Factoring Sparse Bivariate Polynomials Using the Priority Queue

Abstract

Abstract. We revisit the polytope method for factoring sparse bivariate polynomials over finite fields, and address the bottleneck arising from solving the Hensel lifting equations using the sparse distributed polynomial representation. We revise the analysis when polynomials are represented as such, which reveals how performing the polynomial multiplications and ensuing additions in separate (serialised) phases causes the Hensel lifting phase to suffer from poor work, space, and I/O complexity, and hinges on the size of the intermediary output, as size is defined in the sparse distributed representation. We propose to overlap all polynomial arithmetic in one Hensel lifting step using a MAX priority queue. The overlapping approach adapts not only to the growth in the degree of the input polynomial but also to irregularities in the sparsity of intermediary output. It also results in evading expression swell and reducing the overall work and space complexity by an order of magnitude. When the priority queue is implemented as a cache-oblivious data structure, the overlapping approach achieves an order of magnitude improvement in I/O over the serialised approach, even when the latter is using cache efficient structures to assist in polynomial multiplications and additions. We present empirical results for the polytope method using a max-heap implementation of the global priority queue, which demonstrate extremely superior performance, and specifically against Magma, for sufficiently sparse input polynomials of very high degrees.

Cache oblivious sparse polynomial factoring using the funnel heap

“…More specifically and with regard s to the latter algorithm, we will denote by PQ-HL B the version that uses Binary Heap as a priority queue, and by PQ-HL F the version that uses Funnel Heap instead. Our experiments in Abu- Salem et al (2014Salem et al ( , 2015 demonstrate that the polytope method is now able to adapt significantly more efficiently to sparse input when its Newton polygon consists of a few edges, something not to have been observed when employing SER-HL.…”

“…In (Abu- Salem et al, 2014) we revised the analysis associated with the bottleneck in computation arising in Eq. (1), using the sparse distributed representation.…”

“…In the overlapping approach, one handles all arithmetic simultaneously using a single Max priority queue. In (Abu- Salem et al, 2014), we analysed the work, space, and cache complexity, when polynomials are in sparse distributed representation. We derived that the performance of the serialised version in all three metrics is critically affected not only by the degree of the input polynomial, but also by the following factors: (i) the sparsity of each polynomial multiplication, and (ii) the sparsity of the resulting polynomial products to be merged into a final summand.…”

“…In this section we gather further insight into the bottleneck associated with the irregularity in computations as a result of the varying density of the intermediary output arising during Hensel lifting. In (Abu- Salem et al, 2014), we observed that if space is not a concern and one is willing to store all polynomial products during each Hensel lifting step before their final merge into S k , the only cache oblivious competitor at scale for the global priority queue approach would be the following scenario. One would perform each polynomial multplication separately using a dedicated (local) priority queue, followed by merging of all the polynomial products using the static and cache oblivious k-merger of (Frigo et al, 1999).…”

“…Proof. From(Abu-Salem et al, 2014), Alg. PQ-HL F requires the following costs:Reductions in space borne by FH-HL are obvious by comparing θ kḡ…”