the greatest common divisor of two integers of n bits in length with probability 1-o(1) that takes O(n loglog n / log n) expected time using n^{6+\epsilon} processors on the EREW PRAM parallel model of computation. We believe this to be the first randomized sublinear time algorithm on the EREW PRAM for this problem

Two Compact Incremental Prime Sieves

A prime sieve is an algorithm that finds the primes up to a bound $n$. We say that a prime sieve is incremental, if it can quickly determine if $n+1$ is prime after having found all primes up to $n$. We say a sieve is compact if it uses roughly $\sqrt{n}$ space or less. In this paper we present two new results: (1) We describe the rolling sieve, a practical, incremental prime sieve that takes $O(n\log\log n)$ time and $O(\sqrt{n}\log n)$ bits of space, and (2) We show how to modify the sieve of Atkin and Bernstein (2004) to obtain a sieve that is simultaneously sublinear, compact, and incremental. The second result solves an open problem given by Paul Pritchard in 1994

A distributed wheel sieve algorithm using Scheduling by Multiple Edge Reversal

Number of pages: 12This paper presents a new distributed approach for generating all prime numbers in a given interval of integers. From Eratosthenes, who elaborated the first prime sieve (more than 2000 years ago), to the current generation of parallel computers, which have permitted to reach larger bounds on the interval or to obtain previous results in a shorter time, prime numbers generation still represents an attractive domain of research and plays a central role in cryptography. We propose a fully distributed algorithm for finding all primes in the interval $[2\ldots, n]$, based on the \emph{wheel sieve} and the SMER (\emph{Scheduling by Multiple Edge Reversal}) multigraph dynamics. Given a multigraph $\mathcal{M}$ of arbitrary topology, having $N$ nodes, an SMER-driven system is defined by the number of directed edges (arcs) between any two nodes of $\mathcal{M}$, and by the global period length of all ''arc reversals'' in $\mathcal{M}$. The new prime number generation method inherits the distributed and parallel nature of SMER and requires at most $n + \lfloor \sqrt{n}\rfloor$ time steps

A Modular Integer GCD Algorithm

This paper describes the first algorithm to compute the greatest common divisor (GCD) of two n-bit integers using a modular representation for intermediate values U, V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A generalization of the binary GCD algorithm, in: ISSAC \u2793: International Symposium on Symbolic and Algebraic Computation, Kiev, Ukraine, 1993, pp. 111–116; K. Weber, The accelerated integer GCD algorithm, ACM Trans. Math. Softw. 21 (1995) 111–122] when U and V are close to the same size, that replaces U by (U-bV)/p, where p is one of the prime moduli and b is the unique integer in the interval (-p/2,p/2) such that b=UV ^-1(mod p) . When the algorithm is executed on a bit common CRCW PRAM with O(n log n log log log n) processors, it takes O(n) time in the worst case. A heuristic model of the average case yields O(n/log n) time on the same number of processors

Polylog Depth Circuits for Integer Factoring and Discrete Logarithms

AbstractIn this paper, we develop parallel algorithms for integer factoring and for computing discrete logarithms. In particular, we give polylog depth probabilistic boolean circuits of subexponential size for both of these problems, thereby solving an open problem of Adleman and Kompella. Existing sequential algorithms for integer factoring and discrete logarithms use a prime base which is the set of all primes up to a bound B. We use a much smaller value for B for our parallel algorithms than is typical for sequential algorithms. In particular, for inputs of length n, by setting B = nlogdn with d a positive constant, we construct •Probabilistic boolean circuits of depth (log) and size exp[(/log)] for completely factoring a positive integer with probability 1−(1), and •Probabilistic boolean circuits of depth (log + log) and size exp[(/log)] for computing discrete logarithms in the finite field () for a prime with probability 1−(1). These are the first results of this type for both problem

The pseudosquares prime sieve

Abstract. We present the pseudosquares prime sieve

Two Fast Parallel Prime Number Sieves

AbstractA prime number sieve is an algorithm that lists all prime numbers up to a given bound n. Two parallel prime number sieves for an algebraic EREW PRAM model of computation are presented and analyzed. The first sieve runs in O(log n) time using O(n/(log n log log n)) processors, and the second sieve runs in O(root n) time using O(root n) processors. The first sieve is optimal in the sense that it performs work O(n/log log n), which is within a constant factor of the number of arithmetic operations used by the fastest known sequential prime number sieves. However, when both sieves are analyzed on the Block PRAM model as defined by Aggarwal, Chandra, and Snir, it is found that the second sieve is more work-efficient when communication latency is significant