Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

OPI x Soft Decoders

André Chailloux

TL;DR

The paper reframes lattice-based quantum reductions in a code-theoretic setting to address the Optimal Polynomial Interpolation problem (OPI). By translating CH25’s reduction into linear codes and leveraging strong Reed–Solomon decoders (Berlekamp–Welch, Guruswami–Sudan, Koetter–Vardy), it constructs quantum algorithms for inhomogeneous constrained codeword problems ICC that yield improvements for OPI. The authors show that OPI can be solved in quantum polynomial time under various error models and decoder choices, unifying prior results (Jordan et al., CT25) and extending them to broader code and decoding configurations. This work advances code-based quantum algorithm design, enabling more efficient quantum completions of interpolation-like tasks, with potential implications for quantum crypto-analytic and cryptanalytic tasks. The approach highlights the power of combining Fourier-analytic techniques over finite fields, decoder-aware error functions, and code duality to translate lattice insights into practical code-based quantum speedups.

Abstract

In recent years, a particularly interesting line of research has focused on designing quantum algorithms for code and lattice problems inspired by Regev's reduction. The core idea is to use a decoder for a given code to find short codewords in its dual. For example, Jordan et al. demonstrated how structured codes can be used in this framework to exhibit some quantum advantage. In particular, they showed how the classical decodability of Reed-Solomon codes can be leveraged to solve the Optimal Polynomial Intersection (OPI) problem quantumly. This approach was further improved by Chailloux and Tillich using stronger soft decoders, though their analysis was restricted to a specific setting of OPI. In this work, we reconcile these two approaches. We build on a recent formulation of the reduction by Chailloux and Hermouet in the lattice-based setting, which we rewrite in the language of codes. With this reduction, we show that the results of Jordan et al. can be recovered under Bernoulli noise models, simplifying the analysis. This characterization then allows us to integrate the stronger soft decoders of Chailloux and Tillich into the OPI framework, yielding improved algorithms.

OPI x Soft Decoders

TL;DR

The paper reframes lattice-based quantum reductions in a code-theoretic setting to address the Optimal Polynomial Interpolation problem (OPI). By translating CH25’s reduction into linear codes and leveraging strong Reed–Solomon decoders (Berlekamp–Welch, Guruswami–Sudan, Koetter–Vardy), it constructs quantum algorithms for inhomogeneous constrained codeword problems ICC that yield improvements for OPI. The authors show that OPI can be solved in quantum polynomial time under various error models and decoder choices, unifying prior results (Jordan et al., CT25) and extending them to broader code and decoding configurations. This work advances code-based quantum algorithm design, enabling more efficient quantum completions of interpolation-like tasks, with potential implications for quantum crypto-analytic and cryptanalytic tasks. The approach highlights the power of combining Fourier-analytic techniques over finite fields, decoder-aware error functions, and code duality to translate lattice insights into practical code-based quantum speedups.

Abstract

In recent years, a particularly interesting line of research has focused on designing quantum algorithms for code and lattice problems inspired by Regev's reduction. The core idea is to use a decoder for a given code to find short codewords in its dual. For example, Jordan et al. demonstrated how structured codes can be used in this framework to exhibit some quantum advantage. In particular, they showed how the classical decodability of Reed-Solomon codes can be leveraged to solve the Optimal Polynomial Intersection (OPI) problem quantumly. This approach was further improved by Chailloux and Tillich using stronger soft decoders, though their analysis was restricted to a specific setting of OPI. In this work, we reconcile these two approaches. We build on a recent formulation of the reduction by Chailloux and Hermouet in the lattice-based setting, which we rewrite in the language of codes. With this reduction, we show that the results of Jordan et al. can be recovered under Bernoulli noise models, simplifying the analysis. This characterization then allows us to integrate the stronger soft decoders of Chailloux and Tillich into the OPI framework, yielding improved algorithms.

Paper Structure

This paper contains 23 sections, 17 theorems, 71 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notations and probability theory
  4. Finite fields and Fourier transform.
  5. Linear Codes
  6. Code problems and OPI
  7. The CH25 Reduction Theorem and applications
  8. Choice of error function
  9. Applications
  10. The case of binary codes
  11. OPI
  12. Using the Berlekamp-Welch decoder
  13. Using the Guruswami-Sudan decoder
  14. Using the Koetter-Vardy decoder in the case of the infinity norm
  15. Graphical representation of our results
  16. ...and 8 more sections

Key Result

Proposition 1

The characters $\chi_{{\boldsymbol{y}}} : \mathbb{F}_q^n \rightarrow \mathbb{C}$ have the following properties:

Figures (1)

  • Figure 1: Ratio of satisfied constraints $\widetilde{\tau}$ as a function of the code rate $R = \frac{k}{n}$, for $\rho = \frac{|S|}{q} = \frac{1}{2}$.

Theorems & Definitions (39)

  • Claim 1
  • Definition 1
  • Proposition 1
  • Definition 2
  • Claim 2: Parseval's Identity
  • Claim 3
  • Claim 4
  • Definition 3
  • Claim 5
  • proof
  • ...and 29 more