## Parallel Linear Congruential Generators With Prime Moduli (1997)

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Venue: | Parallel Computing |

Citations: | 20 - 6 self |

### BibTeX

@ARTICLE{Mascagni97parallellinear,

author = {Michael Mascagni},

title = {Parallel Linear Congruential Generators With Prime Moduli},

journal = {Parallel Computing},

year = {1997},

volume = {24},

pages = {923--936}

}

### Years of Citing Articles

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### Abstract

. Linear congruential generators (LCGs) remain the most popular method of pseudorandom number generation on digital computers. Ease of implementation has favored implementing LCGs with power-of-two moduli. However, prime modulus LCGs are superior in quality to power-of-two modulus LCGs, and the use of a Mersenne prime minimizes the computational cost of generation. When implemented for parallel computation, quality becomes an even more compelling issue. We use a full-period exponential sum as the measure of stream independence and present a method for producing provably independent streams of LCGs in parallel by utilizing an explicit parameterization of all of the primitive elements modulo a given prime. The minimization of this measure of independence further motivates an algorithm required in the explicit parameterization. We describe and analyze this algorithm and describe its use in a parallel LCG package. 1. Introduction. Perhaps the oldest generator still in use for the generati...

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Citation Context ...nd m, a, and b. We refer to the period of the sequence fx n g as Per(x n ). When m is prime, Per(x n ) = m \Gamma 1 is the longest period achievable, occurring when a is a primitive element modulo m, =-=[9]-=-. 1 With a primitive modulo m, any choice for b gives Per(x n ) = m \Gamma 1. For this reason it is customary to choose b = 0, since the set of m \Gamma 1 elements in the full period of (1) will conta... |

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Citation Context ...Gamma1 m\Gamma1 (k). 4. Exponential Sum Cross-correlations. A very common theoretical measure of the quality of a serial pseudorandom number generator is a metrical quantity known as the discrepancy, =-=[10, 14, 16, 17]-=-. The discrepancy of a sequence measures its equidistribution quantitatively by computing the maximal deviation of the given sequence from the uniform distribution. MICHAEL MASCAGNI 5 This equidistrib... |

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Citation Context ...Gamma1 m\Gamma1 (k). 4. Exponential Sum Cross-correlations. A very common theoretical measure of the quality of a serial pseudorandom number generator is a metrical quantity known as the discrepancy, =-=[10, 14, 16, 17]-=-. The discrepancy of a sequence measures its equidistribution quantitatively by computing the maximal deviation of the given sequence from the uniform distribution. MICHAEL MASCAGNI 5 This equidistrib... |

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Citation Context ...Gamma1 m\Gamma1 (k). 4. Exponential Sum Cross-correlations. A very common theoretical measure of the quality of a serial pseudorandom number generator is a metrical quantity known as the discrepancy, =-=[10, 14, 16, 17]-=-. The discrepancy of a sequence measures its equidistribution quantitatively by computing the maximal deviation of the given sequence from the uniform distribution. MICHAEL MASCAGNI 5 This equidistrib... |

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Citation Context ...ty" pseudorandom numbers in a computationally inexpensive and scalable manner. The difficulty of this problem can be seen by considering the example of Monte Carlo applied to a problem in neutron=-=ics, [21]-=-. Here independent neutron paths are generated based on the outcome of many events whose probabilities are understood. Statistics are collected along the paths, and computation produces estimates for ... |

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Citation Context ... is the linear congruential generator (LCG). This generator is sometimes called the "Lehmer" generator, in honor of its originator, D. H. Lehmer, the father of electronic computational numbe=-=r theory, [13]-=-. The LCG is based on the following modular integer recursion for producing pseudorandom integers: (1) x n = ax n\Gamma1 + b (mod m): Equation (1) defines a sequence of integers modulo m starting with... |

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Citation Context ... LCG pseudorandom numbers. One drawback to this approach is that very little research into the quality consequences of splitting full-period cycles for parallel pseudorandom generation has been done, =-=[2, 12, 5, 6]-=-. Much less research has been done into these results when both splitting and parameterization are used together. MICHAEL MASCAGNI 13 Acknowledgments This work is supported by DARPA/ITO under order nu... |

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Citation Context ... a useful method for the implementation of LCGs on parallel machines is required. There has been some work on the splitting of full-period LCG sequences into shorter subsequences for use in parallel, =-=[7, 5]-=-. This paper takes an altogether different approach to parallelizing LCGs. We seek a parameterization of complete 1991 Mathematics Subject Classification. 65C10, 65Y05. Key words and phrases. pseudora... |

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Citation Context ...NTIAL GENERATORS and distinct full-period LCG sequences so that each new parallel process can use an entirely distinct full-period sequence. To our knowledge this has been examined by only one group, =-=[18]-=-, where the parameterization of power-of-two modulus LCGs was studied by varying the additive constant, b, in the recursion (1). In this paper we will study the consequences of parameterizing full-per... |

21 |
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Citation Context ... LCG pseudorandom numbers. One drawback to this approach is that very little research into the quality consequences of splitting full-period cycles for parallel pseudorandom generation has been done, =-=[2, 12, 5, 6]-=-. Much less research has been done into these results when both splitting and parameterization are used together. MICHAEL MASCAGNI 13 Acknowledgments This work is supported by DARPA/ITO under order nu... |

21 |
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Citation Context ...ct primitive elements that can serve as multipliers, so this method provides at most this number of full-period LCGs. Given m \Gamma 1 and its factorization one can compute OE(m \Gamma 1) explicitly, =-=[8]. Ho-=-wever, it is more generally known that OE(m \Gamma 1) �� m= log 2 log 2 (m). We have implemented this algorithm as part of a parallel linear congruential generation package in a portable manner us... |

17 |
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Citation Context ...uences that are then used on individual parallel processors, [5]. Recently, certain pseudorandom number generators have been parallelized using different seeds to select different full-period cycles, =-=[15, 19]-=-. This is a form of parameterization of the full-period cycle. Another form of parameterization has been used on LCGs with power-of-two moduli, [18]. Here a different additive constant was used to pro... |

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(Show Context)
Citation Context ...uences that are then used on individual parallel processors, [5]. Recently, certain pseudorandom number generators have been parallelized using different seeds to select different full-period cycles, =-=[15, 19]-=-. This is a form of parameterization of the full-period cycle. Another form of parameterization has been used on LCGs with power-of-two moduli, [18]. Here a different additive constant was used to pro... |

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Citation Context ... to compute m 1 (k). 4. Exponential Sum Cross-correlations. A very common theoretical measure of the quality of a serial pseudorandom number generator is a metrical quantity known as the discrepancy, =-=[10, 14, 16, 17]-=-. The discrepancy of a sequence measures its equidistribution quantitatively by computing the maximal deviation of the given sequence from the uniform distribution. 1MICHAEL MASCAGNI 5 This equidistr... |

6 |
Studies of random number generators for parallel processing
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Citation Context ... LCG pseudorandom numbers. One drawback to this approach is that very little research into the quality consequences of splitting full-period cycles for parallel pseudorandom generation has been done, =-=[2, 12, 5, 6]-=-. Much less research has been done into these results when both splitting and parameterization are used together. MICHAEL MASCAGNI 13 Acknowledgments This work is supported by DARPA/ITO under order nu... |

6 |
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Citation Context ... number relatively prime to m \Gamma 1. In the subsequent discussion on computing �� \Gamma1 m\Gamma1 (k) we take advantage of the previous algorithmic work for computing ��(x) for large value=-=s of x, [4, 11]-=-. In our application, computing the kth primitive element modulo the prime m, we need to compute the kth number relatively prime to OE(m) = m\Gamma1, when m is prime. We will assume that for the parti... |

5 |
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Citation Context ... number relatively prime to m \Gamma 1. In the subsequent discussion on computing �� \Gamma1 m\Gamma1 (k) we take advantage of the previous algorithmic work for computing ��(x) for large value=-=s of x, [4, 11]-=-. In our application, computing the kth primitive element modulo the prime m, we need to compute the kth number relatively prime to OE(m) = m\Gamma1, when m is prime. We will assume that for the parti... |

1 |
personal communication
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Citation Context ... modular multiplication can be implemented by performing the full integer multiplication with only the inclusion of bitwise shifting and integer addition required to accomplish the modular reduction, =-=[1]-=-. The reader will be convinced by considering the relationship bewteen modular redution modulo 2 p and 2 p \Gamma 1. Thus in the sequel we will focus on Mersenne prime modulus LCGs to minimize the cos... |