Fast multipoint evaluation and interpolation of polynomials in the LCH-basis over FPr

Abstract

Lin, Chung and Han introduced in 2014 the LCH-basis in order to derive FFT-based multipoint evaluation and interpolation algorithms with respect to this polynomial basis. Considering an affine space of n = 2j points, their algorithms require O(n · log2 n) operations in F2r. The LCH-basis has then been extended over finite fields of characteristic p by Lin et al. in 2016 and an n-point evaluation algorithm has been derived for n = pj with complexity O(n · logp n · p). However, the problem of interpolating polynomials represented in such a basis over FPr has not been addressed. In this paper, we fill this gap and also derive a faster algorithm for evaluating polynomials in the LCH-basis at multiple points over FPr. We follow a different approach where we represent the multipoint evaluation and interpolation maps by well-defined matrices. We present factorizations of such matrices into the product of sparse matrices which can be evaluated efficiently. These factorizations lead to fast algorithms for both the multipoint evaluation and the interpolation of polynomials represented in the LCH-basis at n = pj points with optimized complexity O(n · log2 n · log2 p · log2 log2 p). A particular attention is paid to provide in-place algorithms with high memory-locality. Our implementations written in C confirm that our approach improves the original transforms.