Modular composition via factorization

Abstract

Modular composition is the problem to compute the composition of two univariate polynomials modulo a third one. For polynomials with coefficients in a finite field, 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 article, we explore particular cases of moduli over finite fields for which modular composition turns out to be cheaper than in the general case. In the most favourable cases, our algorithms achieve quasi-linear costs.

Accelerated tower arithmetic

“…For some other particular cases of interest, fast algorithms also exist for modular composition and related problems. For fixed irreducible moduli h ∈ q [x] of sufficiently smooth degree d 1 ⋯ d t , practically faster algorithms for modular composition have been proposed in [39]. When ⊆ ℂ, we have also shown in [38] that modular composition can be achieved in quasi-linear time when computing with a sufficiently high numeric precision.…”

“…Remark 2.3. Special kinds of primitive tower representations have been used before; see [17,39], for instance. Algorithms therein use special routines for conversions between…”

“…A convenient framework for analyzing the "ultimate complexity" of numeric algorithms was introduced in [38]. Concerning the factorization of complex polynomials of degree d, it was shown therein that all roots could be computed at precision p in time O(I(d p) log d) for "sufficiently large precisions" p. The actual precision from which this bound becomes relevant highly depends on the location of the roots.…”

“…Since ℂ is algebraically closed, towers of separable algebraic extensions can be factored into towers of degree one. From the "ultimate bit complexity" point of view from [38], it follows that the ultimate bit complexity of the conversions in Propositions 5.1 and 5.2 becomes O(I(d p) logd) and so does the cost of factoring itself in Proposition 5.3. Using our rule of thumb, we expect that a precision p of the order of d ≔ max (d 1 , …, d t ) should be sufficient in practice in order to observe these complexity bounds.…”

“…Fast computations in towers of finite field extensions q ⊆ q d 1 ⊆ q d 1 d 2 ⊆ ⋯ ⊆ q d 1 ⋯d t were treated quite extensively in [39]. The case when d ≔ max (d 1 , …, d t ) is small is most favourable.…”