Abstract

Polynomial factorization is a classical algorithmic algebra problem with a wide range of applications. Of particular interest is factorization over finite fields, among which fields of order two are probably the most important ones when representing Boolean functions by Zhegalkin polynomials. In particular, factorization of Boolean polynomials corresponds to conjunctive decomposition of Boolean functions given in algebraic normal form. In addition, factorization enables decomposition of functions given in full disjunctive normal form (DNF) and positive DNF, as well as Cartesian decomposition of relational data. These applications demonstrate the importance of developing fast factorization algorithms. In this paper, we consider some recently proposed factorization algorithms of polynomial complexity and describe a parallel MIMD implementation that takes advantage of both task-level and data-level parallelism. We conduct some experiments on logic synthesis benchmarks and synthetic (random) polynomials to demonstrate significant factorization speedup. In conclusion, we discuss results of testing a parallel implementation of the algorithm on a massively parallel multicore architecture (REDEFINE).

中文翻译：

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布尔多项式的因式分解：并行算法和实验评估

摘要

多项式因式分解是具有广泛应用的经典算法代数问题。特别令人感兴趣的是有限域上的因式分解，其中用Zhegalkin多项式表示布尔函数时，二阶域可能是最重要的域。特别地，布尔多项式的因式分解对应于以代数范式形式给出的布尔函数的联合分解。此外，因式分解可以分解以全析取范式（DNF）和正DNF形式给出的函数，以及对关系数据进行笛卡尔分解。这些应用程序证明了开发快速分解算法的重要性。在本文中，我们考虑了一些最近提出的多项式复杂度的分解算法，并描述了一种利用任务级和数据级并行性的并行MIMD实现。我们在逻辑综合基准和综合（随机）多项式上进行了一些实验，以证明分解速度显着提高。总之，我们讨论了在大规模并行多核体系结构（REDEFINE）上测试算法的并行实现的结果。