Fast arithmetic for triangular sets: From theory to practice

Li, Xin; Maza, Marc Moreno; Schost, Éric

doi:10.1016/j.jsc.2008.04.019

Cited by 36 publications

(66 citation statements)

References 15 publications

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“…Consequently, the parallel overhead is potentially large. A first attempt for parallelizing the serial normal form algorithms of [17] reached limited success as reported in [14].…”

Section: Efficient Parallel Computation Of Normal Formsmentioning

confidence: 99%

“…We use the serial C routines for 1-D FFT and 1-D TFT from the modpn library [16]. Our integer arithmetic modulo a prime number relies also on the efficient functions from modpn, in particular the improved Montgomery trick [18], presented in [17]. All our benchmarks are carried out on a 16-core machine with 16 GB memory and 4096 KB L2 cache.…”

Section: Experimentation Frameworkmentioning

confidence: 99%

“…This staircase phenomenon can be smoothened by the so-called Truncated Fourier Transform (TFT) [11]. In most practical cases, the TFT performs better in terms of running time and memory consumption than the radix-2 Cooley-Tukey Algorithm; see the experimentation reported in [17]. However, the TFT has its own practical limitations.…”

Section: Ffts Over Finite Fields and The Truncated Fourier Transformmentioning

confidence: 99%

“…This phenomenon is better controlled by an algorithm proposed by Li, Moreno Maza and Schost in [17]. This latter procedure relies on Cook-Sieveking-Kung's "fast division trick" [5,20,13] which reduces an Euclidean division to two multiplications.…”

Section: Normal Form Computationmentioning

confidence: 99%

“…This latter procedure relies on Cook-Sieveking-Kung's "fast division trick" [5,20,13] which reduces an Euclidean division to two multiplications. The algorithm of [17] proceeds by induction on the number of variables and its pseudocode is shown below. The base case, that is n = 1, is given by the procedure NormalForm 1 which is in fact Cook-Sieveking-Kung's fast division.…”

Section: Normal Form Computationmentioning

confidence: 99%

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FFT-Based Dense Polynomial Arithmetic on Multi-cores

Maza

Xie

2010

High Performance Computing Systems and Applications

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Abstract. We report efficient implementation techniques for FFT-based dense multivariate polynomial arithmetic over finite fields, targeting multi-cores. We have extended a preliminary study dedicated to polynomial multiplication and obtained a complete set of efficient parallel routines in Cilk++ for polynomial arithmetic such as normal form computation. Since bivariate multiplication applied to balanced data is a good kernel for these routines, we provide an in-depth study on the performance and the cut-off criteria of our different implementations for this operation. We also show that, not only optimized parallel multiplication can improve the performance of higher-level algorithms such as normal form computation but also this composition is necessary for parallel normal form computation to reach peak performance on a variety of problems that we have tested.

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“…Consequently, the parallel overhead is potentially large. A first attempt for parallelizing the serial normal form algorithms of [17] reached limited success as reported in [14].…”

Section: Efficient Parallel Computation Of Normal Formsmentioning

confidence: 99%