Fast multivariate multi-point evaluation revisited

Abstract

In 2008, Kedlaya and Umans designed the first multivariate multi-point evaluation algorithm over finite fields with an asymptotic complexity that can be made arbitrarily close to linear. However, it remains a major challenge to make their algorithm efficient for practical input sizes. In this paper, we revisit and improve their algorithm, while keeping this ultimate goal in mind. In addition we sharpen the known complexity bounds for modular composition of univariate polynomials over finite fields.N mod p i are handled recursively, for i = 1, …, s. Modular compositionLet us first discuss the standard modular composition problem when n= 1. Let f , g and h be polynomials in [x] of respective degrees <d, <d and d, with h monic. The naive modular composition algorithm takes O(d M(d)) operations in . In 1978, Brent and Kung [9] gave an algorithm with cost O(d 2 + d M(d)), which uses the baby-step giant-step technique [39]. Their algorithm even yields a sub-quadratic cost O(d + d M(d)) when using fast linear algebra; see [32, p. 185].

On the Complexity Exponent of Polynomial System Solving

“…Another major problem concerns the practical efficiency of algorithms for modular composition. This holds in particular for the recent fast algorithms for finite fields designed by Kedlaya and Umans [42], and revisited in [37]; we refer the reader to the concluding section of [37] for a quantitative discussion. Consequently we do not expect our Theorem 3.3 to be of practical use, and we cannot claim our new main complexity bounds for polynomial system solving to be relevant in practice.…”

“…We also handle fast evaluations of f in a larger algebra of the form p k[e, y, t] / (e 2 , y M , Q(e, y, t)), where M ⩾ 1 and Q is monic in t of degree d n : this is a particular case of composition modulo triangular sets studied in [63], for which we also improve upon the dependency in the coefficient size. These new results are based on our recent advances on multivariate multi-point evaluation algorithms [37]. At a more abstract level, our algorithm applies to any field over which fast multi-point multivariate polynomial evaluation exists.…”

“…For the Kronecker solver presented in this paper, the following particular instances of are needed: p k[y] / (y M ) and ℤ / p ℤ, along with tangent numbers over theses rings. In this section, we design fast algorithms for these cases, on the basis of Kedlaya and Umans' ideas, and new variants from [37]. If = 1, then GR(p , k) is the finite field p k, whereas k = 1 corresponds to the ring ℤ/p ℤ of the p-adic integers truncated to precision .…”

Fast computation of generic bivariate resultants

“…Kedlaya and Umans designed various algorithms for modular composition and multipoint evaluation [12]; see also [11]. They also gave algorithms for the transposed operation, called power projection.…”

“…• Unfortunately, we are not aware of any efficient implementations of Kedlaya-Umans' algorithms; see [11] for a discussion. For the time being, we therefore do not expect Theorem 1 to induce faster practical implementations of bivariate resultants.…”