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SIMDoriented fast Mersenne twister: A 128bit pseudorandom number generator
 and QuasiMonte Carlo Methods 2006
, 2007
"... Summary. Mersenne Twister (MT) is a widelyused fast pseudorandom number generator (PRNG) with a long period of 2 19937 − 1, designed 10 years ago based on 32bit operations. In this decade, CPUs for personal computers have acquired new features, such as Single Instruction Multiple Data (SIMD) opera ..."
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Cited by 46 (4 self)
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Summary. Mersenne Twister (MT) is a widelyused fast pseudorandom number generator (PRNG) with a long period of 2 19937 − 1, designed 10 years ago based on 32bit operations. In this decade, CPUs for personal computers have acquired new features, such as Single Instruction Multiple Data (SIMD) operations (i.e., 128bit operations) and multistage pipelines. Here we propose a 128bit based PRNG, named SIMDoriented Fast Mersenne Twister (SFMT), which is analogous to MT but making full use of these features. Its recursion fits pipeline processing better than MT, and it is roughly twice as fast as optimised MT using SIMD operations. Moreover, the dimension of equidistribution of SFMT is better than MT. We also introduce a blockgeneration function, which fills an array of 32bit integers in one call. It speeds up the generation by a factor of two. A speed comparison with other modern generators, such as multiplicative recursive generators, shows an advantage of SFMT. The implemented Ccodes are downloadable from
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 Primitive Trinomial of Degree 6972593
 Mathematics of Computation
, 2003
"... We describe a search for primitive trinomials of degree 6972593 over GF(2). The only primitive trinomials found were x + 1 and its reciprocal. This completes the search for primitive trinomials whose degree is a Mersenne exponent less than ten million. ..."
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Cited by 16 (10 self)
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We describe a search for primitive trinomials of degree 6972593 over GF(2). The only primitive trinomials found were x + 1 and its reciprocal. This completes the search for primitive trinomials whose degree is a Mersenne exponent less than ten million.
Optimal irreducible polynomials for GF(2 m ) arithmetic. Cryptology ePrint Archive
, 2007
"... Abstract. The irreducible polynomials recommended for use by multiple standards documents are in fact far from optimal on many platforms. Specifically they are suboptimal in terms of performance, for the computation of field square roots and in the application of the “almost inverse” field inversion ..."
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Cited by 9 (3 self)
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Abstract. The irreducible polynomials recommended for use by multiple standards documents are in fact far from optimal on many platforms. Specifically they are suboptimal in terms of performance, for the computation of field square roots and in the application of the “almost inverse” field inversion algorithm. In this paper we question the need for the standardisation of irreducible polynomials in the first place, and derive the “best ” polynomials to use depending on the underlying processor architecture. Surprisingly it turns out that a trinomial polynomial is in many cases not necessarily the best choice. Finally we make some specific recommendations for some particular types of architecture.
An application of finite field: Design and implementation of 128bit instructionbased fast pseudorandom number generator
, 2007
"... (1) SIMDoriented Mersenne Twister (SFMT) is a new pseudorandom number generator (PRNG) which uses 128bit Single Instruction Multiple Data (SIMD) operations. SFMT is designed and implemented on C language with SIMD extensions and also implemented on standard C without SIMD. (2) Properties of SFMT a ..."
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Cited by 8 (0 self)
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(1) SIMDoriented Mersenne Twister (SFMT) is a new pseudorandom number generator (PRNG) which uses 128bit Single Instruction Multiple Data (SIMD) operations. SFMT is designed and implemented on C language with SIMD extensions and also implemented on standard C without SIMD. (2) Properties of SFMT are studied by using finite field theories, and they are shown to be equal or better than Mersenne Twister (MT), which is a widely used PRNG. (3) Generation speed of SFMT is measured on Intel Pentium M, Pentium IV, AMD Athlon 64 and PowerPC G4. It is shown to be about two times faster than MT implemented using SIMD. 1
Redundant trinomials for finite fields of characteristic 2
 Proceedings of ACISP 05, LNCS 3574
, 2005
"... Abstract. In this paper we introduce socalled redundant trinomials to represent elements of nite elds of characteristic 2. The concept is in fact similar to almost irreducible trinomials introduced by Brent and Zimmermann in the context of random numbers generators in [BZ 2003]. See also [BZ]. In f ..."
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Abstract. In this paper we introduce socalled redundant trinomials to represent elements of nite elds of characteristic 2. The concept is in fact similar to almost irreducible trinomials introduced by Brent and Zimmermann in the context of random numbers generators in [BZ 2003]. See also [BZ]. In fact, Blake et al. [BGL 1994, BGL 1996] and Tromp et al. [TZZ 1997] explored also similar ideas some years ago. However redundant trinomials have been discovered independently and this paper develops applications to cryptography, especially based on elliptic curves. After recalling well known techniques to perform e cient arithmetic in extensions of F2, we describe redundant trinomial bases and discuss how to implement them e ciently. They are well suited to build F2n when no irreducible trinomial of degree n exists. Depending on n ∈ [2, 10, 000] tests with NTL show that improvements for squaring and exponentiation are respectively up to 45 % and 25%. More attention is given to relevant extension degrees for doing elliptic and hyperelliptic curve cryptography. For this range, a scalar multiplication can be speeded up by a factor up to 15%. 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.
Almost Irreducible and Almost Primitive Trinomials
 in Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams, Fields Institute
, 2003
"... Consider polynomials over GF(2). We de ne almost irreducible and almost primitive polynomials, explain why they are useful, and give some examples and conjectures relating to them. 2 ..."
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Cited by 4 (2 self)
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Consider polynomials over GF(2). We de ne almost irreducible and almost primitive polynomials, explain why they are useful, and give some examples and conjectures relating to them. 2
Divisibility of Trinomials by Irreducible Polynomials over F2
"... Irreducible trinomials of given degree n over F2 do not always exist and in the cases that there is no irreducible trinomial of degree n it may be effective to use trinomials with an irreducible factor of degree n. In this paper we consider some conditions under which irreducible polynomials divide ..."
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Irreducible trinomials of given degree n over F2 do not always exist and in the cases that there is no irreducible trinomial of degree n it may be effective to use trinomials with an irreducible factor of degree n. In this paper we consider some conditions under which irreducible polynomials divide trinomials over F2. A condition for divisibility of selfreciprocal trinomials by irreducible polynomials over F2 is established. And we extend Welch’s criterion for testing if an irreducible polynomial divides trinomials x m + x s + 1 to the trinomials x am + x bs + 1.
Finding Low Weight Polynomial Multiples Using Lattices
"... Abstract. The low weight polynomial multiple problem arises in the context of stream ciphers cryptanalysis and of efficient finite field arithmetic, and is believed to be difficult. It can be formulated as follows: given a polynomial f ∈ F2[X] of degree d, and a bound n, the task is to find a low we ..."
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Abstract. The low weight polynomial multiple problem arises in the context of stream ciphers cryptanalysis and of efficient finite field arithmetic, and is believed to be difficult. It can be formulated as follows: given a polynomial f ∈ F2[X] of degree d, and a bound n, the task is to find a low weight multiple of f of degree at most n. The best algorithm known so far to solve this problem is based on a time memory tradeoff and runs in time O(n ⌈(w−1)/2 ⌉ ) using O(n ⌈(w−1)/4 ⌉ ) of memory, where w is the estimated minimal weight. In this paper, we propose a new technique to find low weight multiples using lattice basis reduction. Our algorithm runs in time O(n 6) and uses O(nd) of memory. This improves the space needed and gives a better theoretical time estimate when w ≥ 12. Such a situation is plausible when the bound n, which represents the available keystream, is small. We run our experiments using the NTL library on some known polynomials in cryptanalysis and we confirm our analysis.