Accelerated tower arithmetic

Abstract

Nowadays, asymptotically fast algorithms are widely used in computer algebra for computations in towers of algebraic field extensions of small height. Yet it is still unknown how to reach softly linear time for products and inversions in towers of arbitrary height. In this paper we design the first algorithm for general ground fields with a complexity exponent that can be made arbitrarily close to one from the asymptotic point of view. We deduce new faster algorithms for changes of tower representations, including the computation of primitive element representations in subquadratic time.

Directed evaluation

“…The second main aim of this paper is to generalize the directed evaluation strategy from the univariate case, while ensuring that the complexity remains bounded by the number of steps in our algorithm multiplied by d 1+o (1) (which corresponds roughly speaking to the cost of arithmetic operations in the tower [29]). Intuitively speaking, we use the univariate strategy in a recursive fashion, but the analysis becomes far more technical due to the fact that splittings can occur at any level of the tower.…”

“…The corresponding conversions between tower and primitive element representations can also be done using modular composition. All this is a consequence of the transposition principle and Le Verrier's method; we refer to [29,30,33,38,39] for more details. Unfortunately, no softly linear algorithm is known for modular composition over a generic field, although efficient algorithms have recently been designed for specific cases [9,27,28].…”

“…Unfortunately, this bound is far from linear if C is much larger than 1. Lebreton has shown that one may take C = 3; see [35] and [29,Proposition 2.4].…”

“…An alternative approach for efficient computations in a given tower is to produce an equivalent, so-called accelerated tower of small height, such that conversions between both towers are reasonably cheap. This approach was first proposed in [29]: when the i are all irreducible, we have shown how to multiply elements in t with O(d 1+ ) operations in , for any fixed real > 0.…”

“…For arbitrary, not necessarily bounded heights t, the complexity analysis becomes tedious. We need to revisit the construction of accelerated towers of fields, introduced in [29], in the context of directed evaluation. The key idea is as follows: before the directed evaluation of a given computation tree, we perform the directed computation of an accelerated version of the current tower, so the computation tree can subsequently benefit from accelerated arithmetic.…”