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A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377
 Math. Comp
, 2003
"... Abstract. The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r/2+O(1) bits of memory and significantly fewer memory references and bitoperations than the standard algorithm. ..."
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Cited by 24 (15 self)
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Abstract. The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r/2+O(1) bits of memory and significantly fewer memory references and bitoperations than the standard algorithm. If 2 r − 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give primitive trinomials for the Mersenne exponents r = 756839, 859433, and 3021377. The results for r = 859433 extend and correct some computations of Kumada et al. The two results for r = 3021377 are primitive trinomials of the highest known degree. 1.
Random Number Generators with Period Divisible by a Mersenne Prime
 Proc. ICCSA 2003
, 2003
"... Pseudorandom numbers with long periods and good statistical properties are often required for applications in computational finance. We consider the requirements for good uniform random number generators, and describe a class of generators whose period is a Mersenne prime or a small multiple of ..."
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Cited by 16 (5 self)
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Pseudorandom numbers with long periods and good statistical properties are often required for applications in computational finance. We consider the requirements for good uniform random number generators, and describe a class of generators whose period is a Mersenne prime or a small multiple of a Mersenne prime. These generators are based on "almost primitive" trinomials, that is trinomials having a large primitive factor. They enable very fast vector/parallel implementations with excellent statistical properties.
A fast algorithm for testing irreducibility of trinomials mod 2
 pub199.html
, 2000
"... The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r+O(1) bits of memory and Θ(r 2) bitoperations. We describe an algorithm which requires only 3r/2 + O(1) bits of memory and significantly fewer bitoperations than the standard algorithm. Using the ..."
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Cited by 9 (7 self)
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The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r+O(1) bits of memory and Θ(r 2) bitoperations. We describe an algorithm which requires only 3r/2 + O(1) bits of memory and significantly fewer bitoperations than the standard algorithm. Using the algorithm, we have found 18 new irreducible trinomials of degree r in the range 100151 ≤ r ≤ 700057. If r is a Mersenne exponent (i.e. 2 r −1 is a Mersenne prime), then an irreducible trinomial is primitive. Primitive trinomials are of interest because they can be used to give pseudorandom number generators with period at least 2 r − 1. We give examples of primitive trinomials for r = 756839, 859433, and 3021377. The three results for r = 756839 are new. The results for r = 859433 extend and correct some computations of Kumada et al. [Math. Comp. 69 (2000), 811–814]. The two results for r = 3021377 are primitive trinomials of the highest known degree. 1 Copyright c○2000, the authors. rpb199tr typeset using L ATEX 1 1
An Area Time Efficient Field Programmable Mersenne Twister Uniform Random Number Generator
 In Proc of International Conference on Engineering of Reconfigurabe Systems and Algorithms
, 2006
"... Reconfigurable computing offers an attractive solution to accelerating infrared scene simulations. In infrared scene simulations, the modeling of a number of atmospheric and optical phenomena like scintillation, refraction, blurring due to lens optics and photon noise may be implemented in parallel. ..."
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Cited by 8 (3 self)
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Reconfigurable computing offers an attractive solution to accelerating infrared scene simulations. In infrared scene simulations, the modeling of a number of atmospheric and optical phenomena like scintillation, refraction, blurring due to lens optics and photon noise may be implemented in parallel. All of these require simultaneous and continual generation of random numbers. Furthermore, random number generation is only a small component of all of these algorithms. Current software random number generators are too slow whilst current hardware random number generators are plagued by issues such correlations and are not area efficient. We describe a reconfigurable computing based uniform random number generator based on the mersenne twister algorithm that is area time efficient and that does not suffer from correlations. 1.
Fast and reliable random number generators for scientific computing
 PROC. PARA'04 WORKSHOP ON THE STATEOFTHEART INSCIENTIFIC COMPUTING
, 2004
"... Fast and reliable pseudorandom number generators are required for simulation and other applications in Scientific Computing. We outline the requirements for good uniform random number generators, and describe a class of generators having very fast vector/parallel implementations with excellent sta ..."
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Cited by 6 (2 self)
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Fast and reliable pseudorandom number generators are required for simulation and other applications in Scientific Computing. We outline the requirements for good uniform random number generators, and describe a class of generators having very fast vector/parallel implementations with excellent statistical properties. We also discuss the problem of initialising random number generators, and consider how to combine two or more generators to give a better (though usually slower) generator.
Random Number Generation on
 SupercomputersExtended Abstract , Europar98
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Uniform random number generators and primitive trinomials
, 2001
"... In the first part of this talk, we consider the requirements for uniform pseudorandom number generators in largescale simulations. We describe a class of random number generators which have good statistical properties and can be implemented efficiently on vector/parallel computers. To obtain new ..."
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In the first part of this talk, we consider the requirements for uniform pseudorandom number generators in largescale simulations. We describe a class of random number generators which have good statistical properties and can be implemented efficiently on vector/parallel computers. To obtain new generators in this class we need primitive trinomials of high degree. In the second part of the talk (joint work with Samuli Larvala and Paul Zimmermann), we consider the problem of testing trinomials over GF(2) for reducibility. We describe a new algorithm for testing primitivity of trinomials whose degree is a Mersenne exponent. The algorithm has been used to find primitive trinomials of degree 3021377 (the highest previously known was 859433). The corresponding uniform random number generators have extremely long period and good statistical properties in all dimensions less than 3021377.