The complexity of sparse Hensel lifting and sparse polynomial factorization

Monagan, Michael; Tuncer, Baris

doi:10.1016/j.jsc.2019.05.001

Cited by 6 publications

(6 citation statements)

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“…Regarding polynomial factorization, 1,2 Monagan and Tuncer 4,17 reduce factorization of a multivariate polynomial…”

Section: Polynomial Factorizationmentioning

confidence: 99%

“…Regarding polynomial factorization, 1,2 Monagan and Tuncer 4,17 reduce factorization of a multivariate polynomial f in

ℤ [x_{1}, x_{2}, \dots, x_{n}]

to (i) evaluating

f (x_{1}, x_{2}, β^{t})

for 1 ≤ t ≤ T , (ii) doing a computation with the bivariate images, and (iii) recovering the factors of f using sparse interpolation techniques 15,16,18 . See Roche 19 for a recent discussion on sparse polynomial interpolation methods and an extensive bibliography.…”

Section: Modular Polynomial Evaluation In Two Variablesmentioning

confidence: 99%

“…Modern algorithms 4,5 for factorization and gcd computation rely on many polynomial evaluations, which dominate the overall computation cost. These polynomial evaluations have two main features.…”

Section: Introductionmentioning

confidence: 99%

“…We show that the optimized AVX version implementation of van der Hoeven et al 14 can safely be used in our polynomial evaluation, and we then propose the first (to our knowledge) implementation of such modular multiplication algorithm on AVX‐512, as well as the corresponding FP‐based modular addition. With respect to the reference polynomial evaluation implementation of Monagan and coworkers, 4,5 which relies on scalar integer‐based algorithms for modular operations, we detail the performance gains of our SIMD FP‐based modular arithmetic for modular operation microbenchmarks and for the polynomial evaluation. We carefully compare intrinsics and OpenMP compiler directives for programming SIMD modular arithmetic and for their integration within the polynomial evaluation. We detail the relevance, the advantages, the issues, and the performance results of both programming paradigms. We show how to significantly improve the performance of the modular polynomial evaluation by increasing the operational intensity via data reuse, and by filling the pipelines of the floating‐point units.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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High‐performance SIMD modular arithmetic for polynomial evaluation

Fortin

Fleury

Lemaire

et al. 2021

Concurrency and Computation

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Two essential problems in computer algebra, namely polynomial factorization and polynomial greatest common divisor computation, can be efficiently solved thanks to multiple polynomial evaluations in two variables using modular arithmetic. In this article, we focus on the efficient computation of such polynomial evaluations on one single CPU core. We first show how to leverage SIMD (single instruction, multiple data) computing for modular arithmetic on AVX2 and AVX-512 units, using both intrinsics and OpenMP compiler directives. Then we manage to increase the operational intensity and to exploit instruction-level parallelism in order to increase the compute efficiency of these polynomial evaluations. All this results in the end to performance gains up to about 5x on AVX2 and 10x on AVX-512.

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“…Regarding polynomial factorization, 1,2 Monagan and Tuncer 4,17 reduce factorization of a multivariate polynomial…”

Section: Polynomial Factorizationmentioning

confidence: 99%

“…Regarding polynomial factorization, 1,2 Monagan and Tuncer 4,17 reduce factorization of a multivariate polynomial f in