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Fast polynomial computations with space constraints

Bruno Grenet

TL;DR

This work investigates fast polynomial computations under space constraints, first delivering constant-space variants for core algebraic operations in ro/rw and rw/rw models while preserving near-optimal time. It then applies these ideas to sparse polynomials, delivering quasi-linear sparse interpolation over the integers and robust sparse arithmetic including verification, multiplication, division, and factorization. The framework leverages reductions, transposition, and automatic morphisms to reuse output space as working space and to transform bilinear algorithms into constant-space variants. The results have broad implications for computer algebra, with practical benefits, potential quantum-space applications, and meaningful connections to coding theory and cryptography. The research also outlines open questions around lower bounds, unbalanced sparsity, and extending quasi-linear sparse methods to finite fields.

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.

Fast polynomial computations with space constraints

TL;DR

This work investigates fast polynomial computations under space constraints, first delivering constant-space variants for core algebraic operations in ro/rw and rw/rw models while preserving near-optimal time. It then applies these ideas to sparse polynomials, delivering quasi-linear sparse interpolation over the integers and robust sparse arithmetic including verification, multiplication, division, and factorization. The framework leverages reductions, transposition, and automatic morphisms to reuse output space as working space and to transform bilinear algorithms into constant-space variants. The results have broad implications for computer algebra, with practical benefits, potential quantum-space applications, and meaningful connections to coding theory and cryptography. The research also outlines open questions around lower bounds, unbalanced sparsity, and extending quasi-linear sparse methods to finite fields.

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.

Paper Structure

This paper contains 67 sections, 78 theorems, 25 equations, 14 figures, 30 algorithms.

Table of Contents

  1. Preliminaries
  2. Notations
  3. Algebra
  4. Complexity analyses
  5. Polynomial products
  6. Other polynomial and power series computations
  7. Linear recurrent sequences
  8. Polynomial operations as structured linear algebra
  9. Straight-line programs and arithmetic circuits
  10. Transposition principle and reversion
  11. Time- and space-efficient polynomial computations
  12. The computational models -- and why they do matter
  13. Algebraic algorithms and their models of computation
  14. Space complexity of algebraic algorithms
  15. Call stack and tail recursion
  16. ...and 52 more sections

Key Result

Proposition 2.2.3

Let $f$, $g\in \mathsf{R}[x]$ of respective sizes $m$ and $n$. Then $\LowProd(f,g)$ can be computed in $\mathsf{M}(m)$ operations, and $\UppProd(f,g)$ in $\mathsf{M}(n-1)$ operations. The middle product $\MidProd(f,g)$ can be computed in $\mathsf{M}(m,n)$ operations.

Figures (14)

  • Figure 1: Pages from al-Khwārizmī's book containing geometrical solutions to two quadratic equations (public domain, via Wikimedia commons).
  • Figure 2: Conversion between representations of linear recurrent sequences
  • Figure 3: Straight-line program for the polynomial $x^2-1$ (left) and its arithmetic circuit representation (right).
  • Figure 4: \ref{['algorithm:SemiCumulativeProduct']} as a Toeplitz matrix-vector product. The first step is $f_0g$ (purple), then $f_1g_0$ (red) and finally a tail recursive call $f_1g_1$ (blue). Fake padding is used to handle the shaded parts.
  • Figure 5: \ref{['algorithm:LowerProduct']} as a lower triangular Toeplitz matrix-vector product. The first steps correspond to the bottom strip covered by triangular Toeplitz matrices. The tail recursive call corresponds to the top triangular part.
  • ...and 9 more figures

Theorems & Definitions (115)

  • Definition 2.2.1
  • Definition 2.2.2
  • Remark
  • Proposition 2.2.3: HanrotQuerciaZimmermann2004
  • Proposition 2.3.1
  • Proposition 2.3.2: Sieveking1972Kung1974
  • Corollary 2.3.3
  • Proposition 2.3.4: Strassen1973
  • Proposition 2.3.5: BorodinMoenck1974
  • Proposition 2.3.6: BrentGustavsonYun1980
  • ...and 105 more