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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 20 (14 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 14 (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.
Fast and reliable random number generators for scientific computing, Lecture
 Proc. PARA'04 Workshop on the StateoftheArt inScientific Computing
"... Abstract. 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 exce ..."
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Cited by 6 (2 self)
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Abstract. 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. 1
From Mersenne Primes to Random Number Generators ∗
, 2006
"... ∗ Advanced Computation seminar, ANU. Copyright c○2006, the author. AdvCom1t Fast and reliable pseudorandom number generators are required for simulation and other applications in Scientific Computing. Because of Moore’s law, random number generators that were satisfactory in the past may be inadequ ..."
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∗ Advanced Computation seminar, ANU. Copyright c○2006, the author. AdvCom1t Fast and reliable pseudorandom number generators are required for simulation and other applications in Scientific Computing. Because of Moore’s law, random number generators that were satisfactory in the past may be inadequate today. We outline some requirements for good uniform random number generators, and describe a class of generators having very fast vector/parallel implementations. These generators are based on primitive or almost primitive polynomials, and the degrees of the polynomials correspond to the exponents of certain Mersenne primes. We consider how to combine two generators to give a generator with better statistical and/or cryptographic properties, and also discuss the problem of initialization. We also mention some new “xorshift ” generators. 2
CORRIGENDA TO “NEW PRIMITIVE tNOMIALS (t =3, 5) OVER GF (2) WHOSE DEGREE IS A MERSENNE EXPONENT,” AND SOME NEW PRIMITIVE PENTANOMIALS
"... Abstract. We report an error in our previous paper [2], where we announced that we listed all the primitive trinomials over GF (2) of degree 859433, but there is a bug in the sieve. We missed the primitive trinomial X 859433 + X 170340 + 1 and its reciprocal, as pointed out by Richard Brent et al. W ..."
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Abstract. We report an error in our previous paper [2], where we announced that we listed all the primitive trinomials over GF (2) of degree 859433, but there is a bug in the sieve. We missed the primitive trinomial X 859433 + X 170340 + 1 and its reciprocal, as pointed out by Richard Brent et al. We also report some new primitive pentanomials. 1. Corrigendum In [2, Table 1], we claimed that all primitive trinomials with degree 859433 (32nd Mersenne exponent) over GF (2) are X 859433 + X 288477 +1 and its reciprocal, but there was a bug in a code for the sieve. Richard Brent et al. [1] pointed out that there are two more primitive trinomials of this degree: X 859433 + X 170340 +1 and its reciprocal, through their complete search for the primitive trinomials of degrees 756839, 859433, and 3021377 [1]. (The primitivity of the above trinomial was confirmed by our corrected code, too.) Their current search is shown on the website
ANC (ALGORITHMS, NUMBERS, COMPUTERS) ANUINRIA ASSOCIATE TEAM PROPOSAL
, 2007
"... Abstract. We propose to join the research efforts of Richard Brent’s team at ANU (Australian National University, Canberra, Australia) with those of the CACAO team from the “Centre de Recherche INRIA NancyGrand Est ” (Nancy, France) into an INRIA associate team called ANC (Algorithms, Numbers, Comp ..."
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Abstract. We propose to join the research efforts of Richard Brent’s team at ANU (Australian National University, Canberra, Australia) with those of the CACAO team from the “Centre de Recherche INRIA NancyGrand Est ” (Nancy, France) into an INRIA associate team called ANC (Algorithms, Numbers, Computers). We wish to extend in such a way a longterm and successful cooperation between Richard Brent and Paul Zimmermann to all members of both teams, in particular young researchers. The support of the ANC associate team will allow to reinforce that cooperation, especially in the common scientific projects outlined in this document.
* Requirements for RNGs
, 2006
"... Abstract Fast and reliable pseudorandom number generators are required for simulation and other applications in Scientific Computing. Because of Moore's law, random number generators that were satisfactory in the past may be inadequate today. We outline some requirements for good uniform rando ..."
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Abstract Fast and reliable pseudorandom number generators are required for simulation and other applications in Scientific Computing. Because of Moore's law, random number generators that were satisfactory in the past may be inadequate today. We outline some requirements for good uniform random number generators, and describe a class of generators having very fast vector/parallel implementations. These generators are based on primitive or almost primitive polynomials, and the degrees of the polynomials correspond to the exponents of certain Mersenne primes. We consider how to combine two generators to give a generator with better statistical and/or cryptographic properties, and also discuss the problem of initialization. We also mention some new &quot;xorshift &quot; generators.
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.