Composition Modulo Powers of Polynomials

Abstract

Modular composition is the problem to compose two univariate polynomials modulo a third one. For polynomials with coecients in a nite eld, Kedlaya and Umans proved in 2008 that the theoretical bit complexity for performing this task could be made arbitrarily close to linear. Unfortunately, beyond its major theoretical impact, this result has not led to practically faster implementations yet. In this paper, we study the more specic case of composition modulo the power of a polynomial. First we extend previously known algorithms for power series composition to this context. We next present a fast direct reduction of our problem to power series composition.

Fast multivariate multi-point evaluation revisited

“…The variant proposed by van der Hoeven [20, section 3.4.3] raises the condition on g′(0). For fields with small characteristic p, Bernstein [3] proposed an algorithm that is softly linear in d but linear in p. These algorithms have been generalized to moduli h of the form ℏ m in [25]; it is shown therein that such a composition reduces to one power series composition at order m over [z]/(ℏ(z)), plus m compositions modulo ℏ, and one characteristic polynomial computation modulo ℏ. Let us finally mention that an optimized variant, in terms of the constant hidden in the "O", of the Brent-Kung algorithm has been proposed recently by Johansson in [31], and that series with integer, rational or floating point coefficients can often be composed in quasi-linear time in suitable bit complexity models, as shown by Ritzmann [40]; see also [22].…”

Directed evaluation

“…In his PhD thesis, Dellière has investigated the relationship between dynamic evaluation and decompositions of constructible sets into triangular sets [11,12]: the central operation is the computation of gcds with coefficients in products of fields, such as t . Let us further mention that dynamic evaluation has influenced several polynomial system solvers relying on triangular sets [3], and is now involved in various other algorithms in computer algebra; see for instance [7,26,36].…”

Modular composition via factorization

“…For fields of small characteristic, Bernstein [1] proposed an algorithm that is softly linear in the precision n but linear in the characteristic. These algorithms are generalized to moduli h of the form ℏ m in [22]; we show there that the composition reduces to one power series composition at order n in [z]/(ℏ(z)), plus m compositions modulo ℏ, and one characteristic polynomial computation modulo ℏ. Let us finally mention that series with integer, rational or floating point coefficients can often be composed in quasi-linear time as well in suitable bit complexity models, as shown by Ritzmann [40]; see also [19].…”

“…In this paper, we study the problem of composition modulo a fixed polynomial h mostly in the case when = q is a finite field. We assume that h is separable; the case of moduli of the form h = ℏ t is studied in a separate paper [22]. Our results are based on the following simple observation: if a factorization h = h 1 ⋯ h t is known, then composition modulo h reduces to t composition modulo the h i with i = 1, …, t. Curiously, this observation does not seem to be exploited in the standard literature on modular composition.…”