Recovering polynomials over finite fields from noisy character values

2026-01-12

Electron. Colloquium Comput. Complex.

AI Summary

This work addresses the problem of efficiently recovering a square-free polynomial over a finite field from its noisy quadratic residues or additive character evaluations, where the noise affects a constant fraction of the values. The authors propose a polynomial-time algorithm that integrates Stepanov’s method, the Berlekamp–Welch decoding framework, and a novel pseudo-polynomial technique, while effectively leveraging the Weil bound as an algorithmic tool. This is the first algorithm capable of efficiently reconstructing low-degree polynomials from such noisy character data. As a significant byproduct, the approach yields the first polynomial-time decoding algorithm for dual-BCH codes, thereby substantially expanding the algorithmic frontiers of algebraic coding theory.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Let $g(X)$ be a polynomial over a finite field ${\mathbb F}_q$ with degree $o(q^{1/2})$, and let $\chi$ be the quadratic residue character. We give a polynomial time algorithm to recover $g(X)$ (up to perfect square factors) given the values of $\chi \circ g$ on ${\mathbb F}_q$, with up to a constant fraction of the values having errors. This was previously unknown even for the case of no errors. We give a similar algorithm for additive characters of polynomials over fields of characteristic $2$. This gives the first polynomial time algorithm for decoding dual-BCH codes of polynomial dimension from a constant fraction of errors. Our algorithms use ideas from Stepanov's polynomial method proof of the classical Weil bounds on character sums, as well as from the Berlekamp-Welch decoding algorithm for Reed-Solomon codes. A crucial role is played by what we call *pseudopolynomials*: high degree polynomials, all of whose derivatives behave like low degree polynomials on ${\mathbb F}_q$. Both these results can be viewed as algorithmic versions of the Weil bounds for this setting.

Problem

• polynomial recovery
• finite fields
• character sums
• noisy data
• error correction

Innovation

• polynomial recovery
• character sums
• pseudopolynomials
• Weil bounds
• dual-BCH decoding