Faster Modular Composition

Abstract

A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by 𝑛, the algorithm uses 𝑂 (𝑛 1.43 ) field operations, breaking through the 3/2 barrier in the exponent for the first time. The previous fastest algebraic algorithms, due to Brent and Kung in 1978, require 𝑂 (𝑛 1.63 ) field operations in general, and 𝑛 3/2+𝑜 (1) field operations in the particular case of power series over a field of large enough characteristic. If using cubic-time matrix multiplication, the new algorithm runs in 𝑛 5/3+𝑜 (1) operations, while previous ones run in 𝑂 (𝑛 2 ) operations.Our approach relies on the computation of a matrix of algebraic relations that is typically of small size. Randomization is used to reduce arbitrary input to this favorable situation.

High-order lifting for polynomial Sylvester matrices

“…We note however that the general approach we follow, as well as the characteristic polynomial algorithm of the case d = 1, can be interpreted in terms of operations on bivariate polynomials, see [40,Sec. 7] and [34,Sec. 1.6.2].…”

“…This characterization of Sylvester matrices allows us to highlight what makes these matrices special in the class of Toeplitz-like matrices. The fact that their displacement rank does not increase by raising to power makes the connection with the algorithm in [34] for d = 1 (see Eq. ( 1)), and gives an intuition about the displacement structure of high-order components and residues in the next sections.…”

“…It follows that the resultant can be computed generically with respect to a using Õ(n (ω+2)/3 ) arithmetic operations from the characteristic polynomial algorithm of [34,Sec. 10.1].…”

“…10.1]. In this special case, the latter algorithm is the first one that allows to break the barrier 3/2 in the exponent of n. The cost bound can be sligthly improved using rectangular matrix multiplication [34].…”

Newton iteration for lexicographic Gröbner bases in two variables

“…A direct approach takes quadratic time, and Brent-Kung's baby-steps / giant-steps algorithm uses O(n 1.69 ) operations (and relies on fast matrix arithmetic). Bringing this down to a quasi-linear runtime has been an open question since 1978: it is so far known to be feasible only over finite K [53], with the best algorithm for an arbitrary K to date featuring a Las Vegas cost of O(n 1.43 ) [54].…”