Fast polynomial computations with space constraints

TL;DR

This paper investigates the impact of space constraints on the efficiency of polynomial algorithms, developing new time- and space-efficient methods for fundamental operations and sparse polynomial interpolation.

Contribution

It introduces algorithms that are both time- and space-efficient for polynomial computations under space constraints, including the first quasi-linear sparse interpolation algorithm.

Findings

Developed algorithms with improved time-space efficiency for polynomial operations

Introduced the first quasi-linear algorithm for sparse polynomial interpolation

Explored computational hardness in sparse polynomial divisibility and factorization

Abstract

The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For many problems, extremely efficient algorithms have been developed since the 1960s. Here, we are interested in how this efficiency is affected when space constraints are introduced. The first part focuses on the time-space complexity of fundamental polynomial computations - multiplication, division, interpolation, ... While naive algorithms typically have constant space complexity, fast algorithms generally require linear space. We develop algorithms that are both time- and space-efficient. This leads us to discuss and refine definitions of space complexity for function computation. In the second part, the space constraints are put on the inputs and outputs. Algorithms for polynomials assume in general a dense representation for the polynomials, that is storing the full list of coefficients. In contrast, we work with sparse polynomials, in which most coefficients vanish. In particular, we describe the first quasi-linear algorithm for sparse interpolation, which plays a role analogous to the Fast Fourier Transform in the sparse settings. We also explore computationally hard problems concerning divisibility and factorization of sparse polynomials.